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1, this is a consequence of what we've shown about harmonic functions: If f E HP, then Theorem 1.5.1 (or rather, its analogue for the disk) implies that f is the Poisson integral of a function 9 E V(aD) and that furthermore, f converges nontangentially to 9 a.e. and fr((}) = f(re iO ) converges in LP(aD) to g. Then Naf(e iO ) ~ AaMg(eiO) (the analogue of Theorem 1.4.7 for the disk) and thus, IINafilp ~ AallMgllp ~ Ca,pllgllp = Ca,pllfIIHv. Here for the last equality we have used the fact that fr 7 9 in V(aD). We now assume that 0 < p < 00. Let 0 < r < 1 and set fr(z) = f(rz). Then fr is analytic on a neighborhood of D and so Ifr(z)l~ is subharmonic on this neighborhood. Let h(z) be the harmonic extension of Ifr(z)l~ to D. Then Ifr(z)l~  h(z) is subharmonic on a neighborhood of D, identically zero on aD, so that by the maximum principle (Remarks 1.1.8), Ifr(z)l~ ~ h(z) for all zED. Then
IINafrll~ = IlNa(Jr~)II~ ~ IINahll~ ~ CalIMhll~ ~ Callhll~ = Callfrll~ ~ Callflli{p" As r
iI, Nafr(e iO ) i Naf(e ifJ ), so by Fatou's lemma, IINafilp ~ CallfllHv.
Corollary 1.5.5 Suppose f E HP, 1 integral of its nontangential limit.
~
p
~ 00,
on the disk. Then f is the Poisson
Proof: If 1 < p ~ 00 this follows immediately by applying the analogue for the disk of Theorem 1.5.1(a) to the real and imaginary parts of f. For the p = 1 case, the analogue for the disk of Theorem 1.5.1(b) implies f has a nontangentiallimit a.e., call it f(e ifJ ). Fix 0 < r < 1 and set fr(z) = f(rz). This is analytic on D, and thus,
24
1. Basic Ideas and Tools
That is,
By continuity, limrll f(rse i9 ) = f(se it ). Also, since f(re i9 ) ::; C aN a f(e i9 ) and the latter function is in £1, we may apply Lebesgue's dominated convergence theorem to conclude that
as desired. As the proof showed, the case p = I distinguishes analytic functions in HP from harmonic functions which satisfy a condition like that in the definition of HP. And clearly, in this p = I case, the key was Theorem 1.5.3. A slightly alternative approach to the proof of Theorem 1.5.3 is as follows: Let f E HP and let B denote the Blaschke product of the zeros of f· Then 9 = has no zeros in D and since IB(z)1 = I for z E aD then IlgllHP = IlfIIHP. In the interior of D, IBI ::; I so If I ::; Igl. Suppose we can show IlNagllp ::; CallgllHp. Then IINafilp ::; IINagilp ::; CallgllHP. In other words, we have reduced matters to showing the theorem for functions 9 E HP which have no zeros in D. But for such 9 we may consider g~; this is in H2, so we may apply the first part of the proof of Theorem 1.5.3, that is, the p > I case. This alternative approach requires a detailed knowledge of the zeros of an analytic function in HP on the unit disk. But the proof we gave essentially required the same: we used the subharmonicity of Ifl P which itself is a consequence of Jensen's formula. Jensen's formula is simply a statement about the distribution of zeros of an analytic function, and in fact, in most presentations of the theory of one complex variable, Jensen's formula is used to prove that the zeros of an HP function can be used to form a convergent Blaschke product. In any case, what is of decisive importance here is that the zeros of an analytic function are isolated. This is not true if we only assume that the functions we are considering are harmonic. In fact, if u is a real valued harmonic function defined on a domain 0, and if u(a) = 0, then the mean value property and continuity of u imply that u must vanish at least once on aB(a,r) whenever B(a,r) S;;; O. Consequently, the zeros of a real valued harmonic function can never be isolated. It is also not true that analytic functions of several complex variables have isolated zeros. For a number of years, these difficulties thwarted efforts to extend these ideas to higher dimensions. Finally, Stein and Weiss [SWI], [St2], found the correct viewpoint. In one dimension, the CauchyRiemann equations imply that the pair (v, u), consisting ofthe imaginary part and the real part of an analytic function F = u + iv, is locally the gradient of a harmonic function. In several variables, on lR~+1, the gradient of a harmonic function is an n + Ituple of functions. Stein and Weiss found that certain powers of the norm of such an n+ Ituple are subharmonic; this is exactly what is required
i
25
1.5 HP spaces on the upper half space
to take the proof of Theorem 1.5.3 and extend it to higher dimensions. We will now begin to discuss these ideas in detail. Definition 1.5.6 Let F(x,y) = (uo(x,y), ... ,un(x,y)), x E ~n,y > 0, be an n+ 1 tuple of C 1 functions defined on ~~+1. Write Xo = y. We say that F(x, y) satisfies the CauchyRiemann equations (or generalized or SteinWeiss CauchyRiemann equations) if EJ~o ~au: = 0 and ~aau. = ~au. for every j and k. Xj Xk XJ If we consider the matrix M = [~], j, k = 0, ... ,n, the definition simply says that M has zero trace and is symmetric. If n = 2, these are just the usual CauchyRiemann equations. Note that the conditions in the definition can be rewritten as div F = 0 and curl F = O. That curl F = 0 implies F is locally the gradient of a function, and div F = 0 implies this function is harmonic. Hence F is locally the gradient of a harmonic function. But all derivatives of a harmonic function are harmonic, in particular, the Uj are harmonic. 1
Definition 1.5.7 For F
=
(uo, . ..
,Un)
we set
IFI = (Ej=o IUj 12) "2.
For
0:
> 0 we
define NaF = NalFl. The key observation is: Theorem 1.5.8 If F = (uo, ... ,un) satisfies the CauchyRiemann equations and if q 2: n~l, then IFlq is subharmonic. We will not prove this theorem. Its proof is lengthy, mostly computational, but also clever. It would lead us too far off track to present the proof here. See [SWl] or [St4] for a complete presentation of the proof. Definition 1.5.9 We say that F = (uo, ... ,un) is in HP(~~+l) if it satisfies the CauchyRiemann equations on ~~+1 and 1
IIFIIHP: = sup ( y>O
(
J~n
IF(x, y)IPdX) Ii
1 was a consequence of results on harmonic functions, but the p = 1 case required the estimate of the nontangential maximal function, Theorem 1.5.3. (Which in turn was a consequence of the subharmonicity of powers of the modulus of an analytic function.) Now in higher dimensions, the same will be true: the case p > 1 will be a consequence of results on harmonic functions but the case p = 1 will need an estimate of the nontangential maximal function. (Which again, will be a consequence of the subharmonicity of powers of a function satisfying the CauchyRiemann equations.) We do this estimate now.
26
1. Basic Ideas and Tools
Theorem 1.5.10 Suppose F E HP(IR++ 1 ) , n~l < p ::; 00 and a > O. Then NaF E LP(lRn). In fact, there exists a constant Ca,n, depending only on a and n, such that IINaFllp ::; Ca,nIIFIIHP. The proof is similar to that of Theorem 1.5.3 although we need to take a little more care here because of the noncompactness of 1R++1 . Before presenting the proof of the theorem, we prove a lemma that will help us dispense with the difficulties that arise in this noncompact case. Lemma 1.5.11 Let R > 0 and set ER = {(x, y) E IR n+1 : Ixl 2 + y2 = 2R2} n 1R++l. Let v be a Borel measure on IRn with Ivl(lRn) < 00. Let c > 0 be fixed and set
v(x, y) Then as R
i
00,
v(X,Y)IER
>
=
(
llRn
Py+" (x  t)dv(t).
o.
Proof: Let 17 > O. Choose R o so that JlRn\B(o,R o) dlvl let (x, y) E E R . There are two possibilities: Case 1. Ixl > y. Then Ixl > R and thus, Iv(x, y)l::; ( Py+" (x  t)dlvl(t) 1B(O,Ro)
+ (
llRn\B(O,Ro)
< 17· Suppose R > 2Ro, and
Py+,,(x  t)dlvl
= I + II.
For I, note IPy+,,(x  t)1 ::; l~nJrl!!~' which implies I::; Ivl(B(O, Ro)) (1~~~~)"2+1
::;
Ivl(lRn)(Tjl~'~:l (Rko)n < 17 if R is large. To estimate II, we estimate IPy+,,(xt)1 ::; (y~'E)n so that II::; ~lvl(lRn\B(o,Ro)) < C(n,c)17. Thus, in this case, Iv(x,y)1 ::; 17 + C(n, c)17 on ER if R is large. Case 2. Ixl ::; y. Then y ~ Rand
Iv(x, y)l::; { Py+" (x  t)dlvl(t)::; { ( en )n dlvl(t) llRn llRn y + c ::; (y :nc)n Ivl(lRn) ::; 17 if R is large.
Proof of Theorem 1.5.10: Let F = (uo, ... ,un); fix p, n~l < p ::; 00. First note that if p > 1, FE HP(IR++ 1 ) implies that SUPy>o Iluj(·,y)llp < 00 for each j and hence F is the Poisson integral of its nontangentiallimit by Theorem 1.5.1(a). The result then follows from Corollary 1.4.9. So we now assume n~l < p ::; 1. Set r = n;;:l. Let c > 0 and set F,,(x, y) = F(x, y + c) and Uj" = Uj(x, y + c). Since IF(x, yW is subharmonic on 1R++1, then fory~~,
1.5 HP spaces on the upper half space
27
(1.5.1)
We now define ge(x, y) = JlRn Py(x  t)lFe(t, OW dt. We would like to show that lFe(x, yW ~ ge(x, y) for all (x, y) E ffi.~+1. Note that lFe(x, OW = g€(x, 0) and that lFe(x, y)IT is subharmonic and ge(x, y) is harmonic. This is not enough, however, to conclude lFe(x,yW ~ ge(x,y) on ffi.~+1. (For example, consider the functions and y on ffi.~+1.)
°
The above computation yields 1F~(x,y)1 :S C(E)IIFIIHP. Writing 1F~(x,y)1 = IF~ (x, y)IPIF~ (x, y)11P leads to the conclusion that F~ E Hl(ffi.~+1) and therefore for each j E {O, ... ,n}, SUPy>o Iluj. 1. Then by Theorem 1.5.1(a), each Uj(x, y) has a nontangential limit, call it Uj(x) E LP(lRn ), and Uj(x, y) = flR n Py(x  t)uj(x)dt. Hence, setting F(x) = (uo(x), ... ,Un (x)), F(x) is the nontangential limit of F(x,y), IFI E LP(lRn) and F(x, y) =
r Py(x  t)F(t)dt.
JlRn
Now suppose p = 1. Then by Theorem 1.5.1(b), each Uj has a nontangential limit, call it Uj(x) E Ll(lRn), hence F(x, y) has a nontangential limit a.e., call it F(x). Let s > O. Then, estimating as in (1.5.1) (now with p = r = 1), we obtain IFe(x,y)1 :::; C(s)IIFIIHl and hence conclude lFel E H2(lR:;:+l). Then F(x, y + s) = flR n Py(x  t)F(t, s)dt. Fix a > O. By continuity, limc+o F(x, y + s) = F(x,y). Furthermore, since for every s > 0, IF(t,s)1 :::; CaNaF(t) , and IINaFlll :::; Ca,nllFllHl by Theorem 1.5.10, we may let s + 0 and apply Lebesgue's dominated convergence theorem to conclude
F(x, y) =
r Py(x  t)F(t)dt.
JlRn
That IlFy  Flip + 0 as y ! 0 when 1 :::; p < 00 is a consequence of the same result for the Uj, Theorem 1.4.11. Crucial to the last proof was the estimate IINaFllp :::; CallFllHP provided by Theorem 1.5.10. Note that when n = 2, Theorem 1.5.10 states IINaFllp :::; CallFllHP for 0 < p < 00 and this is what we expect in view of Theorem 1.5.3. It would be natural to ask if Theorem 1.5.10 remains valid for a larger range of pi s. Unfortunately, this is not true, although there is a way to repeatedly expand the definition of HP(lR~:+l) so as to obtain a version of Theorem 1.5.10 valid for larger and larger range of pi s. Each successive expansion encompasses a larger range of pi s in such a way that this new definition of HP (lR~+l) agrees with the previous definitions on the previous range of piS. See Fefferman and Stein [FSj. However, it is only when 1 :::; p :::; 00 that we will be able to recover HP(lR~+l) functions as the Poisson integrals of their nontangentiallimits.
1.6
Some basics on singular integrals
In this section, we will discuss the most rudimentary facts concerning singular integral operators. These will be used in the next section when we begin our study of the LittlewoodPaley gfunction and the Lusin area function. Here and throughout we will define the Fourier transform of a function f E Ll(lRn) by j(y)
=
r
JlRn
e2TCix.y f(x)dx.
1.6 Some basics on singular integrals
29
Theorem 1.6.1 Let K(x) E L2(]Rn). Suppose there exists a constant B such that
(a)
IK(x)l::::; B
(b)
K is CIon ]Rn\{o} and IV'K(x)1 ::::; Ixl~+l when x =I o.
Suppose 1 < p
. We will not prove this here. The theorem is due to Calderon and Zygmund [CZ1]. (See either their original paper or Stein [St4] for a proof.) The first step of the proof is to show IITfl12 : : ; Bllfl12; this is an immediate consequence of hypothesis (a) and Plancherel's theorem. This is the only use of hypothesis (a) in the entire proof of the theorem. Thus, instead of (a) we could have merely assumed that T satisfied such an L2 estimate and the conclusion of the theorem would remain valid. We stated the theorem this way simply because it is the most common way to do so. We shall ultimately need a version of Theorem 1.6.1 for Hilbert space valued functions. We recall the necessary concepts and definitions. Let 1i be a separable Hilbert space, that is, 1i contains a countable dense subset, and consequently, a complete orthonormal system {h n } which is countable. A function f: ]Rn + 1i is said to be measurable if whenever h E 1i, (f(x), h) is measurable (as a function of x) on ]Rn. Letting I· I denote the norm of 1i, we see that if f is measurable, then 1
If I = (2:~=1 I(f, hn ) 12» is measurable as a function from ]Rn to R As in the case of complex valued functions we may define a theory of integration of Hilbert space valued functions by first considering simple functions and then taking limits. (Here a simple function has the form 2:~1 hiXEi (x) where Ei ~ ]Rn and hi E 1i.) With this definition IIRn f(x)dx = Y E 1i implies IIRn (f(x), h)dx = (y, h) for every h E 1i since this latter equation holds if f is a function of the form O:XE(X) (0: E 1i, E ~ ]Rn) and both sides are linear. Or, we may simply consider f :]Rn + 1i such that If I E L1(]Rn) and use this relation to define IIRn f(x)dx: We say IIRn f(x)dx = Y E 1i if IIRn (f(x), h)dx = (y, h) for every h E 1i. (If If I E L1(]Rn) it is easy to prove such a y exists.) These definitions are equivalent and in any case, we obtain an integral for 1i valued functions which has all the usual properties of the Lebesgue integral for complex valued functions. Furthermore, for 1 ::::; P ::::; 00 we may define LP (]Rn , 1i) analogously to LP (]Rn): We consider all f: ]Rn + 1i which are measurable and say that such an f E p(]Rn,1i) if IlfIILP(IRn,'H): = IllfIIILP(IRn) < 00. As in the case of complex valued
1. Basic Ideas and Tools
30
functions, we identify two functions if they agree a.e., so that LP (JRn, H) is a set of equivalence classes under this identification. Suppose HI and H2 are separable Hilbert spaces and let B(Hl' H2) denote the Banach space of bounded operators from HI to H 2. If K: JRn + B(Hl' H 2), we say K is measurable if K(x)cp is a measurable H2 valued function for every cp E HI. Let {k n } denote a countable dense subset of HI. Then IK(x)1 = sup{IK(x)knl: Iknl = I}. Thus, IKI is the supremum of a countable set of measurable functions and is therefore measurable if K is. This allows us to define LP(JRn,B(H 1 ,H2)) as the space of all B(H 1 ,H2) valued measurable functions K on JRn such that
Also, using I I, we may define continuity of functions that are B(Hl' H2) valued, and we may define derivatives of such functions in the usual way using difference quotients. Let K E Lq(JRn, B(Hl' H2)), f E LP(JRn, Hd, and suppose 1 :S r :S 00 and ~ = ~ + 1. Then flRn IK(x  y)f(y)ldy :S flR n IK(x  y)llf(y)ldy which is finite for a.e. value of x so that for such X, flR n K(x  y)f(y)dy defines an element of H2 which we denote by K * f(x). Then
*
11K * fllu(lR
n
,H2)
= IIIK * flllu(lRn) :S IIIKIIILq(lRn)lllfIIILP(lRn) =
IIKIILQ(lR n ,B(Hl,H2)) IlfIILP(lRn,H,)
that is, the usual inequalities for convolution are valid in this setting, and are a consequence of similar inequalities for complex valued functions. With these definitions and facts collected, we can now state a vector valued version of Theorem 1.6.1. Theorem 1.6.2 Let HI and H2 be separable Hilbert spaces and suppose that K (x) E
L2(JRn, B(Hl' H 2)) and 1 < p < 00. For f E LP(JRn, HI) n£l(JRn, HI) set T f(x) flR n K(x  y)f(y)dy. Suppose also that there exists a constant B such that (a)
IITfllu(lRn,H2):S Bllfllu(lRn,H')
(b)
K is 0 1 on JRn\{o} and IV'K(x)1
for every f E L2(JRn, Hd
:S
Ixl~+l when x
=
n Ll(JRn, Hd
I O.
Then there exists a constant Ap such that liT fIILP(IR n ,H2) :S ApllfIILP(lRn,Htl. Here Ap depends only onp,B,n but not on the L 2(JRn,B(Hl,H 2)) norm of K. We will not prove this theorem. Although in Theorem 1.6.2 the functions involved take values in Hilbert spaces, its proof is the same as that of Theorem 1.6.1 and is, in fact, essentially typographically indistinguishable from that proof. See [Hor], [Jou] or [St4]. Note that our statement of Theorem 1.6.2 is not quite parallel with the statement of Theorem 1.6.1. This can be readily reconciled in view of the comments made after that theorem.
1.7 The gfunction and area function
31
1.7
The gfunction and area function Consider a G I function h: lRn > lRm , not identically zero, which satisfies: (i)
There exists G > 0 such that for all x
(ii)
Ihl is radial. That is, componentwise.
l\7h(x)1 ::; G(l + Ixl)n2.
E
lRn,lh(x)1 ::;
G(l
+ Ixl)nl,
Ih(x)1 = Ih(y)1 whenever Ixl = Iyl where h is defined
Jo
Note that (ii) implies oo Ih(t~W~, ~ =J 0 is constant and we further require the finiteness of this constant: (iii)
For ~ =J 0,
As usual, set ht(x)
=
t;, h
(f)·
Theorem 1.7.1 Suppose 1 < p
O. Let lRt = {y E lR: y > c}. Set HI =!C, H2 = EB?=I L2 (lRt, ~), the direct sum of m copies of L2 (lRt ) ~). HI and H2 are separable Hilbert spaces. Consider the kernel K (x) = t~ h (f). We will show that for each x, K (x) is an element of H 2 , and hence, since HI =!C, K(x) may be identified as an element of B(HI' H2)' (K(x) acting on a E !C is given by the multiplication aK(x).) We will then show that K E L2(lRn, B(HI) H 2)) and K satisfies the hypotheses of Theorem 1.6.2. This will then show the boundedness of the operator Td(x)
=
r
JlRn
K(x  y)f(y)dy.
This will then be used to show the desired estimate for T f. We remark that we will not try to show that T itself is represented by a kernel which satisfies the hypotheses of Theorem 1.6.2.
32
1. Basic Ideas and Tools
r
Routine computation using the first estimate in (i) shows that
IIK(x)II", :;
I~n (l (u + ~)'n+,dU
1
If Ixl :::; 1 it is easy to estimate IIK(x)II1i2 :::; CE;' Thus, for all x E JRn, K(x) E H2. Furthermore, our estimates show that IK(x)1 = IIK(x)II1i2 E L2(JRn ), which by definition, means K E L2(JRn, B(Hl' H2))' Using the same reasoning, and the second estimate in (ii) we have
IVK(x)l:;
IXI~H
(1
(u+
~)'n+,dUr
Suppose f E L2(JRn) n Ll(JRn). Then
This and condition (iii) shows IITdll£2(IRn,1i2) :::; cllfIIL2(IRn). Consequently, by Theorem 1.6.2, IITd IILP (IRn,1i2) :::; ApllfllLP(IRn) whenever f E LP(JRn) n Ll(JRn). Explicitly, (1.7.2) whenever f E V (JRn) nLl(JRn). Suppose f E LP(JRn ). By choosing fn E V(JR n ) n Ll(JRn ) such that fn  t f in V(JRn) and a.e., Fatou's lemma shows that (1.7.2) continues to be valid for all f E LP(JRn). Let E 1 0 and again use Fatou's lemma to obtain the desired estimate IITfllp :::; Apllfllp. We note that Ap in 1.7.2 depends only on p, n and some constants associated with the function h. Finally, let E 1 0 in (1.7.1) to obtain
IITfl12 = cllfl12 whenever f E L2(JRn) n Ll(JRn). That this is true for all f E L2(JRn) can be seen by choosing fn E L2(JRn )nLl(JRn) with fn ) fin L2(JRn) and noting that the sublinearity of T then implies Tfn ) Tf in L2(JRn).
Hence
33
1.7 The gfunction and area function
We note that c in the end of the proof is just the constant c occurring in (iii) so that by renormalizing, we could have simply required that c = 1, in which case T would be an isometry. The integrability condition (iii) forces h(O) = 0, which is the same as requiring that the integral of h be zero. It is possible to show that if h satisfies (i) above and has integral zero on JRn then 1000 Ih(tl;)I2~ < c for every l; E JRn for some finite constant c. Using (1.7.1), this shows T is bounded on L2 and as in the above proof then yields the LP estimate. Then if we further assume Ihl is radial, we are back in the previous case and T is, up to a constant, an isometry on L2. See Journe [Jou]. Let us again consider a function h satisfying (i)(iii) at the beginning of this section. Fix a> 0 and consider the family offunctions h(y) (x) = h(xy), Iyl < a.
< p < 00
Theorem 1.7.2 Suppose 1
and for f E U(JRn ) set
Then there exists a constant Ap depending only on p, nand h such that liT flip Apllfllp. Furthermore, T is, up to a constant factor, an isometry on L2(JR n ).
:s;
Proof: The proof is essentially the same as the proof of Theorem 1.7.1 so we quickly sketch it, noting only the changes. Fix E > 0, and define Hilbert spaces 111 = C,
112 =
EB7=1 L2 (
(JRt,
~)
x (B(O, a), dY )). Consider the kernel K(x)
=
t~h(y) (~).
Iyl < a, the estimates in (i) above. Thus, as in the previous proof, we conclude that K E L2(JRn , B(111' 112)) and IVK(x)l:s; Ixl 0. Note that h2(X) = nP(x)  2:.7=1 Xj (Compare to formula 2.2.1 below which is similar yet slightly different; h2(X) has a different purpose than the kernel there.) Plainly, h2(X) satisfies the first estimate in condition (i) and routine computation shows that it also satisfies the second condition. Also,
g:,.
10
which is radial. Note that Ih2(~)1 = Ihl(~)1 so for ~ =I 0, 00 Ihdt~W~ h2 also satisfies conditions (i)(iii). For f E LP(JR n ), 1 < p < 00, set u(x, y) = Py * f(x). Then
I(hdy
* f(xW = y21V' xu(x, yW
I(h2)y
* f (x W= y21 ~~ (x, y) 12
Definition 1.7.4 Suppose u is harmonic on JR~+I. For x E JRn we define
=
i. Thus,
(1. 7.4)
1. Basic Ideas and Tools
36
and
When u is the Poisson extension of a function f, we will write g(1), gl (1) and g2 (1) for g( u), gl (u), g2 (u) respectively. The function g(1) is called the LittlewoodPaley 9 function of f. Theorem 1.7.5
(b)
Suppose 1 < p < 00. Then there exists constants ap and A p , depending only on p and n such that if f E LP (JRn),
apllfllp:S: Ilg(1)llp :s: Apllfllp apllfllp:S: Ilgl(1)llp :s: Apllfllp apllfllp:S: Ilg2(1)llp :s: Apllfllp" Proof: We have noted that hI and h2 satisfy (i)(iii) at the beginning of this section and we explicitly computed that in each case the constant c appearing in (iii) is t. Noting (1.7.4), we apply Theorem 1.7.1 with T = gl and T = g2 to conclude that both gl and g2 are bounded on LP(JRn) and, in view of the comment immediately after the proof of that theorem, satisfy IlgI(f)112 = tllfl12 and Ilg2(1)112 = tllfl12' The other LP inequalities: apllfllp :s: IlgI(f)llp and apllfllp :s: Ilg2(1)llp follow from and 1i = en for Lemma 1.7.3. (Using, in the notation there, A = [0,00), v = gl and 1i = e for g2.) Thus, we've shown (a) and (b) for gl(1) and g2(1) and to finish the proof we need to do the same for g(1). Consider h = (hI, h 2) : JRn t JRn+l. Since both hI and h2 satisfy the es
!if
timates in (i) then so does h. Also,
v2lhll.
Then
Ihl
h = (hl,h2)
and so
Ihl = Vlh112+ Ih212 =
is radial and for ~ =I 0
Then by Definition 1.7.4, (1.7.4) and Theorem 1.7.1, Ilg(1)112 = 2~ IIfl12 and 9 is bounded on LP(JR n ). The other LP inequality follows from Lemma 1.7.3.
1.7 The gfunction and area function
37
Definition 1.7.6 Let a > O. Suppose u is harmonic on JR.~+1. For x E JR.n we define 1
A"u(x) = ( (
1ra (x)
lV'u(s, tWt1ndsdt)
"2 1
(At)au(x) = ( (
1ra (x)
(A2)aU(X) =
lV'su(s, tWtlndsdt)
(lra(X) I: (s, t)1
"2 1
2
t1ndsdt) "2
When u is the Poisson extension of a function f, we will write Aaf, (Adaf and (A 2 )af for Aau, (Al)aU, (A2)aU respectively. The function Aaf is called the Lusin area function of f, or the square function of f. It is a consequence of the mean value theorem for harmonic functions that g(u)(x) :S CaAau(x) where Ca depends only on a and n. See Zygmund [Zy2, vol. II, p. 210] or Stein [St4, p. 90] for a proof. We will apply Theorem 1.7.2 to these to prove estimates similar to those of Theorem 1.7.5. First, a calculation: Let B = B(O, a) ~ JR.n. Then for f E V(JR.n),
11 1 liln 00
o
=
=
B
l(h1(y))t
00
* f(xWdy
dt t
2 (h1(y)Mx  W)f(W)dWI dy
~t
{ 1 (x  w) 12 dt 10roo 1B{ 11R.n t n hl  t   Y f(w)dw dy t
(1. 7.5) Here for the next to last equality we have made the substitution s in the step previous we made use of (1.7.4). Likewise,
1 11 00
o
B
(h(y))t
* f(x)1 2 dy
dt = (Aaf(x)) 2 .
t
=
x  ty, and
1. Basic Ideas and Tools
38 Theorem 1.7.7
(a)
Suppose f E L2(1R.n). Then there exists a constant co, depending only on a, n such that
(b)
Suppose 1 < p < 00. Then there exists constants bp and B p , depending only on p, n and a such that if f E LP (JR.n)
bpllfllp :::; IIAofilp :::; Bpllfllp bpllfllp:::; II(Al)ofllp :::; Bpllfllp bpllfllp:::; II(A2 )ofllp :::; Bpllfllp· We will not prove this theorem. Its proof follows the proof of Theorem 1.7.5, using Theorem 1.7.2 instead of Theorem 1.7.1 used in the proof of Theorem 1.7.5. The proof of Theorem 1.7.2 and (1.7.5) reveal that Co = IB(O, a)I!. The approach we have followed, via singular integrals taking values in a Hilbert space, is due to Harmander [HarJ. As we have seen, this approach yields LP inequalities such as Theorems 1.7.1 and 1.7.2 that hold in greater generality than the setting of harmonic functions. However, in most of this text, we will use techniques and ideas from works (many mentioned below) that have considered only the harmonic case. We followed the Harmander approach here only because it was the most expedient way to obtain the LP inequalities of Theorems 1.7.5 and 1.7.7. The disadvantage to this approach, is that, because it uses singular integrals, it seems to constrain us to the range 1 < p < 00 and it does not yield the local estimates we will see throughout the text. See also Question 4.2.16 below. In Theorems 1.7.5(b) and 1.7.7(b) we have made the assumption that f E LP(JR.n). This does not diminish the thrust of the inequalities, Ilg(f)llp :::; Apllfllp, IIAoflip :::; Bpllfllp and the others similar to this, but is an unnecessary apriori assumption for the reverse inequalities. We will now remedy this situation. We do this only for g and Ao since these are of greatest interest to us. Theorem 1.7.8 Let 1 < p as y + 00. Then
< 00.
Suppose u is harmonic on JR.+.+1 and u(x, y)
+
(a)
If g(u) E LP(JR.n) then u is the Poisson integral of a function f E LP(JR.n). Furthermore,
(b)
If Aou E LP(JR.n) for some a > 0, then u is the Poisson integral of a function f E LP(JR.n). Furthermore,
°
1.7 The gfunction and area function
39
Proof: We will show (a); (b) is similar. Fix 0 < c < L < 00 and set v(x, y) = u(x, y +c)  u(x, y+ L). The function v(x, y) is harmonic on JR~+l and continuous on JR~+l. Furthermore,
Iv(x, y)1
(y+L 8u = lu(x, y + c)  u(x, y + L)I ::; Jy+e: 8t (x, t) dt 1
1
::;
r+L 18 12)! ( Jy+e: r+L 1 ) ! ( Jy+e: t 8~ (x, t) dt t dt ::; Ce:,L g(u)(x).
Then by Theorem 1.5.1, v(x, y) is the Poisson extension of its nontangentiallimit, that is, v(x, y) = Py * h(x) where h(x) = u(x, c)  u(x, L) E U(JRn). Also,
(g2(h)(x))2
=
1 tl ~~ 00
: ; 21
00
(x, t) 12 dt
~~ (x, c + t) 12 dt +
t1
21
00
~~ (x, L + t) 12 dt
t1
so that (g2(h)(x))2 ::; 4(g(u)(x))2. This, plus Theorem 1.7.5 gives:
Ilhllp ::; Apllg2(h)llp ::; 2Apllg(u)llp, or
(
JIltn Letting L .......
00
lu(x, c)

u(x, L)lPdx ::; 2PA~llg(u)II~.
we obtain
This is true for every c > O. Hence, by Theorem 1.5.1, there exists an f E LP(lR n ) with u(x, y) = Py * f(x). Since u(·, c) ....... fO in LP(lR n ) (Theorem 1.4.11) the result follows. The above argument is from Stein [Stl]. We remark that if u is harmonic on JR~+l, adding a constant to u leaves g(u) and Aau unaltered. This the reason for the hypothesis on the behavior of u at infinity. The function g(f) first arose in Littlewood and Paley's study [LP], of the decomposition of Fourier series into lacunary blocks. (See Question 4.2.15 below.) They considered a version of 9 defined on the disk and showed that if F E HP on the disk, 0 < p < 00, then Ilg(F) lip::; ApllFllp. As is the case with many of the results we've discussed concerning HP, this inequality, in the range 0 < p ::; 1, depends on the analyticity of F. They also showed the reverse inequality if F(O) = 0 and 1 < p < 00.
1. Basic Ideas and Tools
40
Suppose f E LP(aD), 1 < p < 00, and f has mean value 0 on aD. Set P[J] and let v be a harmonic conjugate of U on D with v(O) = O. Write F = u+iv. By the M. Riesz theorem (see discussion below) Ilulip ~ Ilvllp. Observe that the CauchyRiemann equations imply g(u) = g(F). Thus, Littlewood and Paley's results show that on the disk, Ilg(f)llp ~ Ilfllp, 1 < p < 00. Later, T .M. Flett [FI] gave a partial extension of one of Littlewood and Paley's inequalities. He showed: If FE HP, 0 < p::; 1, and if F has no zeros on the disk, then 1!Fllp ::; Ap(llg(F)llp+IF(O)l). At about the same time, Waterman [Wat] gave analogues for Littlewood and Paley's results in the setting of the upper half plane JR~. Finally, a few years later, Stein [Stl] showed the results on g we've stated here in Theorems 1.7.5 and 1.7.8. Gasper [Gas] later extended one of Stein's inequalities by showing: if FE HP on the ball in JRn, n~l < p < 00, then Ilg(F)llp ::; Apl!Fllp. It was Lusin [Lu] who initiated the study of the area integral. He considered a version of A, defined on the disk and showed that if F is analytic on the disk with F(O) = 0, then IIAa(F)112 = Cal!F112. Marcinkiewicz and Zygmund [MZ2] showed that if F E HP on the disk, 0 < p < 00, then IIAaFllp ::; Cal!Fllp" They showed the pointwise majorization g(F)(x) ::; CaAaF(x) and thus concluded the reverse inequality, 1!Fllp ::; CailAaFIlp if F(O) = 0, 1 < p < 00, as a consequence of Littlewood and Paley's result mentioned above. Stein [Stl] showed the inequalities for the area integral that we've shown here. Calderon [Ca3] completed a last piece that Marcinkiewicz and Zygmund hadn't shown by showing that if F E HP(JR~), 0 < p ::; 1, then 1!Fllp ::; CaiIAaFllp. Shortly thereafter, Segovia [Se] extended this to higher dimensions: if F E HP on JRf+l, n~l < p < 00, then I!FIIHP ~ IIAaFIIp" Gasper [Gas] showed a version of this for functions in HP of the unit ball in JR n . Let f E V(JR), 1 < p < 00, Set u(x, y) = Py * f(x) and let v(x, y) be a harmonic conjugate to u on JR~ with v(x, y)  t 0 as y  t 00. (So U + iv is analytic on JR~). The CauchyRiemann equations imply Aau = Aav pointwise, so that Theorems 1.7.7(b) and 1.7.8(b) imply that v is the Poisson integral of a function in LP (JR); this is usually denoted as H f and is called the Hilbert transform of f. Thus, these theorems show the theorem of M. Riesz: if f E LP(JR) , 1 < p < 00, then IIHfllp ::; Cpllfllp" Similar considerations show an analogous result on the disk. This is not at all surprising; H f can be represented as a singular integral: Hf(x) = lime>o ~ ~xYI>e ~~; dy (see Garnett [Gal, Katznelson [Kat] or Stein [St4] for this) and the theory of singular integrals, used in our proofs of the theorems on the area integral, is a generalization (albeit a deep one) of the study of the Hilbert transform. Similar ideas hold for the notion of conjugacy in higher dimensions. For f E V(JRn), 1 < p < 00, form uo(x, y) = Py * f(x). Given this Uo, it is possible to create Ul, ... ,Un, harmonic on JR+.+ 1 and vanishing at infinity, so that Uo, Ul, ... ,Un satisfy the generalized CauchyRiemann equations. (See Fefferman and Stein [FS, p. 169].) However, here the CauchyRiemann equations do not allow us to conclude equality of the area functions of the Uj as in the case when n = 1, but it can be U
=
1.7 The gfunction and area function
41
shown that A",uj(x) ::=; GA,auo(x) whenever 0' < /3, j E {I, ... ,n}. (Stein [St4, p. 213].) Here G depends only on 0', /3 and n. Thus, fixing 0' < /3 we conclude IIA",ujllp ::=; GIIA,auollp ::=; Gllfllp and hence that Uj is the Poisson integral of a function in LP(lRn). This function is denoted by Rjf and is called the jth Riesz transform of f. Thus, IIRjflip ::=; Gllfllp, Each R j can be represented as a singular integral operator. In fact, the singular integral theory of the previous section can be used to show directly that IIRjflip ::=; Gpllfllp. Then the Uj may be created as Uj(x, y) = Py * Rjf(x). See Stein [St4] for this circle of ideas. In Theorem 1.5.10 we showed that if F E HP(lR~+l), n;;:l < p ::=; 00, then Writing F = (UO,Ul,'" ,un) this immediately implies When n = 1, Burkholder, Gundy and Silverstein [BGS] proved a converse: If U is harmonic on lR~, N",u E LP(lR), 0 < p < 00, then there 1 exists a harmonic conjugate v for U on lR~ such that SUPy2:o (flR Iv(x + iy)IPdx) P ::=; G""pIIN",ullp' (The choice of 0' is irrelevant; see Lemma 4.1.2 below.) Thus, F = U + iv is in HP(lR~) and IIFIIHP ::=; G""pIIN",ullp' This establishes a onetoone correspondence between functions in HP(lR~) and harmonic functions on lR~ which have nontangential maximal function in LP(lR)  with an equivalence of norms.
IIN",Fllp ::=; G""pIIFIIHP' IIN",uollp ::=; G""pllFllm.
o Iluyllp
0 and
U is a harmonic function on lR~ which has N",u E Then by the theorem of Burkholder, Gundy and Silverstein, there exists a harmonic conjugate v with F = u + iv E HP(lR~) and with IIFIIHP ::=; IIN",ullp' But, by a previously mentioned result of Marcinkiewicz and Zygmund (or rather the version of their theorem for lR~, due to Waterman [Wat]) and the fact that A,au = A,av, this then yields: IIA,aullp ::=; IIA,aFllp ::=; GpllFllm ::=; GpIIN",ullp. Conversely, if u is harmonic on lR~ with A,au E LP(lR) and with u(x, y) + 0 as y + 00, we let v be a harmonic conjugate of u which also vanishes at infinity and set F = u + iv. Then IIA,aFlip = 2l1A,aullp and by Marcinkiewicz and Zygmund, and Calderon, IIFIIHP ::=; GpIlA",Fllp = 2GpIlA",ullp' (The apriori assumption in these theorems that F E HP may be removed with the assumption that F vanishes at infinity and a limiting argument which was essentially given in the proof of Theorem 1.7.8. See Fefferman and Stein [FS, p. 165].) Then by the
Suppose
0',
LP(lR), where 0 < p
0 and u is harmonic on ~~+1, then IIA,eullp :S CpllNaull p and, if in addition, u vanishes at infinity, then IINauilp :S CpIIA,eull p. Now consider a harmonic function Uo on ~~+1 with Nauo E LP(~n), n~l < p < 00. Then IIA,euollp :S CpllNauollp. It is possible to form Ul, ... ,Un harmonic on ~~+ 1 , vanishing at infinity, and so that uo, Ul, .•. ,un satisfy the generalized CauchyRiemann equations. As mentioned previously, it is not the case that A,euo and A,euj, j E {I, ... ,n} are equal pointwise, but it can be shown that Ayuj(x) :S CA,euo(x) whenever'Y < j3. (Here C = C(,,!,j3,n).) Thus, for each j, IINaujllp :S CpIIA1'uj(x)llp :S CpIIA,euo(x)llp :S CpllNauollp· Consequently, F = (uo, Ub ... ,un) E HP(~n) and IIFIIHP :S CpllNauollp. Note that Theorem 1.5.10 provides the reverse inequality: IIN",uollp :S CpIIFIIHP. Thus, as in the case of ~~, there is a onetoone correspondence between functions in HP(~~+1), n~l < p < 00 and harmonic functions on ~~+l which have nontangential maximal function in LP(~n)  with an equivalence of norms. We remark that Segovia's [Se] result IIFIIHP ~ IIA",Fllp, n~l < p < 00, follows immediately from these estimates. We further note that as we mentioned after the proof of Theorem 1.5.10, it is possible to successively expand the definition of HP(~~+l) so as to obtain a version of that theorem valid for larger and larger ranges of p. With this definition of HP (~~+1 ), a proof similar to that given above shows that this onetoone correspondence is valid on the entire range 0 < p < 00. See Fefferman and Stein [FS] for details. Burkholder and Gundy [BG2] generalized Fefferman and Stein's result relating the V norms of the area function and nontangential maximal function. Consider a nondecreasing continuous function IP defined on [0,00] with IP(O) = 0, IP not identically zero and which satisfies the growth condition: 1P(2A) :S CIP(A) for every A > 0, where c is a fixed constant. They showed: Theorem 1.7.9 Suppose a, j3 > 0 and IP is as above. If u is harmonic on ~~+1 then
{ IP(Aau(x))dx:S C { IP(N,eu(x))dx. A~n JlRn If, in addition, limy_+<Xl u (x, y)
= 0 for all x
E ~ n, then
Here C depends only on a ,j3, n, and the growth constant c of IP.
43
1.8 Classical results on boundary behavior
A special case of this is (A) = AP, 0 < p < 00. Therefore, this result includes all previously mentioned estimates regarding the area function. Burkholder and Gundy proved this by means of inequalities relating the distribution functions of the nontangential maximal function and area function. These authors had previously developed and used similar inequalities in treating martingale analogues of these functions [BGl]. These inequalities, which came to be known as goodA inequalities, were later sharpened by R. Fefferman, Gundy, Silverstein and Stein [FGSS] and by Murai and Uchiyama [MU]. It is a major goal ofthis book to state and prove sharper and more general versions of these. This is done in Chapter 4. We also include there a discussion of the probabilistic roots of these inequalities. We will not give the proof of Burkholder and Gundy's theorem here, but will present it in Chapter 4 as a consequence of the goodA inequalities we show there. Chapter 4 also contains applications of goodA inequalities to several other estimates.
1.8
Classical results on boundary behavior
Let u be harmonic on lR~+l and bounded there. Then by Theorem 1.5.1, u is the Poisson integral of a function f E £CXl (lRn) and the nontangentiallimit of u is f a.e. In other words, if u is harmonic and bounded on lR~+l, then it has a nontangential limit a.e. In this section we discuss local versions of this result. We recall (Definition 1.4.10) that if u is harmonic on lR~+1 and x E lRn , we say that the nontangentiallimit of u at x is L if for every a > 0, lim u(s, t) = L. (s,t)+x (s,t)Era(x)
Definition 1.8.1 For a > 0, h > 0 we define the truncated cone r~(x) = {(s, t) E lR~+l: Ixsl < at, 0 < t < h}. We say u is nontangentially bounded at x E lR n if u is bounded on some r~(x). Equivalently, if we set N~u(x) = sup{lu(s, t)l: (s, t) E r~(x)}, then u is nontangentially bounded at x E lR n if N~u(x) < 00 for some a, h. In the case of the unit disk we can make analogous definitions by using the cones r a(O). Privalov [Prj (or see Zygmund [Zy2, vol. II, p. 199]) showed that in the case of the disk, a harmonic function u has a nontangentiallimit at a.e. point at which u is nontangentially bounded. Calderon [Cal] extended this to harmonic functions on lR~+1. Succinctly: if u is harmonic on IR~+l then, {x E IR n : u has a nontangential limit at x} = {x E IRn: u is nontangentially bounded at x} a.e. As we noted earlier, Lusin [Lu] showed that IIAaFI12 = calIFI12 whenever F is analytic on the unit disk and has F(O) = O. Marcinkiewicz and Zygmund [MZ2] showed a local version: Suppose that F is analytic on D and E c:;;; aD is the set of points at which F is nontangentially bounded. Then for a.e. 0 E E, AaF(O) < 00 for every a, 0 < a < 1. Later, Spencer [Sp] proved a converse: Suppose that F is analytic on D and E c:;;; aD is a set with the property that for each 0 E E there is an a = 0'(0) such that AaF(O) < 00. Then F is nontangentially bounded at a.e. 0 E E.
44
1. Basic Ideas and Tools
On lR~+l the cones r~(x) allow the definition of a "truncated" version of A,u(x):
Using this, Calderon [Ca2] showed a generalization of the Marcinkiewicz and Zygmund theorem: If u is harmonic on lR~+1 and if E ~ lR n is the set of points at which u is nontangentially bounded, then for a.e. x E E, A~u(x) < 00 for every a, h > O. Stein [St2] showed the converse: Suppose u is harmonic on lR~+1 and E ~ lR is a set with the property that for each x E E there are a = a(x) and h = h(x) such that A~u(x) < 00. Then for a.e. x E E, u is nontangentially bounded at x. We will not prove any of these theorems. For the proofs of the theorems on the disk see Zygmund [Zy2, Chapter XIV]. For the proofs of the versions in lR~+1 see Stein [St4, Chapter VII]. The techniques used in these proofs are not dissimilar to those used in the LP inequalities relating these functions and likewise have much in common with the ideas and techniques used throughout this book. However, it would lead us too far astray to present these proofs here. Nevertheless, these results are of great relevance to this monograph. We will obtain results that quantify, in a very precise way, the behavior of harmonic functions on the complement of the set where A~u(x) < 00. By the above results, at a.e. point where A~u(x) = 00, u fails to be nontangentially bounded; the laws of the iterated logarithm in Chapter 3 will measure the relative growth of these quantities near the boundary. As a corollary of the proof, we will obtain Stein's result mentioned above, but our true focus is to quantify the behavior of a harmonic function on the complement of the sets mentioned in the theorems in this section. See Chapter 3 for details.
Chapter 2 Decomposition into Martingales: An Invariance Principle Our goal in this chapter is to show that arbitrary harmonic functions in the upper half space can be approximated by dyadic martingales so that both the error terms in this approximation and the area function of the martingale are controlled by the area function of the harmonic function. This technique is very similar to the invariance principles discussed in Chapter 6, hence the title for this chapter. Much of what we do here was done in Bafiuelos, Klemes, and Moore [BKM1], and in Banuelos and Moore [BM2]. However, it is to be emphasized that the techniques of these two references are essentially a refinement of those found in Chang, Wilson, and Wolff [CWW]. In order to understand the motivation for the results and techniques here, as well as the historical precedents, we first discuss the ChangWilsonWolff result. We will need some terminology. Throughout this monograph, a cube Q 1, write Q E F as Q = Q1 X ... x Qn, where each Qj has the form Qj = (k2:1, ~). For a subset (J ~ {I, ... ,n} set
2n;3
BU
=
{Q: Qi E B1 if i E (J, Qi E B2 if it/:
(J}.
For such a (J define Xu = (Xl, .. ' ,X n ) E ~n by Xi = ~ if i E (J, Xi = 0 if i t/: (J. Then the 2n sub collections BU and the points Xu satisfy (2.1.9). We direct the reader to Wilson [Wi3] for a refinement of this lemma. Such splittings are also reminiscent of techniques used in the study of interpolating sequences; see Garnett [Ga, p. 416]. Now fix j, 1 ~ j ~ N, and for each Q E FXj let g(j)(x) = {AQo(X)
0
Q
if Q = Q~ for some Qo E Bj otherwise.
(2.1.11)
Then (2.1.12) and
h
(2.1.13)
gg)(x)dx =0.
Thus, (2.1.12) and (2.1.13) put us back in the situation of (2.1.1) and (2.1.2). We then define
AQ(X) 2",29(Q)S2 3
and martingales f(j)
= {f~·e}~=o
(2.1.14)
= QEPXj 2",+1 S£(Q)S1
QEBj
by
(j) IFXj) m > 1 and f(j) = 0 f m(j) = E(A m m' JO .
As before we define
S(f$,{»)(x) = and
(~lld~)XQk(X)II~ )
(2.1.15)
1
'2
(2.1.16)
2.2 Decomposition of harmonic functions
55
(2.1.17) As an immediate consequence of Lemma 2.1.1 we have:
Lemma 2.1.3 There are constants C 1 and C 2 depending only on n such that for each j E {I, ... ,N} and m 2:: 1, (2.1.18) and (2.1.19) We remark here that we have continued the small abuse of terminology in calling the f;,{) of (2.1.15) martingales, even though they are defined on all of ]Rn. We have assumed that each of the ..\Q(x) in (2.1.14) is supported on Q, the cube having the same center as Q but with twice the sidelength. If we fix a dyadic cube Qo of unit sidelength and in (2.1.14) further restrict the sum to those Q E Bj such that Q has nontrivial intersection with Qo, then each of the corresponding f;,{) given by (2.1.15) are martingales supported on a cube in FXj. Furthermore, a straightforward computation shows that each of these cubes is contained in a cube having the same center as Qo but with three times the sidelength. In other words, this procedure is local: if in (2.1.14) we consider only those XiS in a certain cube, then the corresponding martingales generated by (2.1.15) are all supported in a slightly larger cube.
2.2
Decomposition of harmonic functions
In this section we develop a decomposition of harmonic functions into atoms so that we may apply the estimates of Section 2.1. We note that our decomposition is more or less the wellknown Calderon reproducing formula. (See Calderon and Torchinsky [CT]. Also see Frazier, Jawerth and Weiss [FJW] for a detailed explanation of this formula.) However, most presentations of this formula require some global behavior of the functions under consideration, and since we will need local estimates, we will derive a formula suitable for our needs. Let p > 0 be given and fix a function K E coo(]Rn) which is positive, radially symmetric and decreasing, and supported in B(O, p). We also assume JlRn K(x)dx = 1. With such a K chosen we put
P(x) = 
oK LXi ax(x)  (n l)K(x). n
i=l
t
(2.2.1)
Note that P(x) is also supported in B(O, p) and a simple integration by parts shows that JlRn P(x)dx = 1.
2. Decomposition into Martingales
56
Now set q(x) = (XIP(X)
fJK
+ fJ (X), ... Xl
, XnP(X)
fJK
+ fJ (x)) Xn
(2.2.2)
and observe that the support of q is also in B(O, p). By integrating the ith component of q first in the ith variable, it follows that { q(x)dx = (0, ... ,0).
(2.2.3)
JRn
Define Pt(x) = Cnp(f) with a similar definition for qt(x) and Kt(x). Lemma 2.2.1 If u is a harmonic function on ~~+!, and if we define
v(x, t)= (
then for
JRn
°< c < f we have
v(x,f)  v(x, c)
=
Ii JRn(
(2.2.4)
Pt(x  y)u(y, t)dy
qt(x  y) . V' yu(y, t)dydt
(2.2.5)
e;
+ (
JRn
fKt(x  y) fJfJu (y, f)dy t
{ cKe;(x  y/fJtU (y, c)dy.
JRn
We remark that since P has compact support and u(·, t) is locally integrable, the integral defining v makes sense. Proof: We first note that
~(~p(XY)) fJt t n t and
=
~ ~ ~ fJy· i=l
1
fJ tn P (Xy)  t  = fJt
•
(XiYip(Xy)) t n+! t
(1
tnl K (Xy)) t,
both of which are easily proved by straightforward computation. Now apply the fundamental theorem of calculus to v(x, t) and get v(x, f)  v(x, c) =
i 1Rn Ii Ln ! C~ +I 1 Rn t fJ fJ
t
e;
=
i
e;
y)
1 P (x :;;   u(y, t)dydt t t P (x
~ y) )
u(y, t)dydt
1 (x y) fJ fJu (y, t)dydt :;;P  t t t
(2.2.6)
(2.2.7)
2.2 Decomposition of harmonic functions
57
and by (2.2.6) and (2.2.7) this becomes = l ei ~ ~ e
+
IRn i=l
a
t
tn  K
IE
IE
 t
JlRn
1~
l e1 IE
IE
+ [
JlRn
(y t)dydt ot 2 '
eKe(X  y) ~u (y,f)dy 
= le
au at (y, t)dydt
Xi  Yi P (x  y) au ( t)d dt t o  y, Y y,
ei 1K  IRn t n  1 t
+ [
t
~ t n +!
IRn i=l
IE
+
y,
l eJlRn[ at (1 1 (x  y)) 1~ (x  y) 02u l
= le

a_(xy. (xy)) a 'n+l ' P u(y, t)dydt
c:Ke(X  y) °atU (y, c:)dy
Xi  Yi P (x  y) au ( t)d dt t o · y, Y y,
~ t n +1
IRn i=l
IRn
[
JlRn
ut
1
t
nl
K
(xy) L n
t
i=l
eKe(X  y) ~u (y, e)dy 
ou 2 ~ 2 (y, t)dydt uYi [
JlRn
ut
c:Ke(X  y) °atU (y, c:)dy
Here the last equality is just a consequence of the harmonicity of u and the next to last equality is just integration by parts. Another integration by parts in the second term above gives
v(X, e)  v(x,c:)
I e1 y. (x  y) l Ln 1 (OK ~
=
e
IRn
ei

e
TIl>
~n
X· 
~ 'n+!' P i=l t
i=l
[_
t n +1
+ JlRn c:Ke(x 
=
l
e
+ 
y)
t
1 IRn
eKe(X  y) c:Ke(X
1)
(Xy)) (  (y, au t)dydt oy't t toyt
au
_
[
at (y, c:)dy  JlRn c:Ke(x 
ei qt(X  y) . "!l(y, au t)dydt IRn uy
Ln
au uy,
~(y,t)dydt
a;; (y, e)dy au
y)~(y,c:)dy
ut
which completes the proof of Lemma 2.2.1.
y)
au
at (y, c:)dy
58
2. Decomposition into Martingales
Throughout we will use the term caloric function to denote a function u(x, t) defined on jRn+l which satisfies the heat equation ~~ = ~u. Caloric functions are also sometimes called parabolic functions. Here is a corresponding formula for caloric functions.
Lemma 2.2.2 Let u(x, t) be a caloric function on jR~+l and define u(x, t) Let K t (x) be as above and set v(x, t)
=
r
ilR
= u(x, t 2 ). (2.2.8)
Kt(x  y)u(y, t)dy
n
and q(x) Then for 0
< c < if
=
( x1K(x)
8K + 2 8Xl (x), ...
,xnK(x)
8K ) + 2 8x n (x)
.
we have
v(x, c)  v(x, if)
=
if r E:
JJR.n
qt(X  y)'\1 yu(y, t)dydt.
(2.2.9)
The proof of this uses the same integration by parts ideas as the proof of Lemma 2.2.1 and is in fact somewhat shorter; see [BM2] for details. We remark that although we will not define the nontangential maximal function and area function for caloric functions until later, it is natural to use parabolic shaped cones when defining these. The change of variables t > t 2 is used to keep the same dilation structure for the kernels used for caloric functions as those used for harmonic functions. Notice too, that (2.2.9) does not involve the boundary terms that appear in (2.2.5) and this makes the proofs of the results on caloric functions easier than those for harmonic functions. We now show how to decompose harmonic functions into martingales. For Q E F (recall that F is the set of dyadic cubes in jRn of sidelength ::; 1) we define TQ
= { (y, t) : y
E Q,
£~~)
::; t ::;
£~~) }
and AQ(X)
Since supp q
_ C2
(3.0.4)
almost surely on {S (f) = oo}. Actually, Stout's result is proved for the conditioned square function 1
a(fm) =
(~E(d~lFk1))"
in place of S(fm). With this square function the limsup is almost surely equal to 1 as in (3.0.1). As mentioned earlier, a(fm) and S(fm) are pointwise comparable for dyadic martingales and hence (3.0.3) and (3.0.4) follow from Stout's result. Stout's result, first proved in [Sto] using the exponential martingale (see also Neveu [Ne, p. 154] and Chang, Wilson and Wolff [CWW]) can also be derived from the invariance principle in Philipp and Stout [PS2]. See Theorem 6.1.4 in Chapter 6 below, and Hall and Heyde [HH, p. 118] for more in this direction.
3. Kolmogorov's LIL for Harmonic Functions
65
We remark that the condition (K1) in Theorem 3.0.3 is absolutely essential as can already be seen from the sharpness of (Ko) in Kolmogorov's theorem which will be discussed below. This LIL is typical in that the upperhalf requires few hypotheses, but the lowerhalf can only hold with some additional control of the increments. This will also be the case for our LIL for harmonic functions. Finally, in the case of continuous time martingales, we have (here the condition K1 is automatically satisfied), X*
lim sup
t
J2S;(X) loglogSt(X)
t+oo
=1
(3.0.5)
almost surely on {S (X) = oo}. This result follows from the Brownian motion case by time change as in Durrett [Dur, p.77]. However, to illustrate the use of subgaussian bounds, we shall prove the upper half of this LIL below. We remind the reader of the definition of the doubly truncated cone: r~ (x; t) = {(y, s) E 1R~+1 : Ix  yl < as, t < s < h}. With this, we define the doubly truncated area function and doubly truncated maximal function of a harmonic function u on 1R~+1 by:
A~u(x; t) =
(
1
r
Jr~(x;t)
slnIV'u(y, S) 12 dYdS)
"2
and
N~u(x; t)
= sup{lu(y, s) I : (y, s)
E r~ (x;
tn.
(3.0.6)
For typographical convenience, we will write r~(x), A~u(x), and N~u(x) for r~(x; 0), A~u(x; 0) and N~u(x; 0), respectively. We recall the analysis versions (see Chapter 1, Section 8) of the martingale results (3.0.2):
{x
E IRn : A~u(x)
< oo}
a~.
{x
E IRn : N~u(x)
a~.
{x
E
IRn
:
< oo}
lim
(y,t)~(x,O)
(3.0.7)
u(y, t) exists and is finite}.
(y,t)EI'~(x)
Theorem 3.0.4 (UpperHalf LIL for harmonic functions) Fix 0 < (3 < a and o < "I < 1. There is a positive constant C~,/3,,,!,n such that if u is harmonic on IRn +1 then
+ '
lim sup lu(y,t)1 < C1 (y,t)~(x,O) J(Alu(x; "It)) 2 log log(Alu(x; "It))  o.,/3,,,!,n (y,t)EI'~(x)
for almost every x
E
{x
E IRn
:
A~u(x) = oo}.
(3.0.8)
3. Kolmogorov's LIL for Harmonic Functions
66
Let u be harmonic on
Theorem 3.0.5 (LowerHalf 11L for harmonic functions) < t < 1, let KAu)(x; t) be defined by
lR~+1. For 0
I 2 I 2 (A;u(x;t))2 (AI ())2) (Aau(x; t))  (Aau(x; 2t)) = Ka(u)(x; t) 1 1 (
og og
ee
+
a U X;
t
(K2)
and set Ka(u)(x) = 1 + lim sup Ka(u)(x; t). iLO
There is a positive constant C,;,n such that . 1Imsup tLO
for almost every x
E
u(x, t) > C,;,n r~~~7 J(Az,u(x; t))21og1og(Az,u(x; t)) JKa(U) (x) {x
E
lR n : A;u(x) =
00
(3.0.9)
and Ka(u)(x) < oo}.
A few remarks are in order concerning these two theorems. If we consider an 00 and Ka(u)(x) < 00, then (K 2) shows that A;u(x) is asymptotically equal to A;u(x; "It) as t 1 o. Consequently, for such an x, we may take "I = 1 in Theorem 3.0.4. Thus, we also have an upper bound in (3.0.9) for such x's. As is well known (cf. Burkholder [Bu4]), for harmonic functions s'\lu(y, s) plays the role of the martingale difference sequence d k • Thus, if we define da(u)(x; t) = sup{sl'\lu(y, s)1 : (y, s) E f;(x, t)\f;(x, 2t)}, then a more natural analogue of (Ko) would be to require that
x for which A;u(x) =
for almost every x E {x E lRn : A;u(x) = oo}. However, this condition is actually stronger than what we require in Theorem 3.0.5. Indeed, if (K3) holds then by integrating we obtain
Our next result is an example which shows that Theorem 3.0.5 is sharp even under the stronger condition (K3). (This is possible because in this example (K3) and (K 2) are equivalent; this is a consequence of the uniformity of I'\lu I in this example.) We shall do this in the setting ofthe unit disc. First, for a> 0,0 < p < 1 and f) E [0, 27r] define the truncated cone
ra(f);p) = {rei'P: 0 < r < p,
If)  'PI < ~(1 r)}
(3.0.10)
3.1 The proof of the upperhalf
67
and the Lusin area function of the harmonic function u in the unit disc D by 1/2
Aau(B; p)
=
(
( Jr",«(};p)
lV'u(zWdxdy
and set da(u)(B; p) = sup{(1  r)lV'u(re i'!') I : rei'!' we simply write Aau(B) for Ac.u(B; 1).
E
)
r a(B;p)}.
As before, if p = 1
Theorem 3.0.6 Let K > O. There is a harmonic function u in D with Aau(B) = for all B E [0,27r] and such that for all Po S p < 1 and all B E [0,27r],
00
lu(pe i(}) I < Ca . )(Aau (B;p))2loglog A au(B;p)  vIR Furthermore, this function satisfies
for all such p and B. Theorem 3.0.6 shows that the relationship given between the function Ka(u)(x) and the right hand side of (3.0.9) is best possible. Our harmonic function in Theorem 3.0.6 is modeled after the example of Marcinkiewicz and Zygmund [MZ] which shows the sharpness of the Kolmogorov condition (Ko) in Theorem 3.0.1 as well as the sharpness of (K 1 ) in Theorem 3.0.3. Indeed, Marcinkiewicz and Zygmund use Rademacher functions; we use lacunary series.
3.1
The proof of the upperhalf
The traditional way to prove the upper bound in the Kolmogorov 1IL is via subgaussian estimates, which are usually obtained from estimates in the central limit theorem, and BorelCantelli arguments. For those readers not familiar with these BorelCantelli arguments, we present the proof of the upper half of (3.0.5). The crucial estimate for this, as well as for the good>. inequalities below, is an estimate like the following Bernsteintype inequality ([RY, p. 145]): Key Estimate 3.1.1 Let {Xd be a continuous martingale with IIS(X)lloo Then for all >. > 0, P{X*
>'} S Cexp ( 21IS(~)IIZx,) .
Before we discuss the proof of this estimate, let us show how it leads to the upper half of (3.0.5). Let 'T} > 1 and for k = 1,2,3, ... define the stopping times Tk =
inf{t : St(X) >
y0]k}.
68
3. Kolmogorov's LIL for Harmonic Functions
It follows then that the martingale the Key Estimate 3.1.1 imply
Xt = X tATk
has IIS(X)lIoo ::;
..;:;ik.
This, and
P{X;k > J(1 + c)21]k log 10g1]k} ::; C exp( (1 + c) log log1]k) C (k 10g(17))!+c' for any c >
o. Thus 00
LP{X;k > J(1 + c)217 k log log 17k }
ko, X;k (w) ::; V(I+c)217kloglog17k. Pick such an w for which we also have S(X)(w) = 00. Then Tk(W) i 00 and S;k (X)(w) = 17 k . Let t be arbitrary large and pick k > ko such that Tk(W) ::; t < Tk+l(W). Then,
X;(w) < X;k+l (w) ::; v'i'+"€V217k+lloglog17k+l = vI + cy'17J2S;k (X)(w) loglog(1]S;k (X)(w))
::; v'i'+"€y'17J2S1(X)(w) log log 1](Sl (X) (w)) This shows that X* lim sup t < y'17v'i'+"€, t+oo v2S1 (X) log log St (X) almost surely. Since 1] > 1 and c > 0 are arbitrary, this gives (3.0.5). The reader can now see why we have called the above result, "Key Estimate". For continuous time martingales, such an estimate is obtained by time changing to Brownian motion and applying the reflection principle. However, there are at least two other ways to arrive at such estimates without changing to Brownian motion  at least not directly. One such way is to use the exponential martingale
and Doob's maximal inequality; we will do this argument below. Another way is to use the inequality: IIX*lIp ::; CpIIS(X)llp with Cp = O(v'P) as p + 00;
3.1 The proof of the upperhalf
69
this follows from B. Davis [Dav1]. This inequality for p = 2,4, ... and the power series for the exponential function combine to give an exponential square integral estimate, which by Chebychev's inequality yields the Key estimate. (This same computation was done in the introduction to Chapter 2.) Note, however, that this approach fails to give the sharp constant ~ in the exponential. In the setting of dyadic martingales, both approaches can still be used to obtain the following estimate, Key Estimate 3.1.2, which is the appropriate analogue of Key Estimate 3.1.1. Theorem 2.0.1 provides the "exponential martingale" estimate necessary for one approach and the other can be accomplished using LP estimates due to G. Wang [Wa]. Either approach will result in the following inequality. Since we have already proved Theorem 2.0.1 we will use it. Key Estimate 3.1.2 Let {in} be a real valued dyadic martingale starting at 0 with 118(J)1100 < 00. Then for all A> 0, P{f*
A2 ) > A} :::; 2exp ( 2118(J)11~ .
Proof: Fix n and let v > 0 and A > O. Then by Doob's maximal inequality (see [Dur, p. 306]) and Theorem 2.0.1 we have:
P{ sup fm > A} l::;m::;n
=
P{ sup exp(vfm) > exp(vA)} l::;m::;n
:::; e:>' E(exp(vfn)) :::; e:>' :::; exp
exp(~2118(Jn)II~)E (exp (Vfn  ~2 (8(Jn))2))
(VA + ~2118(Jn)II~) .
= A/118(Jn)II~. Then do the same for  fn and add. Finally let n t 00. With Key Estimate 3.1.2 we can mimic the above proof of the upperhalf LIL for continuous martingales, (3.0.5), to provide a proof of the upperhalf LIL for dyadic martingales, Theorem 3.0.2. Such a proof is presented in [CWW]. Note that the constant ~ in the exponential allows us to obtain the upperhalf LIL with the best possible constant. So it seems that what is required to adapt the above proof to harmonic functions is a similar Key Estimate for harmonic functions. Such an estimate is easily obtained by applying Chebychev's inequality to the estimate of Chang, Wilson, and Wolff, Theorem 2.0.3. However, notice that in the proof of (3.0.5), we applied the Key Estimate 3.1.1 to the stopped martingales X Tk • In the case of harmonic functions, these stopping times turn out to be Lipschitz domains (see Section 4.2) and thus, what is really needed to adapt the martingale proof to the setting of harmonic functions is a Lipschitz domain version of the Chang, Wilson, Wolff result. Such a result, as we will discuss at the end of Section 4.2, is only known for Lipschitz domains in two dimensions. Thus, in lR.~ we can prove Theorem 3.0.4, Set v
3. Kolmogorov's LIL for Harmonic Functions
70
the upperhalf 11L for harmonic functions, by following the proof of the upper bound in (3.0.5), but such an approach is at an impasse in higher dimensions. We will bypass this difficulty by reducing Theorem 3.0.4 to Theorem 3.0.2 using the machinery developed in Chapter 2. This is the reason why we called the results in Chapter 2 an "invariance principle". The classical invariance principles, such as those of Philipp and Stout [PSI]' [PS2], and Hall and Heyde [HH], bypass the exponential square bounds, the "subgaussian" estimates, by approximating by Brownian motion. (See Theorem 6.1.4 below.) We are now ready for the proof of Theorem 3.0.4. Using the notation of Chapter 2, we choose p = (Such precision is not really necessary for this proof. It is only important here that p be much smaller than f3 and any value of p less than or equal to 32~ will work. However, for definiteness we will set p to be
3!.;n'
this.) Let Cm set
=
T:;;'2
for m
= 1,2, ... , and again with the notation of Chapter 2 N
Vm(x) = LA~??(x) j=1
(3.1.1)
= QEF
2m2~t(Q)~23
By Lemma 2.2.1 with C = cm and g = 2c1 we have
(3.1.2) The first integral in (3.1.2) is just over B(X,2c1P) since suppK C B(O, p). Since < f3 < a, Lemma 2.3.1 implies that
p
in
2C1K2el (x  y)1
c:;: (y, 2C1)ldy
:::; C,.,.B,')',nA!qU(X; 2/'cd [
Jan K 2q (x 
(3.1.3)
y)dy
The same argument gives (3.1.4) The estimates (3.1.2), (3.1.3), (3.1.4) and Lemma 2.2.3 combine to yield (with
3.1 The proof of the upperhalf
71
the martingales {J~)} of Chapter 2) N
Iv(x, Em) 
L f~)(x)1 ~ IVm(X) 
V(X, Em)1
+ IVm(X) 
j=l
N
L f~)(x)1
(3.1.5)
j=l
+ Ca,/3,/"nA~Clu(x; 2}'Ed
~ Iv(x, 2Edl
N
+ Ca,/3,/"nA~Clu(x; }'Em) + C2 L
A~)u(x),
j=l
where as before (3.1.6) As an immediate consequence of their definitions (compare (3.0.6) and (3.1.6)), it follows that
A(j)u(x) < A 2c I E ) m /3 U(X'' m ~ A!Clu(X;}'E m).
(3.1. 7)
(We remark that it is here where we require p < Then (3.1.7) and (3.1.5) combine to give N
IV(X,E m) 
L
f~)(x)1 ~ Iv(x, 2Edl
3!..;n')
+ Ca,/3,/"nA!CI U(x; 2}'Ed
j=l
(3.1.8) Since both Iv(x,2Edl and A;CIU(X; 2},E1) are finite, Lemma 2.2.3, (3.1.7), (3.0.2), and (3.1.8) combine to show that
{x E JRn: A~u(x)
< oo} at {x
E JRn: supIV(X,Em)1 m
< oo}.
(3.1.9)
Likewise, the finiteness of Iv(x,2Edl and A;CIU(X; 2}'Ed combined with Lemma 2.2.3, (3.1.7), Theorem 3.0.2, and (3.1.8) shows that
. hmsup m>oo
IV(X,Em)1
< Ca
J(A~u(x;}'Em))2loglogA~u(x;}'Em) 
/3 /' n
",
(3.1.10)
for almost every x E {x E JRn : A~u(x) = oo}. We now show that (3.1.9) implies
{x E JRn : A~u(x)
< oo} at {x
E JRn : NJu(x)
< oo},
(3.1.11)
3. Kolmogorov's LlL for Harmonic Functions
72
which is one of the containments in (3.0.7), and that (3.1.10) implies the upper half 1IL, (3.0.8). Towards this end, let (y, to) E rb(x) and assume to::; Cl. Let m be such that c m +! < tq ::; Cm. Since "(to::; ,,(cm, we have (3.1.12) By Lemma 2.2.1,
By (3.1.3), (3.1.4) and (3.1.12), it follows that the second and third integrals above are dominated by Ca,/3",nA;U(X; "(to). We need to estimate the first integral. Applying CauchySchwarz inequality we see that the first integral is dominated by
where we have used the fact that conclude that
Cm+l
< to ::;
Cm,
and p < f3
0,
~ P {f~k  f~k_l > (1 ch/20~ loglogo~l.rrk_l } =
00,
(3.2.5)
almost surely. Now the conditional BorelCantelli lemma (Neveu [Ne, p. 152]) implies that (3.2.6)
3. Kolmogorov's LIL for Harmonic Functions
76
infinitely often with probability 1. From here Stout obtains the same conclusion for the original sequence frk' He finally combines this with the upper bound (3.2.3) to finish the proof of the lower bound. The arguments leading to (3.2.5) (hence to (3.2.6)) are frequently called "conditional BorelCantelli type arguments" . The point is that one does not need independence, as in the usual BorelCantelli lemma, to conclude (3.2.6) from (3.2.5). Such conditional BorelCantelli arguments were first used, Stout notes [Sto], by P. Levy [Le2, p. 263J in proving a version of the LIL for martingales. The proof of the lower bound in the harmonic function setting, Theorem 3.0.5, is based on the following two rather technical lemmas which we will discuss after their statements. We need some notation. For any cube Q C IR n of sidelength f(Q) we let 0 denote the concentric cube of sidelength 2f(Q). Fix constants h = 100VriIa, 0 < T < 1 and 1 < K < 00 and set E
= E(K, T) = {x
E IR n : A~u(x)
= 00,
K",(u) (x; t)
< K for 0 < t < T}.
Lemma 3.2.2 There are four positive constants No(a, n), w(a, n), 8(a, n) and (K, a, n), depending only on the parameters indicated, such that for any cube Q with 2hf( Q) ::::; T if
Ro
IE n 012: (1 w)IOI,
(3.2.7)
(K loglog(eeAQ + AQ) + Ro)
(3.2.8)
and R > No where AQ = sup(A~u(x;hf(Q)))2, xEQ
then Q is the finite disjoint union of cubes {Qj} satisfying the following two properties:
(3.2.9)
For all j and all x E Qj (A~u(x; hf(Qj)))2  (A~u(x; hf(Q))2 ::::; 2R
and: There is a subcollection {Qj;} C {Qj} such that for all ji and all x E Qj;, u(x, hf( Qj;))  u(x, hf( Q)) and
IuQj;l2: 81QI· •
2: 8v'R
(3.2.10)
3.2 The proof of the lowerhalf Lemma 3.2.3 Let No, w, 8, and
suppose that
Ro be as in Lemma
77
3.2.2. Let Q be a cube and
2hi(Q) 5,7,
(3.2.11)
IQ n EI
(3.2.12)
~(1 w)IQI,
A~AQ'
B > A and R
~No ( K log 10g~e + B) + Ro )
(3.2.13) (3.2.14)
and: A
+ 2mR 5,B,
for some positive integer m.
(3.2.15)
Then there is a partition of Q into dyadic subcubes {Qj} == :F and a subcollection £ = {Q ji} c :F such that the following hold: (3.2.16)
Il) I~8mIQI· •
Qj;
(3.2.17) (3.2.18)
Furthermore, each Qji satisfies either (a)
or for all x EQj; we have
(b)
u(x, hi(QjJ)u(x, hi(Q)) ~ 8mVR. As we will momentarily show, these two lemmas easily imply Theorem 3.0.5. Lemma 3.2.3 follows by repeated application of Lemma 3.2.2 with m = 1,2, ... , as long as A + 2mR 5, B. Thus, Lemma 3.2.2 is the key. Even if not so obvious at first reading, this lemma is the analogue in our setting (with much less precise information on the constants) of Stout's lemma above. However, we only wish to obtain a lower bound without regard to sharp constants, and consequently, do not need a lemma with the full strength of his exponential lower bound. (We do not know the best con~tants in Theorem 3.0.5, and finding them seems almost
78
3. KoImogorov's LIL for Harmonic Functions
impossible.) Our Lemma 3.2.2, translated to the martingale setting, essentially says that if T is a stopping time for Um}, fo = 0 and CIR ::; 8 2(fT) ::; C 2R, then (3.2.19) To see (3.2.19), note that since CIR ::; 8 2(h) ::; C2R the BurkholderGundy inequalities imply Ilhll p ~ v'R for all 1 < p < 00. In particular, IIhl12 ~ v'R, IIhl14 ~ v'R and from this it is then elementary to conclude (3.2.19). (See [Zyg2, vol. I, p. 216J.) In fact, this idea is the crux of the proof of Lemma 3.2.2. In this context, the hl(Qj) are a "stopping time" and the passage from (3.2.9) to (3.2.10) uses exactly this idea of comparing two different norms of the "stopped" function u(x, hl(Qj))  u(x, h£(Q)). These "stopping times" turn out to be essentially the graphs of Lipschitz functions so that much of what is needed for the proof of Lemma 3.2.2 are estimates similar (although again much more technical) to those that we will do in the proofs of the goodA inequality in Section 4.2. The reader interested in the proofs can see [BKM2J. We now give the BorelCantelli argument to prove Theorem 3.0.5 assuming the lemmas. Let 1] and 1/ be constants much larger than 1 and for k = 1,2, ... , J is the integer set Ak = 1]k, Rk = 1/K 1]k / log k and mk = [log k /31/ KJ, where [ part function, and K is as above. Notice that mk VRk is of order 1]k log log 1]k as k gets large. (This quantity has already appeared in the proof of (3.0.5) as well as in Stout's argument above.) Fix a cube QI C IR n with 2hl(Qd ::; T. The numbers
J
(A, B, R, m)
= (Ak, A k+1, R k+1, mk+1)
satisfy (3.2.13)(3.2.15) with A Q1 , if 1/ ~ 2No, 1] ~ 3 and k ~ kl for some kl large enough. For k 2:: k 1 , define the partition :Fk of Ql and the sub collection Ck C :Fk inductively by :Fkl = {QI} = Ckl' and given :Fk define :Fk+1 and Ck+1 by taking each cube Q E :Fk and applying one of the following cases: Case 1 If IQ n EI 2:: (1  w)IQI, then apply Lemma 3.2.3 with A = 1]k, B = 1]k+1, R = Rk+1 and m = mk+1. We put the resulting {Qj} into :Fk+1 and the {QjJ into Ck+1. (We may apply Lemma 3.2.3 since (3.2.13) holds for Q E:Fk by (3.2.20) below.) Case 2 IflQnEI < (1w)IQI, then let {Qj} be the 2n dyadic subcubes ofQ with l(Qj) = ~£(Q) and construct :Fk+1 and Ck+1 as follows:
(a)
If AQj ::; 1]k+1 for all j = 1,2, ... , 2n , put all of these dyadic cubes into both :Fk+1 and Ck+I.
(b)
If A Qj > 1]k+1 for some j = 1,2, ... , 2n , put Q into both :Fk+1 and Ck+1.
With this construction we have that if x E Q(k) E :Fk for a fixed x, then f(Q(k)) > 0 as k > 00. To see this, observe that if Case 1 applies then Q gets
3.2 The proof of the lowerhalf
79
subdivided and if not, Case 2 (a) eventually applies since 17k+1 t 00 and thus it also gets subdivided. Also if we abuse notation a little and use Ek + 1 to denote the set of points U Q, then using Lemma 3.2.3 QE£k+l
If Q E F k , then AQ :S 17 k . If Q E h, then
IQ n Ek+11
(3.2.20)
:::: 8rnk + 1 1QI
(3.2.21 )
If Qj E Ek+l, then one of the following is true:
(a)
(3.2.22)
For all x E Qj,
u(x, hC(Qj))  u(x, hC(Q)) :::: 8mk+l JRk+1, where Q is the cube in Fk containing Qj, or (b)
IQj nEI < (lw)IQjl, or
(c)
IQ n EI < (1  w)IQI where Q is the dyadic cube of sidelength 2C(Qj) which contains Q.
Here, (3.2.22) follows by noting that if Qj arose from an application of Lemma 3.2.3, that is, via Case 1, then (a) or (b) holds, otherwise, (b) or (c) holds according as Qj arose from Case 2(b) or Case 2(a), respectively. Next, we choose v = v(a, n) so large that
8rnk
> 8(iogk)/3vK > 8(logk)/3v = k(logl/6)/3v >


_1_.
 Vk
(3.2.23)
Notice that (3.2.21) is really a statement about conditional probabilities so we could apply the conditional BorelCantelli lemma as before. However, we can be more direct. With Q1 \Ek = Ek and with our above notation that Ek is the union of cubes in F k , (3.2.21) and (3.2.23) give that for k 2:: k1
Since
then for almost every x E Q1 there is an infinite sequence k1 < k2 < ... , depending on x, such that x E Eke for all C. Thus, almost every x E En Q1 has this property
3. Kolmogorov's LIL for Harmonic Functions
80
and is a point of density of E. Fix such an x. Since E c {x E IR n : A~ u( x) the upperhalf LIL, Theorem 3.0.4, implies that we may also assume lim sup dO
ju(x, r)j
.J(A~u(x; r/2))210glog A~u(x; r/2)
S Ca.,n.
= oo},
(3.2.24)
But even more is true. Since x E E, K(x, t) < K for 0 < t < T so that (3.2.24) and our definition of K(x, t) gives (see the remarks immediately after Theorem 3.0.5) lim sup rlO
S Ca.,n.
ju(x,r)j
.J(A~u(x; r))210g log A~u(x; r)
(3.2.25)
In addition, since x is a point of density of E, if x E Qj E Ckt, then Qj satisfies (3.2.22), (a) with k + 1 = ke as soon as f is large enough. Hence we obtain two sequences of real numbers (depending on our point x),
and
with (3.2.26) and which satisfy by (3.2.20), (A~(u)(x; re))2
sr/t  1
(A~(u)(x;te))2
srlt.
(3.2.27)
Thus, if f is large enough, (3.2.25)(3.2.27) give that 6mkt yff[i:; ju(x, re)j
u(x, te)
;===============>~:;=:==========:;==.:....:. .J(A~u(x, te))2loglogA~u(x, te) y'",kt log log ",kt
C1
(lOgv'jt,k
t

C2y'",kt1loglog",kr1
>~~~======~
y'",kt log log ",kt
?,C3V vK1 C Y:;;ff 4
>C 
5
VvK1 _ Ca.,n ..fK'
3.3 The sharpness of the Kolmogorov condition
81
if 1J is chosen much larger than vK. (Here the constants C 1 , . .. ,C5 , Co.,n depend only on a and n.) Since Q1 with 2h£( Qd :::; T was arbitrary, we have proved that . 11m sup
u(x, t)
tlO
V(Az,u(x;t))210g10gAz,u(x;t)
Co. n ::::  ' 
VK
for almost every x E E = E(K, T). Apply this now to the sequence E(j, 11k) with j, k = 1,2, ... to complete the proof of Theorem 3.0.5. In [Jo], P. Jones proved what is a very special case of the lower bound, Theorem 3.0.5. Let F be a Bloch function in the unit disc D; that is, F is analytic in D and I!FIIB = !F(O) I + sup{(I Izl)IF'(z)1 : zED} < 00. Suppose I!FIIB :::; 1 and that for all Zo ED, sup p(zQ,z):9/ 2
!F'(z)I(Ilzl) :::: c > 0
(3.2.28)
where p(zo, z) is the hyperbolic distance from Zo to z. Jones showed that then there is a positive constant C = C(c) such that Re F(pe iO )
. hmsup pj1
vlog
l~P log log log l~P
:::: C
(3.2.29)
for almost every () E [0, 21f]. Bloch functions trivially satisfy our condition (K3) and, as we will explain in Chapter 6, (3.2.28) implies that an appropriately truncated area function of Re F is uniformly bounded below by Clog l~P' Hence (3.2.29) will follow from Theorem 3.0.5. However, the reader interested in the full details of the proofs of Lemmas 3 .. 2.2 and 3.2.3 will find it very useful to first read Jones' paper. Indeed, these lemmas are natural extensions of his Lemmas 3 and 4 [Jo, pp. 6163] which are technically much simpler. Remark 3.2.4 It is also most likely the case that the proofs of Lemmas 3.2.2 and 3.2.3 given in Banuelos, Klemes and Moore [BKM2] can be modified to give appropriate versions of these lemmas for caloric functions. These in turn will lead to the corresponding lowerhalf 1IL for caloric functions.
3.3
The sharpness of the Kolmogorov condition
In [MZ] , Marcinkiewicz and Zygmund construct a sequence of random variables to show that the condition (Ko) is best possible in order to obtain a lower bound in Kolmogorov's 1IL, Theorem 3.0.1. Their example is constructed from Rademacher functions. Motivated by their example and the well established philosophy that
3. Kolmogorov's LIL for Harmonic Functions
82
lacunary power series behave very much like independent random variables, particularly lacunary series with large gaps, we will construct an example of the form 00
u(re iO ) = L ak rqk cos(qk(J), k=l
(3.3.1)
where {ak} is a sequence of positive numbers and q is a large positive integer, both to be chosen. Not only is our example motivated by that of Marcinkiewicz and Zygmund, but also some of our computations below are inspired by theirs. Notice that the function u is the real part of the lacunary power series 00
F(z)
= Lakzqk,
zED.
(3.3.2)
k=l Finally, before we proceed with the proof, we observe that our example also shows the sharpness of the condition in the M. Weiss [We] LIL for lacunary series (this LIL will be discussed in Chapter 6) which to our surprise had not been done before. We now come to the proof of Theorem 3.0.6. This will take several steps. We will show: Step 1. With {ak} and q appropriately chosen we have
where B! = L:Z'=1IakI2 and by bm rv Cm we mean below, K is as in the statement of Theorem 3.0.6. Step 1 implies a similar result for Abel means:
lim bm/cm
=
1. Here and
m>oo
Step 2. For all (J E [0,27r] and all p close to 1,
I ~akpqk COS(qk(J) I :::; ~JBpoglogBp where B~
= L: 00
k
a~p2q .
k=l We remark that B'tJ and B~ are traditionally defined with a factor of ~ proceeding the summation; we will also adopt that definition later in Section 3.4 and in Chapter 6. Use of the factor ~ here would just make the computations in this proof more cumbersome. Step 3. If q is large enough,
for all (J E [0, 27r].
83
3.3 The sharpness of the Kolmogorov condition Step 4. The function u satisfies condition (K4).
The next lemma constructs the sequences of am's and Bm's. Lemma 3.3.1 Let B;" = exp(Km/logm) and set am = VB;'  B;"_I and al = Then
a2 ,...., K m
B;" loglogB;,
o.
(3.3.3)
and
(3.3.4) Proof:
a;" = exp(Km/logm)  exp(K(m l)jlog(m  1)) =Kl
m
mI
exp(Kx/logx) (1 1  (1 1 ogx ogx
1
,....,Kexp(Km/logm) logm =K B;"
logm
and log log B;, =log m
+ log K
 log log m ,...., log m.
Thus, 2 ,K. . . B;" a .,. m loglogB;,
For convenience, let us set £(m) =B;, log log Bm. Elementary algebra gives
J C(m)  J C(m _ 1) =
C(m)  C(m  1) JC(m) + JC(m 1) C(m)  C(m  1) ,...., 2JC(m)
using the fact that Bm ,...., B m I .
)2) dx
84
3. Kolmogorov's LIL for Harmonic Functions
Also, £(m)  £(m  1)
= L~l
'" L~l
{K
=
1
Co~x 
m
ml
d d {exp(Kx/ log x) (log X X
(lo;x)2) (lOg l:;X)
+ log K
+~ 
loglogx)}dx
Xl~gx} exp C~:)
dx
Kexp(Kx/logx)dx '" Kexp(Km/logm)
=KB!. Thus, J£(m)  J£(m 1) '"
KB2
m
2Bm yflog log Bm
K
2yflog log Bm
v'K
1
'" 2 a v'K m '" a 2 m' Since am
+ 00
we obtain finally that 2
Lak '" v'K L (J£(k) m
k=2
m
J£(k
1))
k=2
2 2 2 '" v'K v/ Bm log log B m ,
and we have proved (3.3.4) and hence the lemma. We have done Step 1. The estimate for Abel means, Step 2, follows from the estimate for partial sums exactly as in M. Weiss [We, pp. 267268]. All that is required is that for k :::; m, B! + 0 as m + 00. This fact is clear for our ak's and Bm's. We now proceed to Step 3. This type of estimate on the square function will also be used later on in Chapter 6. The philosophy here is that for lacunary series, all norms are comparable to the L2norm. We begin by writing the gradient of u in polar coordinates to obtain
aV
and so
3.3 The sharpness of the Kolmogorov condition
=
I
85
+ II.
The first integral can be easily computed to obtain
r
I=o:fa%q2k r 2qk  1(Ir)dr k=l io =
a2kP2qk [q2k + 2(1  p)q3k] > _O:B 2 2qk(2qk + 1)  5 p.
00
0: """'
~
k=l
In the same way 00
1111
:::;2L j=l 00
j=l
k=j+1
(
J=l
1)
2 2qk 00 ak;_j kj k=J+l q k=j+l q
L L 00
:::; 8Bp
L
00
1/2
~ (~ 2qk~_1 )1/2 ~ akP ~ kj
:::; C=I yqJ.
2
k=l
j=lq
8B 2
0 and Cy < 00 the estimate
(1. +'Y)A 2 ) m{ei8 E T: f(e i8 ) > A} ~ Cyexp ( _ 2 9~
(3.4.6)
fails. Thus (3.4.3) does not hold for any 8 > 1/2. To prove (3.4.5), let B t be Brownian motion in the unit disc D starting from the origin and TD be its first exit time from D. Consider the martingale X t = u(Bt/vrD). By the Ito formula, its square function is given by
The connection between 9* (u) and S (X) is the following. For any positive harmonic function h in D, the Doob hconditional Brownian motion is determined by the transition probability densities 1
D
Ph(t,z,W) = h(z)P (t,z,w)h(w) for t > 0 and z, wED, where pD(t, z, w) are the transitions of the Brownian motion B t killed when it exits D. (See Durrett [Dur], Chapter 3 for more on this.) If we take
ho(z) =
1lz12 Iz _ ei8 12'
which is, up to a constant factor, the Poisson kernel for D with pole at ei8 , we obtain Brownian motion conditioned to exit D at e i8 . If we let E8 denote the expectation with respect to this motion starting from the origin, the expression for the transition probabilities and the fact that the Green's function for the disc is ~ log( 1;1) (we are dealing here with onehalf the Laplacian) imply
89
3.4 A related LIL for the LittlewoodPaley g.function
Using this, the fact that BTD is uniformly distributed on the circle and Jensen's inequality, we obtain
h
~2 9;(1)(8)) dm
exp ()..f(e ili ) 
= E (exp()..f(BTD) _ =E
(exp (EBTD ()"X _
~ E (EBTD =E
~2 EBt
(exp ()..X _
(exp ()..X _
(lTD IV'U(BsWdS) )
~2 S2(X)))) ~2 S2(X))))
~2 S2(X))) .
Since for every).. > 0, the process
M t = exp()..Xt
)..2

"2 S;(X))
is a supermartingale starting at 0 (simply apply the exp()..x t); see Durrett [Dur, p.70]), we see that
;2
Since g.(1)(8)
Ito
)..2 g;(1)(8))dm ~ 1. iTrexp()..f(eili )  "2
~
formula to 0 we may take q large enough depending only on E such that for all 0 < p < 1, (3.4.8)
where B~ = ~
2: 00
k=l
p2. q
k
.
3. Kolmogorov's LIL for Harmonic Functions
90
Proof: We give the major steps of the proof, leaving a few of the easy computational details to the reader. Since
B; = ~ £lup(eiOWdm = £g;(up)(O)dm,
(3.4.9)
the left hand side of (3.4.8) immediately follows. To obtain the right hand inequality we compute much in the same way as in Step 3 of the proof of Theorem 3.0.6 (section 3.3). We have:
g;(up)(O)
(1)
_1171"{11r2 Ireit _ eiol2 log :;
:; 71" io =2 =2
i:1 i:1
1 1
it2 IV'up(re)1 rdrdt
(~) drdt
P(re it , eiO ) 1V'u p(reitWrlog p(reit,eiO){1
~qk(pr)qk1pCOSqktI2
+ I ~qk(prl1pSinqk{}rIOg (~) drdt =
2171" (1 P(reit,eiO)
i: 11 71"
+4 =
I
io
+ II.
f
k=1
q2k(pr)2 qk 2 p2rlOg
(~) drdt r
P(re it , eiO ) k~l qk+j(pr)qk+qj2 cos((l qj)t) p2rlog
(~) drdt
k>i
A straightforward computation shows P(re it , eiO ) = P(re iO , eit ). Then, 1= 2
f; r
2qk  1
L q2k p2qk io 00
k=1 00
(
0
log
U) dr = 4';2k' Also note that
1 r 2qk  1 log dr r
1
= 2 "" q2k lqk __
LJ k=1 =B2p
4q2k
For k > j, the function r qk  qj cos((qk  qj)t) is harmonic on a neighborhood of the disk. This, plus the observation that P(reit , eiO ) = P(re iO , eit ) and Theorem 1.2.3 yield
3.4 A related LIL for the LittlewoodPaley g,function
91
Consequently,
IIII =
4111 k~'
j qk+ j (pr)qk+ q  2
k>j
I:
P(re it , e iO )
cos((l qj)t)dt p2r log
(~) drl
j=l k=j+l
=L
L
00
00
qjk pqk+qj
j=lk=j+1
::: L L 00
00
j=l k=j+l
=
f
qj p2qj
j=l
qj_k/ qJ
f
1k
k=j+l q
B2p
(q  1)" Given
E
> 0, we now choose
q
so that (q~l)
>.}:.22) . 2g*
>. > 0, (3.4.11)
If 0, is a Lipschitz domain, such as those which arise below as stopping times, we can also obtain a version for the radial or nontangential maximal function of u as in Proposition 3.4.2. The proof of Proposition 3.4.4 is as above using the representation
where now Bs is the Brownian motion in 0" Tn is its first exit time and E~o is the expectation with respect to the Doob process starting from Xo and conditioned to exit 0, at ~. Finally, A. Baernstein (personal communication) has given an analytic proof of the inequality (3.4.7). His proof is based on the fact that the functions
are solutions to the boundary value problem
~V + lV'ul { View = ef ,
2V
= 0 in D
which is a Dirichlet problem for a Schrodinger operator.
(3.4.12)
Chapter 4 Sharp Good A Inequalities for A and N In this Chapter, as in Chapter 3, we will prove various sharp comparisons for A and N which are motivated by the corresponding results for martingales. First, let us recall the classical inequalities of Burkholder and Gundy [BG 1] for continuous time martingales.
°
Theorem 4.0.1 Let X t be a continuous time martingale with maximal function X* and square function S(X). Then for all < E < 1, 6> 1 and A> 0, E2
P{X* > 6A, S(X) :::; EA}:::; (6 l)2P{X* > A}
(4.0.1)
and E2
P{S(X) > 6A, X* :::; EA} :::; 8 2 1 P{S(X) > A}.
(4.0.2)
As they are expressed here, these are actually a refinement, due to Burkholder [Bu1], of the inequalities of [BG1]. The usefulness of such inequalities is already amply demonstrated by the following lemma, which is but one of many applications (see Section 4.3 and Section 4.4 for more) of these type of inequalities. For this lemma we consider a nondecreasing continuous function defined on [0,00] with (0) = 0, not identically zero, and which satisfies the growth condition: (2A) :::; c(A) for every A > 0, where c is a fixed constant. The following is from [Bu1].
°
°
Lemma 4.0.2 Suppose that f and 9 are nonnegative measurable functions on a measure space (Y, A, Jl), and 8 > 1, < E < 1, and < 'T < 1 are real numbers
such that
Jl{g > 8A, f:::; EA}:::; 'TJl{g > A}
(4.0.3)
for every A > 0. Let p and v be real numbers which satisfy (8A) :::; p(A),
(E 1A) :::; v(A)
(4.0.4)
for every A > 0. Finally, suppose P'T < 1 and fy(min{l,g})dJl < 00. Then { (g)dJl:::; ~ { (f)dJl. }y 1  P'T }y 93 R. Bañelos et al., Probabilistic Behavior of Harmonic Functions © Birkhäuser Verlag 1999
(4.0.5)
4. Sharp GoodA Inequalities for A and N
94
Note that the existence of p and v satisfying (4.0.4) is a consequence of the growth condition imposed on . Proof: We may assume that Jy (g)dfJ < 00. To see this simply note that if 9 satisfies (4.0.3) then so does min{n,g} for any positive integer n. Also note that since Jy (min{l, g} )dfJ < 00 then Jy (min{ n, g} )dfJ < 00 for every n. If (4.0.5) holds for min {n, g} and f for every n then it is immediate that it holds for 9 and f. Associated to is the LebesgueStieltjes measure d satisfying J[a,b)d()') = (b)  (a) whenever 0 :::; a < b :::; 00. This measure is a positive ITfinite Borel measure on [0, (0). Furthermore, an elementary Fubini theorem argument shows that if h is a nonnegative measurable function on (Y, A, fJ) then jy (h)dfJ Jooo fJ{h > )'}d()'). Inequality (4.0.3) implies fJ{g
Consequently,
> 8),} = fJ{g > 8)', f:::; cA} + fJ{g > 8)', f > cA} :::; "YfJ{g > ),} + fJ{J > c),}.
i
(~) dfJ :::; 'Y
i
(g)dfJ +
i
(~) dfJ·
(4.0.6)
But Jy (g)dfJ = Jy (88 1 g)dfJ :::; p Jy ( ~ )dfJ· This, plus (4.0.6) and (4.0.4) yield the conclusion of the lemma, (4.0.5). In a space of finite measure, the hypothesis Jy (min{1,g})dfJ < 00 is superfluous. In the general case this hypothesis was necessary to assure the finiteness of Jy (min{ n, g} )dfJ so that we could reduce to the assumption that Jy (g )dfJ < 00. This in turn was necessary in order to insured that we were not subtracting infinities in the last step of the proof. In the case of spaces of infinite measure, the lemma is not true without the hypothesis Jy (min{1, g} )dfJ < 00; see Journe [Jou, p. 4], or Miyachi and Yabuta [MY] for examples. In this monograph we will sometimes use this lemma to produce estimates like (4.0.5) and often this will be done on spaces of infinite measure. However, in all situations in which we use the lemma on a space of infinite measure, the inequalities we show are already known, and so we may assume the finiteness of the quantities involved. This seems to say that we will use an inequality like (4.0.5) to prove itself, but there is an advantage here: if we know that an estimate like (4.0.5) holds, then a good)' inequality can lead back to (4.0.5), but with better constants. An alternative in the infinite measure case is to use the good), inequality to show (4.0.5) for a set of functions for which the left hand side of (4.0.5) is finite and then take appropriate limits. It is precisely such an approach that we take in the proof of Theorem 4.4.1 below. Essentially this same idea was used in the proof of Theorem 1.7.8. If 0 < p < 00 then (),) = ),p satisfies all the conditions of Lemma 4.0.2. In this case, Theorem 4.0.1 and Lemma 4.0.2 give the following. Theorem 4.0.3 Let 0 < p < 00. There exists constants Ap and B p , depending only on p, such that if X t is a continuous time martingale starting at 0 with maximal
95
4. Sharp GoodA Inequalities for A and N
function X* and square function S(X) then
(4.0.7) and
IIS(X)II :::; apIIX*II·
(4.0.8)
If in using Lemma 4.0.2 to prove this we set 8 = 1 + ~ and c = 2~ for (4.0.7), and 8 = 1 + ~ and c = 2~ for (4.0.8), we find that Ap = O(p) and a p = O(y'P) as p '; 00. This is the best possible order for ap but not the best possible order for Ap; see (4.4.5) below. 2
2
Observe that, with 15 fixed, the expressions (6':'1)2 and 6L 1 in (4.0.1) and (4.0.2) go to 0 as c '; O. In order to obtain the best possible order of A p , and for various other applications of these inequalities, such as LIL's and estimates on various constants, it is desirable to have a better rate of decay as c '; O. The optimal rate of decay is given by the following result. Theorem 4.0.4 There are constants C 1 and C 2 such that for all 0 < c < 1, 1 < 15 and A> 0,
1
P{X* > I5A, S(X) :::; cA} :::; C exp ( _ (8 ~;)2) P{X* > A}
(4.0.9)
and
(4.0.10) These are the subgaussian goodA inequalities referred to in the preface. This theorem follows from more general goodA inequalities of Burkholder [Bu3, pp. 185187] which relate maximal functions and stopping times for Brownian motion in]Rn. These appear with quantities he designates as Rn (15, c) and Ln (15, c) replacing the exponentials on the right hand side of (4.0.9) and (4.0.10) respectively, but explicit computation of these quantities yields the expressions we have stated here. To illustrate once again the vital role played by Key Estimate 3.1.1, we present Burkholder's proof of (4.0.9), breaking it into its basic parts. In fact, what Burkholder's argument really shows is that scaling and the Markov property are essentially enough for goodA inequalities. (For more on this direction, see Revuz and Yor [RY, p. 155].) The following inequality is easily derived from the Key Estimate 3.1.1 and it makes it very clear what is needed for all the goodA inequalities that follow, including those for harmonic functions.
96
4. Sharp GoodA Inequalities for A and N
Key Estimate 4.0.5 Let {Xt} be a continuous martingale with Then for all A> 0 and 8> 1, P{X* > 8A} :::; Cexp ( Proof: Let T = inf{t : Consequently,
IXtl >
(81)2A2)
2118(X)II~
A}. Then, {T < oo}
118(X)1100
A}.
= {X* > A} and IXrl = A.
P{X* > 8A} :::; P{X* > 8A, T < oo}
:::; P{sup IXt

Xrl > (8  l)A,
T
< oo}
t~r
:::; P{X* > (8 l)A}P{T < oo}
(8  1)2 2) * :::; Cexp (  2118(X)II~ A P{X > A}, where the last inequality follows from the Key Estimate 3.1.1. For the proof of (4.0.9), let T = inf{t : 8 t (X) > cAl. Applying the Key Estimate 4.0.5 to the martingale Xt = Xr/\t which has 118(X)1100 :::; cA, we obtain

(
P{X* > 8A, 8(X) :::; cA}:::; P{X* > 8A}:::; Cexp 
(8 2c  1)2) P{X*  > A} 2
:::; Cexp ( (8 ~21)2) P{X* > A}, which proves (4.0.9). For dyadic martingales we have a similar estimate which follows exactly as above from the Key Estimate 3.1.2. This version is Key Estimate 4.0.6 If {fn} is a dyadic martingale with
118(f)1100 < 00,
then
The Key Estimate 4.0.6 implies the next theorem using exactly the same stopping time argument that showed that the Key Estimate 4.0.5 implies (4.0.9). Theorem 4.0.7 Let {fn} be a dyadic martingale with maximal function f* and square function 8(f). There are constants Ct, C2 such that for all 0 < c < 1 and A> 0,
P{f* > 2A, 8(f) :::; CA} :::; C 1 exp ( 
~~ )
P{f* > A}.
(4.0.11)
4. Sharp GoodA Inequalities for A and N
97
Inequality (4.0.11) is proved in [CWW] by Chang, Wilson and Wolff. For dyadic martingales in one dimension we also have P{S(f) > 2A,
1* : : ; lOA}::::; C 3 exp ( ~;)
P{S(f) > A}.
(4.0.12)
The inequalities (4.0.10) and (4.0.12) follow, with similar proofs, from the corresponding key estimates: Key Estimate 4.0.8 If X t is a continuous martingale with P{S(X) > 8>'}::::; Cexp
(
7f
8
(8 2
1)>'2)
IIX*II~
IIX*lloo < 00.
Then
P{S(X) > >.}
and Key Estimate 4.0.9 If {In} is a dyadic martingale with
111*1100 < 00,
then
C >.2 ) P{S(f) > 2>.} ::::; C 1 exp ( llf~lI~ P{S(X) > >'}. The inequalities in the Key Estimates 4.0.8 and 4.0.9 can be proved by showing that the condition IIX*lloo < 00 (respectively, 111*1100 < (0) implies S2(X) (respectively S2(f)) is in BMO. The JohnNirenberg theorem and a stopping time argument (see Lemma 4.2.3 for the adaptation of this stopping time argument to functions on ~n) then gives exponential inequalities involving S2(X) (respectively S2 (f)), and hence, exponential square inequalities involving S (X) (respectively, S(f)). However, it is not possible to obtain either of the Key Estimates 4.0.5 or 4.0.6 using just BMO techniques. The situation is exactly the same for harmonic functions. An analogue of the Key Estimates 4.0.8 and 4.0.9 will be obtained using BMO techniques; this is (4.2.16) below. The analogue of the Key Estimates 4.0.5 and 4.0.6 will be more difficult to obtain. In the harmonic function setting, the analogue of Theorem 4.0.1 was proved in Burkholder and Gundy [BG2]. Their inequalities, although of a slightly different form, are essentially equivalent to those of Theorem 4.0.1 with the expression 10 2 on the right hand side. The expression 10 2 was improved to 10 k for any k by R. Fefferman, Gundy, Silverstein and Stein [FGSS]. T. Murai and Uchiyama [MU] proved the following inequalities, which have exponential decay: Theorem 4.0.10 Fix 0 < f3 < 0:. There is a constant K > 1 and positive constants C 1 , C 2 , C 3 and C4 , all depending only on 0:, f3 and n, such that for all 0 < 10 < 1 and>' > 0,
I{x
E ~n : N{3u(x)
> K>', A,u(x) ::::; c>'}1
1
::::; C exp ( 
~2) I{x E ~n : N{3u(x)
> >'}I
(4.0.13)
4. Sharp GoodA Inequalities for A and N
98
and I{x E IR n : Aj3u(x) > KA, Nau(x) :::: cA}1
:::: c3 exp (  ~;)
I{x E IR n
:
Aj3u(x) > A}I·
(4.0.14)
Notice, however, that unlike (4.0.9) or (4.0.10), (4.0.13) does not give the full subgaussian decay in c. What is needed to improve the c to c 2 is a version for harmonic functions of the Key Estimate 4.0.5. For harmonic functions on IR~+l this can readily be obtained from the result of Chang, Wilson and Wolff, Theorem 2.0.3. However, for the goodA inequality, the result in IR~+l is not sufficient. As we will discuss below, and as the methods of Section 4.2 will illustrate, what is needed is such an estimate on Lipschitz domains. This is not known at this time. As in the proof of the LIL in Chapter 3, we will overcome this difficulty by directly reducing the desired inequality to the dyadic martingale result (4.0.10) using the techniques from Chapter 2. We divide the results in this Chapter into 4 sections. In Section 4.1 we prove (4.0.13) with 1/c2 in the exponential. In Section 4.2 we prove the inequality (4.0.13) not only for IRn but also for Lipschitz domains. This is obtained in the more "traditional" way using sawtooth regions with an estimate similar to the Key Estimates 4.0.8 and 4.0.9. The Lipschitz domain result is crucial for the applications to the Chungtype LIL proved in Section 4.3. That is, the goodA (4.0.14) just on IR n , as sharp as it is, does not give this LIL. The point to be stressed here is that once we have the inequalities in Lipschitz domains we can literally copy the martingale proofs. Finally, as in [FGSS] and [MD], we were partly motivated to improve the c 2 decay in the goodA inequalities by their applications to the behavior of the LPconstants and to ratiotype inequalities for A and N. In a sense to be made precise later, the better the goodA inequality, the better the information on these quantities. These applications are presented in Section 4.4. It should be mentioned here that just as with most of the results in this monograph, ratio inequalities were first proved for martingales. (See Garsia [Gar] or M. Barlow and M. Yor [BY1], [BY2].)
4.1
Sharp control of N by A
Theorem 4.1.1 Fix 0 < f3 < a. There are positive constants K, C 1 and C2 , with K > 1 and all depending only on a, f3 and n, such that whenever u is harmonic on IR~+l, 0 < c < 1 and A > 0 I{x E IR n
:
Nj3u(x) > KA, Aau(x) :::: cA}1
:::: C 1 exp ( 
~~) I{x E IRn : Nj3u(x) > A}I·
4.1 Sharp control of N by A
99
We first state and prove a lemma from Burkholder and Gundy [BG2]. We will use this not only in the proof of Theorem 4.1.1 but in several other places in the text. Lemma 4.1.2 Suppose, > f3 > 0 and u is continuous on 1R~+I. Then for all A > 0,
Here C depends only on" f3, and n. Proof: For (x,y) E 1R~+1 set B(x,y) = {s E IR n : Ix  sl < y}. If a point (x,y) has lu(x,y)1 > A then B(x,f3y) ~ {s: N{3u(s) > A}. Set E = {s E IR n : N{3u(s) > A}. Suppose x E IR n has Nyu(x) > A. Then there exists (s,y) E 1R~+1 with lu(s,y)1 > A and x E B(s"y). Note that then B(s,f3y) ~ E. Thus, MXE(X) > 1 { XE(t)dt  IB(x, (r + f3)y) I JB(x,b+[3)Y)
> IB(s, f3y) n B(x, (r + f3)y) I IB(x, (r + f3)y) I

IB(s,f3y)1 IB(x, (r + f3)y) I
= Set c =
('!f3)n.
(~) n. Then by Theorem 1.4.3 (the HardyLittlewood maximal theo
rem),
1 e
I{Nyu(x) > A}I ::; I{MXE(X) > e}1 ::; llxEilI
=
1 e
lEI.
The following lemma localizes Theorem 4.1.1 to a cube. We recall that for h > 0, N$u(x) denotes the truncated (at height h) nontangential maximal function. Lemma 4.1.3 Suppose f3 < a. There are positive constants K, C 1 , and C2 , with K > 1 and all depending only on a, f3 and n, such that if Q is a cube in IR n with center q, h = 2£(Q)v'nlf3, and u is harmonic in 1R~+l with u(q, h) = 0, then for all A > 0 and 0 < c: < 1,
I{x E Q : N$u(x)
> KA, Aau(x) ::; cA}1 ::; C1exp ( 
~~)
IQI·
Proof of Theorem 4.1.1: Let us first show how this lemma implies the theorem. Let 'Y be very large and let R = {x E IR n : N"Iu( x) > A}. By the previous lemma,
4. Sharp GoodA Inequalities for A and N
100
IRI ::::: I{x E ~n : Nf3u(x) > A}I with constants depending only on f3,'Y and n. In particular, we may assume IRI < 00. Let {Qj} be a Whitney decomposition of R. Then for each j we can pick a point Xj E RC such that IXj  xl < CR.(Qj) for every x E Qj; here C is a constant which depends only on n. For each j we let qj denote the center of Qj and let hj = 2R.(Qjhrn/f3 be as in the statement of Lemma 4.1.3. Also, for each j we set T;j(x) = rf3(x)  rf3h j (x); this is the "top" part of a cone. If "( is large enough depending only on f3 and C, T;J (x) ~ r ')'(Xj) whenever x E Qj and thus, for every x E Qj, (4.1.1)
Let x E Qj have Nf3u(x) > KA and A,u(x) ::; cA. Then since N')'u(xj) ::; A, (4.1.1) gives KA
< N~ju(x).
(4.1.2)
Since we also have lu(%,hj)l::; A, it follows from (4.1.2) that
Therefore if we choose K = that
(K + 1), where K
is as in Lemma 4.1.3, we conclude
{x E Qj : Nf3u(x) > KA, Aau(x) ::; cA} ~
{x
E
h·

Qj : N/ (Iu  u(qj, hj)I)(x) > KA, Aau(x) ::; cA}.
But, by lemma 4.1.3, for each j
Summing over Whitney cubes proves Theorem 4.1.1. Proof of Lemma 4.1.3: We may assume that Q is the cube [!, !In = Qo and = O. In this case, h = 2vn/ f3. We continue to use the notation of Chapter 2 and the proof of the upperhalf LIL, Theorem 3.0.4: set p = f3 / (32vn) (recall that the kernels K, P, and q used in Section 2.2 were all supported in B(O, p» and cm = (2 m  2 R.(Q»/4p. Thus, with this notation, h = 2c1. Suppose Xo E {x E Qo : N3u(x) > KA, A",u(x) ::; cA}. Then there exist (y, t) E r~(xo) such that q
\u(y, t)\ > KA and an m such that any 0 < 'Y < 1,
cm+!
CA}, W = UXEEc r ",(x), and then estimating a version of the area integral defined on aw. Note that with this strategy it results that aw is the graph of a Lipschitz function. However, if instead, E were a set on a Lipschitz graph then the corresponding aw would still be the graph of a
4.2 Sharp control of A by N
103
Lipschitz function. Consequently, since our estimation of A on oW will make use of techniques from the study of Lipschitz domains, it causes us no extra effort to show the good,\ inequalities in the more general setting of Lipschitz domains. These good'\ inequalities have the added benefit that they lead immediately to LIL's and this is the main reason why we present them in this generality. The techniques we will use in this section have been used by several authors. The idea of working on a "sawtooth" domain goes back to Privalov [Prj (see also Zygmund [Zy2, vol. II, p. 200]). The proofs of the good,\ inequalities of Burkholder and Gundy [BG2]' and the subsequent improvements by Fefferman, Gundy, Silverstein and Stein [FGSS], and Murai and Uchiyama [MU] (together with their martingale versions) provide not only an outline for our proof, but indispensable techniques. These works are themselves descendants of the earlier work of Privalov, Marcinkiewicz and Zygmund, Spencer, Calderon and Stein, who showed that, except for sets of Lebesgue measure zero, the nontangential maximal function and Lusin area function are finite on the same sets. (See Section 1.8.) The work of Dahlberg [Da1], [Da2] will provide us with many of the necessary tools for estimates on Lipschitz domains. We recall that ¢ : IR.n + IR. is called a Lipschitz function if there exists a constant M such that I¢(x)  ¢(y)1 ::; Mix  yl for all x, y E IR. n . The smallest such M for which this remains valid for all x and y will be called the Lipschitz constant of ¢. Consider a Lipschitz graph {(x, ¢(x)) : x E IR.n}. For points P in this graph, we define r a(P) to be the cone r a(O) translated so that its vertex is at P. That is, if P = (x, ¢(x)) then
r a(P)
=
{(s, t) : Is  xl < a(t  ¢(x))}.
We similarly define the truncated cones r~(p)
= {(x, t) : Is 
xl < a(t  ¢(x)), t < ¢(x)
+ h}.
Consider a function u which is harmonic on an unbounded Lipschitz domain D = {(x, y) : y > ¢( x)}. Consider any a > 0 for which there exists a' > a such that r a' (P) 0 then
Q,
Q. There is a constant K > 1 and constants C 1 {J, n and the Lipschitz constant of ¢ such that if
IT{P E aD: A,au(P) > KA, Ncr.u(P)::; CA} ::;
Here IT represents surface measure on aD.
The proof of this is long and much of the work will be done via a sequence of lemmas. These lemmas will also be useful for our proofs, in Chapter 5, of goodA inequalities involving the Dfunctional, so we will write these in sufficient generality to be applicable here and in that chapter. As we have noted, the proof of this theorem involves the construction of a "sawtooth" region and estimates of the area function on the boundary of this region. Specifically, we will estimate the BMO norm of the square of the area function on the boundary of this region. We will first need to collect some general facts about BMO. For the basics on BMO, see, for example, Garnett [Gal, or Bass [Bas2] for a probabilistic point of view. Lemma 4.2.2 Let f : ]Rn t ]R be measumble. Suppose that there exist an s, 0 < s ::; ~ and a AO > 0 such that for every cube Q ~ ]Rn there is a constant aQ such that I{x E Q : If(x)  aQI > Ao}1 < slQI· Then f E BMO and IlfllBMO ::; CAo where C depends only on n. This is proved in Stromberg [Str]. Its proof resembles that of the JohnNirenberg theorem. The following lemma will ultimately be applied to the square of the area function, thus producing the harmonic function analogue of Key Estimates 4.0.8 and 4.0.9 above. This essentially appears in Murai and Uchiyama [MUl, but we shall give a proof since we require a slightly stronger result than what is obtained there. Lemma 4.2.3 Suppose IlfllBMO ::; 1. Then for all A > 0, I{x
E]Rn : If(x)1
> 2A}1 ::; C1 exp(C2 A) I{x
E]Rn : If(x)1
> A}I,
where C 1 , C 2 are positive constants which depend only on n. Proof: Fix A. We may assume that I{x E ]Rn : If(x)1 > A}I < 00. Let {Qi} be the maximal dyadic cubes in ]Rn such that I{x E Qi : If(x)1 > A}I > ~IQil. Then {x E ]Rn : If(x)1 > A} ~ UQi almost everywhere. Since IlfllBMO ::; 1, IfQi  f Qi I ::; 2n for every i ; here fQi denotes the average of f over the cube Qi (similarly for f Qi ) and Qi is the unique dyadic cube containing Qi which has twice the sidelength. The maximality of the Qi and the fact that IlfllBMO ::; 1 implies that for each i, If 2 + A. Thus IfQi I ::; A + "I, where "I = 2n + 2. The JohnNirenberg theorem applied to each Qi then yields
Q.I ::;
I{x E Qi : If(x)1 > 2A}1 ::; I{x E Qi : If(x)  fQ.I > A  "I} I
4.2 Sharp control of A by N
105
Summing over i gives the conclusion of the lemma. We will also need to collect some facts about Lipschitz domains. Since we will want BMO estimates on such domains, we will need to make local estimates. This will be accomplished using auxiliary domains. These domains will be bounded Lipschitz domains constructed above "cubes" on the original unbounded Lipschitz domains. We say that a bounded domain 0 ~ JR~+1 is a starlike Lipschitz domain if it satisfies the following two properties. First, we require that the boundary of 0 is locally Lipschitz, that is, for each P E 00 there is a coordinate system (z, t), z E JRn, t E JR, and a Lipschitz function 't/J such that for some neighborhood V of P, V n 0 = {(z, t) : t > 't/J(z)}. We require the existence of an M so that all such 't/J have Lipschitz constant at most M. Second, we assume that there exists a point P* EO such that for every point P E 00, {tP + (1  t)P* : 0 ::::; t ::::; I} ~ O. In this case we say that 0 is a Lipschitz domain which is starlike with respect to P*. Now consider such a bounded starlike Lipschitz domain 0 which has the additional property that there exists (x' > 0 such that for every P E 00 the cone r~, (P) with aperture (x', vertical axis P P* and height h = IP  P* I is contained in O. If (X < (x' and P E 00 we consider the cone r~(p) with aperture (x, vertical axis PP* and height h = IP  P*I, and for U harmonic on 0 we then define Nau(P) and Aau(P) as before except now we use the cones r~(p). That is, we define Nau(P)
= sUp{IU(8, t)1 : (8, t) E r~(p)}
and (4.2.3) where, in the definition of Aau(P) we have written 't/J for the Lipschitz function defining the boundary in a neighborhood of P = (xo, 't/J(xo)). This Aau(P) is, up to multiplicative constants which depend only on the Lipschitz constant of the domain, equivalent to the coordinate free formula (4.2.4) where for any P E 0 we set d(P) = dist(P, (0). Because these two versions of Aa u are equivalent up to constants, we will be able to do much of our estimation using the version of (4.2.4) even though we are ultimately interested in a version like (4.2.3). We now consider Gn(P, P*), the Green's function for 0 with pole at P* which, for brevity, we will denote as G(P) throughout this section, and let w denote
4. Sharp Good.\ Inequalities for A and N
106
harmonic measure on an taken with respect to the point P*. For P E 0,\ {P*} we let P be the point of intersection of an with {r(P  P*) + P* : r > O} and for P E an we define fl (P, r) = an n {P' : IP'  PI < r}. Because of our requirement that r~, (P) ~ 0, for every P E an, there exists a constant ro > 0 such that B(P*,ro) = {P' E ]Rn+1 : IP'  P*I < ro} ~ n. In fact, there exists a constant C(o;') depending only on a' such that ro ~ C(o;')diam(n). Set 0,* = n\B(p*, ro). The following appears in Dahlberg [Da1], [Da2]. Lemma 4.2.4 There is a constant k > 0 such that for all P E 0,* we have
(4.2.5)
The constant k depends only on the Lipschitz constant of 0" the aperture 0;', and the dimension n. Let (J denote surface measure on an. Dahlberg [Da1] has also shown that w and (J are mutually absolutely continuous and that in addition they satisfy an Aoo condition. More precisely, There are constants b > 0, a > 0, C > 0 depending only on the Lipschitz constant ofn such that if E ~ fl = fl(P,r) ~ an then
Lemma 4.2.5
w(E) < C (dE))b and (J(E) < C (W(E))a w(fl) (J(fl) (J(fl) w(fl)
(4.2.6)
We can now describe the construction of our auxiliary domains which will allow us to estimate the BMO norm of the area function on the boundary of a Lipschitz domain. If D is a Lipschitz domain, then we will call a set Q ~ aD a cube if Q = {(x, 1jJ(x)) : x E Q'} where Q' is a cube in]Rn and 1jJ(x) is the Lipschitz function defining D (either globally or locally). We now consider an unbounded Lipschitz domain D = {(x,y) : x E ]Rn, y E]R, Y > 1jJ(x)} where 1jJ is Lipschitz and we fix Q ~ aD. For u harmonic on D we consider Ai3u(P), defined as in (4.2.2), with P E Q. We will separately consider the contribution to Ai3u(P) from the part of the cone r i3 (P) which is "close" to Q and the part of the cone r i3 (P) which is "far" from Q. To estimate the part which is "close" we will construct an auxiliary domain n. Choose "(' and "( so that "(' > "( > (3 and so that r "I' (P) ~ D for every P E aD. As above we write Q = {(x,1jJ(x)) : x E Q' ~ ]Rn} and let us denote by Xo the center of Q' so that (xo,1jJ(xo)) is a "center" for Q ~ aD. There exists R > 0 with the following properties:
+ Re(Q')).
Then for every P E Q, {tP
+ (1 
t)P* :
(i)
Set P* = (xo, 1jJ(xo) 0:::; t:::; I} ~ D.
(ii)
For every P E Q, the cone r~(p) with aperture ,,(, vertical axis P P*, vertex at P and height h = IP  P* I is completely contained in the cone r'Y"(p), where "(" = =y'~1.
4.2 Sharp control of A by N
(iii)
107
For every P E Q, the cone r2(p) with vertical axis {(x, 1/J(x) + 8) : 8 > O} and height h = 1/J(xo) + R£(Q') 1/J(x) is completely contained in the cone r, (P) with aperture ,,(, vertex at P and vertical axis P P* .
We assume that R is the smallest such constant for which (i), (ii) and (iii) holds for all cubes Q 0, depending only on (3 and the Lipschitz constant of 1/J, such that for every PEon there exists a K ~ KO so that the cone r~(p) with vertex at P, height h = IP  P* I, aperture K, and vertical axis P P* is completely contained in a larger cone which is contained in n. In fact, for P E Q 0 which have no ~ n and such that for every E > 0, P* E no, no ~ n and no has smooth boundary. (See Stein [St4, p. 206] for this construction.) Let Go denote the Green's function for no with pole at P*. We then apply Green's theorem to the functions (u(p))2  (u(p*))2 and Go(P) on no. Since (u(p))2  (u(p*))2 vanishes at P = P*, standard arguments (see, for example, the proofs of Theorems 1.1.3 and 1.2.3) yield
Since u ::; 1 on D "2
As
E
~
0,
no, this implies
no ~ n,
Go(P)
~
JJ
0
G(P) and consequently, we obtain
IVu(PWG(P)dP ::; 1.
For PEn, pI E Q set X(P, PI) = 1 if PErl (PI), 0 otherwise. Then, using the Green's function estimate of Lemma 4.2.4 we obtain:
JJ IVu(PWG(P)dP ?: C JJoIVu(P)12d(p)1nw(A(p, d(P))dP ~ C JJoIVu(P) 12 d(p)1n kx(P, z)dw(z)dP C 10 JJoIVu(P) 12 d(p)1n x (p, z)dPdw(z) ?: C 10 Aiu(z)dw(z),
1 ?:
0
=
which completes the proof of the lemma. The next result controls the contribution to A{3u from the "tops" of the cones, that is, we estimate A 2u. The result is simply that A 2u(P) does not vary much as P ranges over Q. This is similar to lemma 3.1 of Murai and Uchiyama [MU]. This lemma will be useful not only here but also in subsequent sections.
4.2 Sharp control of A by N
109
Lemma 4.2.7 With the notation as above, suppose additionally that for every P E D, dist(P,8D)\V'u(P)\ ::; 1. If there exists a Po E 8D such that A,Bu(Po) < 00, then for every PI, P2 E Q, (4.2.7)
Here, the constant 0 depends only on (3 and the Lipschitz constant of 'IjJ. Proof: Since dist(P,8D)\V'u(P)\ ::; 1 on D, then if (s,t) E r,B(x,'IjJ(x)), (t'IjJ(x))\V'u(s, t)\ ::::; 0 where 0 depends only on (3 and the Lipschitz constant of 'IjJ. Set PI = (x, 'IjJ(x)) , P2 = (y,'IjJ(y)). Then
IJr[
2 (p,)
1
\V'u(s, tW(t  'IjJ(x))l ndsdt  [
Jr
::; G(A) \V'u(s, t)\
2
(p,)
\V'u(s, t)\2(t  'IjJ(y))Indsdtl
2\(t  'IjJ(y))nl  (t  'IjJ(x))nl\ (t 'IjJ()) x n 1(t 'IjJ()) y n 1 dsdt
(4.2.8)
For any two sets E and F, denote by EIIF their symmetric difference. We have for fixed t,
\(r 2 (Pdllr 2 (P2 ))
n {(s, t 
'IjJ(y)) : s E ~n}\ ::; O(t  'IjJ(y))nl\P1  P2 \.
Therefore,
(4.2.9)
4. Sharp GoodA Inequalities for A and N
110
Estimates (4.2.8) and (4.2.9) give the result assuming that one of A 2 u(Pi ), i = 1,2 is finite. But these estimates plus the assumption A,Bu(Po) < 00 imply A 2 u(P) < 00 for all P E Q. Hence, we obtain the conclusion of the lemma. We can now obtain the necessary BMO estimates of area functions on Lipschitz domains. Notice that this proposition, together with Lemma 4.2.3, gives the analogue for harmonic functions of Key Estimates 4.0.8 and 4.0.9. Proposition 4.2.8 Suppose /3 > 0 and u is harmonic on an unbounded Lipschitz domain: D = {(x,y) : x E JRn, y E JR, y > ,¢(x)} where,¢ is Lipschitz. Suppose that for every P E D, lu(P)1 1. Suppose also that there exists a point Po E aD at which A,Bu(Po) < 00. Then IIA~u(x, ,¢(x))IIBMO S C, where C depends only on /3, the Lipschitz constant of'¢ and the dimension n.
s
Proof: Fix a cube Q ~ aD. Define the auxiliary domain corresponding Al u and A 2 u. Lemma 4.2.6 implies w{P E Q:
A~u(P)
> A} S
~
S
Cw~Q)
n
as above and the
,
where w is harmonic measure on n taken with respect to P* and where for the last inequality we have used Lemma 2.1 in Hunt and Wheeden [HWJ. Since wand (1 satisfy the ADO condition (4.2.6) on an, then (1{P E Q : Aiu(p) > A} S C~~Q) for some b > O. Since '¢ is Lipschitz,
I{x E Q': A~u(x,'¢(x)) > A}I S
ClQ'I Ab.
(4.2.10)
Since lu(P)1 S 1, the gradient estimate, Lemma 2.3.1, and Lemma 4.2.7 imply that (4.2.11) whenever x, y E Q'. Let Xo be the center of Q'. Then (4.2.10) and (4.2.11) imply
I{x
E
Q' :IA~u(x, ,¢(x))  A~u(xo, ,¢(xo)) I > A}I
S I{x
E Q':
A~u(x,'¢(x)) > A 
*
C*}I S
(A~~~)b·
If AO is chosen so that (>'O~C.)b = then Lemma 4.2.2 implies that A~u(x, ,¢(x)) is in BMO and IIA~u(x, ,¢(x))IIBMO S C, where C depends only on the choice of AO, and hence only on the Lipschitz constant of ,¢, the aperture /3, and the dimension n. Before finally proving Theorem 4.2.1 we need one more lemma. This lemma will allow us to compare area functions defined on the boundary of a sawtooth region to an area function defined on the original domain.
111
4.2 Sharp control of A by N
Lemma 4.2.9 Let u be a harmonic function defined on the cone r,8(O, 0). Suppose v < (3 and (x, d) E r v(O, 0). Then [ lV'u(s, t)12(t  d)lndsdt :::; L [ lV'u(s, tWt1ndsdt, Jr{3(x,d) Jr{3(o,o) where L is a constant which depends only on (3 and v. In particular, if x may simply take v = ~ and then L depends only on (3.
(4.2.12)
=0
we
We remark that this is obviously true with L = 1 when n = 1. Proof: For (s,t) E r,8(x,d) let B«s,t), r) be a ball with center (s,t) and radius r = eod where eo < 1 is chosen so that B«s, t), r) 3d, [ XB«w,z),r)(s, t)(t  d?n dsdt :::; Cz1nIB«w, z), r)1 :::; Jr{3(x,d)
C~+lzln. (4.2.14)
For
z:::; 3d, [ XB«w,z),r)(s, t)(t  d)ln dsdt ir{3(x,d)
:::; [
1 )dl
Jr {3 (x,d)n{ (8,t):t«3+eo)d}
:::; c
(2+e o
o
Isl A} is finite. Let E = {P E aD : Nau(P) > eA} and set D' = UPEEc r a(P). We
4. Sharp GoodA Inequalities for A and N
112
may also assume E C I 0 so that D' I 0. Then D' is a Lipschitz domain with constant determined by a, say D' = {(x,y) : y > ¢(x)} where ¢ is Lipschitz. On D ' , lui ~ cA. So by Proposition 4.2.8, IIA~u(x,¢(x))IIBMO ~ C(cA)2 where C depends only on the Lipschitz constant of ¢, the aperture {3 and the dimension n. Then, by Lemma 4.2.3, for all 'TJ > 0,
Choosing
'TJ
= A2 yields
I{x E lRn
:
Af3u(x, ¢(x)) > V2A}1 ::; C 1 exp
(c~2 )
I{x E lRn
:
Af3u(x, ¢(x)) > A}I.
(4.2.16)
Set K = V2L where L is the constant appearing in Lemma 4.2.9. Then, using (4.2.16) and Lemma 4.2.9 we obtain:
a{P E aD : Af3U(P) > KA, Nau(P) ~ cA} = a{P E E C : Af3u(P) > KA} ~ a{P E aD' : Af3U(P) > KA} ~ CI{x E]Rn: Af3u(x,¢(x)) > KA}I
~ C1 exp ( c~2 ) I{x E lR ~
C2 ) a{P E C 1 exp ( ~
n :
Af3u(x, ¢(x)) >
~}I
aD : Af3(P) > A}.
This completes the proof of Theorem 4.2.1. For caloric functions the corresponding goodA inequality for the case D lR~+l was proved in M. Kaneko [KaJ.
=
Theorem 4.2.10 Let U be a caloric function in lR++l and 0 < {3 < a. There are constants K, C~ and C~, with K > 1 and all depending on a, (3 and n only, such that for all 0 < c < 1 and A > 0, I{x E lRn
:
PAf3u(x) > KA, PNo.u(x) ::; cA}1 ::;
C~ exp ( ~i) I{x E lR
n :
PAf3u(x) > A}I·
Actually, this is just a special case of Kaneko's result. Kaneko considers certain second order differential operators L and functions u for which Lu is a positive Borel measure, and shows a goodA inequality between an area function and a
4.2 Sharp control of A by N
113
nontangential maximal function defined for these. In the case when u is harmonic, these are just the usual area function and nontangential function of u. In the particular case here when u is a caloric function, the change of variables u(x, t) = u(x, t 2 ) yields a function u(x, t) which satisfies Lu = 0 where L = t..x  ~t1 %t and the area function and nontangential maximal function he considers reduce to the usual parabolic area and parabolic nontangential maximal function. His results also include the area function of subharmonic functions studied by T. McConnell [Mc]. We have not checked whether Theorem 4.2.10 also holds for Lipschitz domains. (Recall, with the change of variables u(x, t) = u(x, t 2 ) one obtains P A,Bu(x) = A,Bu(x) and P NO/.u(x) = NO/.u so that parabolic regions never enter into the proof. It is thus perfectly reasonable to ask for such results for Lipschitz domains.) The reader may also wish to consult R. Gundy and 1. Iribarren [GI] where certain types of goodA inequalities are proved for functions which are not solutions to partial differential operators. However, the inequalities in [GI] do not have the sharpness of those presented here and it is not known whether their inequalities can be improved to inequalities having gaussian behavior. We now elaborate some more on the status of Theorem 4.1.1 on Lipschitz domains. As we have already noted, the missing ingredient to extend this result to Lipschitz domains is the following open problem which is the exact analogue for harmonic functions of our Key Estimate 3.1.1 for martingales. Problem 4.2.11 Let D be a Lipschitz domain, D = {(x, y) : y > if> (x)} , where if> : IRn  t IR is Lipschitz. Suppose u is harmonic in D with boundary values f and that IIAO/.uIILOO(oD) = IIAO/.ull oo < 00. Prove that for any cube Q caD,
for all A > 0 where 0 1 and O2 are constants. One possible approach to this gives rise to another problem. With D as above, we define : 1R~+1  t D by (x,y) = (x,y + if>(x)). If u is harmonic on D, then u 0
0 and 0 < 10 < 1, cr{P E aD : Nf3U(P) > 2A, Aau(P) ~ lOA}
~ C 1 exp (  ~~) cr{P E aD : Nf3U(P) > A}. Remark 4.2.13 Theorem 4.2.12, combined with the BorelCantelli argument of the upperhalf of the Kolmogorov LIL, Theorem 3.0.1, can be used to prove Theorem 3.0.4 in two dimensions. This technique will be clearly illustrated in Section 4.3 below where we prove a Chungtype LIL. Recall that in Section 3.4, we considered another area function, the LittlewoodPaley g* function, and showed, just as we did for the Lusin area function, that if a function has this area function bounded, then it is exponential square integrable. Along these lines, one open problem which has been of some interest concerns the LittlewoodPaley gfunction. By Theorem 1.7.5, for 1 < p < 00, if f E V', then Ilg(f)llp ~ Ilfllp" There are also BMO estimates for the operator f + g(f). (See Torchinsky [To], or Journe [Jou] for these.) But it is not known if any subgaussiantype estimates hold for g. Naturally, since g(u) is pointwise smaller than Aau (see the remark immediately following Definition 1.7.6), it should be more difficult to use g( u) to control u or any maximal function of u. The obvious point to begin such an investigation would be with the work of Chang, Wilson and Wolff discussed in Chapter 2. We recall their theorem (Theorem 2.0.4): Suppose a > O. There exists constants C 1 and C 2 , depending only on a and n, such that if f has Aaf E U X ), then for any cube Q ~ IRn, (4.2.17)
Question 4.2.14 Does such a theorem hold with g(f) in place of Aaf? That is, are there constants C 1 and C 2 depending only on n such that if f has g(f) E £00, then for every cube Q ~ IRn, (4.2.18) To appreciate the complications involved when considering the gfunction, let us revisit some of the estimates of Chapter 2. We recall the estimate of Theorem 2.0.1:
4.2 Sharp control of A by N
115
If {In} is a dyadic martingale on Q ~ ~n with fo = 0 and with limit function f, then for all A > 0,
I~I 10 exp (Af(X)  ~2 (Sf(x))2) dx :::; 1.
(4.2.19)
This can be used to show an analogue of itself for harmonic functions. Consider a harmonic function u on ~~+1. Suppose Q ~ ~n is a cube, and suppose that on Q, u(x, y) has a nontangential limit, call it u(x). Let q be the center of Q and let h be as in Lemma 4.1.2. We claim: There exists constants C 1 and C2 , depending only on 0: and n, such that for every A > 0, (4.2.20) This gives an analogue for harmonic functions of the inequality (4.2.19). The proof of (4.2.20) is an application of the ideas in the proof of Lemma 4.1.3. Using the notation of that proof, we note that limm+oo v(x, Em) = u(x) for almost every x and since v(x, Em) can be approximated by a sum of martingales with the square function of each of these martingales pointwise dominated by Ca,nAau, we can pass from the martingale estimate (4.2.19) to the harmonic function estimate (4.2.20). We leave the details of this to the reader. We now apply the same observations that allowed us to deduce Theorem 2.0.2 from Theorem 2.0.1. We replace u by u in (4.2.20); the resulting inequality combined with (4.2.20) yields a similar inequality which has lu(x)  u(q, h)1 in place of u(x) u(q, h). Then, if Aau E L oo , we substitute A = A' /2C11IAaull~ and rearrange to obtain:
This yields the estimate
which upon integrating yields (4.2.21 ) This is essentially (4.2.17). Since 'P(x) = exp(x 2 ) is increasing and convex, Jensen's inequality applied to (4.2.21) shows that IUQ  u(q, h)1 :::; ClIAauiloo and this and (4.2.21) then gives (4.2.17). Thus, (4.2.17) is a consequence of (4.2.20). Later, in Chapter 5, Lemma 5.2.9 will give another proof of the estimate (4.2.20). This proof will show that (4.2.20) follows from the goodA inequality of
116
4. Sharp GoodA Inequalities for A and N
Theorem 4.1.1. In lR~, Theorem 4.1.1 can be shown by using (4.2.17) together with the conformal mapping arguments of Problem 4.2.11 to produce a version of (4.2.17) on Lipschitz domains and, just as in the proofs of this section, this provides the necessary "stopping time" estimates for Theorem 4.1.1. Thus, in lR~, (4.2.17) implies (4.2.20) and from the discussion above, these are equivalent. As we have noted, we do not know if an estimate like (4.2.17) remains valid on Lipschitz domains in higher dimensions and this was precisely what prevented us from proving Theorem 4.1.1 directly using the martingale stopping time arguments. Our invariance principle was a way to circumvent this difficulty. Thus, in higher dimensions we do not know how to show that (4.2.17) implies (4.2.20). Nevertheless, heuristically, we can say that subgaussian estimates like (4.2.17) combined with stopping time arguments give goodA inequalities like Theorem 4.1.1 and by Lemma 5.2.9 these then imply an estimate like (4.2.20). However, here the subgaussian estimates and stopping time arguments are done in the setting of martingales and merely transferred over to harmonic functions. So, at the very least, we can claim that in higher dimensions, (4.2.17) "philosophically" implies (4.2.20). Given the relationship between (4.2.17) and (4.2.20), it seems clear that to prove (4.2.18), we should try to answer the following question: Suppose u is harmonic on lR~+1, Q ~ lRn is a cube and u(x, y) has a nontangentiallimit, call it u(x). Suppose also that q is the center of Q and h is as in Lemma 4.1.3. Does there exist constants 0 1 and O2 , depending only on n, so that (4.2.22) The answer is no. By a construction of Bagemihl and Seidel. [BS], there exists a harmonic function u(x, y) on lR~ which has g(u)(x) < 00 for almost every x but for which limy!o u(x, y) = 00 for almost every x. (See Stein [St4, p. 238] for the exact details on this application of the work of Bagemihl and Seidel.) Consider a cube Q and the functions ue(x,y) = u(x,y + e). If (4.2.22) were true, then in particular
for every t > 0 with 0 1 and O2 independent of t. Since g(ue)(x) ::; g(u)(x) for every t > 0, Fatou's lemma implies
But the lefthand side is infinite and hence (4.2.22) is false. However, this does not necessarily mean that (4.2.18) is false. For the Lusin area function in lR~ we showed above that (4.2.17) and (4.2.20) are equivalent. For the gfunction, (4.2.22), although false, would imply (4.2.18) if it were true.
4.2 Sharp control of A by N
117
This follows from the same arguments used to show that (4.2.20) implies (4.2.17). But we cannot follow the line of reasoning used to show (4.2.17) implies (4.2.20) to produce a proof that (4.2.18) implies (4.2.22). One of the problems here is that there are simply no natural "stopping times" which we can use. Thus, there is no reason to believe that (4.2.18) and (4.2.22) are equivalent, hence the failure of (4.2.22) does not seem to necessarily indicate the failure of (4.2.18). Another possible approach towards a proof of (4.2.18) involves a reduction of the problem to an investigation of the constants in (4.2.17). Let Vn denote the volume of the unit ball in lRn. We observe that if, say u(x, y) is harmonic on lR~+1 and continuous on lR~+\ then limaLo Vn1a n A;u(x) = (g(u)(x))2. In equation (4.2.17), the constant C 1 depends on a. Let us suppose that we could show C 1 = o(an) asa 1 o. We could then pass to the limit in (4.2.17). This gives (4.2.18) in the case u is harmonic on lR~+l and continuous on lR~+l. For arbitrary u harmonic on lR~+1, (4.2.18) would follow from this case by considering the functions ue(x, y) = u(x, y + c). Unfortunately, a careful detailed examination of the proof of (4.2.17) yields at best C 1 = 0 (a 3n ) as a 1 O. Thus, for this approach, what is needed is a different proof of (4.2.17) which gives better control of the constant C 1 as a 1 O. Also relevant to the study of the gfunction are the examples of P. Jones (personal communication) which further emphasize the difficulty of this endeavor. Jones has shown that there exist harmonic functions u and /1 > /2 such that A)'2 u E Loo but An u ~ L oo . These examples can readily be modified (see Moore [Mol]) to give an example of a harmonic function u for which A,u E Loo but A,IU ~ L OO for every /' > /, and an example of a harmonic function u for which g(u) E Loo but A,u ~ Loo for every / > O. In separate but related direction, let us consider a lacunary sequence of positive integers nj; no = 0, n1 = 1, with n~+l > J1 > 1 for every j. For f E L1([_7l', 7l')) with Fourier coefficients ak, we let Sn(lJ) = L~=n akeik() denote the nth partial sum ofthe Fourier series of f and we set tljf(B) = Snj (B)  Snj_l (B), j = 1,2, ... , and tlo(B) = Sno(B). We then define
Qf(9)
~ (~I~;f(9)1')
, 1
A celebrated theorem of Littlewood and Paley asserts that for 1 < p < 00, IIQfll p ~ Ilfllp' The function Qf(B) is reminiscent of the martingale square function or the Lusin area function and hence, it is natural to ask if any of the above subgaussian estimates hold for Qf. Here, as in the previous problem, it seems that this investigation should begin with an analogue of Theorem 2.0.4. We can ask:
118
4. Sharp GoodA Inequalities for A and N
Question 4.2.15 Does there exist constants C 1 and C 2 such that whenever f E L1([n, n)), (4.2.23) Here we would want C1 and C2 to be independent of f, but do allow these to depend on the sequence {nj }~o, preferably only on the lacunarity constant J.L. If f is a lacunary series of the form 2::;:0 ajeinj(J where the nj are as above, then for each j, !1 j f(B) = aj and hence,
The Khintchine inequality for lacunary series, Zygmund [Zy2, vol. I, Theorem 8.20], gives that
(4.2.24) where Cp depends on p and J.L and Cp = O(.jP) as p  00. We now repeat the argument which appeared in the introduction to Chapter 2. Considering p = 2,4, ... in (4.2.24) and summing the power series for eX, we obtain (4.2.23) for the lacunary series f. Of course, many other more sophisticated subgaussiantype estimates are known for lacunary series, among these, the central limit theorem of Salem and Zygmund and the LIL's of Salem and Zygmund, Erdos and GaI, and M. Weiss, all of which will be discussed in Chapter 6. The above argument remains valid in the general case. That is, (4.2.23) follows from the LittlewoodPaley inequality Ilfllp ::; CpllQfllp, if we can prove this with Cp = 0(.jP) as p  00. Computations with the central limit theorem for lacunary series show that Cp = 0(.jP) is the best attainable. For general f E L1([n,n)), Pichorides [Pi] has shown this inequality with Cp = O(plogp) as p  00 and this seems to be the best known estimate now. Moore [Mol] has shown the inequality II flip ::; C.jPQf where Qf is the variant of Qf defined by Qf = (2::;:0 II!1j flloo 2)1/2. But Qf is larger than Qf so this seems to be of little help in answering Question 4.2.15. Question 4.2.16 Consider a function h which satisfies conditions (i)(iii) stated at the beginning of Chapter 1, Section 7. As in Theorem 1.7.2, this can be used to define an "area function", which was there denoted as Tf. Theorem 1.7.2 together with Lemma 1.7.3 show that for 1 < p < 00, IITfllp ~ IIfllp, Consider another function ¢ which satisfies the conditions of Theorem 1.4.7 and use it to create a "nontangential maximal function" as in Definition 1.4.6. Call this N f. Thus,
4.3 Application I. A Chungtype LIL for harmonic functions
119
for 1 < p < 00, liTflip ~ liN flip· Are there any goodA inequalities relating this area function and this nontangential maximal function? If we do not assume any relation between hand ¢, this seems almost intractable. Essentially we have discussed the only situations where this is known: when ¢ is either the Poisson kernel or the Gauss kernel and h is (respectively) a derivative of the kernel. However, Gundy and Irribarren [GI] have made some progress on this question. They have shown goodA inequalities relating a nontangential maximal function and an area function; their maximal function is created using a ¢ which satisfies a certain relationa "dilation equation", but their area function is created somewhat differently than we have done here. We end this section with some further remarks concerning the boundedness properties of square functions on the space BMO of ~n. If we let Aaf denote the square function of the harmonic function u in ~~+1 with boundary values f, it follows from Proposition 4.2.8 that if f E Loo, then A~J E BMO. As we have already mentioned, several authors, among them Kurtz [Ku] and Qian Tau [Qi], have shown that Aa : BMO 7 BMO. The arguments we presented above can be modified to obtain the stronger result that BMO 7 BMO, with similar results for VMO, the space of functions in ~n of vanishing mean oscillation. There are also bilinear versions of these results constructed for various other types of area functions. For this, we refer the reader to Banuelos and Brossard [BB] and to the subsequent papers related to this topic, [Che1], [Che2], and [Che3].
A; :
4.3
Application I. A Chungtype LIL for harmonic functions
The following 1IL was first proved by K.L. Chung [Chu1] for independent random variables. Theorem 4.3.1 Let X t be a continuous time martingale and recall that X; sup IXsl and St(X) = (X);/2. We have O<s:C;t
1· . f (lOglOg S t(X))1/2 X *  ~ i~~ S;(X) t  y8
(4.3.1)
almost surely on {S (X) = oo}.
To state our result for harmonic functions, we denote by r a(x, t) the cone of aperture a and vertex at (x, t) E ~~+l. For t < 1 the truncated version will be similarly denoted by r;(x, t). Notice that here that these truncated cones are a little different than those in Chapter 3. We make this change simply because these new cones are more adaptable to the Lipschitz methods we will use. The results for both types of truncations are equivalent. Using these cones, we define, as in (4.2.1) and (4.2.2)
Nau(x, t) = sup{lu(y, 8)1 : (y, s)
E
r a(x, tn
4. Sharp GoodA Inequalities for A and N
120
and A"u(x, t)
=
(
r
Jr",(x,t)
(s  t)lnlV'u(y, sWdyds )
l~
,
with a similar definition for the truncated operators N~u(x, t) and A~u(x, t), t < 1. As above, if t = 0 we simply write A~u(x) and N~u(x). The following theorem is motivated by Theorem 4.3.1. Theorem 4.3.2 (LowerHalf Chungtype LIL for harmonic functions) Let u be harmonic in lR~+1. Fix 0 < (3 < a and suppose there are points (xo, to), (Xl, tl) in lR~+1 such that A,B'u(xo, to) < 00 and NaIU(Xl' tl) < 00 for some (3' > (3 and a' > a. Then 1·
. f (IOgIOg(A.a u (x,t)))1/2 N ( ) > C (A .aU ())2 aU X, t X, t
(4.3.2)
1m In tlO
for almost every X E {x E lRn a, (3, a', (3' and n.
:
Aau(x) = oo}. The constant C depends only on
Problem 4.3.3 Prove the upper upperhalf Chungtype LIL for harmonic functions. (The full Chungtype LIL does hold for lacunary series as (6.1.10) in Chapter 6 shows.) Proof of LowerHalf of Theorem 4.3.1: In the introductory chapter we claimed that once we have sharp goodA inequalities on Lipschitz domains we can literally copy the martingale proofs over to the harmonic function setting. To illustrate this point more precisely, we first prove that the lower bound in Theorem 4.3.1 follows from the corresponding goodA inequality (4.0.10) in Theorem 4.0.4. We will then "copy" this proof to show that Theorem 4.3.2 follows from the corresponding goodA inequality, Theorem 4.2.1. Towards this end, fix D > 1, ", > 1 and v > 1. For k large, k 2: [(log",)l] + 1 = k* will do, set Ak = Rand
Define the stopping times (4.3.3) Note that since the function f(r)
=
(IOg[Ogr) 1/2 is increasing :s: Tk
Tk+1.
the martingale inequality (4.0.10) we obtain
P{STk(X) > DAk, X;k
:s: ckAk}:S: C
2
exp(vloglog",k)
C2
Applying
4.3 Application I. A Chungtype LIL for harmonic functions
L
121
00
P{STk (X) > OAk}
ko. Since {X* = oo} a~. {S(X) = oo}, we can pick such an w for which S(X)(w) = X*(w) = 00 and therefore X;k (w) = EkAk, by the continuity of the paths of X t . Since X*(w) = 00, we also have that Tk(W) i 00. For t sufficiently large we can then pick k > ko (w) such that Tk (w) < t ::::: Tk+ 1 (w). Then
We again use the fact that the function f(r) clude
=
( log ro r ) g
1/2
is increasing to con
Thus, we have
..
li~~f
(loglogS;(X)) 1/2 * S;(X) Xt
7r
::::
V8
(0 2 _1)1/2 Oy'rIV ,
(4.3.4)
almost surely on {S (X) = oo}. Since this holds for any 8 > 1, 7] > 1 and v > 1, the lower bound in (4.3.1) follows.
Proof of Theorem 4.3.2: We may assume that Ixl : : : 1, (xo, to) = (0,1) and A,61U(0, 1) = 1. Since (3 < (3', then A,6u(x,t) < 00 for all x E]Rn and t > 0; this is by Lemma 4.2.9 (or at least the ideas in the proof of that lemma). Since NalU(X1' td < 00 and a < a', we also have that Nau(x, t) < 00 whenever t > o. Now let 7] > 1 and set Ak = and for k:::: [(1og7])l] + 1 = k*, set
H
Ek 

(
1 ) v log log 7]k
1/2
4. Sharp Good A Inequalities for A and N
122
where 1/ is a constant we will specify momentarily. (In the proof ofthe lowerhalf of 4.3.1, the E:k'S were carefully chosen so as to obtain the precise lower bound 7f/VS. This precision was possible because we know the exact values of the constants in the goodA inequality (4.0.10). We now intend to use the goodA inequality of Theorem 4.2.1 and since the constants there are not precise, there is no point in being so careful here.) Now define the analogue of the stopping times in (4.3.3) by
The functions ¢k are Lipschitz and since f(r) = COg rO g r)1/2 is increasing, ¢k+l(X) :S ¢k(X), Also, Na.u(x, t) < 00 for every x E ]R.n and t > 0 implies that ¢k(X) ! 0 as k i 00 for each Ixl:S 1. Let (3 < "( < min ((3' , a). Set
u The Dk'S are Lipschitz domains as in Section 4.2 with Lipschitz constant depending only on "( = "((a, (3, (3'). For Ixi :S 1, the boundary of Dk is just the graph of ¢k and we can extend ¢k to all of]R.n by defining it to be the boundary of Dk for Ixi 2': 1. Thus, with this, Dk = {(x, t) : x E ]R.n, t > ¢k(X)}. Since (3 < "( < (3', there is an M depending only on "( and (3' such that for all x 2': M, r,8(x, ¢k(X)) C r,811 (0,1) C r,81 (0,1) where (3" = {Jltr. Thus, by Lemma 4.2.9, A,8u(x, ¢k(X)) :S L if Ixi 2': M. Hence, for k large enough, depending only on L, we have
for some C depending on M. Set "(' = {J~1. Then (3 < "(' < "( < a and r "'(' (z) ~ ~ Dk whenever Z E 8D k . Thus, by Theorem 4.2.1, for all such large k,
r ",((z)
O'{z E 8Dk : A,8u(z) >KAk' Na.u(z) :S E:kAk} :S O'{z E 8Dk : A,8u(z) > KAk, N"'(lu(z) :S E:kAd C :S C 1 exp(C2 1/loglogryk) = k2 ' if 1/
= 2/C2 • As in the case of martingales,
for all
Z
E 8D k and it follows that
L 00
k=k*
I{x E]R.n : Ixi :S 1 and A,8u(x, ¢k(X)) > KAk}1 < 00.
The BorelCantelli Lemma now shows that A,8U(X,¢k(X)) :S KAk eventually for almost every x. That is, for almost every x with Ixi :S 1, there exists ko = ko(x)
4.3 Application I. A Chungtype LIL for harmonic functions
123
such that A,aU(X, ko. Pick such an x for which both A,au(x) and Nau(x) are infinite. Let t be very small and choose k > ko such that (0) = 0, cI> not identically zero, and which satisfies the growth condition: cI>(2A) ~ Ccp cI>(A) for every A > where Ccp is a fixed constant. Here is a more thorough version of Theorem 1.7.9:
°
Theorem 4.4.1 Suppose then
(i)
Q,
(3 >
°
and cI> is as above. If u is harmonic on lR++l,
J~n cI>(Aau(x))dx ~ C J~n cI>(N~u(x))dx.
If the left hand side of (i) is finite, then limy>oo u(x, y) exists and is finite and constant for x E lRn. If u is normalized so that this limit is zero, then the converse inequality holds:
(ii)
J~n cI>(N~u(x))dx ~ C J~n cI>(Aau(x))dx.
Here the constant C depends only on
Q,
(3, n and the growth constant Ccp.
This seems to follow almost immediately from Lemma 4.0.2 and the goodA inequalities of Theorems 4.1.1 and 4.2.1. However, a closer examination reveals that there are a few technicalities. In Theorem 4.4.1, Q and (3 are arbitrary, but each goodA inequality required that one be bigger than the other. This is easy to overcome: since I{x E lRn : N,,/u(x) > A}I ~ I{x E lRn : Nou(x) > A}I for every A > 0, with the constants in the equivalence depending only on 'Y and 8 (Lemma 4.1.2), in our proof we may suppose in (i) that (3 > Q and in (ii) that Q > (3. Also recall that to obtain an estimate like (i) via a goodA inequality and Lemma 4.0.2, it is necessary to have the apriori estimate J~n cI>(min{1, Aau(x)})dx < 00, with a similar estimate involving N~u(x) necessary for (ii). To circumvent this difficulty, we will apply this goodA inequality technique to a related harmonic function which does satisfy the requisite apriori estimates. Then we will take limits to obtain (i) and (ii). This limiting argument is much
4.4 Application II. The BurkholderGundy theorem
125
like the proof of Theorem 1.7.8 and much like a limiting argument in Fefferman and Stein [FS; Theorem 8]. The argument will also borrow much from Burkholder and Gundy's original proof of Theorem 4.4.1. The original proof of this theorem used localized versions of the goodA inequalities which resulted in slightly fewer complications.
Proof of Theorem 4.4.1: (i) We assume a < (3, and we may assume the right hand side of (i) is finite. Fix 0 < c: < L < 00 and set v(x, y) = u(x, y + c:)  u(x, y + L). For convenience we set Xo = Y (and likewise So = t). Then
A~v(x) =
1 t I::. r",(x) j=O
(s,
J
t)1
2
t1ndsdt (4.4.1)
For (s, t) E r Q(x) and j fixed,
8U
18s/ s , t + c:) 
t'
t
8u t + L) I = (L  c:) I8iBs/s, 8 2u , I, 8s/s, t)
t
where is between + c: and + L. In particular, (s, t') E r Q(x). Since a < (3, there exists a constant e = eQ ,{3 such that the ball B = B((s, t'), et') is contained in r{3(x). Let r.p be a radial function supported on B(O,et') ~ ffi.n+l with
r
iIR n +1
r.p(x)dx = 1.
Such a r.p can be chosen so that also
whenever that
lal
= 2, where 0 depends on e and
n. From Theorem 1.1.4 it follows
Thus,
18~~:j (S,t')1 :s (LI~~~j (1J(S,t'))1 2d1J ) 2
(Llu(1J) 12 d1J )
:s OIBI (t')2(n+3) (N{3u(x) )21BI = O(N u(x))2(t')4 {3
< O(N{3U(X))2

(t+c:)4
126
4. Sharp GoodA Inequalities for A and N
This and (4.4.1) yields
J J
By assumption,
,P, P > 0, which relates the LP norms of the nontangential maximal function and Lusin area function. What is of further interest in this case is to find the best constants
128
4. Sharp Good'\ Inequalities for A and N
in these LP inequalities. In general, sharper constants can be obtained by using good'\ inequalities with better decay. We shall now discuss this. The goodA inequality
I{x E]Rn : A,au(x)
> 8A, Nau(x) :::; EA}I :::; Ca,,a,n
(82E~ 1) I{x E ]Rn : A,au(x) > A}I
of Burkholder and Gundy [BG2] and Lemma 4.0.2 shows that IIA,aullp :::; Ca,,a,nJP IINauilp for 0 < p < 00 (Burkholder [Bu4, p. 295], or see the discussion immediately after Theorem 4.0.3). The order of the constant on the right hand side, O(JP), is the smallest possible order of magnitude in this LPinequality. A similar goodA inequality in [BG2] with the roles of A and N reversed yields IIN,aullp :::; CpllAaullp with only Cp = O(p) as p + 00. In fact, even the exponential decay given by Murai and Uchiyama in (4.0.13) does not improve this order in p. We now would like to show how to use the subgaussian decay in the goodA inequalities of Theorem 4.l.1 to show that IIN,aullp :::; Ca,,a,nJPIIAaullp for 1 :::; p < 00, which is again the best possible order. In fact, we shall prove more. Once again, our results are motivated by those first proved for martingales and we recall these next. In [Dav1], B. Davis found the best possible values for the constants a p and Ap in the following LPinequalities for continuous martingales:
IIXllp :::; ApIIS(X)llp { and IIS(X)llp :::; apllXllp,
(4.4.5)
for 1 < p < 00, where we write X for the limit of X t as t ; 00. The constants are zeros of parabolic cylinder functions and confluent hypergeometric functions. They are both of order JP as p + 00. In [Wa], G. Wang proves that for 3 :::; p < 00 these constants are also best possible for discrete martingales Un} (with the more traditional square function S(f) defined by (2.0.4)) whose martingale difference sequence {d n } is conditionally symmetric. This means that for all n and real numbers T, P{dn+l > T Id1 , ... ,dn } = P{dn+ 1 < T Id1 , ... ,dn } almost surely. These martingales include the one dimensional dyadic martingales. We should mention here, however, that for arbitrary martingales Un}, the best constant Cp in the inequality I flip :::; CpIIS(f)llp for 2:::; p < 00 (again with the square function S(f) of 2.0.4) is p  1. This was shown by Pittinger [Pit] for integers p 2 3 and by Burkholder [Bu5] for all 2 :::; p < 00. Thus, for general martingales, the constants can be much larger. In [BY2], Barlow and Yor introduced the operators
4.4 Application II. The BurkholderGundy theorem
129
o < a ::; 1, and proved that with Ga,p = Oa(JP), as p
+ 00.
Motivated by this we have
Theorem 4.4.2 Suppose u is harmonic in lR~+l, not identically zero, 0 and 0 < p < 00. If u(x, t) + 0 as t + 00 then
o < a ::; 1
< (3 < 0:, (4.4.6)
In general, (4.4.7) Furthermore, both G and G' are Oa,cx,(3,nC/p) as P best possible.
+ 00
and this order in p is
The additional normalization in (4.4.6) that u vanish at 00 is necessary as can be seen by considering constant functions. As mentioned at the beginning of this Chapter, the desire to improve the rate of decay in c of the goodA inequalities arose partly in efforts to prove ratio inequalities between A and N. (Ratio inequalities were first proved for martingales by A. Garsia in [Gar].) The following ratio inequalities sharpen those in R. Fefferman, Gundy, Silverstein and Stein [FGSS] and Murai and Uchiyama [MU]. Theorem 4.4.3 Suppose u is harmonic in lR.~+ 1 and 0 < (3 < 0:. There exists constants G 1 and G2 , depending only on 0:, (3 and n, such that for any 0 < p < 00,
(4.4.8) and (4.4.9) We now proceed with the proof of Theorem 4.4.2. Since the proofs of (4.4.6) and (4.4.7) are the same, we just prove (4.4.6). Let M be a large real number to be specified later. We have
4. Sharp GoodA Inequalities for A and N
130
= I
+ II,
where _
Eij  {2
i
2i 
2ij
j  1
< N{3u ::; 2i+1 ,~< Aa u ::; M}·
Estimating the double sum and applying Theorem 4.1.1 with A = 2i I K and c K2 j 1M we obtain I::;
LL
2i+l ( 2 jl M i
)P
=
(2ij)ap
M
IEijl
iEZjEl\!
As functions of j, 2(1a)Jp is increasing and exp( C2~22 3) is decreasing. We estimate the last sum by an integral and successively set w = 22x and v = (C2M2/4K2)w to find "
=
2
2"
1
K 2) (1a)p/2 °O _1_ ( _4__ v((1a)p/2)le v dv 2 log 2 C2M2 C2M2 / K2
Substituting this last estimate back into the estimate for I gives
4.4 Application II. The BurkholderGundy theorem
This gives that for any 0 < p
00 and we have proved (4.4.6) with the correct behavior of the constant in p. The fact that the behavior of the constants in (4.4.6) and (4.4.7) (for a = 1) cannot be better than y'P, as p > 00, follows from the fact that the behavior of the LPconstants for the Riesz transforms cannot be better than p, as p > 00. However, in the unit disc one can even obtain more information on these constants using the central limit theorem for lacunary series and computations similar to those in the proof of Theorem 3.0.6. For more on this, see [BM3], [Ba2] and [Mol]. Proof of Theorem 4.4.3.' Again, the proofs of (4.4.8) and (4.4.9) are the same so we just prove (4.4.8). Define the sets Eij as in the proof of Theorem 4.4.2 with M = 1. The goodA inequality, Theorem 4.1.1, implies
IEijl:::; cexP (C'2 2j
)1 {x E]Rn: NfJu > ~} I,
where K is the constant appearing in that theorem. Thus,
:::; C
in
(NfJu)Pdx +
~
=
I
L L exp (Cl2;~~;~l)) 2(i+ l )PIE
ij l
iEZjEN
+ II.
i
II:::; Cp LLexp((16Cl  C')2 2j ) 2ip {x E]Rn: NfJu
>
~} I
tEZ JEN
:::; Cp
r (NfJu)Pdx,
J~n
if C 1 is chosen small enough relative to C'. This completes the proof of Theorem 4.4.3.
133
4.4 Application II. The BurkholderGundy theorem
In the same way, the goodA inequalities for caloric functions, Theorems 4.1.4 and 4.2.10 give: Theorem 4.4.4 Suppose u is a caloric function in lR~+1 and 0 < (3 < any 0 < a S; 1 and 0 < p < 00,
PNfju I p S; CII(PA",u) a lip II (PA.u)1a
Ct.
Then for
(4.4.13)
and (4.4.14)
with C and C f both Oa,,,,,fj,n(.jP) as p ~ and C4 such that
Ln
exp (C1
00.
Also, there are constants C1, C 2 , C3
(;~::) 2) (PNfju)Pdx S; C
2
Ln
(PNfju)Pdx
(4.4.15)
and (4.4.16) Once again, we could ask for more information than just the best order of magnitude in the above constants. Our methods above do not give any finer information. In this direction, let us consider the g*function in the unit disc defined by (3.4.1). Since, as we showed in Section 3.4, this function is the conditional expectation of the martingale square function, Jensen's inequality and Davis' result, (4.4.5) above, give Theorem 4.4.5 Let a p and Ap be the constants in (4.4.5). Then:
For 2 S; p
. inequalities relating the pair. These inequalities, first proved in Banuelos and Moore [BM4], provide the harmonic analysis analogue of the results of Bass and Davis. In Section 4.2 we proved a good>. inequality between Nau(x) and A,au(x) for a> (3. We did this in the more general setting of Lipschitz domains simply because it caused us no extra effort to do so and because our estimates then lead to LIL's. We do the same here. Recall our definitions: for P = (x, Y) E JR.~+l, we set r a(x, y) = r a(P) = {(s, t) : Ixsl < a(t and define
yn
Nau(x, y) = Nau(P) = sup{lu(s, t)1 : (s, t) E r a(pn and
5. GoodA Inequalities for the Density of the Area Integral
138
Likewise, we will now define
Dau((x, y); r) = Dau(P; r) =
r
JraCP)
(t  y?n f},(u(s, t)  r)+(dsdt)
(5.0.6)
and
DaU(X,y) = Dau(P) = supDau(P;r). rEIR
The following is the analogue for harmonic functions of (a) and (b) in Theorem 5.0.1. ; IR be a Lipschitz function and let D = {(x,y) : x E IR n , y E 1R, y > ¢(x)}. Suppose u is harmonic in D and 0 < (3 < a < where M is the Lipschitz constant of ¢. There are constants K 1 , K 2 , G1 , G2 , G3 and G4 , with K1 and K2 > 1 and all depending only on a, (3, nand M, such that if A > 0, 0 < E < 1 then
Theorem 5.0.2 Let ¢ : IR n
(a)
I{x
E
ir
IR n : N(3u(x,¢(x)) > K 1 A, Dau(x,¢(x)) :::; EA}I :::; G1 exp ( 
(b)
~2)
I{x E IRn
:
N(3u(x, ¢(x)) > A}I
I{x E IRn : A(3u(x, ¢(x)) > K 2 A, Dau(x, ¢(x)) :::; EA}I :::; G3 exp ( 
~; )
I{x E IRn
:
A(3u(x, ¢(x)) > A}I.
Note that (b) involves a subgaussian estimate. Just as before, this estimate will lead to an LIL. The estimate in (a) is not of subgaussian type, however, as we will later show, it is, in some sense, the sharpest possible. (This will be made precise.) The inequality in (b) is in this same sense the sharpest possible. All of this will be discussed in the third section of this chapter. Our second theorem is similar to Theorem 5.0.2 but with the roles of the functions reversed. For this theorem it will be necessary to work with "smoother" versions of the density and maximal density. Let cp(x) be a smooth, positive, radially symmetric function supported on a ball B(O, (3). Then set
Du(x; r) =
r
JIR"j+l
tCPt(x  s)f},(u(s, t)  r)+(dsdt)
Du(x) = supDu(x;r)
r E IR (5.0.7)
rEIR
where, as usual, we have set CPt(x) = t~CP (~). Both Du(x;r) and Du(x) depend on the choice of cP and this should probably be indicated by a notation such as D 0 and any 0 < p < (3 we can always find a Coo function
A}I. The unfortunate aspect of (b) is the restriction on (3 and a. This result should no doubt be true only under the assumption (3 < a, but we have been unable to prove it and we leave it as an open question. If we assume only that (3 < a, then in Banuelos and Moore [BM4], it is shown that (b) holds with the expression C 3 exp (~) replacing the expression C3 exp (~). In fact, this is shown in the more general setting of Lipschitz domains. The proof uses the representation of the density function in terms of the conditional expectation of the local time and the results of Barlow and Yor. Also [BM4] contains a different proof of a version of (b) which has as hypothesis (3 < a. However, the conclusion ofthis result is merely the same goodA inequality with the expression C3 exp(  gCf;3) replacing the expression Furthermore, the proof does not seem to be adaptable to the setting C3 exp ( of Lipschitz domains. In keeping with the general theme of this monograph, we find it more desirable to prove a subgaussian estimate like Theorem 5.0.3(b) rather than these other results, despite the annoying and probably unnecessary restriction on (3 and a. Another advantage of Theorem 5.0.3(b) as written is that it is in some sense the sharpest attainable. This will be made precise and discussed later in the third section of this chapter. Theorems 5.0.3(a) and (b) as written should probably be true in the setting of Lipschitz domains, but our proofs do not seem to be adaptable to that situation. We leave this as another open problem.
*).
140
5. GoodA Inequalities for the Density of the Area Integral
This chapter will be divided into 4 sections. In Section 5.1 we prove Theorem 5.0.2 and in Section 5.2 we will prove Theorem 5.0.3. In Section 5.3 we will show some corollaries, show the sharpness of our results and present some further discussion. Finally, in Section 5.4 we present an application of the techniques of sections Section 5.1 and Section 5.2 to the study of the space L log L within Hl. These results of Brossard and Chevalier [BC2] further reinforce the notion of the D functional as an analogue oflocal time. Much of this chapter, in particular, Theorems 5.0.2 and 5.0.3(a) is taken from Banuelos and Moore [BM4], which in turn, is based on earlier work of Gundy and Silverstein [GS] (see also Gundy [Gu2]). The proof of Theorem 5.0.3(b) is from [M02].
5.1
Sharp control of A and N by D
Our proof of Theorem 5.0.2 follows exactly the strategy of Section 4.2; in both (a) and (b) we will build a "sawtooth" region over {x E ~n : Dau(x, ¢(x)) > c'\}. That is, we set E = {x E ~n : Dau(x, ¢(x)) > c'\} and D' = UXEEc f a(x, ¢(x)). Then aD' is the graph of a Lipschitz function, call it 1jJ(x). For Theorem 5.0.2(a) we will estimate IIN,au(x,1jJ(x))IIBMO and for Theorem 5.0.2(b) we will estimate IIA~u(x,1jJ(x))IIBMO. In both cases these BMO estimates will be obtained as in Section 4.2; for each cube Q ~ aD' we will form an auxiliary domain n above Q and consider the contributions to either N,au(x,1jJ(x)) or A~u(x, 1jJ(x)) from both the part of the cones f,a(x,1jJ(x)) inside n (the part "close by Q") and the part outside n (the part "far away from Q"). Exactly as in Section 4.2, the "close by Q" part will be estimated using Green's theorem arguments on n and the "far away" part will be controlled using gradient estimates. With these BMO estimates in hand, the good'\ inequalities will follow exactly as in Section 4.2. To get this program under way, we first show a lemma that provides the necessary gradient estimates to control boundary terms and other errors that will arise in our approximations. This is an analogue for the Dfunctional of the gradient estimates of Lemma 2.3.1 and will be just as indispensable. Unfortunately, its proof is somewhat longer.
Lemma 5.1.1 Suppose a > f. There is a constant C, depending only on a, 'Y and n such that ifu is harmonic on fa(x) and if (s,t) E f,(x) then
tIV'u(s, t)1 :::; CDau(x). Proof: Fix (8, t) = Po E f,(x). We may assume that u(Po) = 0; otherwise consider u  u(Po). Choose 10 > 0 so that B(Po, 4/0) ~ f a(X); if '0 is chosen as large as possible then '0 ~ Ct where C depends only on a and 'Y. For j = 1,2,3,4 set B j = B(Po,j,o) and set M j = sup{lu(z,y)1 : (z,y) E B j }. By the subharmonicity of IV'ul and the GundySilverstein formula (5.0.4) we have:
t2IV'u(8,t)12:::; C
r IV'u(z,y)1 ylndzdy
lE2
2
5.1 Sharp control of A and N by D
s.CJ M 2
{
M2 JB2
141
~(u(z,y)r)+ylndzdydr
s. CM2 Da u (X). Thus, (5.1.1) using similar reasoning we can also conclude that if (z, y) E B2 then YIV'u(z, y)1 s. C.;M3JD a u(x). Since u(s, t) = 0 and for (z, y) E B 2, Y ~ 2Cro, it then follows that (5.1.2) Now consider B4 and apply Green's theorem to lu(P)rl, P E B 4, r E lR and G(P,Po) = IP  pol1n  (4ro)1n. Technically, we must approximate lu(P)  rl by smooth functions of u(P)  r and then take limits. To do this, we consider a Coo function a(r) which has fIR a(r)dr = 2, a( r) = a(r), supp a ~ [c:, c:] and set a,,(r) = ~a(~). Let b,,(r) be smooth functions satisfying b~(r) = a,,(r) and b,,(O) = b~(O) = O. Then b,,(r) i Irl as c: ! O. We now apply Green's theorem. We
remark that here we are using essentially the same argument as in the proof of the mean value property, Theorem 1.1.3. We obtain:
{
JB4
~(b,,(u(P)r))G(P,Po)dP=
C:ro JaB4 ( b,,(u(P)r))d(J(P)C~b,,(u(Po)r).
However, by assumption, u(Po) = o. Also, ~(b,,(u(P)r)) = a,,(u(P)r)IV'u(P)12, so that by the formula of Gundy and Silverstein, (5.0.4), this last equation becomes:
( a,,(s) {
JIR
JB4
~(u(P) 
r  s)+G(P, Po)dPds =
C:ro JaB4 ( be(u(P)  r))d(J(p)  C~be(r).
The arguments in Brossard [Brl, Lemma 2] show that as a function of r, fB4 ~(u(P)  r)+G(P, Po)dP is continuous. We then let c: t 0 in the above expression to obtain:
2 {
JB4
~(u(P) 
r)+G(P, Po)dP
=
C: (
ro JaB4
lu(P)  r)ld(J(P) 
C~lrl.
We remark that since ~lu(P)  rl = 2~(u(P)  r)+ this last equation is exactly what we would have obtained had we formally applied Green's theorem to lu(P)rl and G(P, Po) on B 4. (See Gundy [Gul], [Gu2], or Gundy and Silverstein [GS] for similar applications of Green's theorem.) Rearranging this last equation gives: (5.1.3)
5. GOOdA Inequalities for the Density of the Area Integral
142
[
lB4
~(u(P) 
r)+G(P, Po)dP
S [
~(u(P) 
r)+G(P, Po)dP +
S [
~(u(P) 
r)+G(P, Po)dP + C [
S [
~(u(P) 
r)+G(P, Po)dP + CDo:u(x).
lBI
l~ lBI
[
lB4\BI
~(u(P) 
l~\~
r)+
~(u(P) 
~l
ro
dP
r)+ pi n dP
This and (5.1.3) gives:
~ [ lulda S ro laB4 Choosing r
Clrl + [ ~(u(P) lBI
r)+G(P,Po) dP
(5.1.4)
+ CDo:u(x).
= M2 in (5.1.4) yields: (5.1.5)
;f;
Simple estimates for the Poisson kernel show that for P E B 3 , lu(P)1 J8B4 lulda. This and (5.1.5) show that:
s
(5.1.6) Substituting (5.1.6) into (5.1.2) we have
Consequently, M2 S CDo:u(x) and this and (5.1.1) complete the proof of the lemma. In Section 4.2, it was necessary to compare A,eu(x, Yl) and A,eu(x, Y2) for Y2 > Yl; this was done in Lemma 4.2.9. Here it will be necessary to compare Do:u(x, Yl) and Do:u(x, Y2) for Y2 > Yl· For n = 1 it is clear that Do:u(x, Y2) S Do:u(x, Yl), but for n ;::: 2 this is no longer obvious. The following lemma will allow us to make the necessary comparisons. However, it will be necessary to.show this in greater generality than we did when we showed the corresponding lemma for area functions. Nevertheless, the proof of this lemma for the Dfunctional will be essentially the same as the proof of Lemma 4.2.9. Let r(p) be a cone in lR~+1, either infinite or truncated, with vertex P. We do not assume r(p) has axis parallel to {(O,y) : Y > O}.
5.1 Sharp control of A and N by D
143
For u harmonic on r(p) we define Du(P;r)
=
Du(P)
=
r
ir(p)
d((s,t),p)lnA(u(s,t) r)+(dsdt)
(5.1.7)
supDu(P;r). rEIR
Note that if r(p) = r",(x,y), then Du(P;r) as defined in (5.1.7) and D",u((x,y);r) as defined in (5.0.6) are equivalent up to constants depending on a and n. Lemma 5.1.2 Suppose u is harmonic on r",(x,y), a> p and r(p) ~ rp(x,y). Then Du(P) ~ LD",u(x, y), where L is a constant which depends only on a, p and
n. Proof: First we note that there exists a constant Co, depending only on a and p such that B(P, 2Cod(P, (x, y))) ~ rp(x, y) where p = For j = 1,2, set B j = B(P, jCod(P, (x, y))). . If (s, t) E r(p)\B l then d((s, t), (x, y)) ~ (1 + 0 )d((s, t), P) so that
Pt"'.
6
r
ir(p)\Bl ~ ( ~
d((s,t), p)lnA(u(s,t) r)+(dsdt)
1 l+C
)lni
o
d((s,t), (x,y))lnA(u(s,t)r)+(dsdt)
(5.1.8)
r(p)\Bl
CD",u(x, y).
Let G(Q, P), Q, P E lR.n + l be the Green's function for B2 with pole at P. Then for (s, t) E r(p) n B l , d((s, t), p)ln ~ G((s, t), P). This and Green's theorem then yields:
r
d((s,t), p)lnA(u(s,t) r)+(dsdt)
ircP)nBl
~C
r
iB2
G((s,t),P)A(u(s,t)r)+dsdt
= CJ(ffB2 )
(5.1.9)
kB2 ((u(s, t)  r)+  (u(P)  r)+)dCJ(s, t).
Here we note that technically, to apply Green's theorem we must approximate u(s, t)  r by a smooth function of u(s, t), then apply Green's theorem, and then pass to a limit. The details of this argument are exactly the same as in a similar application of Green's theorem in the proof of Lemma 5.1.1. As in that case, we ultimately obtain the same formula we would have obtained had we formally applied Green's theorem.
5. GoodA Inequalities for the Density of the Area Integral
144
Now B2 (Y so that r a' (P) 2A, Dau(P) :::; cA}
= o{P
E EC
:::; CI{x E lRn
N{3u(P) > 2A}
:
:
N{3u(x,7{i(x)) > 2A}1
:::; G1 exp ( ~2) I{x E lRn :::; G1 exp ( 
5.2
:
Nf3u(x, 7{i(x))
~2) O"{P E aD : Nf3U(P)
> A}I
> A}.
Sharp control of D by A and N
In this section we will prove the goodA inequalities in Theorem 5.0.3. The proof of this theorem has ideas in common with the proof of the goodA inequalities in Theorem 5.0.2, and the proofs of several of our other theorems. We will still use gradient estimates to estimate contributions from the "far away" parts of cones. For part (a) we will use Green's theorem to estimate the contributions from the "close by" parts of cones and for part (b) we will estimate these "close by" parts using "invariance principle" techniques similar to those used in the proof of Lemma 4.1.3. The problem is that these arguments readily produce estimates for Du(x; r) for each fixed r, but what we want, of course, is an estimate for Du(x) = sUPrEIR Du(x; r). This will make our task more difficult as we will need to estimate Du(x; r) as both x and r vary. We recall that in Theorem 5.0.3 we are considering the "smoother" versions of the density and maximal density given by (5.0.7). Throughout the course of the proof, we will need to consider various versions of these defined on sub domains
5.2 Sharp control of D by A and N
of lR~+l. For k > 0 and a sub domain n
D~u(x;r)= D~u(x)
=
r
c
lR~+l, we set
tCPt(xs)t!.(u(s,t)r)+(dsdt)
inn {t 0 and D~u(x; r) and D~u(x) are defined as above. Suppose also tlV'u(s, t)1 ::; 1 for every (s, t) E where ::J n is a domain which has the property that there exists a Co > 0 such that ((v,w) : dist((v,w),n) < Cow} c Then there exists a constant C = C(Co, n) such that if x, y E lRn ,
n
n
n.
(a)
ID~u(x; a)  D~u(y; a)1 ::;
(b)
ID~u(x)  D~u(y)1 ::;
Clxkyl,
a E lR
Clxkyl
Proof: Fix x, y E lR n and set
We recall that supp cP
R
= (r{3(x) u r{3(Y)) n n n {t 2: k}.
c::;;;
B(O, (3). Then
ID~(x; a)  D~ (y; a)1 =
Il
t(cpt(x  s)  CPt(Y  s))t!.(u(s, t)  a)+(dsdt)
::; Clx 
I
yll Cnt!.(u(s, t)  a)+(dsdt).
!.
We may assume that Co < Then we can find a Coo function ¢(s, t) such that o ::; ¢(s, t) ::; 1 for all (s, t) E lR~+l, ¢(s, t) = 1 on R, supp ¢ c::;;; {(v, w) : dist((v,w),R) < Cow} c::;;; and 1V'¢(s,t)l::; Note that for t < (1 Co)k, I{s : (s, t) E supp¢}1 = 0 and for t 2: (1 Co)k,
n
f.
I{s: (s, t) E supp¢}1 ::; I{s: dist((s,t),r{3(x)) < Cot}1
+ I{s: dist((s,t),r{3(Y))::; Cot}l::; Ctn.
5. GoodA Inequalities for the Density of the Area Integral
150
Therefore,
r rn~(U(s,t)a)+dsdt::;J
In
: ; oj
: ; 01 : ; 01
rn¢(s,t)~(u(s,t)a)+dsdt
supp¢
(tnIV¢(s,t)1 +rnI¢(S,t)) IVu(s,t)ldsdt
supp¢
r n  2 dsdt
supp¢
o
00
r
n  2 1{s:
(s,t) E supp¢}ldt
(ICa)k
::; k' Then (a) follows; (b) follows by taking supremums. For certain subdomains W ~ ~~+1 we will need to create a slightly different version of D{vu(x; r) which approximates it, but is easier to estimate. This new version will be obtained from D{vu(x; r) via numerous integrations by parts. So at this point it is convenient to state and prove a technical lemma that will allow us to control the boundary terms that arise from these integrations by parts. Lemma 5.2.2 Suppose (3 < /, E ~ ~n and W = UXEE f /,(x). Suppose also that p is a function supported on B(O, (3). Let h > and set f(x) = f~(x) n W. There is a constant 0, depending only on (3, /, n such that for x E ~n,
°
r
I Jar(x) rnp
(~) du(s, t)1 t
Here u denotes surface measure on
::; Ollplloo.
of (x).
Proof: Clearly 8r(x) ~ {(s, h) : Ix  sl ::; (3h} U(r f3(x) noW) U{ (s, t) : Ix  sl = (3t}. Since p is supported on B(O, (3), the integral of Pt(xs) vanishes on the third set. The integral of Pt(x  s) over the first set is clearly bounded by Ilplloo. To control the integral over the second set we note that since oW is the graph of a Lipschitz function with Lipschitz constant ~, elementary geometric arguments show that u(f f3(x) noW) ::; O(inf{ t : (s, t) E r f3(x) noW})n where 0 depends on 'Y, (3 and n. Then,
r rnp (~) du(s, t)1 I Jrf3(x)naw t ::; Ilplloou(f f3(x) n W) (inf{ t : (s, t) E f f3(x) n W} )n ::; Ollplloo , which finishes the proof of the lemma.
5.2 Sharp control of D by A and N
151
Now consider a set E ~ IR n , suppose a> "/ > {3 and set W = UXEE f '"'((x). For h > 0 consider D{Vu(x; r) and D{Vu(x) as defined in (5.2.1), set W' = UXEE f ",(x) and define N~hw'u(x) = sup{lu(s, t)1 : (s, t) E f;h(x) n W'}. We now create a new version of D{V'u( x; r) which approximates it. Lemma 5.2.3 With the notation as in the previous paragraph, there exists a vector
valued function (x) = (l(X), ... ,n+1(x)), with the following properties: (a)
Each i(X) is supported on B(O,{3), has mean value zero, and is smooth.
(b)
If we set
J
D{Vu(x;a)=
t(xs)·V(u(s,t)a)+dsdt
r3(x)nw then for
lal ::; N~~w'u(x)
we have
ID~u(x; a)  D~u(x; a)1 ::; CN~~wlu(x). Here C depends only on a, ,,/, {3, n and our original choice of cp. Proof: For typographical convenience, we will write f(x) for f~(x)nW throughout this proof. Then by the divergence theorem,
D~u(x; r) = =
1
1
r(x)
tCPt(X 
r(x)
+
r
s)~(u(s, t) 
a)+dsdt
V (tCPt(x  s))· V(u(s, t)  a)+dsdt
Jaqx)
tCPt(XS):! (u(s,t)a)+d(](s,t) un
= I + II. Here there are several technicalities in this application of the divergence theorem, since neither the functions involved nor the regions involved are smooth. To overcome these, we first note that the region f(x) is of the kind considered by Stein [St4, p. 206]. Stein shows that there exist smooth regions V8 such that V8 ~ f(x) for every 8> 0 and V6 f(x) as 8 ! O. We may then apply the divergence theorem on each V6 as before. (See the proofs of Lemmas 5.1.1 and 5.1.2.) That is, we apply the divergence theorem to smooth approximations of (u(s, t)  a)+ and then take limits. We actually then obtain a similar formula on each V6 and subsequently let
r
8! O.
To estimate II, note that for (s, t) E 8f(x), It tn (u( s, t)  a)+ I ::; tIVu(s, t) I ::; CN~~wlu(x) where the last inequality is essentially just Lemma 2.3.1. (Lemma
152
5. GoodA Inequalities for the Density of the Area Integral
2.3.1 does not apply directly, but simply note that in this situation, there exists a constant Co, depending only on a,{3,"(, such that if (s,t) E af(x), then B((s, t), Cot) 4, Ihl i Q 1i5{Vu(x; a)  i5{Vu(x; b)IPdx < Cia  bl ~ where C is a constant indepen
dent of a and b. Then for almost every x E Q, sup ID{Vu(x; a)1 ~ 8(4~)III!% Br(x) aEr
+ CN~hWlu(x) + 1i5{Vu(x; d)l. '
Proof: Definition (5.2.2) and the hypothesis, combined with Fubini's theorem shows that Br(x) < 00 for almost every x E Q. Furthermore, Kolmogorov's continuity theorem implies the existence of a measurable function f(x, a), x E Q, a E I which is continuous in a for every x and such that for each a E JR, f(x, a) = i5{Vu(x; a)
154
5. GoodA Inequalities for the Density of the Area Integral
for almost every x. A simple Fubini argument shows that if we define ih(x) as in (5.2.2) with f(x, a) in place of i5{vu(x; a) then BJ(x) = ih(x) a.e. on Q. Since f(x, a) is continuous as a function of a, the Garsia, Rodemich, Rumsey inequality 1 1 1 states that If(x,a)  f(x,b)1 8(4 v )la  bl"2 v B J(x) for every a, bE R Thus, there exists a null set A such that if x E Q\A then
s
A
whenever a E I is rational. So if x E Q\A, a is rational, and Lemma 5.2.3 implies that
lal s N~hwlu(x) then '
But this is true even if lal > N~hWlu(x) since then left hand side is O. Since D{vu(x; a) is lowersemicontinuous'as a function of a (see Gundy and Silverstein [GS]), the conclusion follows. We have now finished the preliminaries that will be used in both the proofs of the goodA inequalities in Theorem 5.0.3 and begin the proof of Theorem 5.0.3(a). We let A > 0, 0 < c: < 1 be fixed. We set 'Y = ,B~"', E = {x E IRn : N",u(x) S C:A} and define W = UXEEfy(x) and W' = UXEEf",(X). We may assume E # 0 so that W # 0. Then
lui S C:A on W tl\7u(s, t)1 S CC:A for (s, t)
E
W.
(5.2.3)
The first of these statements is obvious and the second follows from the gradient estimates of Lemma 2.3.1. For x E IRn , a E IR we now consider Dwu(x;a) and Dwu(x) as defined in (5.2.1). As before, we have the corresponding Key Estimate:
If W is as above, and Dwu(x) is defined by (5.2.1), then IIDwullBMO S CC:A where C = C(a,{J,n, C:A Dwu(x; a) = 0 if lal > C:A Dwu(x; a) = D{vu(x; a) + Dwu(x; a) Dwu(x) S D{vu(x) + DWu(x).
(i) D{vu(x; a) = 0 if
(ii) (iii)
(5.2.4)
5.2 Sharp control of D by A and N
155
Lemma 5.2.1 provides a way to estimate D'{;.ru(x; a) and D'{;.ru(x). Our next lemma will estimate D{vu(x; a). Lemma 5.2.6 Suppose
Here C
=
1~p
A}I :::; "2 IQI .
We will show that for such Q, C2 ) I{x E Q : Du(x) > KA, A",u(x) :::; cA}1 :::; C1 1QI exp ( """"€2
.
Summing over Q gives the result. Fix such a Q and an c with 0 < c < 1; we may also assume that there exists an Xo E Q such that A",u(xo) :::; cA. Also we may pick Xl E Q such that DU(X1) < A. Our choice of (3 insures that r/3(x)\r~(x) ~ r~ (xo) whenever X E Q. (This just requires 4(3 < a.) As before, we set DT u(x; a) = Du(x; a)  Dhu(x; a), DT u(x) = sUPaEIR DT u(x; a). Then Lemma 5.2.1 implies
IDT u(x)  DT u(xdl :::; CcA for all X E Q. Thus, IDTu(x)1 :::; CcA + A for all x E Q. Set KI = K  (C + 1). Then
I{x E Q : Du(x) > KA, A",u(x) Set
E = {x Replace u by
c
2 AlU
E
< cA}1 :::; I{x E Q: Dhu(x) > KIA, A",u(x) :::; cA}I·
Q : Dhu(x) > K 1 A, A",u(x) :::; cAl.
in Proposition 5.2.8; this gives
and the result follows by taking K 1 , and hence K, large enough. Thus, we have reduced matters to the proposition. Before proceeding with this we show a lemma that will be used several times throughout the proof of the proposition.
160
5. GoodA Inequalities for the Density of the Area Integral
Lemma 5.2.9 With the notation as above, let q denote the center of Q and suppose u(q,2h) = O. (Here h = 2.Jii,i(Q)f31 as in Proposition 5.2.8.} Then there exists constants C 1 and C 2 , which depend only on 01,13 and n, such that
1 r IQT 1Q exp[N22h,Bu(x) 
2
C 1(Aa u (x)) ]dx ~ C2 .
(5.2.7)
Proof: We use Lemma 4.1.3. In the present situation, however, we have 213 in place of the 13 which occurs there, so that the corresponding h in the statement of Lemma 4.1.3 is h = 2y'rif(Q)/2f3 which is smaller than the 2h = 4y'rif(Q)/f3 we are using in the statement of this lemma. But, as we have noted, Lemma 4.1.3 remains valid for any larger value of h than stated there. (See the remarks after the proof of Lemma 4.1.3.) Thus, for all A > 0,0 < c < 1,
I{x E Q: N?gu(x) > K)", Aau(x)
~ c)..}I~ C3 1QI exp (  ~:).
(5.2.8)
For k,j E {O, 1,2, ... } set Ekj
= {x E Q
: k ~ N:;gu(x)
< k + 1,
j ~ Aau(x)
< j + 1}.
To complete the proof of the lemma, we merely break up the integral in (5.2.7) into a sum of integrals over the E kj , noting that we really only need to consider those E kj for which j2 is (approximately) less than k. Then use (5.2.8) and note that if C 1 is chosen large enough (depending on K and C4) the resulting sum converges. We leave the details to the reader. We remark that here we have really shown that an inequality like (5.2.7) is valid whenever the aperture used in the definition of the nontangential maximal function is less than the aperture used in the definition of the Lusin area function. We did not write the lemma in this generality simply because we will only use it in the context of the proof of Proposition 5.2.8. This Lemma and proof should also be compared to the discussion in Problem 4.2.14 where (5.2.7) is shown in another way. As in the proof of Theorem 5.0.3(a), it is fairly easy to obtain estimates for j)hu(x; a) and we do this in the next lemma. Here we connect with the ideas of Chapters 2 and 3 and exploit the particular form of j)hu(x; a). Lemma 5.2.10 There are constants C 1 and C 2 independent of a E IR such that
Proof: Recall that j5hu (x; a)
=
r
1r~(x)
t(x  s) . ~(u(s, t)  a)+dsdt
5.2 Sharp control of D by A and N
161
where c1? = (c1?1, .•. ,c1?n+1) and each c1?i is supported on B(O,{3), has mean value zero, and is smooth. As in Chapter 2, we write jjhu(x; a), x E Q as the sum of martingales. An examination of the proof there shows that the square function of each of these martingales is dominated by
(See also (3.1.7).) By our choice of {3, 32..fii{3 < Q so this last expression is dominated by Aau(x). The result then follows from the corresponding theorem for martingales, Theorem 2.0.1. This proof should be compared to the proof of inequality (4.2.20) which is essentially the same. Of course, we really need estimates for Dhu(x). To obtain these, we will break up Dhu(x) into "smaller" maximal densities. For j an integer set
Dju(x)
=
sup{Dhu(x; a): a E [j,j + I)}.
Then
Dhu(x) ::; sup{Dju(x):
Ijl::; Ni$u(x) + I}.
We have Lemma 5.2.11 There are C 1 and C 2 independent of j such that
i
Before proving Lemma 5.2.11 we show how it implies Proposition 5.2.8. For set Ei = {x E Q: i::; Ni$u(x) < i + I}. Then
= 0,1,2,3,
I~I
k
::; L
1 IQI
00
exp[Dhu(x)  C 1 (Aa u(x))2]dx
1
i=O
i+2
::; ~ .L 00
.=03=.2
0::
exp[ sup Dju(x)  C 1 (Aa u(x))2]dx Ijl (3 and a' > a. Then .. hmmf tlO
(
loglog(A1u(x, t)) ) (AI ())2
for almost every x E {x E lR.n only on a, (3, a', (3' and n.
fJu x, t
: A~u(x)
1/2
1
Dau(x, t) ~ C
= oo}. Here C
is a constant which depends
This result is motivated by (5.3.2). As in the case of Theorem 4.3.2, the upper bound is an open problem. Also both, the upper and lower half analogues of (5.3.1) for harmonic functions, are open. An upper bound would follow, as in Chapter 3, from a Lipschitz domain version of Theorem 5.0.3(b) which we do not have.
5. GoodA Inequalities for the Density of the Area Integral
166
The inequality S;(X) :::; L;X; shows that the LIL's in (5.3.1) and (5.3.2) are closely related to the Chung LIL and the Kolmogorov LIL, at least in as much as we do not ask for best constants on the right hand side. For example, we showed in Chapter 3, (3.0.5), that lim sup t>oo
X* t
'}
)
d)"
I{x: MDf3u(x;O) > C2 )..}ld)".
This last estimate is just equation (6) in Stein [St3]; here C1 , C2 depend only on n. Set b = C 2 /2C where C is the constant appearing on the right hand side of Lemma 5.4.2. Then I{x : MDf3u(x; 0) > C2 )..} I ~ I{x : Df3u(x) > b)..}1
+ I{x : Mu(x) > b)..}I·
Using the fact that I\Df3UI\1 ~ CI\NaUI\1 (which follows from Theorem 5.3.3(c)) and the weak type 11 estimate for the HardyLittlewood maximal function we obtain:
r
Df3u(x; 0) log+ Df3u(x; O)dx
JR."
~ C11= I{x: Df3u(x) > b)..}ld)" + C1 ~ CI\Df3UI\1
~ CI\NaUI\1
C1=11 +"\ b
C +b
~ CI\Naulh + C
1 b
/\
{x:lu(x)I>H
I{x: Mu(x)
> b)..}ld)"
lu(x)ldxd)"
1
{x:lu(x)I>H
lu(x)1
1=
2IU (X)'
b
1 >.d)"dx
r lu(x)l(l + log+ lu(x)l)dx.
JR"
By Fatou's lemma, the last integral is dominated by
r
sup lu(x, y)l(l y>O JR"
+ log+ lu(x, y)\)dx.
This completes the proof. For various other applications and uses of the Dfunctional, we refer the reader to [BC2], [BC3], [BC4], [BC5], [Che4], and the survey article [Br2].
Chapter 6 The Classical LIL's in Analysis In this chapter we will describe how the LIL's in Chapter 3 are related to the classical LIL's for lacunary series and to the more recent LIL's for Bloch functions.
6.1
LIL's for lacunary series
The unit disc in the complex plane will be denoted, as in Chapter 3, by D and the unit circle by T. Throughout this section, a real trigonometric series with partial sums m
8 m (8)
=L
(ak cos nk8 + bk sin nk8)
k=l
which has nk+1/nk set
> q > 1 will be called a qlacunary series. For such a series we
We remark that the definition of Bm used here differs from that in Chapter 3 by the factor in the parenthesis. With this present definition, B! is the variance of 8 m (using the probability measure d8/27r). This will make the following LIL's resemble the Kolomogorov LIL, (3.0.1). It has been shown over the years that lacunary series exhibit many of the properties of partial sums of independent random variables. (In the modern language of probability, lacunary series are examples of "weakly dependent" random variables.) The following, to the best of our knowledge, is the first LIL in analysis.
!
Theorem 6.1.1 (SalemZygmund [SZ2]) Suppose that, with the notation above, 8 m is qlacunary and the nk are positive integers. Suppose also that Bm + 00 as m + 00 and 8 m satisfies the Kolmogorovtype condition:
173 R. Bañelos et al., Probabilistic Behavior of Harmonic Functions © Birkhäuser Verlag 1999
6. The Classical LlL's in Analysis
174
for some sequence of numbers Km ! o. Then . Sm((}) hmsup oo y'2B;" log log Bm for almost every () E T.
(6.1.1)
As we have clearly demonstrated by now (hopefully!), lower bounds in Kolmogorovtype LIL's are much more difficult to obtain, and this is the case here. Erdos and Gat [EG] were the first to make progress in this direction. They showed that if Sm((}) = 2::=1 exp(ink(}) , and if the nk are integers, then lim sup moo
Sm((}) = 1 ymloglogm
(6.1.2)
for almost every () E T. Later, M. Weiss gave a complete analogue of Kolmogorov's LIL in this setting. Theorem 6.1.2 (M. Weiss [We]) Suppose Sm is a lacunary series with Bm as m + 00 satisfying the Kolmogorovtype condition (Ko). Then lim sup moo
Sm((}) y'2B;" log log Bm
=1
+ 00
(6.1.3)
for almost every () E T. The analogous theorem for Abel means of real lacunary series is a consequence ofthis theorem [We]. To state this, let S(p, (}) = 2:~=1 pnk (ak cosnk() + bk sin nk(}) , where the nk are lacunary and set Bp = (~2:~l(a~ + bDp2nk)L Suppose that the ak and bk satisfy the conditions of Theorem 6.1.1. Then · 11m sup
.J
S(p, ())
=1
pj1 2B~ log log B p for almost every () E T. The modern probabilistic approach to the above theorems is via an invariance principle which gives stronger results. Set Bo = So = O. For () E T and t ::::: 0 define
(6.1.4) Theorem 6.1.3 (PhilippStout [PSI]) Let Sm be a qlacunary series with Bm as m + 00. Suppose
+
00
Mm = O(B;,O), for some 0 < 8 ::; 1. Then, without changing the distribution of {Ut : t::::: O}, we can redefine this process on a richer probability space together with standard Brownian motion {Wt : t::::: O} such that
(6.1.5) almost surely as t
+ 00
for each rt
< 8/32.
175
6.1 LIL's for lacunary series
The invariance principle for martingales is the following: Theorem 6.1.4 (PhilippStout [PS2j) Let {d n } be a martingale difference sequence satisfying condition (Kd of Chapter 3:
Id 12 < K m

a 2 (fm) mloglog(ee + a 2 (fm)) '
almost surely on {a(f) = oo} for some sequence of constants Km a 2(fm) = 2:;;'=1 E(d~IFk1). For t > 0, define
1 0,
where
m
it =
Lik,
k=1
Then the process it can be redefined on a richer probability space together with standard Brownian motion {Wt : t ~ O} such that ft  W t almost surely as t
> 00
= o(tloglogt)1/2,
(6.1.6)
on {a (f) = oo}.
Theorem 6.1.3 and 6.1.4 are the invariance principles discussed in the preface and mentioned several times before. These results, together with the easy LIL for Brownian motion, (equation (**) in the Preface), imply the corresponding results for {8m } and {fm}. Notice, however, that (K1 ) is stronger than (Ko). S. Takahashi [Ta] has obtained results similar to Theorem 6.1.3 with a condition very close to (Ko), and furthermore, with weaker conditions on the lacunarity of the nk. We leave this direction to the interested reader. Besides the Kolmogorov LIL, the above invariance principles have various other remarkable consequences. For example, Theorem 6.1.3 immediately implies the KacSalemZygmund [Zy2, vol. II, p. 269], [SZI] central limit theorem: 8 :::; r } m { () E T : ~ Bn
>
1 I 00 (here m is the probability measure d()/21f on T), and the Chungtype LIL discussed in Section 4.3:
~~~ COg~~Bn ) 1/2 8~(()) =
:s
(6.1.8)
for almost every () E T, where 8;;"(()) = max1~k~m 18k (())I. In fact, there are even functional versions of these results as Theorems CE in Philipp and Stout [PSI] show. The above LIL's extend to the case of lacunary power series in the disc. Consider F(z) = 2:%:1 Ckznk with the nk lacunary and for 0 < p < 1 define
176
6. The Classical LlL's in Analysis
Bp = (E%"=llckI2p2nk )1/2. Note that Bp is the variance of F(pe i ()). M. Weiss [We] proves that under the assumptions of Theorem 6.1.2,
IF(pe i ()) I = 1, lim sup pi! J B~ log log Bp
(6.1.9)
for almost every () E T. There are also corresponding statements as in (6.1.7) and (6.1.8). In particular, if we set F;(()) = sUPO Po. Let 1 > P > Po. Then, computing as in the proof of Theorem 6.2.1, we find that
A~(F)(O, p) ::; allFll~ log (1 ~ po) + ae 2 log (1 ~ p) ::; 2ae2log (_1_) Ip
= 2ae 2 )..(p),
if P > PI > Po, with PI depending on Po, e and the Bloch norm of F. Since e > 0 was arbitrary, Theorem 6.3.1 follows from the upper bound of the 1IL, Theorem 3.0.4. By Theorem 4.4.2, for any 0 < p < 00, (6.3.2) A straightforward computation similar to that in the proof of Theorem 6.2.1 shows that for 0 < p < 1 (6.3.3) In fact, if FE B o, we can argue as in the proof of Theorem 6.3.1 to improve (6.3.3) to: A~(Fp)(O) ::; 2ae2)..(p) , whenever e > 0 and P is chosen sufficiently close to 1. This, plus (6.3.2) shows that for all 0 < p < 00, limsup pll
~(
V)..(p)
(27r IF(pei0)IPdO) lip = 0,
(6.3.4)
io
whenever F E Bo. In [Gil, D. Girela proves that (6.3.1) and (6.3.4) also hold for FEBI . Theorem 6.3.2 (Girela [Gil) Let FE B I 1 limsup /\7:\ pTl v)..(p)
(1
27r
a
.
Then for any 0 < p < . ) IF(pe'°)IPdO
lip
= 0
00,
(6.3.5)
and lim sup pll
J
IF(peiO)1 = 0 )..(p) log log )..(p)
(6.3.6)
for almost every 0 E T. The following lemma proves (6.3.5) with some additional information. This lemma, as we will momentarily see, also implies (6.3.6). Notice that in both (6.3.5) and (6.3.6) it suffices to assume F(O) = 0 which we do for the rest of this section.
6. The Classical LIL's in Analysis
182
Lemma 6.3.3 Let F E Bl with F(O) = 0 and c numbers, 0 < Pm < 1, depending on c and with Pm for each m = 1,2, ... ,
>
O. There is a sequence of 1 as m 4 00 such that
4
(6.3.7)
I!FIIE'
In fact, we may
I!FIIE ::;
1. By Theorem
for all P > Pm, where eo is a constant depending only on take Pm to be of the form
where Me is a constant which depends only on c. Proof: Fix 4.4.2,
a > 0,
and without loss of generality assume
121' (F;(O)) 2m dO ::; em
(v'2mfm 121' A~m(Fp)(O)dO.
Thus, it is enough to show that for P > Pm, (6.3.8) As in Chapter 5, we let Da(Fp)(O) denote the Dfunctional of Fp. By Theorem 5.3.3 there is a constant e l , depending only on a, such that whenever 0 < P < 1, (6.3.9) Since F E B l
,
we may choose Me so large so that if IF(z)1 > Me, then (5.0.4), if 0 < P < 1,
(1lzI)IF'(z)1 < vii. Then, by the change of variables formula,
A~(Fp)(O) =
r
!F'(pZ)12p 2dxdy
J{zaa(&):IF(pz)I<M,,} .
+ =
1
r
IF'(pz)12p 2dxdy
1
J{za a (&):IF(pz)I>M,,}
M"
M"
Da(Fp)(O; a)da +
::; 2MeDa(Fp)(O)
rare)
cp2
(1  plzl)
2
dxdy
(6.3.10)
+ CtcA(p).
Now suppose P > PI = 1  exp(( Me/c)2). Then integrating (6.3.10) and using (6.3.9) and (6.3.3) we obtain
121' A~(Fp)(O)dO ::; 2Me
121' Da(Fp)(O)dO + acA(p)
6.3 LIL's for subclasses of the Bloch space
::; 2cV),(Pl)Cl
183
127r A2a(Fp)(e)de + 2nac),(p)
::; 2cV),(p)Cl2n~v),(p)
+ 2nac),(p)
= (4n~Cl + 2na)c),(p) which proves (6.3.8) for m = 1 with the constant C continue by induction. Suppose
= 4nV2QCl + 2na.
We now
for p > Pml. Then for P > Pm, (6.3.10), (6.3.3), the induction hypothesis, and (6.3.9) give
127r A~m(Fp)(e)de ::; 2Mc 127r A~(ml)(Fp)(e)Da(Fp)(e)de + ac),(p) 127r A~(ml)(Fp)(e)de ::; 2Mcaml(),(p))ml 127r Da(Fp)(e)de + ac),(p)cmlcml(),(p))ml ::; 2cmV),(Pm)aml(),(p))mlCl
127r A2a(Fp)(e)de
+ aCmlcm(),(p))m ::; 2cm a m  l (),(p))m! C l 2nv2a),(p)
=
(4nV2a m!Cl
+ aCmlcm(),(p))m
+ aCml)cm(),(p))m
::; cmcm(),(p))m. Here the last inequality follows by noting that a m ! ::; cmlva and so 4nV2a m!Cl + aC m l ::; 4nV2QCm  l C l + 2naC m  l = cm. With a fixed, C is independent of m, c and F (as long as IIFIIE ::; 1). This completes the proof of (6.3.8) and thus completes the proof of the lemma. Before proceeding with the proof of (6.3.6), we make some observations about (6.3.5). If we define for 0 < p < 00,
then for 0 < p ::; 2, Jensen's inequality gives (6.3.11)
184
6. The Classical LIL's in Analysis
or (6.3.12) Thus, to prove (6.3.5) it is enough to do it for p = 2 and the requisite estimate for this is provided by Lemma 6.3.3. Therefore, estimate (6.3.5) does not require information on the behavior of the constant Cp in the inequality IIF* lip::; CpIIA"llp. However, the fact that Cp ::; Cyfp for p 2: 2 is crucial for the proof of (6.3.6) which we now come to.
Proof of {6.3.6}: Let c > O. Let rm = 1  ee m and Am = ve m logm. We claim that 00
(6.3.13) m=l
for some constant C. For this, let [logm] denote the integer part oflogm. Observe that if m is large enough, rm > P[logmj, where {Pj} is the sequence of Lemma 6.3.3. By Chebychev's inequality and Lemma 6.3.3,
I{O E T: F* (0) > CyeA }I < m
rm
1  C2[logm]c[logm]A m2[logm]
10r
27r
F* (0)2[logmjdO rm
< cgOg m] c[log mj ([log m]) [log m] (A(r m)) [log m] =
C2[logm]dlogm]A~logm]
(Co) [logm] C2
([log m]) [log m] < (logm)[logm] 
(Co) [logm] C2
If we take C = e.JCo, we have (6.3.13). The BorelCantelli Lemma now gives that F:m (0) ::; CyeAm eventually for almost every 0 E T. That is, for almost every 0 E T, there is an mo = mo(O) such that F:m (0) ::; CyeAm for all m 2: mo. Let P > rmo and choose m 2: mo such that rm::; P < rm+1· Since F;(O) is increasing in P and Am = VA(rm)loglogA(rm ),
F;(O) < F;",+l (0) ::; CyeAm+l ::; CyeAm ::; C yeV A(p) log log A(p) , which proves (6.3.6) since c was arbitrary.
6.4 On a question of Makarov and Przytycki
6.4
185
On a question of Makarov and Przytycki
In [Gi], Girela also proved that if the L 2 norms of F(pe iIJ ) grow slowly enough as p i 1 then, with no other assumption on F, (6.3.6) still holds. The following generalizes his result.
Theorem 6.4.1 Let F be a Bloch function. Suppose that
, h(27r IF(peiIJWdB ~ ",(p) log (_1_) Ip
where '" satisfies 00
" ~
",(1 _ ee"')
(logm)/3