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-O and any there exists a constant cA,E such that m(z) 1 `cx,E (1 + Ix1)-x e(b+E)!YI
(4)
holds for all z=x+iy with xERn and yERn Theorem 2. The following two assertions are equivalent: (i) f ES' and supp Ff c{y I Iyi mob}, (ii) /(z) is an entire analytic function of n complex variables and for an appropriate real number A and for any a>-O there exists a constant cE such that /(z)ICc
(5)
(1+IxI)Ae(b+E)Ivl
holds for all z = x + iy with xERn and yERn. Remark. It is clear how to understand these two theorems: If (i) of Theorem 2 holds, then J ES' is a regular distribution &), which can be extended to an analytic function /(Z), which satisfies (5). Vice versa, if (ii) of Theorem 2 holds, then the restriction of /(z) to Rn satisfies (i). Similarly for Theorem 1. Proofs may be found in L. Schwartz [1, p. 272], L. Hormander [2, Theorem 1.7.7] and K. Yosida [1, VI. 4]. The formulation in Hormander's book looks more
handsome but the above version is the direct counterpart of a corresponding theorem for ultra-distributions, cf. 6.1.4.
1.2.2.
L, ,-Spaces and Quasi-Banach Spaces
If It is a Borel measure on Rn and 0
-0.
(1)
Furthermore, the characteristic function of Qb will be denoted by xb(x). Let Z,a be the lattice from 1.2.4 with the lattice-points k=(k1, . . . , k).
1.3. Inequalities of Plancherel-Polya-Nikol'skij Type
21
Proposition. If Q=Qb is given by (1) then any q'ESQ can be represented by n
,r 2
2
( )
}
( k/ ) (F1X5) (_k)
kEZ, 4'
kEZ
(bk) j=1n sin(bxj-kF) bxi-kjr
xER
(2)
"
(absolutely convergent series).
n
Proof. Step 1. I f x = (x1, ... , xn) E Rn and k = (k1, ... , kn) EZn, then xk =
xjki is the usual
9=1
scalar product in R. As a smooth function, Fop can be developed in Qb in an absolutely convergent multiple trigonometric series. Hence,
(Fq,) (x)=Xb(x) 2 ake
-i 'kx b
with
kEZ,y
irkx
ak=(2b)n f (F9') (x) e b dx eb
n
=(2b)- (2)2 [F1(Fq,)]
(.1t)=()2
b-'gq(b k), kEZn.
This proves the first representation in (2) if one takes into consideration that a-1 E- kx
F-1(Xb(x)
b
(y)=(F-1Xb)(Y-
)
i
b k)
Step 2. In the one-dimensional case, i.e. Qb = [- b, b], we have 1
(F-1X5) (x) _
1
(2n) 2 f e1ZYdy = (_ 12 b
-bxbx
/
-b
,
x E R1.
In the n-dimensional case it follows by multiplication that n
r \ 2 2 (F-1X5) (x- k)=()n b
n sinbxj_kjr (bxi- kin)
(3)
=1
where x = (271, ... , xn) E B. and k = (k1, ... , k,) EZ,,. Now the second representation follows from the first one and (3). The absolute convergence of the series in (2) is clear, because
k2 Remark 1. Formula (2) shows that the function p(x) is known if one knows its values at the lattice points b k, where kEZn. This provides a better insight into the lattice structure of the
inequalities in the preceding subsections. Furthermore, one can understand that has
r b
in
Remark 1.3.3: Let b op, where ap has the same meaning as in the theorem. Iteration of (1.3.2./6) (extended to S'(R,,)) yields (F-1MFf) (x) ={F-1[M(b .)
(Ff(b-1 .)) ('))} (bx) .
1. Spaces of Entire Analytic Functions
28
Now, by (4) with to instead of 0 and
instead of /we have
n
IIF-iMFI I LPII =b P IIF-'[M(b ') (Ff(b-1
I LPII
n
cb
(13)
IIM(b') I H2 II -II/(b-1') I LPII=cIIM(b') I H2 II Ii/ I LPII
where c is independent of b and f ELP. 1.5.3.
Convolution Algebras
If I EL, and gEL1 then the convolution f*g is defined by (f*g) (x) = f / (x - y) g(y) dy
(1)
Rn
If c is an appropriate positive number, then (1) can be rewritten as f*g= cF-1[Ff Fg], and one can define f*g by the last formula for more general distributions f and g if F-1 [Ff Fg] makes sense. Proposition. Let Q be a compact subset of Rn and let 0-0, x2I=tx1},
S9 ={xER, I x10 is natural. The norm in (2) with sx-2 >dp,q. This proves (21), where c is independent of the cho-
sen system a= {ak}k_o. Similarly for the other pairs p and q. Hence, the proof of part (ii) is complete.
Step 4. If 00 such that !I/
(26)
I BP,5(Rn)II ncllf I BP,q(Rn)II X
for any admissible system a E ip(Rn) in the sense of (9). The proof follows the same lines as in Step 3. In the counterpart of (22) we need only the scalar case of Theorem 2.4.9/2, i.e. (1.5.2/13) (including p = oo). Instead of (25) we obtain (for 0
- ap, every / E B7,q(Rn) is a regular distribution.
Of course, this is also valid if 15p-_- and s-0, because Bp,q(Rn)cLp(R.), ef. (30) with p instead of 1. By (2.3.2/9) and ap,gzap, we have the same argument for the spaces F8 q(Rn) with s lap q. Let again p-q - I can be treated by the known multiplier theorem from Proposition 2.4.8 (ii)). On the other hand, Theorem 1.6.3 covers the case q=- (in contrast to Theorem 2.4.9/2). If we use Theorem 1.6.3, then we obtain the assertion of part (ii) of the above theorem under the restriction n min (p, q)
(of course, only qis of interest). This follows immediately from Step 2 and Step 3 of the above proof. This result has been proved in [S, 2.2.3]. Remark 3. In a formal way (without the justification in Remark 1), the calculation in Step 1 and Step 2 go through if s-0. What about the values 0-s=ap, reap. 0<snEap,q? If 0s and if a >- in (4)- (6), then (8) - (12) are equivalent quasi-norms in Bp q(Rn).
P n (ii) Let O-Gp,q. If M is an integer with M>s and if a> min (p, q) in (4)- (6), then (13)-(17) are equivalent quasi-norms in FP q(Rn).
Proof. We have to prove that the unnatural restriction M>2Gp,q+s in part (ii) of the theorem can be replaced by the natural restriction M>s. Similarly for part (i) of the theorem. We prove the above claim for the quasi-norm (13). In an analogous way, one proves correspond-
ing assertions for the quasi-norms (8)-(12) and (14)-(17). We needed M>2Gpq+s only in Step 1 of the proof of the theorem. This shows that it is sufficient to deal with the following
inequality: if M>s, then there exists a constant c such that 1I21k8M kl I Lp(Rn, lq)II =cII2skS2-k 1f JLp(Rn, lq)II +cllf I FP,q(Rn)II
(42)
for all f EFp q(Rn) (under the above hypotheses for s, p, and q). If P(z) is an appropriate polynomial then the following identity holds for complex numbers z,
(z-1)M-2M1 (z2-1)M+(z-1) M+1P(z)
(43)
In particular,
(ein _1)M=2M (e2sng-1)M+(e;ni-1) v+1p(e;n)
(44)
E R. and h E Rn. If we apply (44) to F1 and if we take the inverse Fourier transform of the result, then we obtain where
(L1
/) (x)=2
(x)+dv+1
(d -jhl)
(-'al/ (x+lh)) ,
(45)
where 2' is a finite sum (the counterpart of P(z)). Using (2) and (4) we then have
(SMkf)(x)S,yf
(46)
for all xERn and all k=O, 1, 2, ... If 0-g51, then 2"
2skq(SMkf)q
(x)
S2(3_ M)q
2' 2skq(SMkf)q (x) +c
k=0
k=1
k=1
2sk(SM k i f)q (x)
and because M>s 1
1
qsc k=0
2skq(SMkf)q (x))
with
2skq(SMk if)q (x)) k=0 (2
R(x) = sup (1 + Ix-y!a)-'I/(y)l YERn 7*
+CR(x)
(47)
2. Function Spaces on R.
100
Similarly for 1 0 is a given positive number. If we put the last estimate in the last but one, then we obtain the desired estimate by standard arguments. We obtain the corresponding assertion for (17) in the same way as in the first step. Remark 1. References were given in Remark 2.5.10/3. The last part of the corollary was inspired by P. Nilsson [2].
Remark 2. As in Remark 2.5.10/1, one can replace some differences in the above quasinorms by derivatives. The number a from (2.5.10/20) can be chosen in this case as a=a+m, °p,q Up,q if one interprets Il/(...) I F ,,q(Ri)II via Theorem 2.5.11. Finally, in order to have explicit descriptions of (2) and (3) we introduce the differences
(4h;jf)(x)=f(xi,..., xj-1, xj+h, xj+1, ..., xn)-/(x), (Ah,jf)(x)=4j(4h,j1/)(x)
(4)
where h E R1, x E R. and l =2, 3, ... , cf. (2.2.2/5). Then (2) means essentially that Ill
I LP(R,)II
(5)
f hl -Bp (4 f) (x1.... , xj-1, '
+i=1
dh , xj+1, ..., xn) I LP(R1)IIP hl I LP(Rn-1)
Ri
j=i
is an equivalent quasi-norm in BP,P(Rn), provided that M>s. Similarly for (3).
Theorem.1) ( Let 0-s, then Ilf
I Lp(Rn)II
08, and M>.s. We use holds for all f EBP,q(Rn). Of course, 0- (20)' , we determine K in (4) as the largest non-negative integer such that H r-° r
C2 P,
tr . This is possible (and this 2
is also the reason for the above splitting num-
ber (2C)' ). Now it follows from (4) and (5) that IX kT 2krsi (F-14wkFt)(x)I I
_ct2-Ha,} I .
(F-1TkFt)(x)I 'tP0} I dt
(9)
I
I
k=0,1,2,...
We have t2
Hs, ," t
1-a,r+1
i tps-1 (2C)r
f
-tPo So it follows that
(xl k 2
2krs,
dt
Ps
a,
tP1-1 I{x I
sup k=0,1,2,...
1
I
c(2C) T
c f tPo-1
sup
I {x I
0
k=0,1,2,...
(F-19'kF1)(x)I ;t} I dt I
r +
r
1-i
jj Bp,pp '_
(Rn-1)
(46)
j=o if I < p
1
and a > r +-- This statement is also well-known, cf. S. M. Nikol'skij [3] or [I, 2.9.3.]. P
Remark 2. Traces on hyperplanes of dimension 1 , 2, ... , n-2 can be obtained by iterative application of the above theorem. The method is clear, we shall not go into detail.
Remark 3. Some of the main ideas of the proof of the above theorem are due to B. Jawerth [1] (cf. also B. Jawerth [21), who proved the above theorem for r=0. The most interesting assertion is the independence of the trace of Fp,4(Rn) on Rn-1 of q. This fact has
139
2.7. Embedding Theorems
also been observed (independently) by M. L. Gol'dman [4] and G. A. Kaljabin [3] (if 1
-x'', then there exists a positive
constant c such that It I B',4(Rn)Il --+Oc in the case 15p5
either1 --l-a>s p
p1
1
eithern p-1I-a>s or s>p +a in the case 0- j and j =1, 2.... yield p
tiJ f 218PI (E ' /)(x)I Pdxsc
j=1 kL=j Sj
4
f
Rn_i X[-1,11 r=0
2regc*g(x)J J
dx ,
(13)
if one chooses d::- s. Next we consider the case k::- j and l = 0, ... , j. In particular 2,j_i we have l s j. Let x = (x', x,,) (recall that Ixnl s 2-j). Then it follows from the definition of cl*(x) that 2-1
1
p
f
cl*(x)cc:*(x', 0)Sc'2P (2-1-1 ci*P (x', t) dt
(14)
Now (5), (10) and (14) yield
j C2-kdp+jdp
2k8PI
2-G 21
1=0
f ci P(x', t) alt 2-1-'
2k8P
(15)
P 21-1
c2-kdp+jdp2ksp
21-1ep 1=0
f
2-Z-1 r=0
2regc*g(x', t)J dt
Integration of (15) over the above strip Sj (which gives the factor 2-j on the righthand side)) and summation over k j and j=1, 2, ... yield dx ,2 L] 2] f2k8PI (Ekf)(x)IP j=i k=j Sj
5C 1'
G
j=11=0 11
2-1
jj+
Triebel, Function engl.
2-i+7dp21-18p
f f
Rn_1 2-1-1
P
(16)
=
( ..)g dx
2-kP(d-8)
k=7
162
2. Function Spaces on R.
Let d>s. Then the last sum equals c2-jPd2jP8, and so the factor in front of the integral has the form 20-0(8P-1). Because spy 1 we have
f 2] f 2k8PI(Zk;21)(x)IPdx-- c Rn-1X[0,1]1 r=0 j=1 k=j Sj
P q
2Y8QC, g(x))
dx .
(17)
It follows from (13) and (17) that 2] 2] .1 2k8PI (Ek'f)(x)IP dx=cll2r8C* I LP(Rn, lq)IJP . j=1 k=j Sj
(18)
Finally we consider the case k s j. Then we have also l 5 j and (14) is applicable. It follows that k
218PC1*1(x) 2k8P-lap
(Ek'f)(x)IP1.Weremarkthatf-.(a(x)-a(y))g9(x)f is a self-adjoint operator and use (2.11.2/2). It follows that (14) holds if
1-0 is an isomorphic operator in F',q(Rn), and we have if h10. (4) This follows from the density assertion of Theorem 2.3.3. Consequently, we may
assume, without restriction of generality, that f has compact support in R. Let T(x) E S(Rn) with compact support in R; and 92(x) =1 if x E supp f. Let gt -f
in Fvq(Rn) with g1ES(Rn), cf. again Theorem 2.3.3. Then pgf--f in Fv,q(Rn) is the desired approximation. The proof is complete. Corollary. Let m =1, 2, 3, . . . and let v be the outward normal of Q.
(i) Under the hypotheses for s, p, and q of part (i) of the above theorem we have fEBPq-(S2),
f
aQ=0 if j=0, ..., qn-1
I-
(5)
(ii) Under the hypotheses for s, p, and q of part (ii) of the above theorem we have Fv r'(S2) = {f /EFPq (Q)'
ail
w
852=0 if
j=0, ... , m-1}.
(6)
Proof. In any case, the right-hand sides of (5) and (6) make sense, cf. Theorem 3.3.3. We may again assume that f EFv*Q (Rn) with supp /C {y I Y E Rn, 1y1 1,
211
3.4. Further Properties
yn = 0}. and ax. (x', 0) = 0 if j = 0, ... , m -1. We extend / by zero to R. and denote
the extended function also by /. Then D"/ in Rn is the extension by zero of Da/ in Rn, provided that IaI -- m (here we use that f (x', 0) =0 if j=0, ... , m-1). In particular, D"/EF, q(Rn) if IaI mm. Hence we have IEFP (Rn) and IIf
I F' pgm(Rn )II -
II/
I F' gm(Rn)II
a
(7)
The rest is the same as in the proof of the theorem. Remark 2. Essentially we proved that under the hypotheses of the corollary, the extension by zero from 92 to Rn, is a bounded linear operator from 'Br q""(S2) into
BP gm(Rn)
from Fy q'4(S2) into
FS gm(Rn)
and
respectively. We have the same assertion with R,+ instead of Q, including (5) with R,+ instead of S2. This follows from the above proof.
Remark 3. The proof was based on Theorem 2.8.7 which for the spaces B, q(Rn) covers also the limiting cases where p = and/or q = . For density reasons, part (i) of the theorem and (5) cannot be valid for these limiting cases. Let bP q(Rn) be the completion of S(Rn) in BP,q(Rn) (we assume that max (p, q) =-) and let by q(R,+,) and by q(S2) be the corresponding
restrictions to Rn and S2, respectively. Then it follows from the above considerations that parts (i) of the theorem and of the corollary remain valid if one replaces B8 q(S2) by b8 q(S2) and BP Qm(S2) by by q"n(S2) (under the restrictions (1), where now p = and/or q = are included). Similarly, Remark 2 can be extended to these new spaces, including the possibility of replacing
Dby Rn
14*
4.
Regular Elliptic Differential Equations
4.1.
Definitions and Preliminaries
4.1.1.
Introduction
Let Q be a bounded C°°-domain in Rn with boundary O.Q. Let A, (Au)(x)=
aa(x) Dau(x), IoI
xE.Q,
(1)
2m
be a properly elliptic differential operator in D, and let B1, ... , B,n, (Bju) (y) _ 2] b,,,(y) Dau(y), yEaQ, Ialsmi
(2)
be m boundary operators such that {A; B1, ... , Bm} is regular elliptic (cf. the definition in 4.1.2.). The corresponding boundary value problem reads as follows, (Au)(x)=f(x) if xE.Q and (Bju)(y)=ga(y) if yE8Q
(3)
and j =1, . . . , m. This is the well-known boundary value problem for regular elliptic differential operators. Problems of this type have been studied for a long time, in particular in (Holder)-Zygmund spaces e8(Q) with 8>0 and in Lp(0) with 1-0}. As usual, D and R; is the closure of Q and R;, respectively. Finally, C-(D) is the collection of all complex-valued infinitely differentiable functions g(x) in 0 such
that every derivative Dag(x) can be extended continuously to D. Similarly, C-('9D).
Definition 1. (i) The differential operator A, Au= I aa(x)Dau, aa(x)EC-(D) if
jaJ52m
is said to be properly elliptic in 0 if a(x, ) = aa(x)a * 0 for every 0
E Rn
(1)
181s2m
lal52m
and every
x ED
(2)
and if for every couple E Rn and .77E Rn of linearly independent vectors and for every xED the polynomial a(x, +ai7) in the complex variable r has exactly m roots - c + , 17) with positive imaginary part, k =1, . . . , m. (ii) Let 0 in (i) be replaced by Rn and let aa(x) = as be complex constants. Then (i) defines a properly elliptic differential operator A in R,+, (with constant coefficients). Remark 1. We recall that a =
R. and a = (a1, ... ,
if
.j=1
Furthermore,
the roots a....... . ... . zm are enumerated with respect to their multiplicities. In particular, a(x, E+nn) has also exactly m roots rk =Tj (x; $, ij) with negative imaginary part, k =1, ... , m (by (2) there do not exist real roots of a(x, $+2i7)) Let m
a+(x; sc, ?l, z) = II (r- rk) , a (x; k=1
m
r) = H (z- T k-)
(3)
k=1
Remark 2. A discussion of this definition, in particular of the root condition, has been given in [I, 5.2.1.]. If n z 3, the root condition is a consequence of (2).
Definition 2. (i) The system {Bj} 1 of the k differential operators Bju = bj,a(x) Dau, bj,a(x) E C-(8Q) , kI mj
(4)
is said to be normal on OD if
0 m1 <m2 Irk (')I C1I'1 (2')
IIm rk ( ' ) I zc9j '
IIm rk ( ' ) I -e2IE'I,
(ii) The coefficients ak (') and ak (l') in m
m
a+(S', r) = 11 (r-rA k=1
ak
(3)
k=0 m
m
a (E', r) =k=1 11 (r-rk(E'))=k=0 I ak(') rm-k
(4)
are analytic functions of ' ingRn_1- {0}. Furthermore, ak (AV') =Akak ('), ak.(AE') =Akak (cam')
(5)
if k=0, ... , m and ADO. If E'+0 and r complex, then
a- (-c', -r)=(-1)ma+(E', r) .
(6)
Remark 1. A proof may be found in [I, 5.2.2].
Lemma 2. Let 4p(t) be an infinitely differentiable function on [0, -) with q9(t)=0
if
0--t42 and
T(t)=1
1=t0
We need some properties of these functions b;,1(').
217
4.2. A Priori Estimates
Lemma 1. Let (b t(g'))i=t .., ,n be the matrix of the functions b;,1('). 1=0,...,m-1 Then det *0 if ' E Rn-1- {0} .
(5)
l =0,...,m -1
Let (c; 1( '))i=1
be the corresponding inverse matrix. Let T(t) be an infinitely
,,, rn
l =0,...,rn-1
differentiable function [0, -) with qq(t) = 0 if 0 -- t mi-Z
lal
sup
and ip(t) =1 if 1 _< t < oo. Then
(1+I '12) 2 I Da[c;,l( ') (1+15'l2)
(6)
2
f'E Rn_
for every multi-index a. Remark 1. ,This lemma coincides essentially with [I, Lemma 5.2.3/1], where (6) follows from the proof of that lemma. The above lemma is the counterpart of Lemma 4.1.3/2. In particular,
mi-l
it follows from Theorem 2.3.7, that the functions c;.t(E')
2
are Fourier
multipliers for BP,4(Rn_1) and F1,,q(Rn-1)
Temporarily we denote the (n -1)-dimensional Fourier transform with respect xT,)) _ to the n-1 coordinates xt, .. , xn_1 of u(xl, ... , xnxn) by F', i.e.
=(F'u) (s', xn) with 'ERn-i
Lemma 2. Let u(x) be a complex-valued function on R. having classical derivatives
up to order M such that
sup (1+Ixj2)af I IDau(x)It-. If M is sufficiently large and if
xn ?0 ,
aaDau(x) = 0 for
(Au)(x) _ kI=2m then
0)= imi I ba 1(i') al (F'u) (s', xn)Ixn=o il2xn 1
1=0
tt
where j=1..... m.. Remark 2. A proof of this lemma may be found in [I, 5.2.3.]. The functions b',=(i;') have the meaning of (4). This formula is also the basis of the proof of the last lemma.
4.2.
A Priori Estimates
4.2.1.
Introduction and the Spaces F,,' (Rn )
Our main goal in this chapter is the study of regular elliptic boundary value problems in bounded C°°-domains 92 in Rn in the framework of the spaces BP,,(Q)
and F, 5(Q). This problem can be reduced to a priori estimates for regular elliptic systems {A; B1, . . . , B,n} in R; in the sense of Definition 4.1.2/4 in the corresponding spaces on R.I. Spaces of type BP,Q(on Rn, RI or 0) can be obtained by real interpolation of spaces of type F'P.5 and one can reduce the above problems for BP,Q to corresponding problems for F',5 (we shall describe this method in detail later on). In this sense we restrict our attention in Section 4.2. to a priori estimates
218
4. Regular Elliptic Differential Equations
for, regular elliptic systems {A; B1, . . . , Bm} in Rn and in Q in the framework of the space FP q(Rn) and F',q(Q), respectively. As in [I, 5.3], our treatment in this section is based on the paper by L. Arkeryd [1]. This section may also be considered as an extended version of the brief announcements of the main results of this chapter in H. Triebel [10, 14]. In [I, 5.3] we proved a priori estimates for regular elliptic systems {A; B1, . . . , Bm} in R,n and in Q in the framework of the spaces LP(R+) and L,(Q), respectively, with 1-0 is an arbitrary number and c depends on E. However, (14) is an easy consequence of the real interpolation formula (3.3.6/10) and (3.2.4/3). Now the left-hand side of (3) follows from (13) and (14). The proof is complete.
230
4. Regular Elliptic Differential Equations
Corollary. Let {A; B1; .... Bm} be a regular elliptic system in Q in the sense of Definition 4.1.2/4. Let p, q, and or be real numbers with (4.2.2/28). Let 8 = 0, 1, 2, ... Then there exist two positive numbers c1 and c2 such that ccillu
-_IIAu I BU+'(S2)II + B2m+s(S2)II P,g
IIB7u I BP
Q2m+s-
P(aQ)I?
Q2m+s(Q)II
+Ilu I B",9(Q)II =c2IIu I Bp
holds for every Proof. The proof is the same as the proof of the above theorem. Instead of Theorem 4.2.3 we use Corollary 4.2.3.
Remark 1. The above theorem and the corollary are the main goals of this chapter. In the corollary we again restrict (15) to smooth functions u, where u E S(R.n) can be replaced by u EC-(Q),
ef. 3.2.4. If p0 can be treated in the framework of the usual distributions, but not the second one, i.e.Q(x)=e1x1P with 0{#t1. This class leads to the so-called Gevrey ultra-distributions.
Remark S. Essentially, the measures poEKh have a lattice structure, in particular if (4) is satisfied. Of special interest are Lebesgue measure F1o=dx and atomic measures with uo(Qk) =ko({xk}) = 1, where x& EQk Our goal is to extend inequalities of type (1.3.4/1) and (1.4.1/3) to measures of the type v=x(x) po with Yo E Kh and x(x) EKh(Q, ho) in the sense of the above definition. In other words, the general measure v is divided into a measure yo with latticestructure and a weight function x(x), which on the one hand may grow almost as rapidly as an
exponential function and on the other may be zero on a set of infinite Lebesgue measure. Such general measures v include e.g. weight functions of type H ;xjldj with dj>0 and Ixld with d>0. j=1 6.2.2.
Some Inequalities
Our next aim is the extension of Theorem 1.3.1, Proposition 1.3.2, and Proposition 1.3.3. to the weighted case. With a few technical modifications the proofs can be carried over from the unweighted to the weighted case. For details we refer to [F, 1.3.2.-1.3.4.]. If w(x)E9Y and if Q is a compact subset of R, then Stu={99 191ES.,
supp FT cS2} ,
(1)
cf. (1.3.1/1). We recall that M is the Hardy-Littlewood maximal function from 1.2.3. Theorem. Let 01(x) E V and Q(x) E R(w). Let 0 be a compact subset of R. (i) Let 0-. I .Also the "limiting" cases x1(x) = x2(x) = elx I or x1(x) = x2(x) =e -1x! cause trouble. Furthermore, (6.1.2/2)
is rather natural. For details we refer to [F, 1.3.5.].
252
6. Ultra-Distributions and Weighted Spaces
of Analytic Functions
6.3. 6.3.1.
Definition and Main Inequalities
This is the weighted counterpart of 1.4.1., cf. also [F, 1.4.1. and 1.4.2.]. II' I L,,,Il has the same meaning as in 1.2.2. Definition. Let w(x) OR and Q(x) E R(w). Let S2 be a compact subset of R. and let 0
Owith a=[a]+{o}, where [a] is an integer and 0-0, 1 independently by J. Peetre [8, Chapter 10] and in [F, Chapter 3]; cf. also H. Triebel [22]. We describe in very rough terms the basic idea and than add a few comments concerning concrete applications. Abstract Spaces. Let H be a separable complex Hilbert space and let A be a self-adjoint 18*
272
10. Further Types of Function Spaces
positive-definite operator in H. Let
Aa= f t dE(t) a
(2)
0
be the spectral decomposition of A. Let RZ e with
RZ0a=f 11-
t t
dE(t)a, aEH,
(3)
0
be the Riesz means. Let A be a Banach space. We assume that both H and A are continuously embedded in a common Hausdorff space. Furthermore, let An H be dense in H and dense in
A. The main hypothesis of the theory can be formulated as follows: there exist a natural number 1 and a positive number c such that sup IIR1,0a Allsclla I All for all aEHflA.
(4)
After extension by continuity, {R1,p}c,o is a family of uniformly bounded continuous linear operators on A. We assume that RZ ea-'-a if e - (convergence in A) for every aEA. Let 0 be the collection of all systemsT ={q?j(x)}2=0 of infinitely differentiable functions on R1 such that
supp Toc(-2, 2), supp g7jc(2j-1, 2j+1) with j=1,2,3,...; for every non-negative integer k there exists a positive constant Ck such that 2jklT k)(x) 1sck
for all j=0, 1,2,... and all xERI;
and
I gPj(x) =1 j=0 means
dd
for all
x -O
. . P is the counterpart of 4P(Rn) from Definition 2.3.1/1. In the usual way, I
we construct the /operators qj(A), Tj(A) a= f ipj(t) dE(t) a. 0
Let s>-0, 1 s q s - and T E0. Then the spaces Bq(A) are defined by 1
q
((
Bq(A) =1a ; a E A, Il a I Bq(A)II e = (. 2jsq llwj(A) a I AII5)
(5)
These are the abstract Besov spaces which have been studied in the literature cited above. In other words, we define spaces via spectral decompositions. A similar procedure has been described in 2.2.4. as a motivation for the introduction of the spaces Bp,q(R,n) and Fpq(Rn). What about concrete cases if we specialize H, A and A? Let
H=L2(Rn) , A=-4+E, A=Lp(En) ,
(6)
with 15p--, where E is the unit operator. Then one can prove that the above hypotheses, in particular (4), are satisfied. In that case, BQ(A) coincides with the space BP Q(Rn) from Definition 2.3.1/2. In other words, the spaces Bpq(R.n) and Fpq(Rn) from Definition 2.3.1/2 can be obtained via spectral decompositions of the Laplacian. For details and further examples we refer to [F, 3.5]. Orthogonal Expansions. If we replace Rn in (6) by Tn, then we obtain the spaces B,85(Tn)
from Chapter 9 (under the above restrictions for the parameters s, p and q). In particular, (4) is satisfied in that case, too. One can ask whether other classical orthogonal expansions fit in that framework. For instance, if H=L2(RI),
(Af)(x)=-f"(x)+x'-f(x), A=Lp(RI)
(7)
10.3. Abstract Spaces
273
with i 5pz -, then A is a positive-definite self-adjoint operator with pure point spectrum. The corresponding complete orthonormal system {Hk(x)}k=c,of eigenfunctions is given by the Hermite functions x2 dk
Hk(x)=eke2 dxk (e-02) The hypothesis (4) is satisfied and we obtain Besov spaces which look similar to those in (9.1.3/2) if the system (1) is replaced by {Hk(x)}k=c. Similar one can deal with Laguerre functions, Jacobi functions and spherical harmonics, of. [F, 3.5.5.] for details and references. Recently,
T. Runst [1] has made a thorough investigation of such spaces. We refer also to T. Runst, W. Sickel [1], where the problem of strong summability in the sense of 9.2.4. for Jacobi expansions has been studied.
References Adams, R. A. 1. Sobolev Spaces. Academic Press; New York, San Francisco, London, 1975. Agalarov, S. I. 1. A Littlewood - Paley theorem and Fourier multipliers in weighted spaces with mixed norms.
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1. On the LP estimates for elliptic boundary problems. Math. Scand. 19 (1966), 59-76. Aronszajn, N. 1. Boundary values of functions with finite Dirichlet integral. Techn. Report 14, Univ. of Kansas 1955, 77-94. Aronszajn, N.; Smith, K. T. 1. Theory of Bessel potentials. I. Ann. Inst. Fourier (Grenoble) 11 (1961), 385-476. Bagby, R. J.
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Besov, O. V.; Il'in, V. P.; Nikol'skij, S. if. 1. Integral Representations of Functions and Embedding Theorems. (Russian) Moskva: Nauka 1975 [English translation: Scripta Series in Math., Washington, Halsted Press, New York, Toronto, London: V. H. Winston & Sons 1978/79]. Beurling, A.
1. Quasi-analyticity and general distributions. Lectures 4 and 5, A. M. S. Summer Institute, Stanford 1961. Bjorck, G.
1. Linear partial differential operators and generalized distributions. Ark. Mat. 6 (1966), 351-407. Boas, R. P. 1. Entire Functions. New York: Academic Press, Inc. Publishers 1954. Boman, J. 1. Supremum norm estimates for partial derivatives of functions of several real variables. Illinois J. Math. 16 (1972), 203-216. Bui Huy Qui 1. Some aspects of weighted and non-weighted Hardy spaces. Koky uroku Res. Inst. Math. Sci. 383 (1980), 38-56. 2. Results on Besov-Hardy-Triebel spaces. Techn. Report. 3. Weighted Hardy spaces. Math. Nachr. 103 (1981), 45-62. 4. Weighted Besov and Triebel spaces : Interpolation by the real method. Hiroshima Math. J. (to appear) Burenkov, V. I.; Gol'dman, if. L. 1. On the extension of functions of Lp. (Russian) Trudy Mat. Inst. Steklov 150 (1979), 31-51. Calderdn, A. P.
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