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0' we have
l-
(* to' rov')o r'd's x
r{')"* fo*
'
Namely,
/-
(lt't
-: l,' r(r)dr) rdx'- lo* ro{,)*"0, (1.11b)
/ 2 0 with p > l, € < p - L, e * 1. This equivalence was proved N. KRUGLJAK, L. MALIGRANDA a.nd L. E. PERSSON [2] a'nd by for decreasing functions earlier by C. BENNETT, R. DE VOR^E a,nd R. SHARPLEY [1]. In fact, for p - 2 there is equality in (1.115) even for more general weights than o6, see N. KAIBLINGER, L. MALIGRANDA a^nd L. E. PERSSON [1]; see also Chap. 5. for
1.10.14. The proof of the inequality (1.109) is taken from S. KAIJSER, L. E. PERSSON and A. OBERG [1]. Note that the same proof gives that also the inequality
* l,' (i l,'
rc*,)*
=
I,'
o(/(c)) ('-
;)
*
that 0 < 6 S oo provided .f ttd O a,re as in Proposition L.24.ln pa,rticular, by using this inequality for b < oo and with the special cases from Example 1.25 we obtain the following improvements of the Knopp and Hardy inequalities: holds for all b such
(1.114) Ioo
"*o(: I,'
h /(r)dr) a, s.loo
rt'l (r -i)*,
(1.16)
64
Weighted Ineqnlities of
and for
Hadv I\pe
p) l,
l'(;
fo'
otiat)o
a'
( o \p fh ^. , lJ" x,t
=(#)
1,t@-t)/r\ (.'-(;) )*,
Some Weighted Norm Inequalities
(r117)
p' 58' where bs : bp/@-r) and g(r) : 16b-r)/n1x-r/p as on The inelualities (1.116) and (1.117) have been also proved by A. erZrr,rpsile and J. PECARIC [3] (cf. also [1]) bv using a mixed(1'117) are mean inequatities technique. In fact, both (1'116) *d and refsharp and strict inequalities. For more information, results article review the [2] ur"rr"* concerning the technique mentioned see bv the same authors.
2.1.
Preliminaries
A general weighted norm inequality As already mentioned in Chap.
1.,
now we investigate inequalities of
the form
llTfllq," < cllfllp,"
(2.1)
or, mole precisely,
(1,' x,nailqu(c)ac)'/o where
?
is an operator of the form
(ril@): with
b(
lc(o, oo.
. c (1"' van u@'1a,)'/o (z.z)
l,
k(x,t)l$)dt
t) a given kernel, u, u weight
functions and
(2.3)
-m (c(
66
Some Weighted Norm
Weightet Inqrclities of Hodg Tfue
on the Pa^rameters We are interested in conditions on u, u and is satisfied for certain classes of p, g under which inequalitY (2'2) functions /. are provided Example 2.L. (i)The first examples of the oPerator.T is that I[' by the llardy operator I/ and its conjugate
:
(n/Xc)
fb -..
rz
(2'3) with the kernel k(n't) : These operators are of the form a' respectively' Neces: xtr,'l(t), t 1b, or k(c,t) x1",01(t)'1.t of the corresponding validity sary and sufficient "o"aitio* for'tttu and Chapter 1' inequality (2.1) have been given in Introduction (ii) A simple extension f,rovides dne rnoilifedHardy opetator '2f,
formulas (0.8), (0.9)) so that
2.2.
with g;, rlti, i
t?il@),:Q@) The operatot
,ff
{, 8d
'fr (2.5)
is again of the form (2'3) with the kernel
and an easy calculation shows
lltr
a B we have I| rtrttlflttldr
ll"'
,t',ortqorlr-
lu
(r)u.(c)dc
=
"
:
0 and
w,{r)r,'ovr' (t)dt.
Using these estimates in (2.13) (for
(l'v,totcu(ddn)"'
fo l1tdlou-,{*1* JA
= lu Wr4lf@'-t)a!$-P)
lu' b, {*r' I' l
>
ll/llp,' S ll.fllp,o.' Thus, it
lTf(r)lqu(a)dx
, Iu' lrU, l"',t,t0 r tqdt + e,@) I"o,h,1t1fltyatlq u6ya,
ues
:
(r)lp e)t-p' d,t
F\rrther,
and consequently, both the inequalities in (2.12) hold. This implies that conditions (2.8) (for i: 1) and (2'9) (for i:2) hold' (iii) Necessity for general functions qi,*i'suppose again that irrequiity (2.1) holds and define, for e ) 0, a new weight function
a,(r)
wr{r)l'' (tt
er-p'(fl_c:) q
inequality Suppose that, for the operator f defined by (2'7), holds. Thus, f is a untinuous operator'
T t LP(a) -> Lq(a),
(2.21)
Indeed:
u@)ls(x)lqdx)'u
\JO
[" Jo
+ Lq(u), i,:1,2.
A(pr,dr) : m and Az(pz,tltz): *.
,r-r'1t)l{r(t) +,br4)lo'dt) (zo'-,' S -\
(2.20)
We will proceed by contrailiction.lf. conditions (2.20) are not satisfied, then we c:rn show that necessarily
inequalities Now the first estimate in (2.19) follows from the obvious
(
a.
Notice that the first condition in (2.20) is necessary and sufficient for the continuity of the operator ,91 from (2.10) while the second condition in (2.20) is necessary and sufficient for the continuity of the operator Sz from (2.10),
Si: I?(u) where
Az(pz,tbz)
0, r)t)0. : [' eft)dr, x>t>0, Jt
weighted Lebesgue spaces for a wide range of indices. This problem has been extensively studied in the primary literature (see, for exarnple, S. BLOOM and R. KERMAN [t], F.J. MARTiN-REyES and
E. SAWYER [1], R. OINAROV [1,2J, V. D. STEPANOV [1,2,4] and the sources cited there). (ii) Let us mention that an important role will be played by duality.lf. the operator K from (2.24) maps tre(u) into Lc(u),
K:LP(u)+Lq(u)
s(r)(K
f)(4dtl
where we have applied F\rbini's theorem, Hiilder's inequarity and the assumption that
llK
f llc," s cllf llr,,.
(2.28)
p. 13.) (iia) The particular operator .I(,
(See also
with a non-negative function g.
Remark and Notation 2.8. (i) In the sequel, weight functions will be characterized for which the operator K is bounded between
(t) or)''
ll/11","=r
(iu) The integrol kernel
k(a,t)
fo*
l/ 'd::='
a,b < oo.
The particular cases of (i), (ii)
k(x,t) : (r-t)",
Q|_o,y
with the sarne norm C. Indeed: by duality we have
The kernel
O(a.b)
Lp,
(xf)(,).-
#
1""
O
- qafe)dt, o ) -1
is called the Riemann-Liouville operator and its conjugate
(x
il@),:
#
I,*
U _ a)"
f(t)dt
is called the weyl froctionor integur operotor. Flom the results be. low it follows in pa.rticular that weights will be characterized for which these operators are bounded on weighted Lebesgue spaces when a > 0. The case -l < o < 0 is not covered here
k(r:4 =
(s
- f)", -1 < a < 0, is not an Oinarov
because
kernel. (Recall that
in Example 2.7 (iii), this kernel appeared only with o 2 bfy
80
I\pe
Weighted Ineguolities ol Hardg
Some Weighted Norm Inequalities
(iu) In order to find necessary and sufficient conditions on the weight functions u and u under which inequality (2.28)' i.e., the inequality
(1,* x*n@)rqu(r)d,n)"'=
"
(f
gl
Q) A Ks e Lq(u), then f*.-
.
f@
-.
l (Ks)q(r)u(x)dx N JoI Jo
ff@)lpa(r)d,)"o (2.2s)
s@){oi''odq-t1x11xou11a1ar foo
+
|
JO
holds, we need a technical lemma. To simplify the notation, we denote
s@)(frs7t-t1r'11x"11r1a.
fors)0
:: g"n11r7 :: (r"rrX")
lo"
I,*
u'1r,r1n1r1or,
u"rr,flh(t)dt
(2.30)
with /c an Oinarov kernel. Nolle that if s : 0, then (2.30) reduces to the Hardy operators l/ and -f,[, respectively. We will write
Kyh: Kh, krn:
frn.
F\rrthermore, let us emphasize that without loss of generality we can assume that
/>0.
(2.31)
Now, we are ready to formulate our fundamental lemma.
Lemma 2.9.
Suppose that 1
( q( a
and thot Kq,ko oo itefined
b'y (2.30).
(") If Kf e Lq(u),
Prool. since the proof of (b) is quite similar to that of (a), we omit proving the first pa^rt only. Denote the left and right hand sides of (2.J2) by .I and J, respec_ tively. F\-rbini's theorem and obvious estimates vield
it
r-
Io*
: T,* "@ (1,'
{*
ilo{r)u(r)dt
([
k@,t)r(edt)
dn
ry
f* rcl l,* rr,,t)u(r) (lo' ur,,")f (4d")q *
Io*
rc l,
k@,t)u(r)
(1,'
' o,o,
ur,,z)f (4dz)o-'
*0,
=:/o*fr.
then
!
k@,2)r(z)d,z)'-'
: l* ra I,* ur,,t)u(x) [(/'. |")ro,4r@d4o-' o,o,
Since z lo*
u(,) (lo' ur,,t1y1t1at)q ar
fo*
*
4t
1 x in the integral
f {dt*07'10-r(x)(Kqu)(n)dr
Io*
I@)6.7t-t({(fru)@)dn.
.le one obtains
k(r, z) x k(x,t) + k(t, z), (2.32)
and consequently,
from (2.26) that
82
Ineqnlities of llordg fupe
Weighted
.16
Some Weighted Norm Ineqnlities g3
:
x !,* to (1,* ooo,t)u@)ar) (1,' ,@0,)o-'0, * f- tri I,
:
fo*
k@,t)u(r)
(lr'
un,z)f
Io*
@az) *0,
Thus, we have the result for e
It q * 2, let us wrire
f g)(xu)(t)1x yy-r
,r:
1t1at
Thus f x J * f1, and since.Il ) 0, the lower bound of (a) follows. To complete the proof we must show that I1SJ. Consider first the case q: 2. F\bini's theorem yields
foo
f@
r@
f@
:
f,
J"
l@
I
f@
f@
J,
foo
f@
foo
h N Jo I tAllJt f@lJz foo
foo
| f(t) JtI f@ JzI
foo
fz
Io*
2.
f ft)rt(t)dt
16
/f@
I I@ |\JzI JO
+ k(z,t)
and
k'(r,z)u(r)drdzdt
k(x,t)u(x)
I,*
f@
f, *
k(z,t)k(r,z)u(x)d,xdzdt
I,*
(1," ur*,")/(")d")q ' I,' *r,,2)f
k@,
1,
o,
2)k(t,t)u(n)
(z)dzd,x
(lr' *rr,s)/(")ds)q' 0,0,.
f@
l"*
k2(r,z)u(t)
I,* rk) I,*
(lr'
k(n,z)k(z,t)u(t)
rr,,s)/(s)ds)o-' oro,
(1," rrr,")/(")d")q '0,0"
:' 4(r) + 1i(0.
/n@ ^
= Jo I f@lf@dtl Jz Jo +
rawee)gf)(z)d.z
Here we have again used Fhbini,s theorem. But since t < z < r, condition (2.26) yields k(n,t) x k(r,z) + k(z,t) and consequently,
rt|)x
r@
Jo
:
k(o,t)k(r,z)u(t)hdzd't'
Since t < z < r, we conclude that k(r,t) x k(r,z) consequently, by a repeated use of F\rbini's theorem,
+
=
I{t)::fr* k(r,t)u(a) (1,' rrr,z)f (4dz)o-'
: I f@l *@,t)u(x)l t'(r,z)l@)dzdndt Jt JO Jt
:
f*
where
: J.
11
tu)t*of)()(R2e(z)dz*
: J.
'
r$lwou)(t)(Ksf)q-t1t'1at
*
lo*
k'(r,z)u(r)dtd.z
\ rz k@,2)u(x)dol I k(u,t)f(t)dtdz ,/ JO
Thus,
,r=
Io
t(t)te)dt
* Io* l(t)4(t)dt =: rf + {.
(2.33)
84
Weighted Inequolities of Hardy fupe
If q >
Some Weightd Norm Ine4uolildes 8b
2, we apply Hiilder's inequality (with exponents q
l*) t" obtain rflr;
:
lr* rcl t,*
I
s l,* ta(1,*
(lr'
oru,
rr*,s)/(s)ds) ' *)'-""0-" o"
\ (c-z)/(c-t)
k(r,z)u(a)(K7'Y-r@d")
:
dz
and thus, applying F\rbini's theorem (twice) and again Htilder's inequality,
'(l
:
lo*
\ (q-zlk-r)
k@,2)u(x)(Kf)q-t@)d'-)
)q
(d*)k-
to*
)
(q-r)
J@-z)/(o-r)
2, we find that
t{Drlt)a,
tr* rcl f,* f {,)ot,,q I,
lo*
z) /
f)q-' @d")r/(c-r)
, (1,' k(r,s)/(s)d")n-'
:
4 / f@
u@) (K f
Simila"rly, using the fact that e
4:
qu)t/(q-r) Q)
"U
" (I*
: (lr*, t"lturu)(z)(Ks
t@
tt f6
k@,2)u(a)(Kilc-'@)ds) *)(q-z)/(c-t)
q' ur*,")/(r)a")
uor,,z)u(c))t/(c-r)
* (1,* k(x,z)u(r) (1,'
" (1,* r@ (1,
: (lr* tra6,qk)(Ksr)q-'14a,)r/(q-r)
*ot
1Is
+
Jrl(o-z)/(q-t)
*.I1. The indices q - I and
fi *u conjugate, and hence
rft4 s
l,* r{4(1,' ,av,)'-
r@ (1,' r6ya,)o ' (froQe)azd.t
r1o
.16
= J, and thus
h ScsJ +4h, q- L l.e.
hSCs(q
-
1)/.
kq(x,z)u(x)dnd,z,
and F\rbini's theorem vields
(/o+/r) J It{n:wzo-tiG-mq' But we have shown that
I*
:
fo*
n&od@ lo r@(l:
: #I-
f
(z) (K ou) (z) (K s |
r61a,)o 'ata"
| )t- (z)dz,
88
Weighted Ineqnlities of
Hady 1lpe
Some Weighted Norm Inequalities g9
since
f" rat (1,' ,av,)'-' o, :
-*
f*
I,' #r(1,' ,av,)o-'
-...+ l^ fQ)gu)(z)(Kf)q-t(z)dzl rol
o,
: #(1,"11"P")o: frttrflo-'(,).
: ]q- -t. L
This completes the proof of the lemma.
2.4.
Similarly,
rf lr;
since
k(z,t)f
(z)
General Hardy-type Operators. The Case p Sq
*0,
Now we are able to state and prove the first main theorem of this chapter describing necessary and sufficient conditions of the validity of inequality (2.29).
I,* urr,z)u(r) (lr' urr,"y11"yr")o-' oro"
Theorem 2.LO. Let 10 \Jr / \J0 /
, (f
(2.3s)
/ f@
H , i.e. if we take k(x,t) 1, then .4s a,nd A1 coincide and are nothing else than the constant .4 froni (0.8).
Proof of Theorem 2.10.
(i)
and
(Ko,vr-o"11q
,: [' JO
l{'(t,syur-n'(s)do
0'Hence c n5/n
2 c;ro' (
/-
or)"'
o')o)''' > I'
Since
and in particular
lttl ( /- k(r,t)u(r)
(1,'&,u1*od6y
(Ksur-n' 1, ltoo) Tsyur,n'1s;a")
clh' (lr*
q1
or)t'o
6r"lr+t(q-r)/Q4) (r)/(r)
/ ft(Kout-o' \c-r \r/c . (/, o, 1'1too') 1s1ur-n' g)) ) since ^Rou is a decreasing function. F\rrthermore,
fo'
.
'
(t) at - P' (t)'
cilIllp," > llKfllo,"
: (l-
Inqualities
t*
oot-o' 1' / @o) 1s1ur*r' 1s)ds
:
fo'
,r-r'1",
:;rch :
(/"
,'-o'1r,1ar)r/(et') o,
(lo','-o' t,r")
firWoar-P')'/@'d(t)
"'*
l0l
102
Weighted Ineryalities ol Hordg TVpe
Some Weighted Norm
(+)','
"tn
(ii)
u)r+r / rno'' " (l-,4 /
("r+)' (K
x
sur-
rr)
f (txrour -
o'
( !o* {x r'
Y
/ (p' q'
109
Then Bllo' : llfllq,,rrr-o, and we proceed as before, using of course the dual inequality. The result B;SC proves the necessity part.
so that
Cn[/o
Inqualities
\
+'
n1,1t+r
/ @c'
/
)'
t
(nq)+t / rod
I 1t7ut - t'
(*r+)'/o' (lo* 6nu1le
p'
il (il dt)
tp'
Sufficiency
(2.52). Note that by Lemma 2.9,
r :: Jo[* u611rcry@)dr
1t)
s
",
/q
1t1
of.
(1"* f (,)(Kof)q-t(r)(Kou)(r)da
*
at)L
::
p1
Ir*
I @)@ 71t-' 1r11R uy1,ya,)
Cr(Jo +
Ji
.
estimate Js, we apply Hiilder's inequality for the product of three factors with exponents p, p/(q - l) a^nd r/q. We obtain Tlo
x
(K
sur-
n'
Y
h' 1tlar- n' 1t) Or)t'
Jo
= (*r+),n (I* = (",*)'"'
:
J,1t1,6at)'/o
f@
J, lf (r)ur/o 1ty11r[-n')/n 6)(r 1. The next theorem deals with the case
dr
(tror)" /e(t)(Ke'u'-p')"/p'(r)uv"("))
0 o,
we only suppose that ft(r, t) > 0 is decreosing in t, and no other conditions are imposed. on the other ha^nd, the sufficient conditions a.re different from the necessary ones.
d d")
: q'hBilll(Jill(|-sl')' h S C(p,q,s)ll/llp,,Brll K f ll1:,: :
*6,t)f (t)dt, r )
f llq," s cllf llr,, holds for all f > O uith C < Bo. Conaersely, il Q.56) holds for alt f > 0, then Bz S C llK
82
i,
:: ( [\J0
(2.56)
where
Go'ln' lo 7t1ut-n' 1r1or\"' /
Remark 2.18. (i) Observe that in Theorem 2.lT we require only one condition for the sufficiency part, namely the finiteness of the same 86 as in Theorem 2.15, a^nd also only oue condition for the
i.e.
rsll/llt,"(83 + Bl),
necessity is required.
and (2.34) follows with
(ii) For the proof of sufficiency, cS@$ + B!)1tto.
tr
called' leoel
we need the concept of the so, function introduced by I. HALpERIN [r]. The proof of
110
Weighted Ineqnlities ol HotdV
7W
Some Weighted Norm Inqtalihies
the following lemma can be found, e.g., in the Appendix to [OK]. Another proof was given independently by G. SINNAMON [1].
Lemma 2,L9. Let (a,b) be an intentol ond w o weight function on (a,b). Suppose thot 0 < tu(r) 1 m tor 8.e. E € (a,b) ond' [!w@)dn < a. Then for eoch measwvble function f > 0 therc exists o non' negatiae function fo (colleit "leuel function of | ") such thot f, f,
I !@dtS JoI [o(t)dt force Ja
(i)
(ii) ry w@)
(fi) To prove that Bs ( oo implies (2.b6) we assume without thut..I is compactiy supported in (0,m) and that
loss of generality
ar-p' e ^Ll. Property (i) of Lemm a 2'.lg
(xr)(,)
-@)dx
k(x,z)
(o,b),
k(x,e
(ffi)p = l"'
.@)d'*
(K
,to, p> L-
Proof of Theorern 2.17. (i) Suppose that (2.56) is satisfied for all functions / > 0. Then by the reverse Minkowski integral inequality we find
where
: Applying the duality
of.
U(u) it
:
follows that
:
d::=,
= (/-
(fr
lo*
r ax*ou)uc $)dt
=
r l/y' ouye'
/o
111or-t 1fidt
)
ll(Kou)t/qllp,,o,-t,
-
Bz.
lo" tt4o, * fo'
r"t{o"*
I,'
([
,(")d") datk(r,t))
I' ([
/"G)d") d2(k(x,t))
f')(")
/o is the level function of /. Therefore
l* Io*
wnq@)u(r)d,r
u(,)(Kf\c-t@)
s
I'
rrr,t)f"(t)dtdr
f* r"t4 I,* or,,t)u(n)L(1,' . I') uo,s)/"(s)ds]
= fo* f
C
that
fo- wilo{a)u(x)dx
.tc(c,t) for 0 of Lemma 2.lg with u): ut-p' we have
(
o-'
o,o,
o-'
o*0,
drdt
s S t. By property (ii)
LLz
lIody
Weighted Ineqtolities ol
1t l^ f"{,)a, Jo
:
Some Weighted, Norrar. Inequalities
fupe
ft( f@ \rt-o,(r)d" ,,
llf"nl,"
/, \;r+6 /
:
|o*
u(r)(x f)q(r)ax
= |o*
, (lr',r-r'1s)as)q
fft11o,k'-r)(c-rt-q/p$)uq/p$)
foo
r-d' slpl|ld: l(f -L)/q'-t/pl': rlq yields -
(/-u"l . (l-
=l
i- i t.t k
(koQ'tt@)o')o''
(l-,r')e@),(dd,)q/e ( [* " \/o
ovn'@)(Kour-p')'/q'
wherc
k(t,r)
r)0, e)0,
int. If
i,s increosing
/ fe
I (xoor-r"1,/o' = (\"ro
\r/r
1t)(Kou1,/oltyur-n'
llatl' /
}})
1)-
2.7.2. The general Hardy-type operator in the form from Definition 2.5 was introduced and investigated by S. BLOOM and R. KERMAN [1] and independently by R' OINAROV [1]' The result involving the Riemann-Liouville operator (see Example 2.7 (iii) or Remark i.s (iii)) was proved by F. J. MARTIN-REYES and E. SAWYER [1] and independently by V. D. STEPANOV [t]; in fact, in the first mentioned paper, a somewhat more Seneral kernel was considered, namely that given in Example 2.7 (i). The proofs of the results given here follow closely those of v. D. STEPANOV [2,51. More information will be contained in the book by V' D' STEPANOV: Voltena integral operotors on semiutis (in preparation).
2.7.3. The case 0 < g < 1 < p < @ in Theorem 2.17 requires the concept of the leuel lunction and its properties described in Lemma 2.19. This lemma was proved by I. HALPERIN [1] and used first in connection with the Hardy inequality by G. SINNAMON [1'3]'
r/c
tr4o4 1 was considered,
[3]), the conditions
are satisfied, where
*
:
*
kq(x,rk)u(r)Or)
o
118
Weighted,
Ineqnlities of Hailg
7W
as a mapping between nonweighted U and, Zq with 0 ( p,g ( and p ) ma:c(l/o,l) are considered by D. V. PROKHOROV [1].
m
3
2.7.7.Integral operators, in particular fractional integrals, in more general homogeneous spaces (see Sec. 2.6) are investigated by many authors. Let us mention at least the importa,nt Georgian school, rep resented, e.8., by the paper of V, KOKILASHVILI and A. MESHKHI
[l] and the references
The Hardy-steklov Operator
there.
3.1.
Introduction
In this chapter, we will deal mainly with functions defined on the (sta^ndard) interval
(0,
*).
Let us start with an example.
Example 3.1. The classical Hardy operator tions / : f(t)) 0, f € (0,m), as @
f)(r) ,:
g)dt, Io' f
o
0
r
t'
;
L24
Weighted Ine4talitics of
Hody
The HodySteHoa OPemtor 125
IW
finite if (3'13) holds' the last expression (and consequently also 'r{) is we have no oililitionol enn' Hence, forlhe Hardy-Steklov operator Tz
ilition'ona(andB)andthus,abiggerclassofadmissibleweights. Let (iii) In Tireorem 3.? we will deal with the case 1 < p S q < m' with (3'9) is trivial : o, *"rrtioo that for the ca'se p : q f inequality u(c) =
fo-'(t)
l- (/:;' ,(,)*) u@)dr= Io*r,, (fi;',{do)or, :.1.' On the other ha'nd' the i.e., (3.9) becomes equality with C euery droice i*ai inequality with p : q: L and u(c) : * fails for case u(c) = 1' of u, as the following example illustrates for the there is Example 3.4. The foregoing considerations indicate that
the Hardy-Stcklov a substantial distinction Letween the Hardy and again a(r) and operators. To grve another example, let us consider : weighted norm bic) as in (S.ri), but u(c) : o-Q a$ :r(t) 1'. The inequality lla'rdy the inequality ior the op"rtio'? from (3'2) is then for the averaging oPerator:
which
is not sotisfidif
="(1"*
we take p: q:
r'(d*)'
/t@
t c>0 \Jc
(see, e.g., [OK, Lemma
"o*opl"ding
5' ])
and the equality sign, as ca,n be shown by
The Ha,rdy-Steklov operator gives rise
to the moving
auercging
#@ l"l.' ,n or, t
>
o
(3.14)
This operator in its various forms is of considerable importance to the technicnt andysts in the study of equity markets. These technien'l onolysts try to predict the future of the stock price or the future of an equity market solely on the base of the past performauce of the stock price or market valuation, respectively' For example, some analysts consider a,n equity (stock) / whose price at time t is /(t)' a recommended "buy" if (Sl-zoo/)(t) < /(t) while the reversal of this inequality is a "sell". Similarly, if t(4 represents, say, the Dow Jones Industrial Average in New York at time t and if (Sl-zoo/Xr) S t(t)' then it was observed that the return of the market in the year following t gained 12% while a reversal of this inequality showed that the market lost 7% in the year following t. The introduction of weights and the pointwise estimate replaced by mea,n estimates with good control of the constants may therefore yield additional quantitative information in the study of fina.ncial markets.
The Steklov
u(t\dt
\
o, mk = b-1(o(rn1a1)) if &0.
. l. The estimat e of. 12 is similar. The change of variables g
:
where
b(r)
and (3.30) yield
12
: :
(*E t"'.'
(F,
I^'"':"
(ff'
A
r$\u(Ddt)ou{ta')'to
': :
(Io..' r (t)w(qdt)o'u'o'oo)''
*o (/"
'r-r'1s)ds)
"'
U:::)
\r/P' / ro-'(t)
/ fa
/ t6
and consequently, (3.21)
Remark 3.8. (i) We can easily see that
QI)@):
P@)
Jor,,
the operator
f
con-
tr
conjugate to
f(t)dt
then (3.21) holds for
fa-'(a) Ja_,@)
(
oo,
/
> 0 if
Indeed: Using Minkowski's integral inequality, we obtain
:
(/-
( I^'lo'
,n or)"or*)'''
f(t)dt.
(Use the duality formula (f ,Tg) = \f f ,g, and F\rbini's theorem') Since the operator f nas a form similar to the operator ? and maps,La'(ur-c') continuously into If'(ur-n'\ provided the mapping T : U(a) -+ Lq(u) is bounded, we immediately obtain from Theorem 3.7 a cha,racterization of admissible weights rewriting condition (3.23) into
r{
0 denotes the lc times repeated composition, while for ft ( 0, (D-1 o o)ft is to be interpretea as
6'r$*-r.
Suppose 0 < e 1 p, | < p ( oo, i : i - i. Then inequality (3.21) (or equiualently inequality (3.22)) holils lor oll tunctions .f > 0 if and onlg if
Theorem 8.LL.
A:ma:r(Ar,,4z)(m
(3.36)
13tt
The Hody-SteWoo Operctor 139
Ineqnlities of Hordg TVW
Weighted,
Then
where
A1
:: (!,* {1,' ,(,(,)) ( I.'lu''u'o')''' u@)ax)og)or) " (I'u{qa,)'t'
(1,-
:: (l- (
[:' I""o,' (
,(
(3.37)
1,a-'(u('l))
(l::]
Io
{,,a')' t'
Io"'
uu1,1 a
: (21i,
ond,
Az
(
")'
* (1," utta,)'to ut*)o*)"@or)'''
ou)o 10 or)'''
(l^';' (l^"u,-t4a")'/o
.,67a')'t'
*
u o 1r1
/o'otilds) ( 1u""
uo1"'1a")'
*p-' "
o1o
1r1or)"'
(3.38)
m : (b-1 o dk(t) then t : (o-r oD)&(rn) - mk.Hence if. t : Mr+r then Mpal :'trtrks which implies that m: a-r(l). If t: M1, then (o-r ob)k(rn) : M* : (a-r ob)k M6. and since ,1140 : 6-t (t), it follows that rn: A-l(f). Therefore the last sum is equal to If.
withw=ul-P'. Proo!. (i) Necessitg of condition (3.36). Suppose that (3'22) is satisfied. Let {rr,}3r, {r"}i" r be sequences of Zl-weights such that un(a) < u(r), wn(t) < .(r) and u,. f u, un t ur as n -r oo' If -- un(b-r(ilXb-t)'(y) and u6(3r) : "(b-1(v))(6-r)'(v), then "u.,"(r) q,.(u) < uufu). ' Fix rn > 0 and define {*x}x.z as in Definition 3'5' For D(rn1) and D(rn1) from (3.34) define
D(m)
: (r*"o' r*r,) "',
o1*1
:
(Pru'
r^r)'"
Dt-),
ilgz(-)}
0 with d independent of t.
A Cauchy problem
,,
,a2{,,0a*)'/'
A two-dimensional weight
Proof. We apply Corollary 3.13
It
69,
A/n/ry1
" (l: ,r-t 1fldr)o''' or)''o . * th.en in,equali.fu (3.46) h.olds
for aII fim.etion,.s q )
0.
152
Weightett Ine4volities
Conaerselv,
of
i/ (3'46)
Hardv
hokls
The Hardy-SteHou Opemtor L33
I\pe
for
oll g
20
(3.a6) with
then
U:'t U:'"'("'t)d' r eB \ 1/P' . (f ;-c' Plar) < oo '
o"':F'-
Proof' Fix Then
it
t € (0,1) and apply
C
o)'"
Conversely,
s@)
Corollary 3'13 with o
: f' $ = l'
cntl(1,* taw{da*)oto at, fo'
(/' (/- w(r,t) (1" or"ta")"") o')''o (/' "olryor)'n (/-n't"l,
@o)
=
,,',t)a")'/n
(!'
''l-''
tl")
\
/
k(p,s) (0."11?.,,
lr"t'' .t"'t)a"J
(/u
( rP,r-n'1s)ds) ' te/r'1r/q '(/: ) :
is change g 0' = alti the last expre*sion Here we have made the to 1S.aZ1. Consequently, we have obtained finite for a.e. €10,
t
rjl,r"
0, then for fixed
xp,B1@)ut-n' @)dr.
For this function 9, (3.46) yields
;-r'6ya,)''' : (1,* tav@) (l: " " - (/'u- w(r, t) (l*, t"la,)' r') or)''
:
where
s k(p,q),-.,prr, (f
:
9)
o
. (1,"'u (l;" w(r,t) (l**-'r")xr.,pr(s)ds)'*)
l.e.
c(t)
(3.46) is satisfied for all
(3'48)
l' (/- w(r't)(1" tav')"')"
=
kbt,q)A.
a,0,0 < o < p, let
follows that
s
if
:
r
(1,"'u
(t" w(r't)(l!*-"")'")
: (!:,r-o'1";a") (1,"'u (t" and (3.48) follows by dividins Or
o')'''
o')o')'"
w(x,t)ar) 0,)'''
(f ,t-r'G)a")'/' .
tr
Remark 3.16. Let us point out that (3.48) is not sufficient for (3.46) a,nd that (3.47) is not necessary for (3.46). For details, see H. P. HEINIG a.nd G. SINNAMON [U. The following result is an easy consequence of Theorem 3.15. It answers the question raised in Sec. 3.1 and complements some of the results which will be dealt with in Chap. 5.
The Hardy-Stekloo
Weighted Ineryatities ol Hordg TVpe
154
Corollary 3.L7.
ondf €C1(0,m)' Thenthe
Letlcp39-0 if anil onlY / ral
fte)
\q
1(p< oo' Theninequality (3'52)
satisfi,ed,
/ 16
\1/q
rt,la")'uplat) Itl/ / / \Jo\J"to is
(3.53)
\ ="(/.
lor allF > 0' The constonts
\1/P
P(t\v(t)dt) trsal
: (i) Sufficiencv of (3'54)' Fix / f (*)' ' .rr(r) :
fr,f
€ RN' and define
.
Necessity of (3.54). Suppose
F
by
(t )
:
and define
f
(1" ;-r'(to)tN '.)
t)0, r€Eiv. lr,f {rr)r*-tdr: ar-e'
(tr)tN-'o)''''
Now the change to polar coordinates (3.54) yield, together with (3'53)'
fi
: tT' U :
so and inequality
\ r/c
u(tr)tN-rdrdt)
(l"ir''r"u")
o'
F(t)
o
ov')''
: (/- ,,,) (l.i:' f,.rt ,),*-'a"a")o o,)"0 :
/(s)d,)'t"'*)
(fi1f"/t'o)'" ^*o)
and hence
(1,-
: (1r,, unu(tr)tN -t dr)'to v-'ro {t)'
(l-L
F(t)ur-o'(tr,
that (3.52) is satisfied for J
{rr)r*-'0".
F(t)
=
ro1t1v1t1at)'/o
"
and by (3'53)' Then by lliilder's inequality
([" [.',.,/n*',
u@o')''o
(tr){ -r dr) or)"' =, (I* (!,, r u"ta : (l*,!p(r)u(r)dt)"' (ii)
C in (3'5a) onil (3'52) orc
the sarne.
Prmf'
t" (I*
'r")'")o
U,* U,^
ulto1tv-Lao)
" (,fr: l,,tt"\'*-'a"a")o
o')"0
)
0,
158
The Hardy-SteHoo Operutor lS9
Weighted Inequalities of Hordg Typc
(i) tor L < p 0 and described in Definition 3.2' For c € RN and c r € Ery, we have lxl = t' > 0' Corollary 3.2O. Inequal,ity (3'52) is sotisfied for all | = f (n) c € RN, if onil onlY if
\"/p' I \r/"
-,.
(3'57)
au)an) < oo J .u'-P'@z)daz) ly,lca(lrl)
are satisfied. Here,S is o mtLiol Junction defined by
,9(c)
and u for which
1
:,s(l'l)
:
lclr-trofl'l)/lrrl
witho the normolizing function defined by (a.Bs) ond lDryl the sur-
foce arca of the uni.t sphere in lRN.
Prmf. By Theorem g.1g it suffices to show that conditions (8.5b), (3.56) and (3.57) are equivalent to the conditions of Theorems 3.2 and 3.1'1 with u, u replaced by I),v defined by (8.b3). But this follows at once via changes to polar coordinates. For example, if we change
160
TW
Weighted Ineqtalities of HordV
The Hordy-SteHou Operotor 16l
: 8P' yL: srpr'' u2 = : to polar coordinates in (3'56)' taking r tT' 9 then we obtain (0'oo)' € s2p2 with T,P,Pt,P2€ Erv and t,s,s1,s2
I*
_t
L,tN
s
(t)d,r[11,,",r,
/ ft |
" (/" /"" "fl-'"(
" (fi:'
: (l-"r,r
_,
lr,sN
s1P)dP1ds1
[1-,,",,,,
Weighted inequalities for this operator are studied in A. GOGATISHVILI and J. LANG [1]. Let us mention one particular result which for lc(o, t) : t reduces to Theorem B.Z.
u(s dd,o
Theorem 3.21. Let 1 < p < g < oo. Then the inequality
\'/P
)
1,,'l-"'-o'(s2P2)dP'0") u(")
"]
*)""
(!,hokls
(1":;)
",',";
for oll functions f >0
xf orul only
(r/X') :
if
\Uc I P@) \ rlP' u(s)dsl 1ro'1r,s)ur-e'(s)dsl .*. I - "yp-. ..1\"/c cJs,c(y)
0.
L74
Higher Order Hordy Inegalities l7S
Weighted Ineqtalities ol Hodg Type
Let us again emphasize that we suppose that (Ms,M1) satisfies the P6lya condition and that lMol + lMil : k, Mo * 0, Mr * 0.
where
K1(r,t)
:
K2@,t)
:
l"'
A
-")--t(t - r)--k-td',
ft
t" -")--1(t Jr-
.
--r.- s)--e-lds,
0
O' Then the fKs(c,t)l N sA(l -c)bt"(l
k' lMol >
k- 1-4, so i 4f ' Abo' both zi andzi-larezero'Itfollowsthat( '40)holdsinthiscaseaswell' reducing to 0 2 -1 + 1Y; (i). 11trl(l)
Since the degree of
tr.r
is at most
lc
-
L we must have
0: zi.
(4'41)
Adding inequality (4.39) and inequalities (4'40) obtain zfi
for 1 S i < lc - I
+ zi+ "' + zi-r2 zi + "' * "i-z-
(k
we
- 1)
&-1
+ f(xr(t)
+ xv;(;))
and, by virtue of (4.41), this simplifies to 0
u -(k
-
1)
-(k - 1) + (k 1) - 0' in ( '39) and in ( 'a0) for I S i S lc - l'
+ (lrl + lMil)
:
Thus we have equality is Equality in (4.39) proves that u has no interior zeros which assertion (i). -- ii ,(0 'f O, equdity in (4.40) means that tr(i) has no boundary zeros except those accounted for by the terms 1a(i) and xu;$)' Hence the onlg i/pa.rts of (ii) and (iii). hold when u'(t) # 0' 0' As If u(i) o, ti"" of couise both u(t(O) : 0 and
'(t)11;.= wehaveseen,i>k-L_ainthiscasesotheonlyitpartof(ii)holds : 1' vacuously. Equality in (4.40) when tr(i) = 0 means that 1v;(1) tr so i € Mi and the only if pafi of (iv)"holds as well'
=
then uo"
f
k, thenwrc_r
*
0, and i!
aIc (
lc,
0.
Proof. If. a * e : k, then Lemma 4.1g (iii) states 11ru1 rrl(c-t)1O) 0. # This means, due to the definition of u, that woc_r + 0. As was mentioned above, a * c : lc 1 cannot hold (see (a.2b)), so if a*c < ft we have c < k-!-a. Now by the definition of a* (which is c) we have c e M6, so Lemma 4.19 (ii) states .G)10; ; O, which means that wo" * 0. tr Now we are able to derive the estimates of the Green function. First, let us introduce another boundary value problem which will reduce the number of estimates required. The symmetric boundary value probtem For our pair (Mo,
Mr), let Ms
=
Mt, Mt:
Mo.
The boundary value problem
g(h) :
fin (0,1); e(i)(0) : 0 for t efro, eu)(l) : 0 for j efrr is called syrnmetri,c ro BV? (4.q(fq(a,b) : (0,1)). The incidence matrix E of (Ms,Mq) simply means that we interthe rows of the incidence matrix E of (Ms,M1). Thus the P6lya condition still holds and it is easy to see that cha.nge
A:b,6:a,e:d, i=c and indeed that
A:8, E:
A, 0
:
D,
D:
C.
The Green function G@,t) of the symmetric BVp is given by
G1x,t1: G(l - z,l
- t).
Higher Otder Hordy Ineqnlities 189
188
TVpe Weighted Ineryalities ol Hardv
the enough to prove this theorem under Proof of Theorem 4'11'i|.is restricted are (4'29) iflhe estimates restriction that r + t ! f since become of function?(r,t) the symmetric BVP they
,o ,ft" Green
lc(l-r,1-t)l
* sb(l- flAtD1t-t)c for 0<sctll-r'
for 0
il -jl
i=o j=c
r,'(s) > 0 when 0 < s
/ 0 and obtain the following tripte of necessary and sufficient conditions for the validity of inequality (a. a) (for (a,b) : (0,1)) under conditions (4.49) with our choice of B(: 0) and C(> 1); i.e. under the conditions
e(o):0, .ys'(o)+(t-r)g'(t)=0, l)
1
:
196
Weighted Inequalities
ol
IIigher Order HardA Ineryotities 19?
Hordy Typ'
(i) Let us suppose Mo - 0, Mr = Nr. As we have seen (see Theorem 4.3, Remark 4.4), the Hardy inequality
forl( pSqlcx.' .,'l/q / r'
/ i
\1/P'
't-n'
( [' O" - t)c"(qdt) U, -sup. \Jc O 0, and it can be easily Hiilder's inequality and shown trrat lgii t e .ri.'rurthermore, using
where
f
rf-r)n1r1u(ddr)
lf a
" U,'
(4.66), we have
rlhllo,,
s
or,r,
I,lg(t)lar
JO
Definition 4.26.
(1, ave'$)dt)''o' (lo' ls(t)leu(t)dt )
Since g S lSl + h, the positivity of
llTslln,,
s
T implies
llr(bl + h)lln,, < cll lgl+ hllo,'
that 1z ot-n'(t)d" : [) positive integer n set Suppose
hn(x)
z: anuf,-n'1r1( "\J'
['
vayo,
t:r-p'(x)d'o: oo' For each
- Jo [^' btt)lot)/ x1o,,r(")
€ (0'1) : ut-r' where ,1,-o'(r) - ur-p'(x)'.xs,.(") with '9n - {c u (r) . ,,i, ind, Llan :'ft ul-e' (r)dr' Again' h' ) 0' lgl + n" e and 9 ! lgl + h' so that for every n we have llrgilo,"
s@)
:
Io' "o-r rr,t)h(t)dt
(4.7s)
.
Now, the solution of BVp
g(k):f, g(i)(g):0
for
i€Ms, eU)(l)=0 for [email protected])
can be combined from the BVP (4.77) and from the (overdetermined) BVP h'
: f in (0,1),
t!(0)
: h(l) :
0
(4.80)
whose solution is, according to Lemma 4.22, given by
7r: S$
-
Szf
with,91,52 from Definition 4.2ti, provided there is a z (0,1) € such that (4.66) is satisfied.
208
If
Weighted Inequalities ol Hardy Typ
Eigher Ord,er Hatdy Ine4uatibies
we denote
F(r) :
(x1o,,y(t)
we have h
:
-
xf,,rt(s)).f(t)
:
( I@) for n € (0,2), { t -/(o) for r €. (2,1),
SF where 5 = Sr * 52, and the solution of BVP
(
2Og
Pruf.Suppose, .,!!y) _+ Lq(u) 1nq,/ and g sarisfy (a.29). Set F = (xp,"1 - x1,g)f . since. / 9G., = h satisfy (4.g0) we may apply Lemma 4.2T to get g(,t-l)3"4 if where .g is the op"ruto" fro_ = Definition 4.26, and due to (4.2g) wcfinally obtain
'79)
:
g(t)
is given by
Io' "*-rrr,t)(sF)(t)dt= (r.F)(c).
Thus
s@)
: t;
G&-r(r,
1,,
t)(.9
F)(t)dt
which is (4.82) since
Gft-1(c,t)
lrtrrl " fxro,,l(t) Jo
:
llsllo,"
fz/
r{')a' * x(,,r)(t)
J,
r{')a'lat
(4'81)
fr / f" t--'(" (l
J,
,t)dt
\ _.
"rra
.
)F(s)ds
f
kt 0 < g 1a ondl < p ( 6, letu,a be weight functions on (0, l) and let z €. (0,1). satisly (4.66)' Suppose tnoi i[-r, Mf-r ore subsets o/ N*-r, lMf-tl + lMf-tl : k - L and (M[-r , U!-) satisfies the P6lga condition' Then the Hardy
Lemma 4.gO.
inequality
i=
(4.82)
cllg(u) llo,,
functions g e ACk-r(Mo,Mr) with M. 0,1, if onil only if T : IP(a\ + Lq(u).
(xp,4@)
-
X1,,r.1@))h(z), s@)
:
cx-r6,q6h)(t)dt.
lt
:
Then the following assertion will be useful:
for
(r):
Similarly as above we obtain that 9(c) (Th)(r). The definition of g shows rhat 'sh and g satisfy nviiia.zzl so that g satisfies the boundary conditions
(r/X').
s
c;-i";;*';;
< Cllhlln,, forJunction y, e i(qsatisfying (4.62) in order " that ? : r?(a) _+ Lq(u). Fix such *; ;; a"doJl ;o#nclude
where we have used F\bini's theorem. Let us denote the last expression by
holds
g(hl. that (4.g2) holds. Since ? is a positive op erator (recat that due to Theorem 4.1r, lilill" on (0' 1) x (0,1)), Lemma 4.2b shows that it is enough to prove "iso that llThllq,"
\
fz
llgllq,"
f:
Conversely, suppose
to-'(",t)dt)F(s)ds J" (/" *
= lffFllq,,, < CllFllp,o: Cllfllp,,,
= M:-'
U {e
- l},
o(i)19;:0 for ieMt-r, go)(r):o for ieu!_r. We have also
g(e-t; :,SA so that using Definition 4.26 weobtain
e(&-r)(o)
:
(s1ft)(o)
: g,
Finallg differentiation yields / satisfy BVp (4.29). Thus
ItThllq,,:
llsllq,u
which completes the
gG)
s(e-t)(r)
-,f
(s2nx1)
:
g.
and we have shown that g and
< cllg@ilo,o:
proof.
:
Cllfllp,o
=
Cllhllp,o,
tr
2]iO
Weighted Inquo'lities ol
The caseMo
:
Hody
Nl, M1= {k
Higher Otder Hardy Ineqnlities
Type
Thus (k
- l}
We start with this simpler case since conditions are more transparent' Here' satisfy
the necesriaxy a,nd sufficient I € ACk-r(Ms,M1) has to
: 0 for i = 0,1,...,ft - 1, r(*-t)111 = 0, (4'83) : ft-r' Ul-t :9' According i.e., we have Mf-l : {0, 1, ' ' ' ,k .2} g(i)(0)
has the simple form to s"". 4.2, the-Green function G&-l(c,t;
Gn-' (r, r)
:
G5!
@
-
: L' ll"' ffiro,')(')d']
h(s)ds
x1o,,'1@)
Q2h)(x)
:
X1",r1(x)
(Tsh)(x)
:
x1",r,1@)
(Tah)(x)
:
X1,,rjr;)@
Ifr 1' if we allow, roughly
"o*"
The case Mi:
-,r,
*ro(1
#tn
M!-r u {k - LI' M!-t *
0' i
:
(Mf-t,M!-t).
^L_'r
fz/ fz
L'
-dB I"'rt(r-t)ddt if s1z1z1
- d' I: r(t - t)Ddt ir
ti
I,' Gh-r(r,t)dt
ro(L-r)u
I,/,e-t)ddt if rSz;
rA(r
I:
E
l"' Gk-r(x,t)dt A,
- d,
r(L
-
*o.(t - 48 tc{t f," t,"
ck-r{r,t)at
we obtain
N
,A(r-do
I"
z
0 and Aq*a *L - o4*a+l > 0. (/( is the consta,nt arising from step (ii); it depends onbq+ g and d.) similarly we can verify the remaining conditions even in the case p > q.lf. A = a - I or B = b l, we have to modify the estimates on a,0,'f ,6- It ca,n be shown that these restrictions are also necessaxy for (4.82) to hold.
(4.82) holds provided Then the k-th order Hardy inequality
4.8.
d+1>0, c*1* 4)0' P+L+ bq>0 u in Definition 4'9' where a,b depend on Mf-1, U!-'
&
'y*1)0,
- t)ht,(t
Overdetermined Glasses (Another Approach)
Deffnition 4.35. A couple (Mo, M) of subsets of N1 - U will be calld standond if it satisfies the p6lya
{0, l,
. . .,
"*ditioo
222
Weighted Ineqnkties ol Hordg
(see Sec. 4.1) and
Higher Order Hordy
lVF
if
lMol+lMl:k' we have established necessary Recall that for standard couples validity of the lc-th order Hardy inand sufficient conditions of the and that the Green (a.30) for functions I e ACk-r(Mo'Mr\ "oJit,, function G(x,t) of the BVP
true for (Mo,M).) We will show that necessary and sufficient conditions of the validity of (4.80) for g € ACk-r(M6,Mr) a,re the some as the corresponding conditions for g e ACk-r(fro,fr) prculded the weights u a,nd u sati.sty some additional ossumptions. A particular case We will explain our idea on the particular standard couple
Necessary and sufficient conditions of the validity of (4.30) for e are described in Theorem 4.3 (see (a.i) and/or
g.
!Ck-t(6o,fr)
(4.e)). (4.e7)
of the Hardy inequality is uniquely determined' The investigation of the weighted norm in.ii tt tlien equivalent to the investigation
To the boundary conditions described by
:9'(0) : " ' -
g(0)
:0,
(4.100)
where
M is a nonempty
jeM,
(4.101)
subset of N1. Then we have the situation
described above, with
1l
(r/x") = Jo
(t)dt.
(4.ee)
"@,t)f
problems for the Up to now' we have investigated overdetermined
Mo: fr0, Mt = M. For the operator 16.\t
special case
M;: M!-r {k - 1}' i : 0' 1' uuple' but with respect to a where (Mf-t, MI-\ was a stondord we will deal with i"Jo,rAity of order k - 1' In this section il;; -i"*"rAoverdetermined i'e' couples satisfying U
couples (Mo,
M),
lMol+ lMrl >
r'
add some neu' boundor! con' which are constructed as follows: We couple 1frs'fri' Uy ditionsto the conditions described "t*a*d " the same is since (fi0, fi1) satisfies the P6lva condition'
id;i;o
nu)11;:o for
(4.e8)
for/Z0,where
9(tc-r)(0)
fry,fr1, i.".,
we add conditions
inequalitY
ll"/llq," < cllfllp"
229
fr : ryt, fr, :0.
9(&) : I in (0' 1) ' 9(0(0) : 0 for i€Ms,
90)(t) - 0 for ieMr,
Inepalities
\
T from (4.9g) we have
QI)@)=
|
I
J, @ -t)k-tf(t)dt,
ffi
r € (0,1).
: ?/ obviously satisfies conditions (4.100) and the : / in (0,1). Moreover, since sulr)- (k,, (-l)' tn/,fr (r - 11*-t-' f(t)dt,, : 0,1,"',ft - l, - i-
The functiol 9 equation 9(&)
.
conditions (4.101) lead to the assumptions
I' r, -
tyk-i-rlqqat
:o
for i
eM
.
(4.102)
224 Weightd
Ineqvalities
of
Hardg
IW
Eigher Otder Hardy Ineqntitir-c 226
which satLet us denote by Fu the set of all functions t € I/(u) isfy(4.102).Rrrthermore'supposethattheweightfunctionusatisfies
rl
for i e M,
(fldt c a ['l Jo' -11(t-j-r)r'rt-f and denote by
(4'103)
Vu the linear hull of the functions
Pi(t): (1- 11t-r-tu-r(t)' i
e M'
V-u is a finiteObviously, due to condition (4'103), 9i e U'(a) and dimensional subspace of. It'(u) (of dimension 'n: lMl)' If we define a duality (',')., between Zp(o) and I?'(u) by (g,
h),
: l-1r JO
s61n1t)u(t)dt, s e
lt(a)'
Thus, if we denote bV Vi i.e. the set of all g e U(u) such that
we have shown
Jo
- qh-i-rdt- o
for
i € M'
inequarity under the overdetermined conditions (4.100), (4.101) Uu ,irr"ua to the investigatio" ..f_,i:3"dity ( .l0 on the """, subset Vft of Ln@) ) provided u satisfies (4.108).
(ff) Obviously conditions (4.108) may be replaced by a singte condition
f ft -
t)G-io-t!p'or-e'(t)dt < @
with lb:maxfi:jeM\.
(4.105)
-
for the particular choice .Ds
:
ry1,
te'-t)at-p,(t)at)q u@)tu < @. I' (I" O- t;t-r1r - r)(&-io-r) (4.106)
m' ard Fu is a closed subspace ot' U(a) of finite codimension So we have proved the following assertion'
f}' Lemma 4.g6. Lct M be o nonempty subset of {0'1'"''k closs Then the Honty inequality $'30) lor the oaerdetermineil .ACk-r(N*, M) it equiaolent to the weighted norm inequality
all f e Fu
=_ Mt:0-asfollows
:
Fu:Vi
ffiduality
Fy:ii.
Theorem 4.18. Letl-
q;
O(oltr) ff-"1 Hiilder's
:
vs,
F(c)
:
inequality that
g(k)(t), and since /
( -,
we have by lp
and consequently, due to Theorem 4.4b, inequality (a.fZA)
q,
(l"" (lo''roo*)''o *(0d) fot
P> q,
- tlP' Moreouer, the cnnstont C in (4'128) sotisfies CxA.
oo,
lo,, ( [* e - x1G-Dn'rt-p' r]* " b)dr\ ,'t"' (t{ - rrt|ltt'.' \"/" ) and fr(c) -+ 0 for o -+ oo, i.e. 9(m) : 0. But also 9(m) = 0, and hence, g : A. Moreover, g(m) = y'(*) = ... : |ti-il1*; :
u{da')'^
r;(k-r)n'rr -o'61ar)'/o'
(
:"+_ L)l J,I,* u - x)k-tF(z)d,2, r > o.
l,i(")l s
, (lr* oL/r = rlq
if it
i@)
r;(e-r)r'rr -o' p1a*)tto ro, p : ' (/-,' -
A&,rt:
< m.
(if)
(l- (l',' - r;(t-r)c,1 +a,)''o " (/*
.,4
g(oo)
ar-P'(x)d*] tor p9q, "[\JtI /
Ak,o :=
and, only
o
for g satisfying (4.129), and hence ,4 < m.
r l/P'
/ 16
if
(a.i.J2g) holds
P'oof'(i) The necessity of condition (4.180) folrows from Theo rem 4.45: If (4.128) holds for g satisfying (4.rg1), then it holds also
where
with
The key result reads as follows:
Then inequolity (a''28) hokls
i:
g.
O
ioia" fo, CI
Remark 4.47. (i)
It follows from the foregoing theorem that in the case of the interval (0, oo), only one zero condition at infinity for the least derivative is important. v. D. srEpANov calls this phenomenon rhe heuristi.c principre. Flom Theorem 4.46 it follows that necessary and sufficient conditions of the of inequality 'aridity (4.128) for functions g satisfying (a.181) [i.e. with the second line of the incidence matrix of the form 1,0, 0,. . . ,0, 0l are the same as for
functions satisfying
g(m) = g'(m)
: o
(2nd line
l,
1,0, . . . ,0, o)
240
Weighted Inequolities of Hordy fupe
Higher Order Hordy Ine$alities 241
or
S(m) : g'(-) :
9"(m) : 0
(2nd line
:
9(m)
(ii)
g(0)
(iii)
g(0):e,(0)=0
(ia)
g(m)
(")
g(0):g,(oo):0
(oi)
g'(0) = g(oo)
(uii)
s(0):y'(O):g(oo):0
:
etc. up to
g(m)
:
g'(m)
:...
:g(ft-2)(co)
: 0
(2nd line
1,1,...,1,0)
or finally (Theorem 4.46)
g(oo):g'(m) =... - 9(e-1)(*):0
(2"o line
1,1,1,...,1,1).
Instead of. starting with (4.131), we can also start with functions g satisfying go)(co)
: o
(2"d line
with afixed j, 0 2 some special choices of the "boundary conditions" a,re investigated. Here, we will shortly deal with the case
(*)
y'(0):g(m):e,(m):0
(ri)
g(0)
(ii) This
ls:2. The second order HardY inequalitY
:
g(m)
c,(0)
:
:
g,(m)
s(m)
:
(lt), (l i), (l t)
:0
g,(m)
:0
Tfrl_case (fff) is solved by Theorem 4.3, the case (fu) by Theo rem 4'4s and the case (u) bv Theorem 4.g. The case (i) t. r'"rih it is equivarent to (fu) due to Theorem a.ao. tt u heuristic "i""" principre
(Theorem 4.46) indicates that
Let us deal with the inequalitY llgllo," S Cllg"llo,,
(l ;), (; l),
(aiii) g(0):g'(0)=y'(m)=o
:
(ii) ("i) (uii)
(4.132)
on the interval (0, -). This inequality can be investigated under one of the following nontrivial bounda'ry conditions on g (we list them together with the corresponding incidence matrices):
; ),
and in M'
is equivalent to (ir), is equivalent to (a), is equivalenr to
(ri),
NASyRovA [2], these equivarences ( .$2) in ali cases *" ei";.
of rralidity of
are proved and criteria
242
Higher Otder
Weighted Inequolities ol HodV Tbpe
Example 4.48. In Example 4.?, the case (ui), i'e' g'(0) :9-(T] was considered for the special weights u(c)
: r", u(r):
:0
sc-zP
-tO forp_q.Thecriteriamentionedintheforegoingsubsectionshorl thai inequality (4.132), i.e. inequality ( '17) holds for a) 2p - l'
4.10. Comments and Remarks pa'rticular 4.1.0.1. The P6lya condition mentioned in Sec' 4'1 is a very of Birkhoff case of a general P6lya condition appearing in the theory
interpolaiion. For details, see R' A' LORENTZ [1]' 4.LO.2. Theorem 4.3 is in fact due to v. STEPANOV [1]. It is a consequence of his more general results concerning Riemann-Liouville operators.
4.10.3. The results summarized A. KUFNER and H. P. HEINIG
in
Theorem 4'8 ale due to
[ll'
4.10.4. Necessary and sufficient conditions for the validity of the lcth order Hardy inequality in the general (well-determined) case (see A' WANNEBO Sec. 4.3) have been investigated by A' KUFNER a'nd gave a proof for : 3. Then A' KUFNER [2] : [1] first for lc 2 and /c
g"rrur"t/ceN\f.MsnM1-0andformulatedaconjecturefor proved this !"n"rtl index sets Mo,Mt. Finally, G' SINNAMON [4] conjecture. In Sec. 4-3, we follow his approach'
4.10.5. The sPecial A. KUFNER [6].
cases described
in
Sec. 4.4 can be found in
4.1.0.6. The reduction of conditions (Sec. 4.5) was proposed by A. KUFNER in [3]. 4.LO.7. Condition (4.65) was derived by P' GURKA in an unpub lished paper for the case p S q. Then B' OPIC modifred his approach and extended the results also to the case p > q'For details, see
Hady Inequalities
249
[OK, Sec. 8]. The approach to overdetermined classes for the case when the weight function u satisfies (4.66) for k: 1 (Theorem 4.2g) as well as for the special overdetermined classes if & > 1 (Sec. a.Z) is essentially due to G. SINNAMON and can be found in A. KUFNER and G. SINNAMON [1]. The idea of sptitting the intentol (0,1) by sorne z was suggested by R. OINAROV.
4.10.8. The approach to general overdetermined problems described in Sec. 4.8 is due to A. KUFNER and H. LEINFELDER [1]. partial results can be found in A. KUFNER and C. G. SIMADER [1]. See also M. NASYROVA, V. D. STEPANOV {21 where the ma.:cimal overdetermined case (lMsl + lMl : 2k), k -- 2, p : q: 2, is
characterized.
4.10.9. The higher order Hardy inequality on dealt with in T. KILGORE [l].
(0,
m)
has been also
5 Fractional Order Hardy lnequatities
5.1. Introduction As mentioned in Chap. l, now we will investigate froctional order Hordy inequolities, i.e., inequalities of the form llgllc,"
0). using this inequality and F\rbini's theorem in (5.6) we obtain
v1t+'rP
:2r/cs-r6(r - ,lr;;-tlr
roorl'
0:
f@
fo*
=
n,
Now we use the Hardy inequality for the interval (r, oo) with a
The following result is a counterpart of inequality (5.4) and a special case of inequality (5.2). It was derived in Chap. 3 (see Example 3.18) but we will give here a different proof.
Theorem 5.3. Let I < p
21/p
(b.6)
(u n)
of inequality (5.1): 1. It was derived inde pendently by G. N. JAKOVTEV [1] and P. GRISVARD [1]' but it might have been known ea,rlier (see 5.6.1 in Sec. 5.6, Comments and Remarks). Below, in Theorem 5.9, we generalize inequality (5.4).
Remark 5.2. Inequality (5.4) is a special We take e : p, u(c) : s-)p and a(s,y) :
.AC(0,
: ('I"* J,f*lg@)-gfu)lq, . \r/P
247
-E Jo ls'fu)l1/-^Pav.
( r, inequality (b.5) hords. The constant is the best possible because of the sharpness of the constant in the classical Hardy inequality. Thus, due to the fact that A
c
Proof. Using F\rbini's theorem and the symmetry of the integra,nd, we rewrite the left ha,nd side in (5.5) into the form
(1,*
J,
l"W
Remark 5.4. If boththe inequalities (b.a) and (b.b) hold,
a refinernent of the classical Hardy inequality (1.2b):
fP#a'au)'/
( f* f'lg(") - s@)lp d,ydn * =\/'
tr
llgllp,"
g(s)
Io*
-
g(v)lp
l,* lr- tir5'-
dytu
\ l/p
with (a,6) = (0,rc), u(o)
)
the assumptions on g
a,re
=
0. After calculating the corresponding function g(a) : g.(t) by (5.12) for the case c > 0 and by (5.13) for the case a ( 0, i.e.
l'*)"'
,
1
. ,r . t
: a,nd gA 1 lo" n{r)a,
o
(5.14)
se (55.8 ), we have from (5.13) that while in case
il*l9l'+)"' (l-l9l' +)'''. (l- l*" !.* *"1'*)''' = = =
o 1 or I ( 0, then the integral on the right hand side in (5.16) diverges e.g. for each nonzero function g from C6"(0,oo). Moreover, if A : llp, L ( p ( m,
I,*
n@ I
-l' a, a
cx,p
(1,*
l,*
with Cs,p fum (5.r7).
ffid,dv)'
Remark b'r2' If the right ha"nd side is finite, then g is equivalent to a continuous function
f
on [0, oo) and
IL
:
O$).
Weighted, Ineryolities ol Hadg
5.3.
7W
Fluctional Oder
The General Weighted Case
Now we consider the case when the two.variables weight u(c, y) is not identically equal to 1. Also the interval (o,b) ca'n be arbitrary finite or infinite. First, let us note that by adopting the methods used in sec. 5.2, fractional order Hardy inequalities can be obtained also for some weighted cases. For instance, if we choose w(try) : {,'f € R, we can easily see that by using the methods of the proof of Theorem 5.9, we obtain the following generalization'
Theorem 5.L3. Let 1 < p < oo, .\ ) -t/p onill * thot [{ g(t)dt exists for eaery a > O and thot either
I
0. ri*-
267
/(1) : ry, which is satisfied due to condition (b.2b). Thus we l* have that
Assume thot
1 /" s(t)dt:0. JO
,-+u g,
Moreoaer, let
u(r)
ond
be weight
\r/P / r,
\Jr
,/
u(t)dt)
.*,
then inequality (5.23) hokls with C
Jo
:
[* Ilo(,) - |zJo [" " ro
g@aulo --l
*p10,
* r--out(z) : *-Fa(r). This implies inequatity (5.23) with C from (b.24) since by H6lder's inequality,
(5.24).
lrrrw
lg(,)-)lstu)dvl &Jo | I
Proof. As in the proof of Lemma 5.6, define
h(r) .
ls(x)leu(r)dt
with W(c) = u(x)
x9 -L-P w(r),
furn
l" Jo
(5.25)
: Apf (p - l)r-p. If a(r) = r0-r u(r) +
f@
functions on (0, oo) sotisfging
\ l/P' (/ ur-p'(qat} / \Jo
/ f@
/::sup{/ c>0 and denote B
w(z)
:: s@) - L s}at fiJo["
s
(: l,' b(') |
grilvv)'
1," ual - s(iltpdv
and find that
g(a)
-
:
h(r) +
,'^9)0,
Jo =,
due to the fact that for
see (5.10) and (5.12). Hence
lo*
zp-r
s2p-r
(/- nt'tt
r9lr-Vl-p > l
with 0 0, it folloq's from (5'35) a'nd (5'34) that
(f
vov,1,1o,)''o = " "'
. (l'
loo
{(,*,
/' ts(dte ar)
bt ) - g(v)l'.,(l, - nn*on)
with C independent of
g, b
and e.
\
,:
olllo
(1,' ffior)''o (lo'
,,-o,rt)t^o,at) 'o
ond,
K then
:: -.-A8a (p - t)'tt
for g e I?(u) ue haue
(1,'van',@dr)
1
.*
1u.ru;
264
Weighted Inequolities of
(ii)
Hady Tlpe
frzctional Otder Hailg Ineryalities
Denote
Hiilder's inequality yields
n(il: (i l,' ,'-o'(",iltu)
ll,'tnal -
ntuDdrlo
u(il:6(ilv-b.
q.
I
1.
0 < q < oo. Let w(r,y) ond (0, x (0, b) arul assurne thot u b) on be weight functions -{r,y) 'i.s symrnetri,c: w(t,y) : w(y,t). Assume that the function Bn,o@) defined aboae is
fm
x(q,p):I Q+fr)tlng+$1rt' for p q. Then we estimate the right hand side in (5.42) by Hiilder's inequality with parameters s : p/q ) 1 and s' : pl(p - g), and we find that, by F\rbini's theorem,
(0,c) (with c < 6 fixed) and modifying appropriately the definitions of. Bo*@) and V(y). (Now we are dealing with the conjugate Ha^rdy operator for { : g'(t)dt.) The formulation of the corresponding ff theorem is left to t-he reader. (ii) It is also possible to omit the assumption of the symmetry of u, but in that case, we have to replace w(c,A) by fr(r,y) :: u(r,y)* -(y,.r) in (5.a1) and in the definition of the functions Bo,o(o) and V(y). Indeed: as in the proof of Theorem b.3, see (S.O), we have that
b,@ [' f fl9)]q ta(r,v)dxd,v Jo Jo lr - y1t+'^n
f Jo f 4t"l lr-cr.-
--,?{v,)-l:
Jo (P-
s
2kq(s,
p) (1"'
"fJr-c)
first lines of the proof of rheorem 5.25). consequently we
can modify the proof using now the Hardy inequality on the interrral
(')dr)
il lP
:
" (lr' (l,u v'ot,wr(x,y)dy) o*)0"
*
:
2kq(q,p)q
(1"' (1,' wr@,v)dx) b'(v)l'dv)
:
2kq(q, p)co
(lo'
W,
tolt,v
This is again (5.41), and the proof is
Remark 5.26. (i) Notice that
tilda)
:
complete.
o
due to the symmetry of
ut(r,y),
can also write
- g(y)o w@,y)drdy fJo lo f E(") l, - yr "''
:'fo'(1," ff&11""
I"' (1,'
we
1,'
u(r,y)d.xdy
ffill,'
I,' (1,'
ffil!,'
n'
ruo'1' aa) a,
where fr is symmetric. Now, we proceed as rem 5.25.
in the proof of
Theo_
b.2S with b = oo, Q : p, yl-r-^p+p, we obtain inequality (S.5) lc but with a difierent constant. Notice that in this case, ao,olry i, ,
w(c,y): I and ,{r,y):
or)*'
roo'1' aa) a,
(I' ffill"'tul0,1'0,)*
Example 5.27. (i) If we apply Theorem
n'av'lo
n'
constant!
274
Weighted Inequalities of
(ii) If
Hady Tlpe
hactionol Ord,er Hardy Inequalities 275
we apply Theorem 5.25 with g
w(s,y) = lr we obtain that
- ylr+N-(lx -
:
5.4.
p,
yl) and -r(r,y) --
b(') - s(illpw{r lr' lr'
=o
|oo
-
.{lr -
yl)
vl)drd,s
ls'(r)lPu(x)dr
(5.43)
:
lo'
Some basic facts
First we recall some concepts and results from the real rnethod, of intetpolation (cf. e.g. J. BERGH and J. lOfSfnOU 14;. Let (As,.41) denote a compatible couple of Banach spaces (i.e., both are continuously imbedded into a Hausdorff topologicar vector space). A Banach space .A is called an interrnediote spore, between
if
As and.41
where
u(x)
Hardy-type Inequalities and Interpotation Theory
ug*{r)*1(fu
- tl)d,t
AofiATCACAo*Ar.
with
Bl'e(t):
,."r1?*,
\ p-l rb-t / fy-t -!-o'1rya") -(")0" Ju-,
\lo
Obviously (5.43) may be regarded as a counterpart to inequality (5.30) - see Theorem 5.19. We close this section by stating an elementary result. The proof is left to the reader.
Now, let (Ao,A) and (Bs,B1) be two compatible couples. Then spaces .4 and B arc said to be interpolation spou,s with respect to (,4o,Ar) and (86,^8l) if .4 and B are intermediate spaces to the respective couples and if, for any bounded linear operator ? such that ? : As -+.86 and T : At 1 Br,we have T : A +^B. In terms of inequalities this can be expressed as follows:
Theorem 5.28. Letl1p,9 ( m,0 < b !x, ondletv(r) ond w(r,y) be weight functions on (0, b) ond on (0, b) x (0,b), respectiuely.
If
Moreoaer. denote
v
(r)
:
fo'
,'-o'
o': Then
Ioo
lv(") - v(illsle' 'ffiw(a,s)d.ady p-r. Jz
p and we have
(5.48)
(We call I/o the Hardy aueraging operator.) In order that these two operators coincide, it is necessary that
t" I"' (l_,rywa,)at.
1fr116 :rJo ffu)dv: frJz ffu)dv,
I
This is a modified version of the fractional order Hardy inequality (5.1) with the special weight u(r) : 1. There are many examples of inequalities which hold except for one or more values of the parameters involved. Sometimes, this phe' nomenon can be explained via interpolation' Let us give two simple
-: I
l.e.
fo*
f{ilor:
o'
N the set of locally integrable functions satisfying this condition. In order to interpolate the Hardy operator on weighted Lebesgue spa€es, we have to consider not the spaces themselves but their intersection with /V. Denote by
examples.
Example 5.30. The (classical) Hardy inequality
(l- l*ro \" ,ua.)''o ' c (/- u't"l Yr'dn) holds, with p
2
1 and 9 € Cff(O,oo), for every
p
* p-
1
(5'45)
but
does
nothoklfor P:p-L. In order to be able to understand this phenomenon we first note that inequality (5.45) ca,n be rewritten as
(/- l:
1. When (5.a5) is written
ut'lr
impossible to interpolate between 0 < p- 1 (see (5.46)) and 0 > (see (5.42)) to obtain the inequality for p: p-1. The reason is we have in fact two d,ifferent operotors involved:
thenforany6,0 1) and
f..(t),:+LJO['
I'(ildy.
More exactly they proved that the norm in the Lorentz Lp,,I-spwe,
( r*lso) - I It
Ul * l;) =
i.e.
stu)aYlo'"\ "o
(/-lgl" *)'''.
ff-l#
[
llflln nrildnlo
*)'''
s(r+*) ff-l9/l'+)"' Hence we have the following result about equivalent norms in
I?(t-an-r'1
.': (/-,r'
ro
y. 1tyo4)'/0,
(5.5e)
is equivalent to
lll/lllp,c
that l liml-a6/*' (t) = 0. provided
,: (lr* Gtle U,. e)- /.(,)))o+)'/o t/ @,
lSq<mand
.f'-(oo) =
286
fractional Order Hordg Ineqtalities 287
Weighted Inequalities of Hordy fupe
Using the estimate (5.58) a,nd Lemma 5.6 with g - .f' and a: -Ilp and p replaced by q, we obtain the following somewhat more precise result.
f0
t7
lr'vt xou+ l,r(,)l =+ Ir' f.(v)dv+ l/(41,
Let
lll/lllp,c
(ii)
Since
l/+(t)l
1. (ii) Proposition 5.49 implies that the following sharper version of
Corollary 5.47 holds for q
.
"tX[o,U(t)
z'+B+t \'/' :((-:_1'*i@@)
(5.68)
S-ince
: :
ils
belong to L2(xF) and
Q,
obtain that
llgll3,"'
s
9r(r)::
we have
llg
ilsnz,,o
and (5.67) follows. For r ) (-B - l)/2, the functions
g@)ay. Both the ine4uolities are
Proof. For p < 1 and B
2gs
If.
f
:21
e LP,2,1 < p < oo, and .f-.(m)
:0,
then
we obtain that
(t
-
f)llg
-
H.,gllz,,u
: s :
ll
- gs+e -
l9l
llsllz,"u
(1
+ llg -
(L
-L) vp-r/
-
\
ItrVt/ef.(t)lr+-)
P)H"sll2,,o
YP-L/
(01+ r)llgllz,,,
The lower estimate is the best one for 1
and
(t-0llg-H"sllz,,o
*in (t,
p)H"sllz'u
-
estimatefor2(p y}.
is a norm on Ap(u) if and only
if u is decreasing.
But the expression ll/-.llp,., with
f,.(,)::1""
(/- (i l,' rav")o '$)0,)'/o . c (1,* r,aw@dr) holds whenever
(6.3)
f,(t)dt
/
(6.6)
2 0 is d,ecreasing.
Remark 6.1. (i) Although problems of the type just mentioned were studied by B. STECKIN
is a norm which is equivalent to ll/'llp,- (see, e.9., C. BENNETT and R. SHARPLEY [U). In what follows, we take the measure in ,\y(y) to be the Lebesgue rneasure.
Recall that the rearrangement of the Hardy-Littlewood ma; 0 is decreasing, the set E(f ,t) is the interval (0,,\r(t)). Hence (6.9) holds. tr
The main result of this chapter is the following (SAWYER's) duality principle for decreasing functions /:
Theorem 6.3.
Suppose 1 < p < oo. Let g,a be non-negatiae measurable functions on (0,a) with u locally integrable. Then
[f, f(x)s(x)dn
uqP
o-in (Jo fn(r)u(x)dx)t/e
fo(r)r@)a,
\/o
\1 is the distribution
/
functi'on defineil oboue'
Proof. Denote
E(l,t)
:
* (l- (1,' nr,.to,)'' (1," u(t)dt)-o' ,t )a,)'/o'
./rtrfu\.\ (p( oo, :p'JorF l-- {'-t - I l'- u(fldrldy, 0 where
308
- {'€ (0,-) f (') > t}'
*' If e@)a' (6.e)
(ff
a@)ax1l/n'
(6.10)
-
Remark 6.4. (i) Note that integration by parts shows that the first term on the right hand side of (6.10) can be replaced by
(1,*
{[ s*dt) (1,",av,1'-'' n@)*)
306
Integral Opemtors on the Cone o! Monotone Frncliotts
Weightetl Ineryalities ol HordA fupe
gO?
oo, then by convention the second term on the right hand side of (6.10) is taken to be zero.
But, since (t - pXt - d) : I and p(l - d) = -y', integration by parts shows that the last supremum is finite. Consequently,
Proof of Theorcm 6'3' First choose f (r) : C a positive constant' and denote the left hand side of (6.10) bv tk). Then J is decreasing
L(g)
(ii) If ff, u(r)da:
I ;c suP ;tb
and
f? f(r)g(r)dr
supL L(o\>=;:"" u\s) Ur- yn(x)u(x)dr)r/P
[f, s(r)dr
(6.11)
:
1
c
=SUD nio
ff q(,) E h(t)dtdt (Itr tr (s)Iv (n) / u(*)ln u14ax)ue /o- h(t)G(t)dt ([tr tr @)lv (a) / a(t)ln p(r)dr)' to
Next. let
f where h
(r): Jz[*
> 0 is arbitrary. Then
/
(6.r2)
n1t1a,
is again decreasing and
L(d
: ,uo
Jo g(") '['- h(t)dtdn
: *(1,- [ffi]",,v,)'o : I (1,- (/'nt"u";'' (|"'u(s)ds)-o' ot'a,)'to'
,;ilM
Denote
G(')
7t
= JO[' s@)a" v(t): /o o(s)ds.
The conjugate HardY inequalitY (cf. Introduction, formula (0.10)) shows that
(l-,,',
h(t)
(1.* nav,)' o*)'''
t, (I*
hp
(x)vp (x)o'
-o
p1a*)"
Here we have applied (6.7) with / : Y, g : g a,nd dp(r) : a(t)dt. The lower bound for (6.10) follows now from (6.11) and (6.12). To prove the upper bound of L(g) in (6.10), let
:
-'
/,*
{[e1,1a,)o
(lo' ,uro")
r |.fi'g(")e 1n'-r1r/n' *-'LF;(FJ
o
J
holds
if and only if with C >
-]-. - 1-r
o
,61*
30E
Weighted Inequalities of
Hody
Integrol Operators on the Cone of Monotone Flrnctioru 309
Type
But this expression is the right
By Hiilder's inequalitY we have
ha,nd side
of (6.10). Hence if
we
show that f@ : J, fu)su)h(t)h-t{,)dt
foo
t{ilnl)a, Jo'
(lr* r,aln-p(u)s(ed,a)"' =, (lr* yo1,y,1,)d,)'/' (6.14)
\l/p
/ f@
s (/. !n?)h-o(t)s(t)dt)
(6'13)
for 0
( / J, then the upper bound for L(g) follows from (6.18) by
dividins
. (l-
n''
1r1n1r',or)
''
integration yields
n-p(t)
Now, by F\rbini's theorem,
/ r@
(/,
w' (t)s@at
\
yn'-r 1 1x f roo / fE \ -p' : lrl, (/, ot')a") (/o ,t"lo")' ,1,1a,
1/P'
*
) p
=
(f'nt,la")''-' (1,",(,)a")
(l-r(,)
[/ ."[##]"-']
: (l-
o
(l- (1"'
' ff e(s)ds +(J;@"o
'
{
(|,"t'p')
"
(1,*01";a")o
(/'o1,va")o
-'
(f*,,",r")'*']'" o
[*
((/',1";a")'
- (f't"r'")'*')]
(1,' noo')'-' (1," '(s)as)
nuro,1''
'@)dr
o,)''' u@)
(1,'
."LffiF])'" =
fp(r)u(t)dx)t". to prove (6.14) observe first that
* (/-
sr'sa')
a'
*
: -o'
'@)a')'/o
" (1,'n1"10") (/-,,",*)'*')'*
{(/',1"v,")o
[*
- h(1,*,uro")'-''
([',u,0,1'-o
*" (1,*'t"8")'-']
)'" (6.15)
310
Weighted Ine4nlities of 1
since C
Integrol Operators on the Cone ol Monotone Frnctiorts
Hady fupe
0. Now Lemma 6.2 with
- F2
:
w: h-pg yields
J, f
Thecase0 0
u that
ay
>
0 < p l- Let g,u be non-negatiae = loully integroble. Then (0,m) withu
Suppose
swoble lunctions on
mea-
ff, f(r)s(r)dr
.lPr
fn@)h-v(t)s@)tu
f-, (\J0l'",(,p")/
/ rt
th(t) and p - | F'(0) : g n and (6.17) follows for t: b.
r(")'")
r(r)dr.
sp(/- rf-r J0 [-
(6.12)
that F'(r1
Substituting into (6.16), we conclude by Lemma 6.2 with tr
J"
7t
we have
(')
lo^t{o
xo-tnnqyac
F(t)::p I rn-r6n@)d"-( | n1*1a"1 Jo \"ro /
e(s)ds
,(")0")
Do;
-
P
1
is satisfied. The proof is easy: If we define
Jo
= t;n@)
1P
/ rb
I\"ro I h(a)dr) / sp JoI
,.rrfu\
3
L(g). tr
@)h-e(r)g(r)ds
\ / t^tfu) :p f* ,"-' (/" "" n-p@)s@tu)da' Jo
I
- l)r-r JO/@ !p(x)u(x)dn.
W
Hence (6.14) is established a,nd so is the upper bound for
foo
3ll
g;*Ft'l,t'lr'lm ::lB
(1,'
sau,)
(l'
,6)a*)-'to
(6.18)
3L2
Weighted Ineqnlities of Hadg
Proof.
If.
l(*) = X(0,')(o), r )
[f,
f
Integnul
TW
Remark 6.6. If f (r)
0, then
@il.\n- -
II
g@)a'
{li,{dd'o
fo*
bound for (6'18) follon's' p To establish the upper bound, we apply (6'9) with = P = 1. Then
lo*
r > 0, hence the lower
I
:
Jo U'
:/,f* s
sup
il'@
\
:::E
qp
= XE
o@)d'r)aa
0, then h
0 independ'ent "f f 'f and only it C1 < a' the inequalitv Leto <ma:c(l,p) S g ( oo anilT11: I' Then
/ r@
I,* (lr,'rut,,tdoi(il)
where
Then the inequolity
hokts with
iirl
foo foo : Jo-Jo I t@)aoifu): I oi(v : !fu)>tl)dt
-
l/P
--\ ynP)dn"e)
whichis(6.38).Thesharpnessoftheestimatefollon,sbyinserting trI I@):x10,"1(r),0< o < oo.
negatiae ond decreosing' I'et0 < p S 1 q
Then the inquatity
ll\ro
norng)a,olo
=
m anilTl-?2: I.
=
Io*
sc,
(l-,t Io*
r,"1t)q@)dur(*r)"
(l-,t
r,urt)p@)duz(d)'
326
Weighted Inequalities of
Haily
Integml Operotors on the Cone o! Monotone Rnct:ioru
Type
, r, (l^"(lo* v,*uorX")at)"
:
:
or,@))''o
:
rl/P
I f
"' lJ-"Qz
f)P (r)d'p2@)
)
need to prove the case As in the proof of Lemma 6.2 we see that
(ii) In view of (i), we only
!
< p 1 g < oo'
pr(E(t))
lq@)d'p1@)l
/ o'o
[(;)
: rt'' (lo
(l^,t
: s, pl
/o
/e
no,o-'
(1,*
wr' (E(t)))'/o
+)''')"'
o-r rnhln(Dor)''o
I-, ( Ir*
l tp - (Tzx
p
(t' 1'
1
"
p')
o
"
(1,* rr,lor,("))" cr,pun
d,1t
r@)
to (i), it suffices to prove the ( l. we use again Lemma 6.13 twice and
case, again according
I (where we now have in mind functions defined on IRN so that f e I](w) mea,ns that
I r(,ildv) ,6va*)'/'
q
(.i,, (r, i))'
Comments and Remarks
(Jfl (min (r, l))q u(r)ac)'/q <m. (Jfl (min (r,;))o u@)ar)t/P
1 and p
'av')"' (I*
6.6.1. The boundedness of the Hardy-Littlewood maximal operator M (see (6.4)) on U, p ) 1, had a considerable significance for the development of analysis, specifically harmonic analysis. one of the reasons is that a large class of convolution operators are dominated by the Hardy-Littlewood maximal operator and hence the boundedness of M on u, p 2 l, implies the boundedness of the class of convolution
rtoou)n u14a')'/o
if and onlv if Cg
[" f(ildv,
Jo
< g < oo. Then the inequality
=" (I* (: holds
6.5.
/ |, the following
(1,'
if
The constant c : cs is in all cases the best possible. Note that for the case (ai) a more general result was stated in Theorem 6.g but without the sharp consta,nt.
ki@,y): it for 03y'-x, ki@'Y) : o for a)t' then
"o
::lt
Rnctiolr g:tl
. *.
roou)o ,@)a,)'/o
c:
s;n
(h l;,r(ddt)''' (h
to the secalled
fo,,-o'1,1a,)'/o'
.*
where Q are cubes in RN with edges parallel to the coordinate axes). operators which are ma-
It follows from this result that convolution jorized by M are also bounded on lp(w).
6.6.2. with the introduction, popularity and use of weighted Lorentz spaces (see (6.1)) in the early fifties and sixties the question a.rose whether the weight characterization of the maximal operator M on
332
Weighted Ineqtalities of
Hodg
Integral Operotors on the Cone of Monotone
Type
U(.)
would carry over to a corresponding result for the weighted Lorentz spaces np(tr). (For u.r : 1 this is trivial!). Since the spacc Ap(u) are defined in terms of the equimeasurable decreasing rearrangement f * of f , it became necessary to relate this quantity with M/ (namely with (Mf).).In fact,
(Mf).(r)
*: I,
f. (t)dt
.
(6.40)
The estimate j in (6.40) was established by F. RJESZ lll as a consequence of his nice sunri,se lemma. Moreover, C. HERZ [1] (under the influence of a paper of E. M. STEIN) proved the inequality J in (6.40). Some further information about these inequalities (and about the corresponding inequalities in terms of the distribution function of M t) can be found in the paper I. ASEKRITOVA, N. KRUGLJAK, L. MALIGRANDA and L. E. PERSSON [1]. In view of (6.40), it is clear that M Ap(u) -+ Aq(r) is bounded if ' and only if the Hardy averaging operator fI" defined on decreasing functions is bounded ftom IP(u) into lp(u). In 1990, M. ARIITO and B. MUCKENHOUPT [1] characterized the weights u : r.r for which l,t : ltp(u) -+ Ap(u) is bounded (p > 1), or equivalently, for which the operator llo defined on decreasing functions is bounded on I'o(u). Shortly after the result mentioned, E. SAWYER [3] established an explicit duality theorem for weighted .LP spaces on decreasing functions. This duality theorem was used to show that the mapping prop erties ofan operator defined on decreasing functions are equivalent to the mappirrg properties of an operator defined on orbitrory functions but with different weights. Hence he was able to characterize weights u, u for which ? : Ap(u) -+ Aq(u) with 1 1 p,8 1 oo is bounded, in particular, if ? is the Hardy-Littlewood ma:rimal operator M. In this chapter, we have proved Sawyer's duality theorem (see Theorem 6.3), but our proof follows closely that of V. D. STEPANOV [3] which seems to be more elementary. A multidimensional version of the duality principle was recently proved by S. BARZA' H. P. HEINIG and L. E. PERSSON [U.
Rnctiorc
333
6.6.3. The case 0 < q < p, p ) 1, of Theorem 6.8 is also known and is due to G. SINNAMON [1,3], V. D. STEPANOV [B] and others. 6.6.4. The duality theorem extends in a natural way to inaeasing functions so that weight characterizations for operators defined on increasing functions can also be given, see Remark O.1l (iii).
6.6.5. Among spaces related to the Lorentz spaces Ap(r), let
us
mention the spaces
rP(-): -fE
roo,)' ,61a,)"". {r, (l- (; 1,"
-}
)
.f-(u), it is clear that fP(ru) c Ap(u). More over, if ut € Ap, then Ap(ro) C fp(ur), p ) l, so that for such weights and p ) 1, these spaces are equivalent. A d,uality theorem for funcSince
f-JoI
f.Q)at
tions in such spaces has been obtained by M. L. GOL,DMAN [1] and
by M. L. GOL'DMAN, H. P. HEINIG and V. D. STEPANOV [1] with subsequent weight characterizations for the ma>rimal operator and Hilbert transform in these spaces.
6.6.6. Inequality (6.17) (cf. also (6.38)) was probably first discovered by G. G. LORENTZ [1, p. 39]; various other proofs can be found in literature, see e.g. J. BERGH, V. I. BURENKOV and L. E. PERSSON [U, H. P. HEINIG and L. MALIGRANDA [2], V. c. MAZ,JA [lJ and E. M. STEIN and c. WEISS [U.
6.6.7. The proof of Theorem 6.14 is a simplified form of that presented in S. BARZA, L. E. PERSSON and J. SORIA [l]. Let us note that (r) this proof has the advantage that it can be ca"rried over to the multidimensional ca.se where, in comparison to the one.dimensional case, some new problems appear - see the paper just mentioned and the Ph.D. thesis of S. BARZA [1J; (ii) special cases of Theorem 6.14 have been proved (in other ways) by several authors; see M. J. CARRO and J, SORIA [1,2];
334
Weighted Ineqnlit:i.es of Hady Tlpe
H. P. HEINIG and L. MALIGRANDA [2]; Q. LAI [2]' L. MALIGRANDA [1] and E. A. MYASNIKOV' L. E. PERSSON and v. D. STEPANOV [1]; (iii) for the case q > p fairly little is knownl however, for the one dimensional case, V. D. STEPANOV [4] proved a result corresponding to Theorem 6.14 (iv), and this result was generalized in a multidimensional setting in S. BARZA, L. E. PERSSON and v. D. STEPANOV [1].
References
6.6.8. As far as it concerns multidimensional generalizations, we refet to the Ph.D. thesis by S. BARZA [1] where also a fairly complete list of references and some open questions can be found. 6.6.9. Extensions of the results mentioned, e.g. to modular inequalities and weighted Orlicz function and sequence spaces, were given by H. P. HEINIG and A. KUFNER [2]; H. P. HEINIG and L. MALIGRANDA [1] a,nd Q. LAI [1,3]. The following three books carr serve as standard reference sources for integral inequalities (with a,nd without weights). They are mentioned in the text in the abbreviated forms [HLpl, [MpF] and [OK]:
[HLPJHARDY, G.H., LITTLEWOOD, J.E. ana p6lye, G.,Inequalities,2nd ed., Cambridge Univ. Press Lg52 (first ed. 1g34).
[MpF] MITRINOVIC, D.S., pEeARJe, J.E. and FINK, A. M., In_ equalities inaoluing lunctions and their integrols and d,eriuotiaes, Kluwer Academic Publishers Group, 1991. MR 93m:26036
[OK] OPIC, B. and KUFNER, A., Hardy-type inequolities, pitman Research Notes in Mathematics Series, Longman Scientific & Technical, Harlow, 1990. MR 92b:20028
336
Beternes
Weighted Inequolities of HardV TVpe
FURTHER REFERENCES: ADAMS, R. A.:
Vol' 65' Acade' 9247 56 MR # mic Press, New York-London, 1975'
p1 Soiol",
Spaces, Pure and' Appli'eil Mathematics'
ADAMS, R., ARONSZAJN, N' and SMITH, K'T': potentials, Part II, Ann' Inst' Fourier (Greno[1] Theory of Bessel ble) 17 (1967), 1-135' MR 37 # 428r ALZER, H.:
J' Approt' 1i1A t"n""ment of Carleman's inequality,
Theory 95
(1998), 497-499. MR 99h:26023
ANDERSEN, K. and MUCKENHOUPT, B': [1]WeightedweaktypeHardyinequalitieswithapplicationstoHilberttransformsandmar
