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On Some Fractional Order Hardy Inequalities
HANS P. HEINIG a, ALOIS KUFNER
andLARS-ERIK PERSSON
cDepartment
ofMathematics&
Statistics,McMasterUniversity,Hamilton, Ontario, L8S 4K1, Canada
bMathematicalInstitute, CzechAcademyof Sciences,
itn
25, 11567Praha 1, Czech RepubficDepartment
of Mathematics, Lule& University, S-97187Lule&, Sweden(Received6May 1996)
Weighted inequalitiesfor fractional derivatives(=fractional orderHardy-type inequalities)have recentlybeenprovedin[4]and 1]. Inthispaper,newinequalitiesof thistypeareprovedand applied.Inparticular,thegeneralmixednorm caseandageneraltwodimensionalweightare considered.Moreover,anOrlicznormversionandamultidimensional fractional orderHardy inequality areproved.Theconnections torelated results arepointedout.
Keywords: Hardy inequality; weightfunction; fractional order derivative.
AMSSubjectClassification: 26D15, 46E30
1 INTRODUCTION
Using the notation
Ilflls,o
for the norm in the weighted Lebesgue spaceL (0,
cx;w) L (w),
(fo0 )l/s
llfll,,,o If(t)lSw(t)dt
with 1 < s < cx and w
w(t)
aweightfunction on(0,
x), theHardy inequalitycanbeexpressed
in the formIlullq,o0 <_ Cllu’llp,, (1.1)
25
with a constantC > 0 independentofu,i.e., under certain conditions, the weightedLq-norrnofthe
function
ucanbe estimatedbya suitableweightedL P-norm
of its(firstorder)
derivative uI.
The natural question arises whether it is possible to extend inequality
(1.1)
to"fractionalorder derivatives" u(z) with0 < ) < 1,i.e.,under what conditions wecan deriveinequalitiesof thetype(1.2)
and/or
IluXllq,0 Cllu’llp,w. (1.3)
The first problem which arises is how to understand the expression
Ilu(Z)llq,w. Let
us mention that for the "non-weighted" case (i.e., forw(t)
=_1)
the following definitionofIluZlls,1 Ilu<Zlls
iscommonlyused:
(fo fo lU(x)
u(y)lS)
a/s[[u(X) lls=
Ix yll+Zs
dxdy ,1 < s < cxz,O
< ) < 1.(1.4) For
thisspecial case,thefollowing inequalityholds:lu(x)lPx-XPdx
< Cplit
Ix y[l+Zp
dxdywhich isinequality
(1.2)
forthe specialcase(1.5)
p q,
wx(x) =--
1,wo(x)
x-xp.
Inequality
(1.5)
was derivedbyGrisvard[2] provided 1 < p < cxz,,#--,
1u6C(O,o).
P
In
fact,he rediscovered an earlier result of Jakovlev[3]
who has shown thatfo
cfofo
[u(x)-u(y)[plu(x) u(O)IPx-)Pdx
< CpIx y]l+p
dxdy.(1.6)
Noticethatthe additional term u
(0)
attheleft hand side of(1.6)
is essential sincethe integralonthe left hand side of(1.5)
divergesifuis continuous at zero,u(0) -
0andL >.
The inequality
(1.5)
was extended in the seventies by Kufner and Triebel[5]:
Roughly speaking, theyderived instead of(1.2)
theinequalityIlullp,m0 C([lu(Zllp,o -t-Ilullp,’)
wherenow
Ilu(X>llp’
Ix
ylI+)plip
w(x)dxdy
(1.7)
N
isa certain additionalweightandwo(x) w (x)
x-zp.
Thedoubleintegralin
(1.7)
offersonepossibilityofcharacterizing thestill undeterminedexpression for[lu(X) IIp,w . But
inspiteofthesymmetryofthe"non-weighted" expressionfor
lu(Zlls
in(1.4),
we wouldprefera certain symmetryin x andy also inthe weighted case, i.e., we are looking for a general inequalityof the formfo fofo lu(x)-u(y)lp
lu(x)lPw(x)dx <- CP
Ix yll+Lp
W(x, y)dxdy,(1.9)
possibly(butnotnecessarily)withW (x,
y) W (y,x). In
Section4of this paperwewillprovesome newinequalities ofthetype(1.9).
Our investigationswere motivatedby results ofKufnerand
Persson
[4]and Burenkov and
Evans [1]. In [4]
e.g. the case whenW(x,
y)w(x)
or W(x, y) w(y) is handled, and in[1],
the case when W(x, y)w(Ix Yl)lx
ylI+zp is treated.For
the reader’s convenience and for laterpurposes,
some ofthese results are briefly discussed and compared in Section 2.Up
tonow, we dealt with the case p=qand allinequalities mentionedabove can be derived more orless directly.
Another approach using the theory ofinterpolation ofBanach
spaces
was used in[4]
andledtoaninequality ofthetype(fo
cxlu(x)lqwo(x)dx t
1/q(fo(fo lu(x)-u(Y)IP
<
C
Ix
ylI+zpq/p
)l/q
w(x)dx)
dy(1.10)
i.e., to anexpressionwith a mixed norm on therighthand sideforp
5
q.Moreover,
it wassupposed that 1 < p < q < cxz [whilethe case q < p is stillopen]and theauthorshave notbeenabletoremove the mixed norm forp5
q.In
Section3,we willproveanew inequalityof thetype(1.10),
moreover, with a measurev(y)dyinstead ofdy,but againonlyfor p < q.
Some
of the results obtained in thispaper
can be extended in various directions.Here
weonly present and provean Orlicznorm version of the inequality(1.5) [and
ofits extension to thepower
weight case see,e.g., (2.6) with/3
1+ Zp
andgiveanexampleof a multidimensional fractional orderHardy inequality;see Sections5 and 6, respectively.We
close this introductory ection by noting that in all cases we have consideredhere wehave found that the "fractional"weights w0 inthe left integralarealwaysof thetype(1.8)
for some suitableweightw.
Maybethisissupported by the following inequalitywhich is an
easy
consequence of(1.5)
and of thefactthatubelongstotheweightedspace LP (w)
if andonly if uw1/pbelongs
tothe(non-weighted)space LP.
lu(x)lPw(x)x-ZPdx
fo lu(x)w
1/p(x)
u(y)w1/p(y)lp< C
Ix
ylI+xp dxdy.(1.11)
Also thefighthand side in
(1.11)
could serve as a definitionoftheexpressionIlu()llp,w,
2 SOME PRELIMINARY RESULTS AND DISCUSSIONS
For
thespecialcasewhen theweightfunctionW (x,
y)inthefighthand side of(1.9) depends
onIx Y I,
Burenkov andEvans [1]
recentlyproved
the following interestingresult:THEOREM2.1
define
Let
0 < p < o, let w beaweightfunction
on(0, o)
andv(x) w(t)dt.
Suppose
thatvsatisfies
the A2-condition, i.e., there existsa constant c > 0 such thatv(2t) _< cv(t)
forallt > 0.Then
for
all uLP (O,
cx;v)
lu(x)lPv(x)dx <_
Cplu(x) u(y)lPw(lx
yl)dxdy.For
laterpurposes
wealso state thefollowing slight improvementof the recentresultbyKufnerandPersson
mentioned intheintroduction:THEOREM 2.2
Let
1 < p < o and) >_ -1/p. Furthermore,assume that thefunction
usatisfies
lim 1
fo
xu(t)dt
O.x--cx X
Let wo
andwl beweightfunctions
on(0,e)
satisfying(fo
xt
B"
supwo(t)dt
w(t)dt
x>0
with
p’
p_lP__e_ Then,for
every >O,
< cx
(2.1)
fo fofo xlu(x)-u(y)lp
[u(x)lPwo(x)dx
< CpIx
yl W(x)dydx(2.2)
where
W(x) x-l wo(x) + x-l-Ptol (X)
andCp 2p-1max(l, Cp)
withCp <_ B pP (p 1)
1-p.The
proof
ofTheorem2.1 is similar to that ofProposition1 in[4]
butfor thereader’sconveniencewepresenthere the details.Proof
Firstwe use an ideabyGrisvard[2] (cf.
alsop. 261in Triebel[9])
and defineg(x)
u(x)
u(y)dy[u(x)
u(y)]dy.x x
(2.3)
Obviously,g()
u(z)
and 1for
u’(y) u’(y)-
u(y)+
u(x)dx+
y
--
g(Y)y g(y)- dxxand weconcludethat
u(y) g(y)
g(x)
dx.x
Therefore by using the inequality
la + bl
p <_2p-I(IaIP
-q-IblP), the assumption(2.1)
andHardy’s inequality (see, e.g.,[7])
weobtain thato
]u(x)lPwo(x)dx
(fo fo f )
< 2p-1
[g(x)lPwo(x)dx + wo(x)
g(Y)dyPdx
Y
(fo o
< 2p-1
[g(x)lPwo(x)dx -I- Cp 11)1(x) g(x_) Pdx
x
2P-1 Ig(x)lPWo(x)dx
where
Wo(x) wo(x) + CpX-Pll)I (x).
Therefore,by denotingW1 (x) wo(x) +
x-pWl(x)
andusing(2.3)
we findthat[u(x)lPwo(x)dx < C [g(x)lPWl(x)dx
fo(fo
x)
<
C lu(x)
u(y)ldyWl (x)dx. (2.4)
Furthermore,byH61der’sinequalityandtheassumption/3 >0,we findthat( f0
xlu(x)
u(y)ldy <__x-PxP-1
xlU(X) u(y)[Pdy
x -1
f,x [u(x) -xU(Y)IP
dy<
X/-1 fx lu(x)
u(y)lpJo Ix y[
dy.Theestimate
(2.2)
follows by combining(2.4)
and(2.5).
COROLLARY 2.3
Let
1 < p < x,13
>_O,
>_ --1/p andt > )p 1.If
(t)dt
O, thenlimx-c
2f
ufo [u(x)[Pxa-Pdx
<C
pfoCfoX]U(X)-u(Y)[Px#-l-ZP+adydx.
(2.6)
Proof
ApplyTheorem2.2withwo(x)
xa-zp andtO1(X)
Xt-’kp+p.Let
us notethat applyingTheorem 2.1 withw(t)
a-zp-1 we findthat fora < )p we havefo
c[u(x)IPxa-ZPdx
< CpfoCfoClu(x)-u(Y)lP
ix yl+X
pIx yldxdy. (2.7) Moreover,
Theorem 2.1 cannot be used for any ot >_p
(since thenv(x)
=_o). But
using Corollary 2.3 we see that(2.7)
holds also if.p
_< c < )p / 1 This fact follows from(2.6)
puttingthere/3
0andnoting that
x--XP+
_<Ix y[-1-LP+a
for all y,0 < y _< x and-1-)p
+
ot <0.More
generally, usingTheorem 2.2with/3
0 and withW(x)
strictly decreasingwe obtain aninequalityof thetype[u(x)lPwo(x)dx <_ C
plu(x) u(y)lPW(lx
yl)dxdy,(2.8)
and thisinequalitycannotbe obtained ingeneral using Theorem2.1e.g.inthe casethattheintegralfx w(t)dt
isdivergent.Anotherinequalityof thetype(2.8)
canbeobtainedby usingourTheorem4.1 withw(x, y)w(lx
y[)(see
Remark4.5).
3
THE GENERAL (MIXED NORM) CASEp
qLet
toand v denote weightfunctions on(0, o). Moreover,
definey
V(y)
v(x)dx,
In(x)
u(y)l(Au)z(x,
y):Ix
yl xdx
fo v(x)dx v(y),
The main resultofthis section reads asfollows:
THEOREM 3.1 and
Let
1 < p < q < o,) > l/p,wz (x)
O)p,q(X)X
-zqfrOO )
1/qCp,q
sup wL(x) V
-q(x)dx
r>0
(fo
[wx(x)v
-q(x)]l-q’dx )l/q’
< cx.(3.1)
Then
for
uL
q(0,x; tox)
c
lu(x)lq tox(x)dx)
1/qI(Auz,)(x,
y)lp(3.2)
to(x)dx]
q/pv(y)dy)1/q
provided
K-- Cp,qq
< 1.
(3.3)
(q-
1)l/q’
Remarkinequality3.2
(1.10) Note
withthatforthe case v(y)---
1 weobtainthe mixednormx
WO(X to,(X)
X(-’k+l/p’)qto
1-p’(t)dt)
-q/p’Proof
HSlder’sinequality yields(f0
ylu(x)l-
u(y)lv(x)dx)q
(foy )q/P (JoY (uP(x)) e-
<
[u(x) u(y)[Pw(x)dx
\ w(x)
Thereforeq/p’
0
cx Y>
y_q/p_)q vp (X)
\ w(x)
jo
yu(x)v(x)dx]
qv(y)dy wx(y)lu(y)dx)
-q/P’ lu(y) yv(x)dx
1
foy u(x)v(x)dxlqdy"
V(y)
Usingthis estimate
together
with the Minkowski andHardy inequalities [the latter one canbe useddue to(3.1)]
wefindthat(oC ]u(y)lqwz(y)dy )
1/q(jo
ijo
y),q
<_
wx(y)lu(y)V(y)
u(x)v(x)dxlqv(y)dy
(fOOC (for )q )l/q
-Jr-
wz(y)V-q
(y)lu(x)lv(x)dx
dyIx Yl w(x)dx
v(y)dy+ K w)(y)lu(y)lqdy
Inequality
(3.2)
followsby subtractingandusing(3.3).
Consideringthe casep qin Theorem3.1we obtain
Colot,al,3.3
If
l < p <,
-l/p,wx(x) Op,p(X)X
-xp andCp
supwz (x) V
-p(x)dx
r>0
(zr (wz (x)v-P (x))-P’dx )alp’
<,
then
fo c
lu(x)lPwx(x)dx)
lip< 1
(fofolU(X)-u(y).
p1-K Ix-ylI+zp
w(x)v(y)dxdy)
liP(3.4)
provided
K= Cpp
< 1.
(p-
1)l/p’
Consequently, (3.4)
mayberegardedas a fractionalorderHardy inequality ofthetype(1.9)
withthe weightW(x, y) w(x)v(y) ontherighthand side.
Remark 3.4 Applying Corollary 3.3 with
w(x)
1,v(y) 1 andwith u(x)
replaced byu(x)
u(0)
wefindthat if1 < p < cx,) > 1/p,then(f0 lu(x)-u(O)lPx-ZPdx)
1/p< )p_t_ p l
(fo fo lU(x)
u(y)lp)lip
)p 1
Ix
ylI+xp dxdy(3.5)
cf. formula
(1.6). Moreover,
it is notdifficult to see that(3.5)
holds with 0 <)< 1 and with the constant1/2) / (1 ))
forp 1.Remark 3.5 Theorem 3.1 requires a certain integrability of v and
(vPw-1)1--r-
on(0,
y)for y > 0. Thisrequirementcanbereplaced by an integrabilitycondition on(y,cx),if one modifies theproofof Theorem 3.1 asfollows:Let
fy
V(y)
v(x)x-1/P-’-3dx
with somereal parameter 3,and
wx
(y) v(y)w(x) ,I x-e’dx vq
(y)"Now,
we usetheestimatefoC (fo l(Au)z(x,Y)l ifr l
pw(x) )q/P
v(y)dyfo (fy lU(x)-u(Y)’
p)q/P
>
w(x)dx
v(y)dyand denote the last doubleintegral by
J.
SinceH61der’sinequality yields(fy lu(x)
x1/p+.+6u(y)lv(x)dx
<
(fy
lu(X)-u(y)Ipx+@ w(x)dx )q/P(fyCX:(l)P(x)) \ w(x) @f- x-’P’dx )
q/p’weobtainthat
Then we
proceed
as in theproof
of Theorem 3.1, using the Minkowski inequality andtheHardy
inequality thelast oneassuming thatCp,q
< cwhere
(fo
rt
1/qCp,q sup x (xl -q (x)dx
r>0
.(fr(L(x)v-q(x)xq/P+,q+3q)l-q’dx)
1/q’Finally,we obtainagain inequality
(3.2)
withwz (x) replaced
byz (x)
provided
K Cp,qq
< 1.
(q-
1)l/p’
4
A GENERAL WEIGHT IN TWO VARIABLES FOR
p=
qIn
this section, wewillprove
thefollowingassertion:THEOREM 4.1 Letto(x,
y)
beanon-negativemeasurablefunction
on(0,o) (0, o),
locally integrableinbothvariablesseparately.Let
1 < p < oand)
>_
-lip.(i)
Denote
(xl_. f0
xW(x) col-p’(x, t)dt
andWx (x) W (x)x -zp. If
Cp"
suppO+l---ydx wl-p’(x)x)P’dx
<r>0 x
and
K= Cpp
<1,(p- 1)l/p’
then
foru LP(O,
o;Wz)
(f0 Wx(x)lu(x)lPdx)
1/p1
(foOfolU(X)-u(y)l
p1-K
Ix-yll+xp(4.1)
(4.2)
co(x,
y)dxdy)
liP(ii)
Denote
(f0y
W(y) col-p’(t,
y)dt and(x) "(x)x-zp. If
Cp"
=sup dyr>0
yp(Z+l) (fo
rW
1-p’(y)yXP’dy)lip’
and
Cpp
< 1,(p- 1)l-p’
(4.3)
then
(4.2)
holdswithWz
andK
replacedbyWz
andK,
respectively.Proof
Obviouslyfo lu(x) Ix
ylu(y)ll+xppco(x,
y)dxdy_fofoXlU(X)-u(y)l Ix
yl+xppoo(x, y)dydxfx lu(x)
u(y)lp+ ...ix...2. yl./.p.., w(x,
y)dydxfo
xlu(x)
u(y)lp> xl+.p co(x, y)dydx
fx lU(X) --..u(Y)l
p+ yl+Xp co(x,
y)dydxI1 + I2.
H61der’sinequality yields
X(u(x) u(y))dy <
(fo
xlu(x) u(y)IPco(x, y)dy )
(fo
xcol-P’(x,t)d
tand
consequently
foX W(x)xI-p fox
I1 >_
xl+,p(U(X)
u(y))dylpx
xl+,p
IXU(X)
u(y)dylpfo W(X)
x plu(x)
xlfo
xu(y)dylpHence,
bythe Minkowski andHardy inequalities,wehaveand since
K
< 1,(4.2)
followsatonce.Part
(ii)canbeprovedcompletely
analogously, estimatingnowfrombelow theintegral 12,which can be rewrittenbyFubini’stheorem asf0
yItt(X)
u(y)lpI2 yl+,p
og(x,y)dxdy.But
it follows also directly from part (i) using the following symmetry argument: Since forh(x,
y)lu(x)
u(y)lPlx yl --xp we haveh(x,
y) h(y, x),thefighthand side in(4.2)
satisfiesh(x, y)w(x, y)dxdy h(x, y)w(y, x)dxdy.
If we denote for tOl(X, y) w(y,
x)
byWl(x)
the function, which correspondsto tO1 asW corresponds
to tO, wehave thatWl(x) W(x),
and now weproceed
as in theproof
ofpart (i)withW1
insteadofW.
Remark 4.2 The
proof
of Theorem4.1 showsthat infactwecan derive a better inequality provided(4.1)
and(4.3)
hold simultaneously:(foOO )lip
(1 K) Wx(x)lu(x)lPdx
+ (1 K) Wx(x)lu(x)lPdx
<
(fo fo
lU(x)_u(y)lp)l/p
x
yll+X
po)(x,
y)dxdyCOROLLARY
4.3Let
1 < p< o,>_
-1/p and <)p- 1. Thenlu(x) u(O)IPxa-XPdx
<
Zp +
p-ot l(fo fo lU(X)
u(y)lP;p
ot 1Ix
y I+zpxadxdy
(4.4)
Proof Apply
Theorem 4.1 (i) withu(x)
replacedby u(x) u(O)
and with o(x, y)w(x)
x.
ThenW(x)
xa and(4.1)
holds withCp
(p1) 1/p’/(zp +
p ot1)
providedct p) p < -1 and-tpt/p + ,kp’ +
1 > 0,i.e., ifot+
1 < p(.+ 1). But
this isthe case ifot
+
1 < )p.Moreover, K
p/(Zp+
p-t1)
< 1 ifot+
1 < )p.Remark 4.4
For
the casect 0,the statement inCorollary
4.3 follows alsoatonce from ourCorollary 3.3(see
Remark3.4). Moreover,
according toCorollary
1in[4] (cf.
ourTheorem2.2),
the followingcomplement
also holds:If1 < p < c,Z
>-1/p
andct > )p 1, then(fo )l/p (foC fo lu(x)_u(y)lp
)lip
lu(x)[Px-XPdx <_ C
Ix yll+Xp xadxdy
(4.5)
whereC
2p-1(1 +
p/(ot;p + 1)). For
ct 0, the inequalities(4.4)
and(4.5)
coincide withtheinequalities(1.6)
and(1.5)
mentioned in Section 1.Remark 4.5
By
applying Theorem4.1 withog(x, y)
w(Ix Yl)lx y1-1-xp
we obtain another inequality of the Burenkov-Evanstype(see
Theorem2.1). In
particularby using thisresult withw(x)
x we rediscoverthe inequality(2.7).
Remark 4.6
In
Theorem4.1,we in fact used theintegrability ofcol-p’
(x, y) either withrespectto x or withrespecttoyonintervals(0, z),z
> 0.Ifthis condition is notfulfilled,wecanproceedin a similarwayas inRemark3.5:We
estimatetheintegralIe
fromtheproof
ofTheorem4.1,fofxlU(x)-u(y)l
p12 yl+Zp
co(x, y)dydx,from belowusing the following H61der inequality:
c
u(x)
u(y) Pyl/p+)+
CO(X, y)dy(fxlU(X)__u(y)lp )(fx )p-1
<
CO(X
y)dyy-P’CO(x
y)dyyl+.p
with a suitableparameter6.
Here,
the role of the functionW (x)
isplayed by(fxCXZ
W* (x) y-P’CO(x,
y)dyi.e., we need theintegrability ofCO(x,
y)y-aP’
withsome 6 intheneighbour- hood ofinfinity.Theremaining stepsare similar as in theproof
of Theorem 4.1 (i) and areleftto thereader(seealsoRemark3.5). Let
usmention that condition(4.1)
isthenreplaced
by(fo
r)lip
C"
supr>OW*(x)dx
(fo
cW *I-p’(x) xl/p’-P’+’+’dx)
the function
Wz (x) W(x)x
-zp in(4.2)
isreplaced byW*(x).
X1-1/p-L-3and theparameter6has tosatisfy6 > 1 1/p
A
similarresult can be obtainedusingtheintegrabilityofx-aP’CO(x,
y) (as a functionofx!)
intheneighbourhoodofinfinityifweproceed analogously withI1
rewritten(byFubini’stheorem)
asi1 f0c fy
c lu(x)-u(y)lpxl+.p
CO(x, y)dxdy.5 A GENERALIZATION TO THE ORLICZ NORM
In
this section we willmodify inequalities(1.5)
and(2.6),
i.e., the case with powerweightsx,
usingthe norm in a suitable Orliczspace.For
thispurpose,
let Pbe aYoung
functionsatisfyingtheA2-condition.
Then it iswell-known thatthere exists
a/
>0such that forallK > 1,P(ct)
<_tc P(t). (5.1)
If
H
istheHardy (averaging)operator definedbylf0X
(Hu)(x) u(t)dt
x
andboth
P
and itscomplementaryfunctionP
satisfythe A2-condition, then Palmieri[8](cf.
also[6, Corollary4])
provedthatIltVHulle lltuullp
1v (5.2)
where the norm in
(5.2)
isthe Orlicz norm.[Ifv < 0 then the condition thatP A2
canbeomitted.]It
iswellknown that theOrlicznormI1" II
Pand theLuxemburgnormI1" 117o
definedby
( )
Ilgll,"
inf[ tk
> 0"e !g(x)l
k dx satisfyIlgll,
IlgllP211gll,.
Let0 < ) < 1 and denote
v(x)
(Au)z(x, y)u(x)
u(y)vz(x)=
xX Ix-
ylzfurther, let
I1" IIe(z)
denote the twodimensionalOrlicznorm on (0, o) x(0, cx)
withrespecttothe measured/x
dxdyIx-ylThemainresultof this section reads asfollows.
THEOREM 5.1
l+X,8 < 1,
If P
andP
satisfy(5.1)
withtheconstant >O,
thenfor
Iluxlle CIl(Au)xlle( (5.4)
where
2(1 +
1
+/3()- 1)
Moreover, if
) > 0the condition thatP satisfies (5.1)
may be omitted.Proof
TheconvexityofP
yieldsfofo P(l(Au)zl)dlz fofo P (,u(x)-u(y),)dxdy
Ix
yl zIx
ylfo fo
x( lu(x)
u(y)l) dY
>- P
xz x
x
e
x( fo
xlu(x)
u(y)ldy)
dxP (luz(x) (Hu)z(x)l)dx
>_ P
xx
The Minkowski inequality, the estimates
(5.3)
and the (Hardy) inequality(5.2)
with v -)yieldIluxllP Ilux (Hu)) + (Hu)xll,
Iluz (Hu)zllP + II(Hu)zlle
_< 211ux (nu)zllo + IIt-znulle
, t
_<
211(zXu)zllp(u
/ 1+/3) IluzllP
<_
211(Au)zllP(
4-1
+/, Iluxlle
and
(5.4)
follows immediately.Ofcourse
(5.3)
shows that(5.4)
holdsalso with Orlicz normsreplaced
by theLuxemburgnorms.6
AN N-DIMENSIONAL FRACTIONAL ORDER HARDY
INEQUALITY
We
arealso abletoprovesome N-dimensional versionsoftheinequalities mentioned. First some notation:For
xN N, B(Ixl)
will denote the ball {yNN;
lyl _<Ixl}
andIB(Ixl)l
itsvolume.It
isIB(Ixl)l IxlNISN-11/N
where
S
u-1 istheunitsphere
inN
uandIsN-I
itsarea.THEOREM 6.1
Let
1 < p < cxz,N
> 1 andp
> 1. ThenixlXP---
dx2N(l+Xp/PN1/p [p(l /,k)_ l]
(f f
lu(x)_u(y)lp)liP
IsN-111/P ,p
1 v vIx ylN(+Xp) dydx/(6.1)
Proof
ObviouslyJ" = fr lu(x)-u(Y)lP
N Ix-ylN(l+xp)dydx
(Ixl)
IX
ylN(I+xp)dydxl
flII-N(I+Z’P) fB
> 2N(I+Xp)
IX lU(X) u(y)lPdydx
N (Ixl)
sincefor y
B(Ixl),
it isIx Yl
<21xl,
and 1+ Lp
> 0.But
H61der’sinequality yields
(u(x)
-u(y))dy(Ixl)
<-(fB
(Ixl)lU(X)-u(y)lPdy) IB(Ixl)l
p-1andconsequently
J
>2N(I+Lp)
ixlNO+Xp (u(x) u(y))dylPdx
(Ixl)
l
f ’B(IxI)II-p
2N(+Xp)
ixlN(+Zp) lu(x)lB(lx[)l- u(y)dylPdx
(Ixl)
1
fe B(Ixl) lu(x) u(y)dylPdx.
2N(I+Lp)
Ixl
N<I/zpIn(Ixl)l
(Ixl)(6.2)
Therefore,
by
Minkowski’sinequality(f
N----ff-dx lu(x)l ]xlXP
p)alp
u(y)dy
ixlzpNlU(x)--In(Ixl)l
(lxl>l
u(y)dylPdx }alp
u(y)dylPdx ixlXpN lu(x)-
In(Ix])l
(Ixl)IB(lxl)l-Pf ixlXPN u(y)dyl
pdx}/P
u (Ixl)
lip
--=11+12.
(6.3) It
follows from(6.2)
that2N(I+)p)/PN1/P
I1
<j1/p (6.4)
]sN-111/p
and theHardy inequalitywith
power
weights yields P(flIIu(x)IP)
lip12 <_ Cp
(P_ 1)l/p, [x[PN
dxprovided
Cp.--sup(fl
x’B(IxI)I-P
)l/P(f
xr>0 l>
Ix [ZpN
dx <rYr
sup N-I-LpN-Np
NP
r>0 N-1
ISN-1IP
(fsu_l for tN-X+XP’Ndtdcr)
1/p’r(N-Np-)pN)/p
N[sN-11-1 [sN-1 [1/p+a/p’
supr>0(Np(1
+ ,) N)
1/pr ()p’N+N)/p’ (p-
1)I/P
(,p’N
+ N)I/P
p(1+ ))
1IxI-LpN(1-P’)dx)
dtdcr)
1/p1/p’
P"
(fRo [u(x)lPdx)
1/p and inequality(6.1)
follows byHence 12
< p(l+Z)-I IxlXPcombiningthe last estimate with
(6.2)-(6.3)
andsubtracting.Remark 6.1 Inequality
(6.1)
isan N-dimensionalcounterpartofinequality(1.5),
i.e., with weight 1 on the right hand side. Of course, also more generalcases canbeconsidered.Let
us mention atleastthe followingmore dimensional extensionofTheorem 4.1;we omit theproof
since thearguments arequitesimilar tothe onedimensionalcase.THEOREM
6.2Let co(x,
y) bea non-negativemeasurablefunction
onN
xN,
locally integrable inboth variablesseparately.Let
1 < p < cx andIx col-p’ (X, t)dt)
1-pIf
-lip and denote
W(x) (IB(
i)1fa(Ixl)
Cp"
supr>0{ fx
i>rIx W(x) [.pN
P N1/p
and
K Cp
(p_l)l/p, iSN_l < 1,thenixlXP, wl_P,(x)dx ]
1/p’W(x) lu(x)lPd
xN
IxlXp
g< 1
N1/P2N(I+Xp)/P(f
e ]u(x)--u(y)lp1-
K ]sN-1]
lip uIx- y]U(l+kp) co(x, y)dydx)
X/pAcknowledgements
The workofthesecondauthorwaspartially
supported
bytheGrant Agency
of CzechRepublic,
Grant No.
201/94/1066.References
[1] V. Burenkov and W.D. Evans, Weighted Hardy’s inequalities for differences and the .extensionproblemforspaceswith generalized smoothness(to appear).
[2] P.Grisvard,Espacesinterm6diaires entreespacesdeSobolevavecpoids,Ann.Scuola.Norm.
Sup.Pisa.,23(1969), 373-386.
[3] G.N.Jakovlev,Boundary propertiesof functions from thespace Wpt) on domainswith angularpoints(Russian),Dokl.Akad.NaukSSSR, 140 (1961),73-76.
[4] A. Kufner and L.E, Persson, Hardy inequalities offractionalordervia interpolation, WSSIAA (1994), 417-430.
[5] A.Kufner andH.Triebel, Generalizations ofHardy’s inequality,ConfSem. Mat. Univ.
Bari,156(1978), 21 pp.
[6] L.Maligranda, GeneralizedHardy inequalitiesinrearrangementinvariantspaces, J.Math.
puresetappl.,59(1980),405-415.
[7] B.OpicandA.Kufner, Hardy-type inequalities,LongmanScientific&Technical, Harlow (1990).
[8] G.Palmieri,Unapprocio alia teoria degli spaziditraccia relativiaglispazidiOrlicz-Sobolev, Bol.U.M.L, (5),lli-B(1979),100-109.
[9] H. Triebel, Interpolation Theory, FunctionSpaces, Differential Operators. 2na edition,
Johann Ambrosius BarthVerlag, Heidelberg-Leipzig,1995.