J. of Inequal. & Appl., 1997, Vol. pp. 25-46
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On Some Fractional Order
Hardy Inequalities
HANS P. HEINIG a, ALOIS KUFNER and LARS-ERIK PERSSON c
Department of Mathematics & Statistics, McMaster University,
Hamilton, Ontario, L8S 4K1, Canada
b Mathematical
Institute, Czech Academy of Sciences,
itn 25, 115 67 Praha 1, Czech Repubfic
Department of Mathematics, Lule& University,
S-971 87 Lule&, Sweden
(Received 6 May 1996)
Weighted inequalities for fractional derivatives (= fractional order Hardy-type inequalities) have
recently been proved in [4] and 1]. In this paper, new inequalities of this type are proved and
applied. In particular, the general mixed norm case and a general twodimensional weight are
considered. Moreover, an Orlicz norm version and a multidimensional fractional order Hardy
inequality are proved. The connections to related results are pointed out.
Keywords: Hardy inequality; weight function; fractional order derivative.
AMS Subject Classification: 26D 15, 46E30
1 INTRODUCTION
Using the notation Ilflls,o for the norm in the weighted Lebesgue space
L (0, cx; w) L (w),
llfll,,,o
(fo0
If(t)lSw(t)dt
)l/s
with 1 < s < cx and w
w(t) a weight function on (0, x), the Hardy
inequality can be expressed in the form
Ilullq,o0 <_ Cllu’llp,,
25
(1.1)
H.E HEINIG et al.
26
with a constant C > 0 independent of u, i.e., under certain conditions, the
weighted Lq-norrn of the function u can be estimated by a suitable weighted
L P-norm of its (first order) derivative u I.
The natural question arises whether it is possible to extend inequality
(1.1) to "fractional order derivatives" u (z) with 0 < ) < 1, i.e., under what
conditions we can derive inequalities of the type
(1.2)
and/or
IluXllq,0 Cllu’llp,w.
(1.3)
The first problem which arises is how to understand the expression
Let us mention that for the "non-weighted" case (i.e., for
w(t) =_ 1) the following definition of IluZlls,1
Ilu<Zlls is commonly
used:
Ilu(Z)llq,w.
[[u(X) lls=
(fo fo
lU(x)
u(y)lS
Ix yll+Zs
dxdy
)
a/s
,1 < s < cxz, O < ) < 1.
(1.4)
For this special case, the following inequality holds:
lu(x)lPx-XPdx
lit
< Cp
Ix y[l+Zp
dxdy
(1.5)
which is inequality (1.2) for the special case
p
q, wx(x)
=-- 1,
wo(x)
x -xp.
Inequality (1.5) was derived by Grisvard [2] provided
1
1 < p < cxz, ,#--,
P
u6C(O,o).
In fact, he rediscovered an earlier result of Jakovlev [3] who has shown that
fo
c
-
lu(x)
u(O)IPx-)Pdx < C p
.
fofo
[u(x)-u(y)[p
Ix y]l+p
dxdy. (1.6)
Notice that the additional term u (0) at the left hand side of (1.6) is essential
since the integral on the left hand side of (1.5) diverges if u is continuous at
zero, u(0) 0 and L >
FRACTIONAL ORDER HARDY INEQUALITIES
27
The inequality (1.5) was extended in the seventies by Kufner and
Triebel [5]: Roughly speaking, they derived instead of (1.2) the inequality
Ilullp,m0
C([lu(Zllp,o -t-Ilullp,’)
where now
lip
Ilu(X>llp’
yl I+)p
Ix
w(x)dxdy
(1.7)
N is a certain additional weight and
wo(x)
w (x)
x -zp.
The double integral in (1.7) offers one possibility of characterizing the still
undetermined expression for [lu(X) IIp,w But in spite of the symmetry of the
"non-weighted" expression for lu(Zlls in (1.4), we would prefer a certain
symmetry in x and y also in the weighted case, i.e., we are looking for a
general inequality of the form
.
fo
lu(x)lPw(x)dx
<- CP
fofo
lu(x)-u(y)lp
W(x, y)dxdy,
(1.9)
possibly (but not necessarily) with W (x, y) W (y, x). In Section 4 of this
paper we will prove some new inequalities of the type (1.9).
Our investigations were motivated by results of Kufner and Persson [4]
and Burenkov and Evans [1]. In [4] e.g. the case when W(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 yl I+zp is treated. For the reader’s convenience and for
later purposes, some of these results are briefly discussed and compared
Ix yll+Lp
in Section 2.
Up to now, we dealt with the case
p=q
and all inequalities mentioned above can be derived more or less directly.
Another approach using the theory of interpolation of Banach spaces was
used in [4] and led to an inequality of the type
H.P. HEINIG et al.
28
(fo
cx
t
(fo(fo
1/q
lu(x)lqwo(x)dx
< C
lu(x)-u(Y)IP
Ix
yl I+zp
w(x)dx)
q/p
dy
)l/q
(1.10)
i.e., to an expression with a mixed norm on the right hand side for p 5 q.
Moreover, it was supposed that 1 < p < q < cxz [while the case q < p
is still open] and the authors have not been able to remove the mixed norm
for p 5 q. In Section 3, we will prove a new inequality of the type (1.10),
moreover, with a measure v(y)dy instead of dy, but again only for p < q.
Some of the results obtained in this paper can be extended in various
directions. Here we only present and prove an Orlicz norm version of the
inequality (1.5) [and of its extension to the power weight case see, e.g.,
(2.6) with/3 1 + Zp and give an example of a multidimensional fractional
order Hardy inequality; see Sections 5 and 6, respectively.
We close this introductory ection by noting that in all cases we have
considered here we have found that the "fractional" weights w0 in the left
integral are always of the type (1.8) for some suitable weight w. Maybe this
is supported by the following inequality which is an easy consequence of
(1.5) and of the fact that u belongs to the weighted space LP (w) if and only
if uw 1/p belongs to the (non-weighted) space LP.
lu(x)lPw(x)x-ZPdx
<C
fo
lu(x)w 1/p (x)
Ix
u(y)w 1/p (y)l p
dxdy.
yl I+xp
(1.11)
Also the fight hand side in (1.11) could serve as a definition of the expression
Ilu()llp,w,
2 SOME PRELIMINARY RESULTS AND DISCUSSIONS
For the special case when the weight function W (x, y) in the fight hand side
of (1.9) depends on Ix Y I, Burenkov and Evans [1] recently proved the
following interesting result:
FRACTIONAL ORDER HARDY INEQUALITIES
29
THEOREM 2.1 Let 0 < p < o, let w be a weight function on (0, o) and
define
v(x)
w(t)dt.
Suppose that v satisfies the A2-condition, i.e., there exists a constant c > 0
such that
v(2t) _< cv(t) for all t > 0.
LP (O, cx; v)
Then for all u
lu(x)lPv(x)dx <_ C p
lu(x)
u(y)lPw(lx
yl)dxdy.
For later purposes we also state the following slight improvement of the
recent result by Kufner and Persson mentioned in the introduction:
THEOREM 2.2 Let 1 < p < o and ) >_ -1/p. Furthermore, assume that
the function u
satisfies
lim
1
x-- cx X
fo
x
O.
u(t)dt
Let wo and wl be weight functions on (0, e) satisfying
B"
sup
x>0
with p’
fo
P__e_
p_l
(fo
x
wo(t)dt
Then, for every
> O,
[u(x)lPwo(x)dx < C p
fofo
w
(t)dt
t
xlu(x)-u(y)lp
Ix
yl
< cx
W(x)dydx
(2.1)
(2.2)
where
W(x)
and C p
x-l wo(x) + x-l-Ptol (X)
2 p-1 max(l, Cp) with Cp <_ B pP (p
1) 1-p.
The proof of Theorem 2.1 is similar to that of Proposition 1 in [4] but for
the reader’s convenience we present here the details.
H.P. HEINIG et al.
30
First we use an idea by Grisvard [2] (cf. also p. 261 in Triebel [9])
and define
Proof
u(x)
g(x)
u’(y)-
-
[u(x)
x
u(y)]dy.
(2.3)
u(z) and
Obviously, g()
u’(y)
u(y)dy
x
u(y)
y
+
and we conclude that
1
u(y)
for
u(x)dx +
g(y)
g(Y)
y
dx
x
g(y)-
g(x) dx.
x
Therefore by using the inequality la + bl p <_ 2p-I(IaIP -q-IblP), the
assumption (2.1) and Hardy’s inequality (see, e.g., [7]) we obtain that
o
]u(x)lPwo(x)dx
< 2 p-1
< 2 p-1
2P-1
(fo
(fo
[g(x)lPwo(x)dx +
fo
[g(x)lPwo(x)dx -I- Cp
wo(x)
o
fg(x_)
g(Y)
dy
Y
11)1(x)
x
Pdx
)
Pdx
Ig(x)lPWo(x)dx
where Wo(x)
wo(x) + CpX-Pll)I (x). Therefore, by denoting W1 (x)
-p
x
wo(x) +
Wl (x) and using (2.3) we find that
[u(x)lPwo(x)dx
<C
fo(fo
<C
[g(x)lPWl(x)dx
x
lu(x)
u(y)ldy
)
Wl (x)dx.
(2.4)
Furthermore, by H61der’s inequality and the assumption/3 > 0, we find that
FRACTIONAL ORDER HARDY INEQUALITIES
( f0
x
31
x
lu(x)
<__ x-PxP-1
u(y)ldy
<
u(y)[Pdy
lU(X)
x -1
f,x [u(x) -xU(Y)IP dy
X/-1
lu(x)
fx
Jo
Ix
u(y)l p
y[
dy.
The estimate (2.2) follows by combining (2.4) and (2.5).
COROLLARY 2.3 Let 1 < p < x, 13 >_ O,
limx-c 2 u (t)dt O, then
>_ --1/p and t > )p
1.
If
f
fo
[u(x)[Pxa-Pdx < C p
foCfoX]U(X)-u(Y)[Px#-l-ZP+adydx.(2.6)
Proof Apply Theorem 2.2 with wo(x)
x a-zp and tO1 (X)
a-zp-1
Let us note that applying Theorem 2.1 with w(t)
X t-’kp+p.
we find that
for a < )p we have
fo
c
[u(x)IPxa-ZPdx < C p
Ix
foCfoClu(x)-u(Y)lP
ix yl+X
p
yldxdy. (2.7)
Moreover, Theorem 2.1 cannot be used for any ot >_ p (since then
v(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) putting there/3 0
and noting that x--XP+ _< Ix y[-1-LP+a for all y, 0 < y _< x and
-1 -)p + ot <0.
More generally, using Theorem 2.2 with/3
decreasing we obtain an inequality of the type
[u(x)lPwo(x)dx <_ C p
lu(x)
0 and with W(x) strictly
u(y)lPW(lx
yl)dxdy,
(2.8)
and this inequality cannot be obtained in general using Theorem 2.1 e.g. in the
case that the integral
w(t)dt is divergent. Another inequality of the type
(2.8) can be obtained by using our Theorem 4.1 with w(x, y) w(lx y[)
(see Remark 4.5).
fx
H.P. HEINIG et al.
32
3 THE GENERAL (MIXED NORM) CASEp
q
Let to and v denote weight functions on (0, o). Moreover, define
y
V(y)
v(x)dx,
fo
dx
(Au)z(x, y):
In(x)
v(x)dx
v(y),
u(y)l
yl x
Ix
The main result of this section reads as follows:
THEOREM 3.1 Let 1 < p < q < o, ) >
and
Cp,q
sup
r>0
Then for u
c
l/p,
)
wz (x)
frOO
(fo
[wx (x)v -q (x)]l-q’dx
(x)dx
-zq
1/q
wL (x) V
-q
O)p,q (X)X
)l/q’
< cx.
(3.1)
L q (0, x; tox)
lu(x)lq tox(x)dx) 1/q
(3.2)
I(Auz,)(x, y)l p
to(x)dx]
q/p
v(y)dy)1/q
provided
K--
Cp,qq
(q- 1)l/q’
Remark 3.2 Note that for the case v(y)
inequality (1.10) with
- 1-p’(t)dt)
1 we obtain the mixed norm
x
WO(X
to,(X)
X
(-’k+l/p’)q
(3.3)
< 1.
to
-q/p’
FRACTIONAL ORDER HARDY INEQUALITIES
Proof
(f0
33
HSlder’s inequality yields
y
lu(x)l- u(y)lv(x)dx
(foy
<
[u(x)
)q
u(y)[Pw(x)dx
)q/P (JoY (uP(x))
e-
q/p’
\ w(x)
Therefore
>
0
cx
y_q/p_)q
vp (X)
Y
dx)
\ w(x)
-q/P’
y
lu(y)
v(x)dx
y
jo u(x)v(x)dx]
q v(y)dy
foy u(x)v(x)dxlqdy"
1
V(y)
wx(y)lu(y)
Using this estimate together with the Minkowski and Hardy inequalities [the
latter one can be used due to (3.1)] we find that
(oC
]u(y)lqwz(y)dy
)
1/q
<_
(jo
wx(y)lu(y)
-Jr-
(fOOC
wz(y)V-q (y)
i
V(y)
(for
jo u(x)v(x)dxlqv(y)dy),q
y
lu(x)lv(x)dx
)q )l/q
dy
H.P. HEINIG et al.
34
w(x)dx
Ix Yl
+K
v(y)dy
w)(y)lu(y)lqdy
Inequality (3.2) follows by subtracting and using (3.3).
Considering the case p q in Theorem 3.1 we obtain
Colot,al, 3.3
Cp
If l
<p<
,
-l/p, wx(x)
Op,p(X)X -xp and
wz (x) V -p (x)dx
sup
r>0
(zr
(wz (x)v-P (x))-P’dx
)alp’
<
,
then
c
fo
lu(x)lPwx(x)dx) lip
<
1
1-K
p
(fofolU(X)-u(y). w(x)v(y)dxdy)
Ix-yl I+zp
liP
(3.4)
provided
K=
Cpp
(p- 1)l/p’
< 1.
Consequently, (3.4) may be regarded as a fractional order Hardy inequality
of the type (1.9) with the weight
W(x, y)
w(x)v(y)
on the right hand side.
Remark 3.4 Applying Corollary 3.3 with w(x)
1 and with
1, v(y)
u (x) replaced by u (x) u (0) we find that if 1 < p < cx, ) > 1/p, then
(f0
lu(x)-u(O)lPx-ZPdx) 1/p
<
)p_t_ p l
)p 1
(fo fo
lU(x)
Ix
u(y)l p
dxdy
yl I+xp
)lip
(3.5)
FRACTIONAL ORDER HARDY INEQUALITIES
35
cf. formula (1.6). Moreover, it is not difficult to see that (3.5) holds with
0 < ) < 1 and with the constant 1/2) / (1 )) for p 1.
Remark 3.5 Theorem 3.1 requires a certain integrability of v and
(vPw-1)1--r- on (0, y) for y > 0. This requirement can be replaced by an
integrability condition on (y, cx), if one modifies the proof of Theorem 3.1
as follows: Let
fy v(x)x-1/P-’-3dx
V(y)
with some real parameter 3, and
wx (y)
v(y)
w(x) ,I
vq (y)"
x-e’dx
Now, we use the estimate
l(Au)z(x,Y)l p
foC (fo ifr l
fo (fy
w(x)
)q/P
lU(x)-u(Y)’ p
>
v(y)dy
w(x)dx
)q/P
v(y)dy
and denote the last double integral by J. Since H61der’s inequality yields
(fy
(fy
<
lu(x)
u(y)l
x1/p+.+6
v(x)dx
lu(X)-u(y)Ip
we obtain that
x+@
w(x)dx
)q/P(fyCX:(l)P(x))
\ w(x)
@f-
x-’P’dx
)
q/p’
H.P. HEINIG et al.
36
Then we proceed as in the proof of Theorem 3.1, using the Minkowski
inequality and the Hardy inequality the last one assuming that Cp,q < c
where
(fo
t
.(fr(L(x)v-q(x)xq/P+,q+3q)l-q’dx)
r
Cp,q
sup
r>0
x (xl -q (x)dx
Finally, we obtain again inequality (3.2)
provided
with
Cp,qq
K
1/q
(q- 1)l/p’
1/q’
wz (x) replaced by z (x)
< 1.
4 A GENERAL WEIGHT IN TWO VARIABLES FOR p = q
In this section, we will prove the following assertion:
THEOREM 4.1 Letto(x, y) be a non-negative measurablefunction on (0, o)
(0, o), locally integrable in both variables separately. Let 1 < p < o and
) >_ -lip.
(i) Denote
x
(xl_. f0 col-p’(x,
W(x)
and Wx (x)
Cp"
t)dt
W (x)x -zp. If
sup
r>0
wl-p’(x)x)P’dx
x pO+l--------ydx
<
(4.1)
and
K=
then foru
Cpp
(p- 1)l/p’
<1,
LP(O, o; Wz)
(f0 Wx(x)lu(x)lPdx)
1/p
(4.2)
p
y)dxdy)
1-K (foOfolU(X)-u(y)l
1
Ix-yl l+xp
co(x,
liP
FRACTIONAL ORDER HARDY INEQUALITIES
37
(ii) Denote
(f0y col-p’(t,
W(y)
and
y)dt
"(x)x -zp. If
(x)
(fo W (y)yXP’dy)lip’
r
Cp" =sup
yp(Z+l)
r>0
dy
1-p’
(4.3)
and
Cpp
(p- 1)l-p’
then (4.2) holds with
Proof
< 1,
Wz and K replaced by Wz and K, respectively.
Obviously
lu(x)
fo
Ix
u(y)l p
co(x, y)dxdy
yl l+xp
_fofoXlU(X)-u(y)l
Ix
p
oo(x, y)dydx
yl +xp
fx
+
fo
>
+
x
u(y)l p
lu(x)
...ix...2. yl./.p.., w(x, y)dydx
u(y)l p
lu(x)
xl+.p
fx
co(x, y)dydx
lU(X) --..u(Y)l p co(x, y)dydx
yl+Xp
I1
+ I2.
H61der’s inequality yields
x
X(u(x)
u(y))dy
<
(fo
(fo col-P’(x,t)d
lu(x)
u(y)IPco(x, y)dy
x
t
)
H.P. HEINIG et al.
38
and consequently
I1 >_
foX
W(x)xI-p
xl+,p
W(X)
lu(x)
xp
u(y))dyl p
(U(X)
x
u(y)dyl p
IXU(X)
xl+,p
fo
fox
l
x
fo
x
u(y)dyl p
Hence, by the Minkowski and Hardy inequalities, we have
and since K < 1, (4.2) follows at once.
Part (ii) can be proved completely analogously, estimating now from below
the integral 12, which can be rewritten by Fubini’s theorem as
I2
f0
y
Itt(X)
u(y)l p
yl+,p
og(x, y)dxdy.
But it follows also directly from part (i) using the following symmetry
argument: Since for h(x, y)
lu(x) u(y)lPlx yl --xp we have
h(x, y) h(y, x), the fight hand side in (4.2) satisfies
h(x, y)w(x, y)dxdy
h(x, y)w(y, x)dxdy.
If we denote for tOl (X, y)
w(y, x) by Wl(x) the function, which
corresponds to tO1 as W corresponds to tO, we have that Wl(x)
W(x),
the
of
of
instead
W.
and now we proceed as in
proof part (i) with W1
FRACTIONAL ORDER HARDY INEQUALITIES
39
Remark 4.2 The proof of Theorem 4.1 shows that in fact we can derive a
better inequality provided (4.1) and (4.3) hold simultaneously:
(1
K)
(foOO
+ (1
<
Wx(x)lu(x)lPdx
Wx(x)lu(x)lPdx
K)
(fo fo
lU(x)_u(y)l p
x
COROLLARY 4.3 Let 1 < p < o,
yll+Xp
o)(x, y)dxdy
>_ -1/p and
)l/p
< )p- 1. Then
u(O)IPxa-XPdx
lu(x)
<
)lip
Zp + p -ot
;p
ot
l
1
(fo fo
.
(4.4)
lU(X)
Ix
u(y)lP
xadxdy
y I+zp
Proof Apply Theorem 4.1 (i)
with u(x) replaced by u(x)
u(O) and
x Then W(x)
x a and (4.1) holds with
Cp (p 1) 1/p’/(zp + p ot 1) provided ct p) p < -1 and
-tpt/p + ,kp’ + 1 > 0, i.e., if ot + 1 < p(. + 1). But this is the case if
ot + 1 < )p. Moreover, K
p/(Zp + p- t 1) < 1 if ot + 1 < )p.
with o(x, y)
w(x)
Remark 4.4 For the case ct
0, the statement in Corollary 4.3 follows
also at once from our Corollary 3.3 (see Remark 3.4). Moreover, according
to Corollary 1 in [4] (cf. our Theorem 2.2), the following complement also
holds: If 1 < p < c, Z > -1/p and ct > )p 1, then
(fo
lu(x)[Px-XPdx
)l/p (foC fo
<_ C
lu(x)_u(y)lp
Ix yll+Xp
xadxdy
)lip
(4.5)
where C
+ p/(ot ;p + 1)). For ct 0, the inequalities (4.4) and
(4.5) coincide with the inequalities (1.6) and (1.5) mentioned in Section 1.
2 p-1 (1
Remark 4.5 By applying Theorem 4.1 with
og(x, y)
w(Ix Yl)lx y1-1-xp we obtain another inequality of the
Burenkov-Evans type (see Theorem 2.1). In particular by using this result
with w(x) x we rediscover the inequality (2.7).
H.P. HEINIG et al.
40
Remark 4.6 In Theorem 4.1, we in fact used the integrability of col-p’ (x, y)
either with respect to x or with respect to y on intervals (0, z), z > 0. If this
condition is not fulfilled, we can proceed in a similar way as in Remark 3.5:
We estimate the integral Ie from the proof of Theorem 4.1,
fofxlU(x)-u(y)l
yl+Zp
12
p
co(x, y)dydx,
from below using the following H61der inequality:
c
u(x)
u(y)
yl/p+)+
<
P
CO(X, y)dy
(fxlU(X)__u(y)lp
yl+.p
CO(X y)dy
)(fx
y-P’CO(x
y)dy
)p-1
with a suitable parameter 6. Here, the role of the function W (x) is played by
(fxCXZ y-P’CO(x,
W* (x)
y)dy
i.e., we need the integrability of CO(x, y)y-aP’ with some 6 in the neighbourhood of infinity. The remaining steps are similar as in the proof of Theorem
4.1 (i) and are left to the reader (see also Remark 3.5). Let us mention that
condition (4.1) is then replaced by
r
C"
sup
r>O
(fo
(fo
W*(x)dx
)lip
c
W *I-p’(x)
xl/p’-P’+’+’dx)
the function Wz (x) W(x)x -zp in (4.2) is replaced by W*(x). X 1-1/p-L-3
and the parameter 6 has to satisfy 6 > 1 1/p
A similar result can be obtained using the integrability of x-aP’CO(x, y) (as
a function of x !) in the neighbourhood of infinity if we proceed analogously
with I1 rewritten (by Fubini’s theorem) as
c
i1
f0c fy pxl+.p
lu(x)-u(y)l
CO(x, y)dxdy.
FRACTIONAL ORDER HARDY INEQUALITIES
41
5 A GENERALIZATION TO THE ORLICZ NORM
,
In this section we will modify inequalities (1.5) and (2.6), i.e., the case with
power weights x using the norm in a suitable Orlicz space.
For this purpose, let P be a Young function satisfying the A2-condition.
Then it is well-known that there exists a/ > 0 such that for all K > 1,
P(ct) <_
tc P(t).
(5.1)
If H is the Hardy (averaging) operator defined by
lf0X
(Hu)(x)
x
u(t)dt
and both P and its complementary function P satisfy the A2-condition, then
Palmieri [8] (cf. also [6, Corollary 4]) proved that
lltuullp
IltVHulle
1
(5.2)
v
where the norm in (5.2) is the Orlicz norm. [If v < 0 then the condition that
P A2 can be omitted.]
It is well known that the Orlicz norm I1" II P and the Luxemburg norm I1" 117o
defined by
Ilgll,"
inf
[tk > 0"
( )
e !g(x)l
k
dx
satisfy
Ilgll,
Let 0 <
) <
IlgllP
211gll,.
1 and denote
vz(x)=
v(x)
xX
(Au)z(x, y)
u(y)
Ix- yl z
u(x)
further, let I1" IIe(z) denote the twodimensional Orlicz norm on (0, o) x
dxdy
(0, cx) with respect to the measure d/x Ix-yl
The main result of this section reads as follows.
H.P. HEINIG et al.
42
If P and P satisfy (5.1) with the constant
THEOREM 5.1
l+X,8
> O, then for
< 1,
(5.4)
CIl(Au)xlle(
Iluxlle
where
2(1 +
1 +/3()- 1)
Moreover, if ) > 0 the condition that P satisfies (5.1) may be omitted.
Proof
The convexity of P yields
fofo
>- fo fo (
fofo (,u(x)-u(y),)dxdy
Ix
Ix
P
P(l(Au)zl)dlz
x
P
x
>_
P
u(y)l
lu(x)
xz
e
yl z
yl
) dY
x
x
( fo
x
u(y)l
lu(x)
xx
)
dy dx
P (luz(x)
(Hu)z(x)l)dx
The Minkowski inequality, the estimates (5.3) and the (Hardy) inequality
(5.2) with v -) yield
IluxllP
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( 41
+/, Iluxlle
and (5.4) follows immediately.
Of course (5.3) shows that (5.4) holds also with Orlicz norms replaced by
the Luxemburg norms.
FRACTIONAL ORDER HARDY INEQUALITIES
43
6 AN N-DIMENSIONAL FRACTIONAL ORDER HARDY
INEQUALITY
We are also able to prove some N-dimensional versions of the inequalities
N N, B(Ixl) will denote the ball
mentioned. First some notation: For x
{y NN; lyl _< Ixl} and IB(Ixl)l its volume. It is IB(Ixl)l IxlNISN-11/N
where S u-1 is the unit sphere in N u and IsN-I its area.
THEOREM 6.1 Let 1 < p < cxz, N > 1 and p > 1. Then
ixlXP------ dx
2N(l+Xp/PN1/p [p(l /,k)_ l]
,p 1
IsN-111/P
(f f
v
lu(x)_u(y)l p
v
Ix ylN(+Xp)
)liP
dydx/(6.1)
Proof Obviously
J"
=
N
lu(x)-u(Y)lP
fr
Ix-yl N(l+xp)
(Ixl)
l
2N(I+Xp)
>
IX
yl N(I+xp)
dydx
IX
flII-N(I+Z’P)
fB
N
since for y
B(Ixl), it is
inequality yields
(u(x) -u(y))dy
(Ixl)
dydx
Ix Yl
u(y)lPdydx
lU(X)
(Ixl)
< 21xl, and 1
+ Lp
> 0. But H61der’s
<-(fB lU(X)-u(y)lPdy)
IB(Ixl)l p-1
(Ixl)
and consequently
J > 2N(I+Lp)
l
2N(+Xp)
1
2 N(I+Lp)
(u(x)
ixlNO+Xp
f
fe
’B(IxI)II-p
ixlN(+Zp)
B(Ixl)
Ixl N<I/zp
u(y))dylPdx
(Ixl)
u(y)dylPdx
lu(x)lB(lx[)l(Ixl)
lu(x)
In(Ixl)l
(Ixl)
u(y)dylPdx.
(6.2)
H.P. HEINIG et al.
44
Therefore, by Minkowski’s inequality
(f
lu(x)l p
N
----ff-dx
]xlXP
)alp
u(y)dy
ixlzpNlU(x)--In(Ixl)l
l
(lxl>
}alp
u(y)dylPdx
lip
ixlXpN
u
lu(x)-
IB(lxl)l-Pf
ixlXPN
u(y)dylPdx
In(Ix])l
(Ixl)
u(y)dyl pdx
(Ixl)
}/P
--=11+12.
(6.3)
It follows from (6.2) that
2N(I+)p)/PN1/P
I1 <
]sN-111/p
j1/p
(6.4)
and the Hardy inequality with power weights yields
12 <_ Cp (P_P1)l/p,
(flIIu(x)IP)
[x[PN
lip
dx
provided
Cp.--sup(fl
r>0
’B(IxI)I-P
dx
Ix [ZpN
x l>
sup
r>0
N-1
Yr
)l/P(f
x <r
NP
dtdcr)
ISN-1IP
N-I-LpN-Np
(fsu_l for tN-X+XP’Ndtdcr)
IxI-LpN(1-P’)dx)
1/p’
1/p
1/p’
r(N-Np-)pN)/p
+ ,) N) 1/p
N[sN-11-1 [sN-1 [1/p+a/p’ sup (Np(1
r>0
r ()p’N+N)/p’
(,p’N + N)I/P
Hence 12 <
P"
(p- 1)I/P
p(1 + )) 1
1/p
(fRo [u(x)lPdx)
IxlXP
and inequality (6.1) follows by
the
last
estimate
with (6.2)-(6.3) and subtracting.
combining
p(l+Z)-I
FRACTIONAL ORDER HARDY INEQUALITIES
45
Remark 6.1 Inequality (6.1) is an N-dimensional counterpart of inequality
(1.5), i.e., with weight 1 on the right hand side. Of course, also more
general cases can be considered. Let us mention at least the following more
dimensional extension ofTheorem 4.1; we omit the proof since the arguments
are quite similar to the onedimensional case.
THEOREM 6.2 Let co(x, y) be a non-negative measurable function on N x
N, locally integrable in both variables separately. Let 1 < p < cx and
-lip and denote W(x) (IB( i)1 fa(Ixl) col-p’ (X, t)dt) 1-p If
Ix
Cp"
sup
r>0
and K
{ fx
<
i>r
P
Cp (p_l)l/p,
W(x)
N
W(x)
ixlXP, wl_P,(x)dx
Ix [.pN
N 1/p
iSN_l
1/p’
< 1, then
IxlXp g
lu(x)lPdx
1
1- K
N1/P2N(I+Xp)/P(f
]sN-1]
lip
]
e
]u(x)--u(y)l p
u
Ix- y]U(l+kp)
co(x,
y)dydx)
X/p
Acknowledgements
The work of the second author was partially supported by the Grant Agency
of Czech Republic, Grant No. 201/94/1066.
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H.P. HEINIG et al.
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