Weighted inequalities for the Sawyer two-dimensional Hardy operator and its limiting geometric mean operator
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We consider T f = 0 1 0 2 f (t1 ,t2 )dt1 dt2 and a corresponding geometric mean operator x x G f = exp(1/x1 x2 ) 0 1 0 2 log f (t1 ,t2 )dt1 dt2 . E. T. Sawyer showed that the Hardy-type inequality T f Luq ≤ C f Lvp could be characterized by three independent conditions on the weights. We give a simple proof of the fact that if the weight v is of product type, then in fact only one condition is needed. Moreover, by using this information and by performing a limiting procedure we can derive a weight characterization of the corre´ sponding two-dimensional Polya-Knopp inequality with the geometric mean operator G involved. 1. Introduction The following remarkable result was proved by Sawyer in [3, Theorem 1]. Theorem 1.1. Let 1 < p ≤ q < ∞ and let u and v be weight functions on R2+ . Then ∞ ∞ x1 x2 0
0
0
≤C
0
∞ ∞ 0
q
f t1 ,t2 dt1 dt2
0
1/q
u x1 ,x2 dx1 dx2
p
(1.1)
1/ p
f x1 ,x2 v x1 ,x2 dx1 dx2
holds for all positive and measurable functions f on R2+ if and only if ∞ ∞
sup y1 ,y2 >0
y1
y2
y1 y2 0 0
sup
1/q y1 y2
u x1 ,x2 dx1 dx2
y1 ,y2 >0
y1 ,y2 >0
y1 y2
0
1− p
1/ p
dx1 dx2
q 1/q 1− p x1 x2 dt1 dt2 u x1, x2 dx1 dx2 0 0 v t1 ,t2 1/ p 1− p y1 y2 dx1 dx2 0 0 v x1 ,x2
= A1 < ∞,
(1.2)
∞ ∞ ∞ ∞
sup
0
v x1 ,x2
x1 x2
u t1, t2 dt1 dt2
∞ ∞ y1
p
v x1 ,x2
y2 u x1, x2 dx1 dx2
1− p
1/q
Copyright © 2005 Hindawi Publishing Corporation Journal of Inequalities and Applications 2005:4 (2005) 387–394 DOI: 10.1155/JIA.2005.387
dx1 dx2
= A2 < ∞,
(1.3)
= A3 < ∞.
(1.4)
1/ p
388
Two-dimensional Hardy inequality
´ However in [4] it was proved that to characterize the two-dimensional Polya-Knopp inequality ∞ ∞
exp 0
0
≤C
∞ ∞ 0
x1 x2
1 x1 x2 f
0
0
q
log f t1 ,t2 dt1 dt2
0
p
1/q
u x1 ,x2 dx1 dx2 (1.5)
1/ p
x1 ,x2 v x1 ,x2 dx1 dx2
for 0 < p ≤ q < ∞, only one condition was needed. An interesting observation is that this inequality can be characterized by just using one integral condition even if the inequality seems to be a natural limiting inequality of the Sawyer result mentioned above. The aim of this paper is to find a two-dimensional weight characterization that allow us to perform a limiting procedure (as in [2, 4]), and receive a weight characterization of ´ the corresponding two-dimensional Polya-Knopp inequality (1.5). From the corresponding result in one dimension (see [2, 4]), we know that this requires special homogeneity properties of the conditions that for instance the condition (1.2) doesn’t have. On the ´ other hand the fact that (1.5) is equivalent to a one-weighted Polya-Knopp inequality makes it possible for us to use an Hardy inequality where we allow one weight to be of product type and thus characterize the Hardy inequality with only one condition and with the special homogeneity properties (see Section 2). In Section 3 we will also show that with that condition and th
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