Sharp bounds for Hardy type operators on higher-dimensional product spaces
© Lu et al.; licensee Springer. 2013
Received: 11 July 2012
Accepted: 21 March 2013
Published: 3 April 2013
A new class of Hardy type operator defined on a higher-dimensional product space is discussed. It includes two different kinds of the classical Hardy operators. In addition, we also consider the fractional Hardy operator . The bound of operator from to is explicitly worked out. Especially, the bound of operator from to is sharp.
MSC:26D10, 26D15, 42B99.
holds for , and the constant is best possible.
where the function f is a nonnegative measurable function on and , .
where f is a nonnegative measurable function on .
The boundedness of operator are discussed in many papers (cf. [3–6]). , the norm of , is and obviously depends on the dimension of the space. However, , the norm of ℋ, is still , and does not depend on the dimension of the space. The reason of generating the difference of the two kind of operators is, roughly speaking, that each variable can independently dilate by itself in the operator , nevertheless, for the operator ℋ, all variables dilate by the same scale simultaneously. Generally speaking, the spherical averaging operator has better properties than the rectangle averaging operator does, such as the Hardy-Littlewood maximal function. A detailed account of the history of the topic can be found in the book ; see also Kufner and Persson’s book .
For the operator , we note that every variable is defined on the one-dimensional space. In this paper, we shall extent the definition of so that every variable is spherical average defined on a higher-dimensional space.
Next, we will give the definition of Hardy type operator on higher-dimensional product spaces as follows and discuss the corresponding properties.
where with .
Our first aim in this paper is to provide transparent treatments of multivariate inequalities of higher-dimensional Hardy type both for the rectangle and for the ball case. Our results subsume those of  and . In fact, if , then the operator will become ℋ defined by (2); if , then will become defined by (1). Consequently, the operator includes both and ℋ. It is much significant to discuss the properties of .
where f is a measurable function.
- (a)If , then
- (b)if , then, for any ,
Throughout the paper, we use the following notation. The set denotes a ball with center at the original point and radius , and denotes the volume of the ball ; .
2 Sharp bounds for the Hardy type operator on product space
Proof of Theorem 2.1 We merely give the proof with the case for the sake of clarity in writing, and the same is true for the general case . We adapt some ideas and methods used in .
where and , . Obviously, g is a nonnegative radial function with respect to the variables and , respectively. In the following, we briefly call this function is a radial function on product space.
holds provided that . In addition, clearly if f is a nonnegative radial function, then we have . This means that the norm of the operator is equal to the norm that restricts to the set of nonnegative radial functions. Consequently, without loss of generality, it suffices to fulfil the proof of the theorem by assuming that f is a nonnegative radial function.
where is the volume of the unit ball in , .
Next, we need to prove the converse inequality.
This finishes the proof of the theorem. □
3 Explicit bounds for the fractional Hardy operator
- (i)If , then we have
- (ii)If , then for any ,
As above, this means that the norm of the operator from to is equal to the norm that restricts to radial functions. Consequently, without loss of generality, it suffices to carry out the proof of the theorem by assuming that f is a radial function.
Proof of (ii) of Theorem 3.1
Next, we estimate two sets A and B, respectively.
Consequently, we have . This implies that .
always holds for every . Letting , this forces that . This means that the constant 1 is sharp. □
where f is a measurable function on . In fact, the operators and ℋ enjoy the same boundedness property for and , but they do not have the same property involving the Hardy space . For instance, the operator is bounded from to ; but the operator ℋ is not since ℋ is nonnegative (cf. ).
The authors would like to express their thanks to the referees for valuable advice regarding a previous version of this paper. This research was supported by NNSF of China (Grant Nos. 11271175, 10931001, 10771221, 11271162 and 11201287), NSF of Beijing (Grant No. 1092004) and NSF of Zhejiang (Grant No. Y6110074). This work was also supported by the FCDU in Shanghai and the KLMCS (Beijing Normal University), Ministry of Education, China.
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