Abstract
In this paper, we introduce the p-adic Hardy type operator and obtain its sharp bound on the p-adic Lebesgue product spaces. Meanwhile, an analogous result is computed for the p-adic Lebesgue product spaces with power weights. In addition, we characterize a sufficient and necessary condition which ensures that the weighted p-adic Hardy type operator is bounded on the p-adic Lebesgue product spaces. Furthermore, the p-adic weighted Hardy-Cesàro operator is also obtained.
Highlights
In this paper, we introduce the p-adic Hardy type operator and obtain its sharp bound on the p-adic Lebesgue product spaces
1 Introduction p-adic numbers were introduced by Hensel at the end of the th century, they constitute an integral part of number theory, algebraic geometry, representation theory and other branches of modern mathematics
Theories of functions and operators from Qnp into R or C play an important role in the p-adic quantum mechanics, in p-adic analysis
Summary
· · · × Qnpm , |x|αp ), where |x|αp := |x|αp × |x|αp × · · · × |x|αpm and αi < (q – )ni, the p-adic Hardy type operator Hmp is bounded on Lq(Qnp × Qnp × · · · × Qnpm , |x|αp ), the norm of Hmp can be obtained as follows:. When αi = , the sharp estimate of the p-adic Hardy type operator will be easy to get on the p-adic Lebesgue product spaces, so we only provide the proof of Theorem. Without loss of generality, it suffices to fulfil the proof of the theorem by assuming f is a radial function. Using the definition of the norm of the operator and letting ε → , we conclude that This finishes the proof of the theorem
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