Sharp bounds for Hardy type operators on higher-dimensional product spaces

2018/04/02 by He, Qianjun, Yan, Dunyan
#Classical Analysis and ODEs (math.CA) #FOS: Mathematics

paper · doi:10.48550/arxiv.1804.00468

Abstract

In this paper, we investigate a class of fractional Hardy type operators \mathscrH_β₁,⋯,β_m defined on higher-dimensional product spaces ℝ^n₁×ℝ^n₂×⋯×ℝ^n_m. We use novel methods to obtain two main results. One is that we obtain the operator \mathscrH_β₁,⋯,β_m is bounded from L^p(ℝ^n₁×ℝ^n₂×⋯×ℝ^n_m,|x|^γ) to L^q(ℝ^n₁×ℝ^n₂×⋯×ℝ^n_m,|x|^α) and the bounds of the operator \mathscrH_β₁,⋯,β_m is sharp worked out. The other is that when α=γ=(0,⋯,0), the norm of the operator \mathscrH_β₁,⋯,β_m is obtained.

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