$L^p$ estimates for multilinear convolution operators defined with spherical measure

TL;DR

This paper establishes precise conditions on Lebesgue space exponents for the boundedness of multilinear convolution operators with spherical measures, extending previous results in harmonic analysis.

Contribution

It provides necessary and sufficient conditions for the boundedness of multilinear spherical convolution operators on Lebesgue spaces, generalizing earlier work.

Findings

Derived exact exponent conditions for boundedness

Extended previous results to multilinear setting

Unified framework for spherical measure convolution operators

Abstract

Let $\sigma=(\sigma_{1},\sigma_{2},\dots,\sigma_{n})\in \mathbb{S}^{n-1}$ and $d\sigma$ denote the normalised Lebesgue measure on $\mathbb{S}^{n-1},~n\geq 2$. For functions $f_1, f_2,\dots,f_n$ defined on $\R$ consider the multilinear operator given by $$T(f_{1},f_{2},\dots,f_{n})(x)=\int_{\mathbb{S}^{n-1}}\prod^{n}_{j=1}f_{j}(x-\sigma_j)d\sigma, ~x\in \R.$$ In this paper we obtain necessary and sufficient conditions on exponents $p_1,p_2,\dots,p_n$ and $r$ for which the operator $T$ is bounded from $\prod_{j=1}^n L^{p_j}(\R)\rightarrow L^r(\R),$ where $1\leq p_j,r\leq \infty, j=1,2,\dots,n.$ This generalizes the results obtained in~\cite{jbak,oberlin}.