Convolution integral restricted on closed hypersurfaces

Abstract

The classical convolution integral on Euclidean space is given as follows. For $f\in {{L}^{1}}\left( {{\mathbb{R}}^{n}} \right)$ and $g\in {{L}^{p}}\left( {{\mathbb{R}}^{n}} \right)$ , Tf(g) is defined as $ {{T}_{f}}\left( g \right)\left( x \right):=f*g\left( x \right)=\int_{{{\mathbb{R}}^{n}}}{f\left( x-y \right)g\left( y \right)\text{d}}y. $ It has many applications in analysis and engineering. Young's inequality demonstrates that ${{T}_{f}}:{{L}^{p}}\left( {{\mathbb{R}}^{n}} \right)\to {{L}^{p}}\left( {{\mathbb{R}}^{n}} \right)$ is a bounded operator for 1 ≤ p ≤ ∞. In this study, we have obtained the estimation of the Lp norm of convolution integral restricted on closed hypersurfaces. More precisely, we have established Young's inequality on closed hypersurfaces.