Carleson type theorems for certain convolution operators (Q2501052)

Let \(B(x,\rho)\) be the open ball in \({\mathbb R}^n\) centered at \(x\) and of radius \(\rho\). Set \(TB(x,\rho)=\{(y,t)\in{\mathbb R}_+^{n+1}:\;| y-x| <\rho-t\}.\) For any given \(\beta>0\), a positive measure \(\mu\) on \({\mathbb R}_+^{n+1}\) is called to be a so-called \(\beta\)-Carleson measure if \(\sup_{(x,\rho)\in{\mathbb R}_+^{n+1}}\rho^{-n\beta}\mu(TB(x,\rho))<\infty\). In this paper, the authors establish some mapping properties for some convolution operators, acting from Sobolev spaces in \({\mathbb R}^n\) to Lorentz spaces defined on \({\mathbb R}_+^{n+1}\) with a \(\beta\)-Carleson measure. As an application of the major theorems, the authors give some a priori estimates for the solutions of certain elliptic equations.