Boundedness properties of some convolution operators with singular measures
Let \(Q=[-1,1]\times [-1,1]\) and let \(\varphi: Q\to{\mathbf{R}}\) be given by \(\varphi(x_1,x_2)=|x_1|^\alpha+|x_2|^\beta\) with \(2\leq\alpha \leq\beta\). Suppose \(\mu\) is the measure on \({\mathbf{R}}^3\) defined by \(\mu(E)=\int_Q\chi_E(x,\varphi(x)) dx.\) Let \(T_\mu f(x)=(\mu\ast f)(x)\) and let \(E_\mu=\{(1/p,1/q): \|T_\mu\|_{L^p,L^q}<\infty, 1\leq p,q\leq \infty\},\) where \(L^p\) and \(L^q\) are the usual Lebesgue spaces on \({\mathbf{R}}^3\). The authors obtain a complete description of \(E_\mu\) in the cases \(\alpha =2, \) \(2\leq\beta\leq 4\) and \(2\leq\alpha= \beta\leq 4\) by means of some estimates for the Fourier transform \(\widehat\mu\). The authors also characterize the interior of \(E_\mu\) and obtain some additional information about its boundary in the other cases.
scientific article; zbMATH DE number 1374306