Norm inequalities for convolution operators
Let \(D\) and \(\Omega\) be measurable sets in \(\mathbb{R}^n\), and let \(K\) be a locally integrable function on \(D-\Omega\). There is considered the convolution operator \(A\), where \[ (Af)(y)= (K* f)(y)= \int_D K(x- y)f(x)\,dx, \] \(y\in\Omega\), both in Lebesgue spaces and Lorentz spaces. In case of Lebesgue spaces, there are extended estimations of the norm \(\| A\|_{L_p\to L_q}\), \(1< p< q<\infty\) obtained by Young and O'Neil. In particular, there are obtained estimations from below of this norm. Next, the Young-O'Neil inequalities in Lorentz spaces are extended for the limit value parameters, i.e., \(\| K* f\|_{L(p,h_1)\to L(p,h_2)}\). There are also given estimates in the case of weighted Lorentz spaces.
scientific article; zbMATH DE number 5651240
scientific article