The local trace inequality for potential type integral operators (Q1935450)

The local trace inequality for the following operators of potential type \(T_{\rho}\) is obtained in the context of Morrey spaces: Let \({\mathcal D}\) be the set of all dyadic cubes \(Q \in \mathbb{R^N}\) which intersect the support of the locally finite nonnegative Borel measure \(\sigma\) on \(\mathbb{R^N}\) and let \(\rho:{\mathcal D}\longrightarrow \mathbb{R}^+\) be a map. For a locally \(\sigma-\)integrable function \(f\) on \(\mathbb{R}^N\), the potential type \(T_{\rho}\) is defined by \[ T_{\rho}f(x)=\sum_{Q\in {\mathcal D}} \frac{\rho(Q)}{\sigma(Q)} \int_{3Q} f(y) \;d\sigma(y) \chi_Q(x) \quad x\in \mathbb{R^N}. \] Some applications of the main results are obtained such as an improvement of the Olsen inequality in this context. Also a sufficient condition is obtained for the equivalence between the Kerman-Sawyer condition and the Adams condition that are involved to obtain the local trace inequality.