Nonlinear potentials and two weight trace inequalities for general dyadic and radial kernels

Abstract

We study trace inequalities of the type $$ \| T_k f\|_{L^q(d\mu)}\leq C \|f\|_{L^p(d\sigma)}, \qquad f \in L^p(d\sigma), $$ in the ``upper triangle case'' $1 \leq q<p$ for integral operators $T_k$ with positive kernels, where $d\sigma$ and $d\mu$ are positive Borel measures on $\R^n$. Our main tool is a generalization of Th. Wolff's inequality which gives two-sided estimates of the energy ${\mathcal E}_{k, \sigma} [\mu]=\int_{\R^n} (T_k [\mu])^{p'} d \sigma$ through the $L^1(d\mu)$-norm of an appropriate nonlinear potential $W_{k, \sigma}[\mu]$ associated with the kernel $k$ and measures $d\mu$, $d \sigma$. We initially work with a dyadic integral operator with kernel $K_{\mathcal D}(x, y) = \sum_{Q\in{\mathcal D}} K(Q) \chi_Q(x) \chi_Q(y)$, where $\mathcal D=\{Q\}$ is the family of all dyadic cubes in $\R^n$, and $K: {\mathcal D}\to \R^+$. The corresponding continuous versions of Wolff's inequality and trace inequalities are derived from their dyadic counterparts.