article

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

Authors: Carme Cascante, Joaquim M. Ortega and Igor Verbitsky

Issue: Volume 53 (2004), Issue 3, 845-882

Abstract:

$We study trace inequalities of the type \[ \|T_kf\|_{L^q(d\mu)}\leq C\|f\|_{L^p(d\sigma)},\quad f\in L^p(d\sigma), \] 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 $\mathbb{R}^n$. Our main tool is a generalization of Th. Wolff's inequality which gives two-sided estimates of the energy $\mathcal{E}_{k,\,\sigma}[\mu]=\int_{\mathbb{R}^n}(T_k [\mu])^{p'}\,d\sigma$ 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 $\mathbb{R}^n$, and $K:\mathcal{D}\to\mathbb{R}^{+}$. The corresponding continuous versions of Wolff's inequality and trace inequalities are derived from their dyadic counterparts.$