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

Abstract

We study trace inequalities of the type $$\Vert T_{k}f{\Vert }_{L^q(d\mu )}\le C\Vert f{\Vert }_{L^p(d\sigma )},f\in L^p(d\sigma ),$$ in the ``upper triangle case'' $1\le q<p$ 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 ${\mathcal{E}}_{k,\sigma }[\mu ]={\int }_{{\mathbb{R}}^n}(T_k[\mu ]{)}^{p^{\prime }}d\sigma$ 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.

to appear in Indiana Univ. Math. J. (33 pages)