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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 trianglecase'' $1 \leq q<p$ 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 ageneralization of Th. Wolff's inequality which gives two-sided estimates of theenergy ${\mathcal E}_{k, \sigma} [\mu]=\int_{\R^n} (T_k [\mu])^{p'} d \sigma$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 $K_{\mathcalD}(x, y) = \sum_{Q\in{\mathcal D}} K(Q) \chi_Q(x) \chi_Q(y)$, where $\mathcalD=\{Q\}$ is the family of all dyadic cubes in $\R^n$, and $K:  {\mathcal D}\to \R^+$. The corresponding continuous versions of Wolff'sinequality and trace inequalities are derived from their dyadic counterparts.

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