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

We study trace inequalities of the type parallel toT(k)fparallel to L-q(dmu) less than or equal to C parallel tofparallel to L-p(dsigma), f is an element of L-p (dsigma), in the "upper triangle case" 1 less than or equal to q < p for integral operators T-k with positive kernels, where dsigma and dmu 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 E-k,E-sigma [mu] = f(R)(n)(T-k[mu])p' dsigma through the L-1(dmu)-norm of an appropriate nonlinear potential Wk,sigma[mu] associated with the kernel k and measures dmu, dsigma. We initially work with a dyadic integral operator with kernel K-D (X, Y) = (Qis an element ofD)Sigma K(Q)X-Q (x) X-Q (y), where D = {Q} is the family of all dyadic cubes in Rn, and K : D --> R+. The corresponding continuous versions of Wolff's inequality and trace inequalities are derived from their dyadic counterparts.