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 (dμ) ≤ C∥f∥ L p (dσ) , f ∈ L p (dσ), in the upper triangle case 1 ≤ q < p for integral operators T k with positive kernels, where dσ and dp 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,σ [μ] = R n(T k [μ]) p' dσ through the L 1 (dμ)-norm of an appropriate nonlinear potential W k,σ [μ] associated with the kernel k and measures dp, dσ. We initially work with a dyadic integral operator with kernel K D (x,y) = Σ Q ∈ D K(Q)X Q (x)X Q (y), where D = {Q} is the family of all dyadic cubes in R n , and K: D → R + . The corresponding continuous versions of Wolff's inequality and trace inequalities are derived from their dyadic counterparts.