Nonlinear equations and weighted norm inequalities

We study connections between the problem of the existence of positive solut ions for certain nonlinear equations and weighted norm inequalities. In par ticular, we obtain explicit criteria for the solvability of the Dirichlet p roblem [GRAPHICS] on a regular domain Omega in R-n in the "superlinear case" q >1. The coeffi cients v, w are arbitrary positive measurable functions (or measures) on Om ega. We also consider more general nonlinear differential and integral equa tions, and study the spaces of coefficients and solutions naturally associa ted with these problems, as well as the corresponding capacities. Our characterizations of the existence of positive solutions take into acco unt the interplay between v, w, and the corresponding Green's kernel. They are not only sufficient, but also necessary, and are established without an y a priori regularity assumptions on v and w; we also obtain sharp two-side d estimates of solutions up to the boundary. Some of our results are new ev en if v = 1 and Omega is a ball or half-space. The corresponding weighted norm inequalities are proved for integral operat ors with kernels satisfying a refined version of the so-called 3G-inequalit y by an elementary "integration by parts" argument. This also gives a new u nified proof for some classical inequalities including the Carleson measure theorem for Poisson integrals and trace inequalities for Riesz potentials and Green potentials.