Nonlinear Potential Analysis on Sobolev Multiplier Spaces

Abstract

We characterize preduals and Kothe duals to a class of Sobolev multiplier type spaces. Our results fit in well with the modern theory of function spaces of harmonic analysis and are also applicable to nonlinear partial differential equations. As a maneuver, we make use of several tools from nonlinear potential theory, weighted norm inequalities, and the theory of Banach function spaces to obtain our results. After characterizing the preduals, we establish a capacitary strong type inequality which resolves a special case of a conjecture by David R. Adams. As a consequence, we obtain several equivalent norms for Choquet integrals associated to Bessel or Riesz capacities. This enables us to obtain bounds for the Hardy-Littlewood maximal function on Choquet spaces associated to Bessel or Riesz capacities in a sublinear setting. Finally, we extend those maximal function bounds to full range of exponents, which allow us to deduce Sobolev type embeddings on certain Choquet spaces.