Hierarchical Construction of Bounded Solutions in Critical Regularity Spaces

Abstract

We construct uniformly bounded solutions for the equations div U = f and U = f in the critical cases and , respectively. Criticality in this context manifests itself by the lack of a linear solution operator mapping . Thus, the intriguing aspect here is that although the problems are linear, construction of their solutions is not. Our constructions are special cases of a general framework for solving linear equations of the form , where is a linear operator densely defined in Banach space with a closed range in a (proper subspace) of Lebesgue space , and with an injective dual . The solutions are realized in terms of a multiscale hierarchical representation, , interesting for its own sake. Here, uj's are constructed recursively as minimizers of urn:x-wiley::media:cpa21575:cpa21575-math-0010 where the residuals are resolved in terms of a dyadic sequence of scales with large enough . The nonlinear aspect of this construction is a counterpart of the fact that one cannot linearly solve in critical regularity spaces.© 2016 Wiley Periodicals, Inc.