On the Helmholtz decomposition in Morrey and block spaces

TL;DR

This paper extends the Helmholtz decomposition to vector fields in Morrey, Zorko, and block spaces on bounded or exterior domains, using advanced analysis tools and PDE techniques.

Contribution

It introduces new methods to establish Helmholtz decomposition in these complex function spaces, extending classical analysis tools to novel contexts.

Findings

Helmholtz decomposition established in Morrey, Zorko, and block spaces

Extension of classical analysis tools like Stein extensions and Poincaré inequalities

Potential applications to PDEs in these function spaces

Abstract

In this work, we obtain the Helmholtz decomposition for vector fields in Morrey, Zorko, and block spaces over bounded or exterior $C^{1}$ domains. Generally speaking, our proofs rely on a careful interplay of localization, flattening, and duality arguments. To accomplish this, we need to extend some classical tools in analysis and PDE theory to those spaces, including Stein extensions, compact embeddings, Poincar\'{e} inequalities, Bogovskii-type theorem, among other ingredients. Some of these findings may be of independent interest and applied to the study of a number of PDEs.