The Helmholtz decomposition in weighted $L^p$ spaces in cones

Abstract

In this paper we prove the Helmholtz decomposition in conical domains of $\mathbb{R}^3$ in weighted $L^p_\beta$ spaces under a spectral condition on $\beta,p$. The basic elements are the transformation of the original problem into a problem set in cylindrical domains and the combination of a priori bounds from [5] with the vector-valued multiplier theorem [29].