Weighted $L^q$-Helmholtz decompositions in infinite cylinders and in infinite layers

Abstract

The aim of this paper is to present a new joint approach to the Helmholtz decomposition in infinite cylinders and in infinite layers $\Omega=\Sigma {\times} {\mathbb R}^k$ in the function space $L^q({\mathbb R}^k;L^r({\Sigma}))$ using even arbitrary Muckenhoupt weights with respect to $x'\in \Sigma\subset {\mathbb R}^{n-k}$ and, if possible, exponential weights with respect to $x''\in {\mathbb R}^k, \;1\leq k\leq n-1,\;n\geq 2.$ For $n=2$, we get the Helmholtz decomposition for a strip, for $n=3$ in an infinite cylinder or an infinite layer and for $n>3$ in some (non--physical) unbounded domains of cylinder or layer type. The proof based on a weak Neumann problem uses a partial Fourier transform and operator--valued multiplier functions, the ${\mathcal R}$--boundedness of the family of multiplier operators and an extrapolation property in weighted $L^q$--spaces.