New Generalizations and Results in Shift-Invariant Subspaces of Mixed-Norm Lebesgue Spaces \({L_{\vec{p}}(\mathbb{R}^d)}\)

Abstract

In this paper, we establish new generalizations and results in shift-invariant subspaces of mixed-norm Lebesgue spaces $L_{\overrightarrow{p}}\left({\mathbb{R}}^d\right)$. We obtain a mixed-norm Hölder inequality, a mixed-norm Minkowski inequality, a mixed-norm convolution inequality, a convolution-Hölder type inequality and a stability theorem to mixed-norm case in the setting of shift-invariant subspace of $L_{\overrightarrow{p}}\left({\mathbb{R}}^d\right)$. Our new results unify and refine the existing results in the literature.