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

Junjian Zhao,Wei-Shih Du,Yasong Chen

doi:10.3390/math9030227

Junjian Zhao, Wei-Shih Du + Show 1 more

Open Access

https://doi.org/10.3390/math9030227

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Abstract

In this paper, we establish new generalizations and results in shift-invariant subspaces of mixed-norm Lebesgue spaces Lp→(Rd). 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 Lp→(Rd). Our new results unify and refine the existing results in the literature.

Highlights

A mixed-norm Lebesgue space is a natural generalization of the classical Lebesgue space L p (Rd ), in which independent variables that may have different meanings are considered
By applying Lemma 5, we extend the traditional Minkowski inequality to the mixednorm case as follows
The main goal of the current study is to give some generalizations of inequalities under mixed-norm Lebesgue spaces L~p (Rd ) with the help of L~p (Rd )

Summary

Introduction

A mixed-norm Lebesgue space is a natural generalization of the classical Lebesgue space L p (Rd ), in which independent variables that may have different meanings are considered. Lebesgue space L~p (Rd ) has important results and properties, such as Hölder inequality, stability property, etc., for shift-invariant subspaces of L p (Rd ) which were established in [13]. The characterization study on the revelent mixed-norm space [7,14] is troublesome because of the noncommutative order of the integral. We will consider these properties for shift-invariant subspaces of mixed-norm Lebesgue spaces L~p (Rd ). Our research in this paper promotes the existing conclusions of shift-invariant subspaces, such as [6,13] and can be used in the study of characterization of a mixed norm space in the future. Our new results unify and refine the relevant existing results in the literature [6,13,15,16]

New Mixed-Norm Inequalities and Generalizations

New Convolution-Type Inequality and Stability Theorem in Shift-Invariant

Conclusions