Equivalent norms on analytic subspaces of \(L^ p\)
type: Item
instance of: scholarly article
title: Equivalent norms on analytic subspaces of \(L^ p\)
zbMATH Open document ID: 0626.46017
DOI: 10.1016/0022-247X(87)90089-8
author: Zachariah Sinkala
author: John P. Nolan
published in: Journal of Mathematical Analysis and Applications
publication date: 1987-01-01T00:00:00Z
review text: Let M be a positive integer, let \({\mathbb{R}}^ M\) denote the Euclidean space consisting of M-tuples of real numbers, and let \({\mathbb{C}}^ M\) denote the space consisting of M-tuples of complex numbers. If \(x=(x_ 1,x_ 2,...,x_ M)\in {\mathbb{R}}^ M\), \(h>0\), then the h-cube centred at x is \(Q(x,h)=\{t=(t_ 1,t_ 2,...,t_ M):x_ j-h\leq t_ j\leq x_ j+h\), \(j=1,2,...,M\}.\) In the main result of this paper the authors show that if \(0<p<\infty\), \(\mu\) is a Borel measure in \({\mathbb{R}}^ M\), there is a positive constant \(\nu =\nu (p,M)\) and finite constants \(k_ 1,k_ 2\), such that \[ (*1)\quad k_ 2^{-1}| Q(x,h)| \leq \mu (Q(x,h))\leq k_ 2| Q(x,h)| \quad for\quad | x| >k_ 1, \] then there is a finite constant c such that \[ (*2)\quad c^{-1}\int_{{\mathbb{R}}^ M}| f(x)|^ pdx\leq \int_{{\mathbb{R}}^ M}| f(x)|^ pd\mu (x)\leq c\int_{{\mathbb{R}}^ M}| f(x)|^ pdx \] for f in \(E^ p(T,M)=\{F\in L^ p({\mathbb{R}}^ M):F\) has extension from \({\mathbb{R}}^ M\) to \({\mathbb{C}}^ M\) as a function of exponential type \(T\}\). Results involving \(H^ p\)-spaces are also derived.
reviewed by: George O. Okikiolu
Mathematics Subject Classification ID: 46E30
zbMATH DE Number: 4017617
zbMATH Keywords: analytic subspaces of \(L^ p\)
zbMATH Keywords: equivalent norms
zbMATH Keywords: \(L^ p\)-spaces
zbMATH Keywords: function of exponential type
zbMATH Keywords: \(H^ p\)-spaces
MaRDI profile type: Publication
cites work: Q5833361
cites work: EQUIVALENT NORMS IN SPACES OF ENTIRE FUNCTIONS
cites work: On equivalent norms in the space of square summable entire functions of exponential type
cites work: Stationary Gaussian Markov fields on \(R^d\) with a deterministic component
cites work: Fonctions entières et intégrales de Fourier multiples. II
description: scientific article; zbMATH DE number 4017617