Integral representations, differentiability properties and limits at infinity for Beppo Levi functions
type: Item
instance of: scholarly article
title: Integral representations, differentiability properties and limits at infinity for Beppo Levi functions
zbMATH Open document ID: 0885.31004
DOI: 10.1023/A:1017996900877
author: Yoshihiro Mizuta
published in: Potential Analysis
publication date: 1998-04-02T00:00:00Z
review text: For an open subset \(G\) in \(\mathbb{R}^n\) and \(1\leq p <\infty\), the set of Beppo Levi functions on \(G\), denoted \(BL_m(L^p_{\text{loc}}(G))\), is the set of functions \(u\in L^p_{\text{loc}}(G)\) for which \(|\nabla_m u(x)|= (\sum_{|\mu|=m}|D^{\mu}u(x)|^2)^{1/2}\) is in \(L^p_{\text{loc}}(G)\), where \(D^{\mu} u\) is the distributional derivative of \(u\) of order \(\mu\). In Theorem 2.1 it is proved that if \(u\in BL_m(L^p_{\text{loc}}(\mathbb{R}^n))\) satisfies \(\int_{\mathbb{R}^n}|\nabla_m u(y)|(1+|y|)^{m-n-l-1}dy <\infty\) for some integer \(l\geq -1\), then there exists a polynomial \(P\) such that \[ u(x) = \sum_{|\mu|=m}a_{\mu}\int K_{\mu,l}(x,y)D^{\mu}u(y)dy + P(x) \] a.e. on \(\mathbb{R}^n\), where the \(K_{\mu,l}\) are modified kernel functions derived from \(K_{\mu}(x) = x^{\mu}\big/|x|^n\). As applications of the above integral representation, the author studies continuity and differentiability properties of Beppo Levi functions \(u\) satisfying \(\int \Psi_p(|\nabla_mu(x)|)dx<\infty \), where \(\Psi_p(r) = r^p\psi(r)\), with \(\psi\) a positive non-decreasing function on \((0,\infty)\). In the second part of the paper the author studies the existence of limits at infinity for Beppo Levi functions. The author also considers the existence of fine-type limits at infinity with respect to Bessel capacities, which yields the radial limit result at infinity.
reviewed by: Manfred Stoll
Mathematics Subject Classification ID: 31B10
Mathematics Subject Classification ID: 46E35
Mathematics Subject Classification ID: 31B15
zbMATH DE Number: 1045346
zbMATH Keywords: Beppo Levi function
zbMATH Keywords: integral representation
zbMATH Keywords: differentiability properties
zbMATH Keywords: limits at infinity
zbMATH Keywords: continuity
MaRDI profile type: Publication
full work available at URL: a:1017996900877
OpenAlex ID: W231950206
description: scientific article; zbMATH DE number 1045346