On boundedness and compactness of a certain class of kernel operators
type: Item
instance of: scholarly article
title: On boundedness and compactness of a certain class of kernel operators
zbMATH Open document ID: 1250.47032
DOI: 10.1155/2011/531683
author: E. P. Ushakova
published in: Journal of Function Spaces and Applications
publication date: 2011-11-18T00:00:00Z
review text: \noindent The author considers the integral operator \[ Kf(x) = w(x) \int_{a(x)}^{b(x)} k(x,y) f(y) v(y) \, dy \] \noindent on \(\mathbb R^+ = [0, \infty)\), where \(a(x), b(x)\) are differentiable and strictly increasing in \((0, \infty)\), \(a(0) = b(0) = 0\), and \(a(x) < b(x)\) for \(0 < x < \infty\), \(a(\infty) = b(\infty) = \infty\). The kernel \(k > 0\) on \(\{(x,y)\mid x>0\), \( a(x) < y < b(x) \}\), and satisfies Oinarov's conditions \(\mathcal{O}_a\) and/or \(\mathcal{O}_b\), where \(k \in \mathcal{O}_b\) if there exists a constant \(D \geq 1\), independent of \(x, y, z\), such that \[ \frac1D k(x,y) \leq k(x, b(z)) + k(z,y) \leq D k(x,y),\;z \leq x, \;a(x) \leq y \leq b(z), \] \noindent and where \(k \in \mathcal{O}_a\) if there exists a constant \(D \geq 1\), independent of \(x, y, z\), such that \[ \frac1D k(x,y) \leq k(x, a(z)) + k(z,y) \leq D k(x,y),\;z \geq x,\;a(z) \leq y \leq b(x). \] \noindent These operators have been studied and sufficient conditions that \(K: L_p \mapsto L_q\) have been given in a paper of Stepanov and Ushakova, which involve a double supremum of the kernel, e.g., one needs the finiteness of terms like \[ A_{b,0} = \sup_{t > 0} A_{b,0}(t) \] \[ = \sup_{t > 0} \sup_{b^{-1}(a(t)) \leq s \leq t} \left( \int_s^t k^q(x, b(x))w^q(x) \, dx \right)^{1/q} \left( \int_{a(t)}^{b(s)} v^{p^{\prime}}(y)\, dy \right)^{1/p^{\prime}} , \] \noindent and other terms like it to show that \(K\) is bounded above and below in terms of the given constants, with other conditions on the constants implying that \(K\) is compact. \noindent The author gives a criterion that only involves one supremum, but requires the introduction of new functions related to \(a, b\) called fairway functions that express boundedness of \(K\) satisfying one or both of Oinarov's conditions. The conditions are as long as the above sample; we refer the interested reader to the paper for details. \noindent The author gives examples of several cases to which the result applies,…
Mathematics Subject Classification ID: 47B34
Mathematics Subject Classification ID: 47B38
Mathematics Subject Classification ID: 26D15
Mathematics Subject Classification ID: 47G10
zbMATH DE Number: 5974888
zbMATH Keywords: integral operators
zbMATH Keywords: Lebesgue spaces
zbMATH Keywords: weights
zbMATH Keywords: boundedness
zbMATH Keywords: compactness
reviewed by: Raymond Johnson
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1155/2011/531683
OpenAlex ID: W2088786660
description: scientific article; zbMATH DE number 5974888