Fractional integral operator on spaces of homogeneous type
type: Item
instance of: scholarly article
title: Fractional integral operator on spaces of homogeneous type
zbMATH Open document ID: 1067.43005
author: Liya Jiang
author: Ming Xu
published in: Applied Mathematics. Series B (English Edition)
publication date: 2005-02-24T00:00:00Z
review text: A space of homogeneous type \((X,d,\mu)\) is a set \(X\) with a quasi-metric \(d\) and a positive measure \(\mu\) such that \(\mu(B(x,r))<\infty\) for all \(x\in X\) and all \(r>0\). Here \[ B(x,r)=\{y\in X:\, d(x,y)<r\}. \] The measure \(\mu\) satisfies the doubling property: there exists a universal constant \(A>0\) such that \[ \mu(B(x,2r))\leq A \mu(B(x,r)) \] for \(x\in X\) and all \(r>0\). For \(f\in L_{loc}(X)\), one defines the fractional integral operator as follows: \[ I_\alpha(f)(x)=\int_X \frac {1}{\rho(x,y)^{1-\alpha}}f(y)dy,\quad 0<\alpha<1, \] where \[ \rho(x,y)=\inf\{\mu(B):\, B\,\, {\text{is a ball containing \(x\) and \(y\)}}\}. \] In this paper, the authors prove that if \(\max \Big\{\frac{1}{1+\varepsilon}, \frac{1}{1+\varepsilon+\alpha} \Big\}<p,q<\infty\), then \[ \| I_\alpha(f)\| _{\dot F_p^{\alpha,q}} \leq C\| f\| _{\dot F_p^{0,q}}, \] where \(0<\alpha<\varepsilon\). Here \(\dot F_p^{\alpha,q}\) is a Triebel-Lizorkin type space.
Mathematics Subject Classification ID: 43A85
Mathematics Subject Classification ID: 42B20
Mathematics Subject Classification ID: 47G10
zbMATH DE Number: 2138471
zbMATH Keywords: fractional integral
zbMATH Keywords: space of homogeneous type
zbMATH Keywords: Triebel-Lizorkin space
reviewed by: Der-Chen Chang
MaRDI profile type: Publication
full work available at URL: s11766-004-0055-4
OpenAlex ID: W2087107350
cites work: Q2548398
cites work: Lipschitz functions on spaces of homogeneous type
cites work: On fractional differentiation and integration on spaces of homogeneous type
cites work: Embedding theorem on spaces of homogeneous type
cites work: Opérateurs de Calderón-Zygmund, fonctions para-accrétives et interpolation. (Calderón-Zygmund operators, para-accretive functions and interpolation)
cites work: A T(b) theorem with remarks on analytic capacity and the Cauchy integral
cites work: Discrete Calderón-type reproducing formula
cites work: Plancherel-Pôlya type inequality on spaces of homogeneous type and its applications
DOI: 10.1007/S11766-004-0055-4
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description: scientific article; zbMATH DE number 2138471