Differential and Integral Equations

Weighted Sobolev inequalities for the Riesz potentials and transforms

Chérif Amrouche

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Abstract

Thanks to isomorphism results for Laplace's operator in weighted Sobolev spaces, we derive some applications. Various related spaces, intersection of Hardy spaces and weighted Sobolev spaces, are also considered. We show that the Riesz potentials spaces, consisting of all Riesz potentials $I_\alpha$ of order $\alpha$ of $L^p$ functions, with $1<p<\infty$, are equivalent to some appropriate weighted Sobolev spaces. It is known that if $ 1< p\le q<\infty$ and if $w$ and $v^{1-p'}$ satisfy the reverse doubling condition, then $I_\alpha$ satisfy the following continuity property: $$ \Big (\int_{\mathbb R }|I_\alpha f(x)|^q w(x) dx\Big )^{1/q}\le C \Big (\int_{\mathbb R }|f(x)|^p v(x) dx\Big )^{1/p}, \tag*{$(\star)$} $$ if and only if the weights satisfy the $(A_{p,q})$ condition $$ \frac {1}{|Q|^{1-\alpha/ {\rm N}}} \Big (\int_Q w dx\Big )^{1/q}\Big (\int_Q v^{1-p'} dx\Big ) ^{1/p'}\le C \ \ \ \hbox{ for all cubes } Q\subset\mathbb R . \tag*{$(A_{p,q})$} $$ A similar property holds for the Riesz transforms $R_j$ with $j = 1,\ldots , {\scriptstyle N}$: $$ \int_{\mathbb R }|R_j f(x)|^p u(x) dx\le C \int_{\mathbb R }|f(x)|^p u(x) dx, \tag*{$(\star\star)$} $$ if and only if the weight $u$ satisfies the Muckenhoupt $A_p$ condition: $$ (\dfrac{1}{|Q|}\int_Q u dx)^{1/p}(\dfrac{1}{|Q|}\int_Q u^{1-p'} dx) ^{1/p'}\le C \ \ \ \hbox{ for all cubes } Q\subset\mathbb R . \tag*{$(A_p)$} $$ Here, $f$ belongs to a weighted Sobolev space $W$. We show that even if these conditions are not satisfied, the continuity property of $I_\alpha$ and $R_j$ still holds in appropriate subspaces of $W$. By interpolation, a weighted version of the Sobolev imbedding theorem is also proved.

Article information

Source
Differential Integral Equations, Volume 18, Number 6 (2005), 699-720.

Dates
First available in Project Euclid: 21 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.die/1356060177

Mathematical Reviews number (MathSciNet)
MR2136706

Zentralblatt MATH identifier
1212.42022

Subjects
Primary: 42B20: Singular and oscillatory integrals (Calderón-Zygmund, etc.)
Secondary: 26D15: Inequalities for sums, series and integrals 42B30: $H^p$-spaces 46E35: Sobolev spaces and other spaces of "smooth" functions, embedding theorems, trace theorems

Citation

Amrouche, Chérif. Weighted Sobolev inequalities for the Riesz potentials and transforms. Differential Integral Equations 18 (2005), no. 6, 699--720. https://projecteuclid.org/euclid.die/1356060177


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