Sobolev-BMO and fractional integrals on super-critical ranges of Lebesgue spaces

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• 作者：Lucas Chaffee^a ; ^{lucas.chaffee@wwu.edu" class="auth_mail" title="E-mail the corresponding author} ; Jarod Hart^b ; ¹ ; ^{jvhart@ku.edu" class="auth_mail" title="E-mail the corresponding author} ; Lucas Oliveira^c ; ^{lucas.oliveira@ufrgs.br" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Fractional integral operator ; Sobolev-BMO ; Sobolev inequality ; Bilinear operator
• 刊名：Journal of Functional Analysis
• 出版年：2017
• 出版时间：15 January 2017
• 年：2017
• 卷：272
• 期：2
• 页码：631-660
• 全文大小：507 K

文摘

In this article, we explore the mapping and boundedness properties of linear and bilinear fractional integral operators acting on Lebesgue spaces with large indices. The prototype ν  -order fractional integral operator is the Riesz potential I_ν$I_{\nu }$, and the standard estimates for I_ν$I_{\nu }$ are from L^p$L^p$ into L^q$L^q$ when $1<p<\frac{n}{\nu }$ and $\frac{1}{p}=\frac{1}{q}+\frac{\nu }{n}$. We show that a ν  -order linear fractional integral operator can be continuously extended to a bounded operator from L^p$L^p$ into the Sobolev-BMO   space I_s(BMO)$I_s(BMO)$ when $\frac{n}{\nu }\le p<\infty$ and 0≤s<ν$0\le s<\nu$ satisfy $\frac{1}{p}=\frac{\nu -s}{n}$. Likewise, we prove estimates for ν  -order bilinear fractional integral operators from L^p₁×L^p₂$L^{p_1}\times L^{p_2}$ into I_s(BMO)$I_s(BMO)$ for various ranges of the indices p₁$p_1$, p₂$p_2$, and s   satisfying $\frac{1}{p_1}+\frac{1}{p_2}=\frac{\nu -s}{n}$.