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 &#xA0;-order fractional integral operator is the Riesz potential 96&_mathId=si1.gif&_user=111111111&_pii=S0022123616303196&_rdoc=1&_issn=00221236&md5=8e07cf170075e093857e2fac110ee52e" title="Click to view the MathML source">I_ν$I_{\nu }$, and the standard estimates for 96&_mathId=si1.gif&_user=111111111&_pii=S0022123616303196&_rdoc=1&_issn=00221236&md5=8e07cf170075e093857e2fac110ee52e" title="Click to view the MathML source">I_ν$I_{\nu }$ are from 96&_mathId=si2.gif&_user=111111111&_pii=S0022123616303196&_rdoc=1&_issn=00221236&md5=75faae3c94ea2f4c169ab8b9d78be339" title="Click to view the MathML source">L^p$L^p$ into 96&_mathId=si3.gif&_user=111111111&_pii=S0022123616303196&_rdoc=1&_issn=00221236&md5=f3cdd959a191d1f5b61df2383d595453" title="Click to view the MathML source">L^q$L^q$ when 96&_mathId=si4.gif&_user=111111111&_pii=S0022123616303196&_rdoc=1&_issn=00221236&md5=e9e6c000b85fd9bd767a409f221053a0">96-si4.gif">$1<p<\frac{n}{\nu }$ and 96&_mathId=si32.gif&_user=111111111&_pii=S0022123616303196&_rdoc=1&_issn=00221236&md5=5965d40a472d112f022525becde64ccb">96-si32.gif">$\frac{1}{p}=\frac{1}{q}+\frac{\nu }{n}$. We show that a ν  -order linear fractional integral operator can be continuously extended to a bounded operator from 96&_mathId=si2.gif&_user=111111111&_pii=S0022123616303196&_rdoc=1&_issn=00221236&md5=75faae3c94ea2f4c169ab8b9d78be339" title="Click to view the MathML source">L^p$L^p$ into the Sobolev-BMO   space 96&_mathId=si54.gif&_user=111111111&_pii=S0022123616303196&_rdoc=1&_issn=00221236&md5=b95ab6851a303e49a0aa16b7b09c57ed" title="Click to view the MathML source">I_s(BMO)$I_s(BMO)$ when 96&_mathId=si7.gif&_user=111111111&_pii=S0022123616303196&_rdoc=1&_issn=00221236&md5=f8d572a69a02d03fe733edcb49a579b0">96-si7.gif">$\frac{n}{\nu }\le p<\infty$ and 96&_mathId=si8.gif&_user=111111111&_pii=S0022123616303196&_rdoc=1&_issn=00221236&md5=f312c8b3339a849ec7e3d09273afcf06" title="Click to view the MathML source">0≤s<ν$0\le s<\nu$ satisfy 96&_mathId=si58.gif&_user=111111111&_pii=S0022123616303196&_rdoc=1&_issn=00221236&md5=fc8182b771c8cf17c2eaa6320d3645a2">96-si58.gif">$\frac{1}{p}=\frac{\nu -s}{n}$. Likewise, we prove estimates for ν  -order bilinear fractional integral operators from 96&_mathId=si10.gif&_user=111111111&_pii=S0022123616303196&_rdoc=1&_issn=00221236&md5=bf874ff9081478b59b6451ec436a2ab6" title="Click to view the MathML source">L^p₁×L^p₂$L^{p_1}\times L^{p_2}$ into 96&_mathId=si54.gif&_user=111111111&_pii=S0022123616303196&_rdoc=1&_issn=00221236&md5=b95ab6851a303e49a0aa16b7b09c57ed" title="Click to view the MathML source">I_s(BMO)$I_s(BMO)$ for various ranges of the indices 96&_mathId=si11.gif&_user=111111111&_pii=S0022123616303196&_rdoc=1&_issn=00221236&md5=002fae8aa964dabde87c643f0155f61d" title="Click to view the MathML source">p₁$p_1$, 96&_mathId=si12.gif&_user=111111111&_pii=S0022123616303196&_rdoc=1&_issn=00221236&md5=c530fad0e79aa63a6786fd8d9bad09b6" title="Click to view the MathML source">p₂$p_2$, and s   satisfying 96&_mathId=si103.gif&_user=111111111&_pii=S0022123616303196&_rdoc=1&_issn=00221236&md5=4693c118743e88503574a05eb50fa113">96-si103.gif">$\frac{1}{p_1}+\frac{1}{p_2}=\frac{\nu -s}{n}$.