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 class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022123616303196&_mathId=si1.gif&_user=111111111&_pii=S0022123616303196&_rdoc=1&_issn=00221236&md5=8e07cf170075e093857e2fac110ee52e" title="Click to view the MathML source">I_νclass="mathContainer hidden">class="mathCode">$I_{\nu }$, and the standard estimates for class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022123616303196&_mathId=si1.gif&_user=111111111&_pii=S0022123616303196&_rdoc=1&_issn=00221236&md5=8e07cf170075e093857e2fac110ee52e" title="Click to view the MathML source">I_νclass="mathContainer hidden">class="mathCode">$I_{\nu }$ are from class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022123616303196&_mathId=si2.gif&_user=111111111&_pii=S0022123616303196&_rdoc=1&_issn=00221236&md5=75faae3c94ea2f4c169ab8b9d78be339" title="Click to view the MathML source">L^pclass="mathContainer hidden">class="mathCode">$L^p$ into class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022123616303196&_mathId=si3.gif&_user=111111111&_pii=S0022123616303196&_rdoc=1&_issn=00221236&md5=f3cdd959a191d1f5b61df2383d595453" title="Click to view the MathML source">L^qclass="mathContainer hidden">class="mathCode">$L^q$ when class="mathmlsrc">itle="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022123616303196&_mathId=si4.gif&_user=111111111&_pii=S0022123616303196&_rdoc=1&_issn=00221236&md5=e9e6c000b85fd9bd767a409f221053a0">class="imgLazyJSB inlineImage" height="18" width="71" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022123616303196-si4.gif">class="mathContainer hidden">class="mathCode">$1<p<\frac{n}{\nu }$ and class="mathmlsrc">itle="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022123616303196&_mathId=si32.gif&_user=111111111&_pii=S0022123616303196&_rdoc=1&_issn=00221236&md5=5965d40a472d112f022525becde64ccb">class="imgLazyJSB inlineImage" height="22" width="73" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022123616303196-si32.gif">class="mathContainer hidden">class="mathCode">$\frac{1}{p}=\frac{1}{q}+\frac{\nu }{n}$. We show that a ν  -order linear fractional integral operator can be continuously extended to a bounded operator from class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022123616303196&_mathId=si2.gif&_user=111111111&_pii=S0022123616303196&_rdoc=1&_issn=00221236&md5=75faae3c94ea2f4c169ab8b9d78be339" title="Click to view the MathML source">L^pclass="mathContainer hidden">class="mathCode">$L^p$ into the Sobolev-BMO   space class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022123616303196&_mathId=si54.gif&_user=111111111&_pii=S0022123616303196&_rdoc=1&_issn=00221236&md5=b95ab6851a303e49a0aa16b7b09c57ed" title="Click to view the MathML source">I_s(BMO)class="mathContainer hidden">class="mathCode">$I_s(BMO)$ when class="mathmlsrc">itle="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022123616303196&_mathId=si7.gif&_user=111111111&_pii=S0022123616303196&_rdoc=1&_issn=00221236&md5=f8d572a69a02d03fe733edcb49a579b0">class="imgLazyJSB inlineImage" height="17" width="80" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022123616303196-si7.gif">class="mathContainer hidden">class="mathCode">$\frac{n}{\nu }\le p<\infty$ and class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022123616303196&_mathId=si8.gif&_user=111111111&_pii=S0022123616303196&_rdoc=1&_issn=00221236&md5=f312c8b3339a849ec7e3d09273afcf06" title="Click to view the MathML source">0≤s<νclass="mathContainer hidden">class="mathCode">$0\le s<\nu$ satisfy class="mathmlsrc">itle="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022123616303196&_mathId=si58.gif&_user=111111111&_pii=S0022123616303196&_rdoc=1&_issn=00221236&md5=fc8182b771c8cf17c2eaa6320d3645a2">class="imgLazyJSB inlineImage" height="22" width="59" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022123616303196-si58.gif">class="mathContainer hidden">class="mathCode">$\frac{1}{p}=\frac{\nu -s}{n}$. Likewise, we prove estimates for ν  -order bilinear fractional integral operators from class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022123616303196&_mathId=si10.gif&_user=111111111&_pii=S0022123616303196&_rdoc=1&_issn=00221236&md5=bf874ff9081478b59b6451ec436a2ab6" title="Click to view the MathML source">L^p₁×L^p₂class="mathContainer hidden">class="mathCode">$L^{p_1}\times L^{p_2}$ into class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022123616303196&_mathId=si54.gif&_user=111111111&_pii=S0022123616303196&_rdoc=1&_issn=00221236&md5=b95ab6851a303e49a0aa16b7b09c57ed" title="Click to view the MathML source">I_s(BMO)class="mathContainer hidden">class="mathCode">$I_s(BMO)$ for various ranges of the indices class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022123616303196&_mathId=si11.gif&_user=111111111&_pii=S0022123616303196&_rdoc=1&_issn=00221236&md5=002fae8aa964dabde87c643f0155f61d" title="Click to view the MathML source">p₁class="mathContainer hidden">class="mathCode">$p_1$, class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022123616303196&_mathId=si12.gif&_user=111111111&_pii=S0022123616303196&_rdoc=1&_issn=00221236&md5=c530fad0e79aa63a6786fd8d9bad09b6" title="Click to view the MathML source">p₂class="mathContainer hidden">class="mathCode">$p_2$, and s   satisfying class="mathmlsrc">itle="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022123616303196&_mathId=si103.gif&_user=111111111&_pii=S0022123616303196&_rdoc=1&_issn=00221236&md5=4693c118743e88503574a05eb50fa113">class="imgLazyJSB inlineImage" height="22" width="102" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022123616303196-si103.gif">class="mathContainer hidden">class="mathCode">$\frac{1}{p_1}+\frac{1}{p_2}=\frac{\nu -s}{n}$.