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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

   

 

New weighted estimates for bilinear fractional integral operators


Author: Kabe Moen
Journal: Trans. Amer. Math. Soc. 366 (2014), 627-646
MSC (2010): Primary 42B20, 26A33
Published electronically: July 26, 2013
MathSciNet review: 3130311
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Abstract: We prove a plethora of weighted estimates for bilinear fractional integral operators of the form

$\displaystyle BI_\alpha (f,g)(x)=\int _{\mathbb{R}^n}\frac {f(x-t)g(x+t)}{\vert t\vert^{n-\alpha }}\,dt, \qquad 0<\alpha <n.$

When the target space has an exponent greater than one, many weighted estimates follow trivially from Hölder's inequality and the known linear theory. We address the case where the target Lebesgue space is at most one and prove several interesting one and two weight estimates. As an application we formulate a bilinear version of the Stein-Weiss inequality for fractional integrals.

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Additional Information

Kabe Moen
Affiliation: Department of Mathematics, University of Alabama, Tuscaloosa, Alabama 35487-0350

DOI: http://dx.doi.org/10.1090/S0002-9947-2013-06067-9
Keywords: Bilinear operators, fractional integration, weighted inequalities
Received by editor(s): November 7, 2011
Published electronically: July 26, 2013
Additional Notes: The author was partially supported by NSF Grant 1201504
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.