Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS

Remote Access
Green Open Access
Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)


Weighted norm inequalities for fractional integrals

Author: G. V. Welland
Journal: Proc. Amer. Math. Soc. 51 (1975), 143-148
MSC: Primary 26A86; Secondary 26A33
MathSciNet review: 0369641
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: A simpler proof of an inequality of Muckenhoupt and Wheeden is given. Let $ {T_\alpha }f(x) = \smallint f(y)\vert x - y{\vert^{\alpha - d}}dy$ be given for functions defined in $ {{\mathbf{R}}^d}$. Let $ \upsilon $ be a weight function which satisfies

$\displaystyle (\vert Q{\vert^{ - 1}}\int_Q {{{[\upsilon (x)]}^q}dx{)^{1/q}}(\vert Q{\vert^{ - 1}}\int_Q {{{[\upsilon (x)]}^{ - p'}}dx{)^{1/p'}} \leq K} } $

for each cube, $ Q$, with sides parallel to a standard system of axes and $ \vert Q\vert$ is the measure of such a cube. Suppose $ 1/q = 1/p - \alpha /d$ and $ 0 < \alpha < d,1 < p < d/\alpha $. Then there exists a constant such that $ \vert\vert({T_\alpha }f)\upsilon \vert{\vert _q} \leq C\vert\vert f\upsilon \vert{\vert _p}$. Certain results for $ p = 1$ and $ q = \infty $ are also given.

References [Enhancements On Off] (What's this?)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 26A86, 26A33

Retrieve articles in all journals with MSC: 26A86, 26A33

Additional Information

PII: S 0002-9939(1975)0369641-X
Article copyright: © Copyright 1975 American Mathematical Society

Comments: Email Webmaster

© Copyright , American Mathematical Society
Contact Us · Sitemap · Privacy Statement

Connect with us Facebook Twitter Google+ LinkedIn Instagram RSS feeds Blogs YouTube Podcasts Wikipedia