Remote Access Proceedings of the American Mathematical Society
Green Open Access

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
DOI: https://doi.org/10.1090/S0002-9939-1975-0369641-X
MathSciNet review: 0369641
Full-text PDF

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?)

  • [1] R. R. Coifman and C. Fefferman, Weighted norm inequalities for maximal functions and singular integrals, Studia Math. 51 (1974), 241-250. MR 0358205 (50:10670)
  • [2] L. I. Hedberg, On certain convolution inequalities, Proc. Amer. Math. Soc. 36 (1972), 505-510. MR 47 #794. MR 0312232 (47:794)
  • [3] B. Muckenhoupt and R. L. Wheeden, Weighted norm inequalities for fractional integrals, Trans. Amer. Math. Soc. 192 (1974), 261-274. MR 0340523 (49:5275)
  • [4] A. Zygmund, Trigonometric series. Vols. I, II, 2nd ed., Cambridge Univ. Press, New York, 1959. MR 21 #6498. MR 0236587 (38:4882)
  • [5] -, A note on the differentiability of integrals, Colloq. Math. 16 (1967), 199-204. MR 35 #1732. MR 0210847 (35:1732)

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

DOI: https://doi.org/10.1090/S0002-9939-1975-0369641-X
Article copyright: © Copyright 1975 American Mathematical Society

American Mathematical Society