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

 

A two weight weak type inequality for fractional integrals


Author: Eric Sawyer
Journal: Trans. Amer. Math. Soc. 281 (1984), 339-345
MSC: Primary 26A33; Secondary 26D10, 26D15, 42B25
MathSciNet review: 719674
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Abstract: For $ 1 < p \leqslant q < \infty ,0 < \alpha < n$ and $ w(x),\upsilon (x)$ nonnegative weight functions on $ {R^n}$ we show that the weak type inequality

$\displaystyle \int_{\{ {T_\alpha }f > \lambda \} }\,w(x)\;dx \leqslant A{\lambda ^{ - q}}{\left( \int \vert f(x){\vert^p}\;\upsilon (x)\;dx \right)^{q/p}}$

holds for all $ f \geqslant 0$ if and only if

$\displaystyle \int_Q\,[{T_\alpha }({\chi_Q}w)\,(x)]^{p'}\upsilon (x)^{1 - p'}\,dx \leqslant B\left( \int_Qw \right)^{p^{\prime}/q^{\prime}} < \infty $

for all cubes $ Q$ in $ {R^n}$. Here $ {T_\alpha }$ denotes the fractional integral of order $ \alpha ,{T_\alpha }f(x) = \int \vert x - y{\vert^{\alpha - n}}f(y)\,dy$. More generally we can replace $ {T_\alpha }$ by any suitable convolution operator with radial kernel decreasing in $ \vert x\vert$.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1984-0719674-6
PII: S 0002-9947(1984)0719674-6
Article copyright: © Copyright 1984 American Mathematical Society



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