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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
DOI: https://doi.org/10.1090/S0002-9947-1984-0719674-6
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 \[ \int _{\{ {T_\alpha }f > \lambda \} } w(x)\;dx \leqslant A{\lambda ^{ - q}}{\left ( \int |f(x){|^p}\;\upsilon (x)\;dx \right )^{q/p}}\] holds for all $f \geqslant 0$ if and only if \[ \int _Q [{T_\alpha }({\chi _Q}w) (x)]^{p’}\upsilon (x)^{1 - p’} dx \leqslant B\left ( \int _Qw \right )^{p’/q’} < \infty \] for all cubes $Q$ in ${R^n}$. Here ${T_\alpha }$ denotes the fractional integral of order $\alpha ,{T_\alpha }f(x) = \int |x - y{|^{\alpha - n}}f(y) dy$. More generally we can replace ${T_\alpha }$ by any suitable convolution operator with radial kernel decreasing in $|x|$.


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Article copyright: © Copyright 1984 American Mathematical Society