A two weight inequality for the fractional integral when $p=n/\alpha$
Authors:
Eleonor Harboure, Roberto A. Macías and Carlos Segovia
Journal:
Proc. Amer. Math. Soc. 90 (1984), 555562
MSC:
Primary 26D10
DOI:
https://doi.org/10.1090/S00029939198407334050
MathSciNet review:
733405
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Abstract: Let ${I_\alpha }$ be the fractional integral operator defined as \[ {I_\alpha }f(x) = \int {f(y){{\left  {x  y} \right }^{\alpha  n}}dy.} \] Given a weight $w$ (resp. $\upsilon$), necessary and sufficient conditions are given for the existence of a nontrivial weight $\upsilon$ (resp. $w$) such that \[ {\left \ {\upsilon {\chi _B}} \right \_\infty }\frac {1}{{\left  B \right }}\int _B {\left  {{I_\alpha }f(x)  {m_B}({I_\alpha }f)} \right } dx \leqslant C{\left ( {\int {{{\left  f \right }^{n/\alpha }}w} } \right )^{\alpha /n}}\] holds for any ball $B$ such that ${\left \ {v{\chi _B}} \right \_\infty } > 0$.

E. Harboure, R. A. Macías and C. Segovia, Boundedness of fractional operators on ${L^p}$ spaces with different weights, preprint.
 Benjamin Muckenhoupt and Richard Wheeden, Weighted norm inequalities for fractional integrals, Trans. Amer. Math. Soc. 192 (1974), 261–274. MR 340523, DOI https://doi.org/10.1090/S00029947197403405236
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© Copyright 1984
American Mathematical Society