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A two weight weak type inequality for fractional integrals. (English) Zbl 0539.42008
If \(1<p\leq q<\infty\), \(\omega\) and \(\mu\) are positive Borel measures on \(R^ n\), and if \(T(f\mu)(x):=\int_{R^ n}K(x-y)f(y)d\mu(y),\) where the kernel K(x) is a positive lower semicontinuous radial function decreasing in \(| x|\) and satisfying an additional condition if \(n\geq 2\), the author gives a necessary and sufficient condition in order that \(\int_{\{T(f\mu)>\lambda \}}d\omega \leq A \lambda^{- q}(\int_{R^ n}| f|^ pd\mu)^{q/p}\) for all \(f\geq 0\), \(\lambda>0\). In the special case \(K(x)=| x|^{\alpha -n} (0<\alpha<n)\), \(d\omega(x)=w(x)dx\), \(d\mu(x)=v(x)^{1-p'}dx\) (p’,q’ are the conjugate indices to p,q, and w,v are non-negative weight functions on \(R^ n)\), and replacing f by \(fv^{p'-1}\), \(T(f\mu)\) becomes the fractional integral operator \(T_{\alpha}\), where \(T_{\alpha}f(x):=\int_{R^ n}| x-y|^{\alpha -n}f(y)dy.\)
The theorem then shows that in order that \(\int_{\{T_{\alpha}f>\lambda \}}w(x)dx\leq A\lambda^{- q}(\int_{R^ n}| f(x)|^ pv(x)dx)^{q/p}\) for all \(f\geq 0\), \(\lambda>0\), it is necessary and sufficient that, for all cubes \(Q\subset R^ n\), \[ \int_{Q}[T_{\alpha}(\chi_ Qw)(x)]^{p'}v(x)^{1- p'}dx\leq B(\int_{Q}w(x)dx)^{p'/q'}<\infty. \]
Reviewer: D.C.Russell

42B25 Maximal functions, Littlewood-Paley theory
Full Text: DOI
[1] David R. Adams, On the existence of capacitary strong type estimates in \?\(^{n}\), Ark. Mat. 14 (1976), no. 1, 125 – 140. · Zbl 0325.31008
[2] R. R. Coifman, Distribution function inequalities for singular integrals, Proc. Nat. Acad. Sci. U.S.A. 69 (1972), 2838 – 2839. · Zbl 0243.44006
[3] Björn E. J. Dahlberg, Regularity properties of Riesz potentials, Indiana Univ. Math. J. 28 (1979), no. 2, 257 – 268. · Zbl 0413.31003
[4] Charles Fefferman, Inequalities for strongly singular convolution operators, Acta Math. 124 (1970), 9 – 36. · Zbl 0188.42601
[5] Kurt Hansson, Imbedding theorems of Sobolev type in potential theory, Math. Scand. 45 (1979), no. 1, 77 – 102. · Zbl 0437.31009
[6] R. Kerman and E. Sawyer, Weighted norm inequalities of trace-type for potential operators, preprint. · Zbl 0673.47030
[7] V. G. Maz’ya, On some integral inequalities for functions of several variables, Problems in Math. Analysis, No. 3, Leningrad Univ. (Russian) · Zbl 0284.26011
[8] Benjamin Muckenhoupt and Richard Wheeden, Weighted norm inequalities for fractional integrals, Trans. Amer. Math. Soc. 192 (1974), 261 – 274. · Zbl 0289.26010
[9] Eric Sawyer, Multipliers of Besov and power-weighted \?&sup2; spaces, Indiana Univ. Math. J. 33 (1984), no. 3, 353 – 366. · Zbl 0546.42011
[10] Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. · Zbl 0207.13501
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