Two weight norm inequalities for fractional one-sided maximal operators

Authors:
F. J. Martín-Reyes and A. de la Torre

Journal:
Proc. Amer. Math. Soc. **117** (1993), 483-489

MSC:
Primary 42B25

DOI:
https://doi.org/10.1090/S0002-9939-1993-1110548-9

MathSciNet review:
1110548

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Abstract: In this paper we introduce a new maximal function, the dyadic one-sided maximal function. We prove that this maximal function is equivalent to the one-sided maximal function studied by the authors and Ortega in *Weighted inequalities for one-sided maximal functions* (Trans. Amer. Math. Soc. **319** (1990)) and by Sawyer in *Weighted inequalities for the one-sided Hardy-Littlewood maximal functions* (Trans. Amer. Math. Soc. **297** (1986)), but our function, being dyadic, is much easier to deal with, and it allows us to study fractional maximal operators. In this way we obtain a geometric proof of the characterization of the good weights for fractional maximal operators, answering a question raised by Andersen and Sawyer in *Weighted norm inequalities for the Riemann-Liouville and Weyl fractional integral operators* (Trans. Amer. Math. Soc. **308** (1988)). Our methods, avoiding complex interpolation, give also the case of different weights for the fractional maximal operator, which was an open problem.

**[1]**K. F. Andersen and E. T. Sawyer,*Weighted norm inequalities for the Riemann-Liouville and Weyl fractional integral operators*, Trans. Amer. Math. Soc.**308**(1988), 547-557. MR**930071 (89h:26006)****[2]**F. J. Martin-Reyes, P. Ortega, and A. de la Torre,*Weighted inequalities for one-sided maximal functions*, Trans. Amer. Math. Soc.**319**(1990), 517-534. MR**986694 (90j:42047)****[3]**E. T. Sawyer,*A characterization of a two-weighted norm inequality for maximal operators*, Studia Math.**75**(1982), 1-11. MR**676801 (84i:42032)****[4]**-,*Weighted inequalities for the one-sided Hardy-Littlewood maximal functions*, Trans. Amer. Math. Soc.**297**(1986), 53-61. MR**849466 (87k:42018)**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1993-1110548-9

Keywords:
One-sided fractional maximal functions,
weighted inequalities

Article copyright:
© Copyright 1993
American Mathematical Society