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Sufficient conditions for one-sided operators. (English) Zbl 0984.42010
The author considers maximal one-sided operators on \(\mathbb R\) such as \[ M_{L^p}^+ f(x) := \sup_{r > 0} \Biggl(\int_0^1 |f(x+rt)|^p dt\Biggr)^{1/p} \] or more generally, for a Banach space \(X\) \[ M_X^+ f(x) := \sup_{r > 0} \|f(r\cdot) \chi_{[x, x+r]}(r \cdot) \|_X. \] The authors study the question of whether such operators are bounded from one weighted \(L^p\) space \(L^p(v^p dx)\) to another \(L^p(w^p dx)\). Their first result is that if \(M_X^+\) is bounded on \(L^p(dx)\) and \(v\), \(w\) obey a certain \(A_p\) weight condition (adapted to \(X\)), then \(M_X^+\) is also bounded on the weighted spaces. Some necessary conditions for \(M_X^+\) to be bounded on \(L^p\) are then given. Weighted estimates are then given for certain one-sided singular integrals (i.e., Calderón-Zygmund operators whose kernel is supported on the half-space \(\{ (x,y): x > y\}\)), in terms of the above maximal operators.

MSC:
42B25 Maximal functions, Littlewood-Paley theory
42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
26A33 Fractional derivatives and integrals
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[1] Andersen, K.F. and Sawyer, E. (1988). Weighted norm inequalities for Riemann-Liouville and Weyl fractional integral operators,Trans. Am. Math. Soc.,308, 547–557. · Zbl 0664.26002 · doi:10.1090/S0002-9947-1988-0930071-4
[2] Aimar, H., Forzani, L., and Martín-Reyes, F.J. (1997). On weighted inequalities for one-sided singular integrals,Proc. Am. Math. Soc.,125, 2057–2064. · Zbl 0868.42007 · doi:10.1090/S0002-9939-97-03787-8
[3] Bliendtner, J. and Loeb, P. (1992). A reduction technique for limit theorems in analysis and probability theory,Ark. Mat.,30, 25–43. · Zbl 0757.28006 · doi:10.1007/BF02384860
[4] Córdoba, A. and Fefferman, C. (1976). A weighted norm inequality for singular integrals,Studia Math.,57, 97–101. · Zbl 0356.44003
[5] García Cuerva, J. and Rudio de Francia, J.L. (1985). Weighted norm inequalities and related topics,North-Holland Math. Studies, vol. 116, North-Holland, Amsterdam. · Zbl 0578.46046
[6] Lorente, M. and de la Torre, A. (1996). Weighted inequalities for some one-sided operators,Proc. Am. Math. Soc.,124(3), 839–848. · Zbl 0895.26002 · doi:10.1090/S0002-9939-96-03089-4
[7] Martín-Reyes, F.J., Ortega, P., and de la Torre, A. (1990). Weighted inequalities for one-sided maximal functions,Trans. Am. Math. Soc.,319(2), 517–534. · Zbl 0696.42013 · doi:10.2307/2001252
[8] Martín-Reyes, F.J. and de la Torre, A. (1993). Two weight inequalities for fractional one-sided maximal operators,Proc. Am. Math. Soc.,117(2), 483–489. · Zbl 0769.42010
[9] Pérez, C. (1995). On sufficient conditions for the boundedness of the Hardy-Littlewood maximal operator between weightedL p -spaces with different weights,Proc. London Math. Soc.,71(3), 135–157. · Zbl 0829.42019 · doi:10.1112/plms/s3-71.1.135
[10] Pérez, C. (1994). Weighted norm inequalities for singular integral operators.J. London Math. Soc.,49, 296–308. · Zbl 0797.42010
[11] Pérez, C. (1994). Two weighted inequalities for potential and fractional type maximal operators.Indiana Univ. Math.,43, 1–28. · Zbl 0812.35006 · doi:10.1512/iumj.1994.43.43001
[12] Sawyer, E. (1982). A characterization of a two-weighted norm inequality for maximal operators,Studia Math.,75, 1–11. · Zbl 0508.42023
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