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Necessary and sufficient conditions for weighted Orlicz class inequalities for maximal functions and singular integrals. II. (English) Zbl 0858.42011
This is an interesting paper on integral operators on homogeneous type spaces. For singular kernel operators \(Kf(x)=\lim_{\varepsilon\to 0} \int_{d(x,y)>\varepsilon} k(x,y)f(y)d\mu\), criteria are found for weak-type Orlicz class inequalities. Calderón-Zygmund operators are treated as well. Of particular interest are the characterizations of strong type inequalities for singular integrals and maximal functions. Applications to classical operators such as the Hilbert transform are pointed out.
[See also Part I, same journal 2, No. 4, 361-384 (1995; Zbl 0836.42012)].
Reviewer: L.Pick (Praha)
MSC:
42B25 Maximal functions, Littlewood-Paley theory
26D15 Inequalities for sums, series and integrals
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
47G10 Integral operators
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References:
[1] A. Gogatishvili and V. Kokilashvili, Necessary and sufficient conditions for weighted Orlicz class inequalities for maximal functions and singular integrals. I.Georgian Math. J. 2 (1995), No. 4, 361–384. · Zbl 0836.42012
[2] R. A. Macías and C. Segovia, Singular integrals on generalized Lipschitz and Hardy spaces.Studia Math. 65 (1979), No. 1, 55–71.
[3] R. A. Macías, C. Segovia, and J. L. Torrea, Singular integral operators with not necessarily bounded kernels on spaces of homogeneous type.Advances in Math. 93 (1992), 25–60. · Zbl 0795.42010
[4] R. A. Macías and C. Segovia, A well-defined quasi-distance for spaces of homogeneous type.Trab. Mat. Inst. Argentina Mat. 32 (1981), 1–14.
[5] S. Hoffman, Weighted norm inequalities and vector-valued inequalities for certain rough operators.Indiana University Math. J. 42 (1993), No. 1, 1–14.
[6] F. J. Ruiz-Torrea and J. L. Torrea, Vector-valued Calderon-Zygmund theory and Calderon measures on spaces of homogeneous nature.Studia Math. 88 (1988), 221–243. · Zbl 0639.42015
[7] G. David, Opérateurs intégraux singuliers sur certaines courbes du plan complexe.Ann. Sci. École Norm. Sup. (4)17 (1984), No. 1, 157–189.
[8] Qian Tao, Weighted inequalities concerning the Radon measure of the arc length of curves on the complex plane.J. Syst. Sci. Math. Sci. 6 (1986), No. 2, 146–152. · Zbl 0613.42012
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