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Maximal and singular integral operators via Fourier transform estimates. (English) Zbl 0568.42012
In this paper $$L^ p$$ inequalities are proved for singular integral operators written as $$Tf=\sum_{k}\sigma_ k*f$$ and maximal operators defined by $$Mf=\sup_{k}| \mu_ k*f|,$$ where $$\sigma_ k$$ and $$\mu_ k$$ are Borel measures with uniformly bounded total variation, the $$\sigma_ k's$$ have zero integral and the $$\mu_ k's$$ are positive. The inequalities follow from the regularity at zero and the decay at infinity of the Fourier transforms of $$\sigma_ k$$ and $$\mu_ k$$. Several applications are given: lacunary maximal functions (which generalize lacunary spherical means), homogeneous singular integrals with size conditions on the kernel and some of their variants, including weighted inequalities for kernels bounded on the unit sphere, maximal functions and Hilbert transforms along different types of curves (homogeneous, approximately homogeneous, convex plane curves). Some of them are old resuls with easier proofs, others are new. For all the singular integral operators the almost everywhere convergence of the truncated kernels is also obtained.

##### MSC:
 42B25 Maximal functions, Littlewood-Paley theory
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##### References:
 [1] Calderón, A.P., Zygmund, A.: On singular integrals. Am. J. Math.78, 289-309 (1956) · Zbl 0072.11501 · doi:10.2307/2372517 [2] Calderón, C.: Lacunary spherical means. Ill. J. Math.23, 476-484 (1979) [3] Chen, L.K.: On a singular integral (Preprint) [4] Christ, M., Duoandikoetxea, J., Rubio de Francia, J.L.: Maximal operators related to Radon transform and the Calderón-Zygmund method of rotations (To appear in Duke Math. J.) · Zbl 0656.42010 [5] Coifman, R., Weiss, G.: Analyse harmonique non commutative sur certains espaces homogènes. Lect. Notes Math.242. (1971) · Zbl 0224.43006 [6] Coifman, R., Weiss, G.: Review of the book Littlewood-Paley and multiplier theory. Bull. Am. Math. Soc.84, 242-250 (1978) · doi:10.1090/S0002-9904-1978-14464-4 [7] Fefferman, R.: A note on singular integrals. Proc. Am. Math. Soc.74, 266-270 (1979) · Zbl 0417.42009 · doi:10.1090/S0002-9939-1979-0524298-3 [8] García-Cuerva, J., Rubio de Francia, J.L.: Weighted norm inequalities and related topics. Amsterdam: North-Holland: 1985 · Zbl 0578.46046 [9] Greenleaf, A.: Principal curvature and harmonic analysis. Indiana Univ. Math. J.30, 519-537 (1981) · Zbl 0517.42029 · doi:10.1512/iumj.1981.30.30043 [10] Jawerth, B.: Weighted inequalities for maximal operators: Linearization, localization and factorization (To appear in Am. J. Math.) · Zbl 0608.42012 [11] Kurtz, D.S.: Littlewood-Paley and multiplier theorems on weightedL p spaces. Trans. Am. Math. Soc.259, 235-254 (1980) · Zbl 0436.42012 [12] Littman, W.: Fourier transform of surface-carried measures and differentiability of surface averages. Bull. Am. Math. Soc.69, 766-770 (1963) · Zbl 0143.34701 · doi:10.1090/S0002-9904-1963-11025-3 [13] Nagel, A., Stein, E.M., Wainger, S.: Differentiation in lacunary directions. Proc. Natl. Acad. Sci. USA75, 1060-1062 (1978) · Zbl 0391.42015 · doi:10.1073/pnas.75.3.1060 [14] Nagel, A., Wainger, S.: Hilbert transform associated with plane curves. Trans. Am. Math. Soc.223, 235-252 (1976) · Zbl 0341.44005 · doi:10.1090/S0002-9947-1976-0423010-8 [15] Rivière, N.: Singular integrals and multiplier operators. Ark. Mat.9, 243-278 (1971) · Zbl 0244.42024 · doi:10.1007/BF02383650 [16] Stein, E.M.: Singular integrals and differentiability properties of functions. Princeton N.J.: Princeton University Press 1970 · Zbl 0207.13501 [17] Stein, E.M.: Oscillatory integrals in Fourier analysis (Preprint) · Zbl 0618.42006 [18] Stein, E.M., Wainger, S.: Problems in harmonic analysis related to curvature. Bull. Am. Math. Soc.84, 1239-1295 (1978) · Zbl 0393.42010 · doi:10.1090/S0002-9904-1978-14554-6 [19] Weinberg, D.A.: The Hilbert transform and maximal function for approximately homogeneous curves. Trans. Am. Math. Soc.267, 295-306 (1981) · Zbl 0484.42005 · doi:10.1090/S0002-9947-1981-0621989-4 [20] Zygmund, A.: Trigonometric series, I & II. London, New York: Cambridge University Press 1959 · Zbl 0085.05601
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