# zbMATH — the first resource for mathematics

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
Full Text:
##### References:
  Calderón, A.P., Zygmund, A.: On singular integrals. Am. J. Math.78, 289-309 (1956) · Zbl 0072.11501 · doi:10.2307/2372517  Calderón, C.: Lacunary spherical means. Ill. J. Math.23, 476-484 (1979)  Chen, L.K.: On a singular integral (Preprint)  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  Coifman, R., Weiss, G.: Analyse harmonique non commutative sur certains espaces homogènes. Lect. Notes Math.242. (1971) · Zbl 0224.43006  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  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  García-Cuerva, J., Rubio de Francia, J.L.: Weighted norm inequalities and related topics. Amsterdam: North-Holland: 1985 · Zbl 0578.46046  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  Jawerth, B.: Weighted inequalities for maximal operators: Linearization, localization and factorization (To appear in Am. J. Math.) · Zbl 0608.42012  Kurtz, D.S.: Littlewood-Paley and multiplier theorems on weightedL p spaces. Trans. Am. Math. Soc.259, 235-254 (1980) · Zbl 0436.42012  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  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  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  Rivière, N.: Singular integrals and multiplier operators. Ark. Mat.9, 243-278 (1971) · Zbl 0244.42024 · doi:10.1007/BF02383650  Stein, E.M.: Singular integrals and differentiability properties of functions. Princeton N.J.: Princeton University Press 1970 · Zbl 0207.13501  Stein, E.M.: Oscillatory integrals in Fourier analysis (Preprint) · Zbl 0618.42006  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  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  Zygmund, A.: Trigonometric series, I & II. London, New York: Cambridge University Press 1959 · Zbl 0085.05601
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.