Some inequalities for singular convolution operators in $L^ p$-spaces
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- by Andreas Seeger PDF
- Trans. Amer. Math. Soc. 308 (1988), 259-272 Request permission
Abstract:
Suppose that a bounded function $m$ satisfies a localized multiplier condition ${\sup _{t > 0}}||\phi m({t^P} \cdot )|{|_{{M_p}}} < \infty$, for some bump function $\phi$. We show that under mild smoothness assumptions $m$ is a Fourier multiplier in ${L^p}$. The approach uses the sharp maximal operator and Littlewood-Paley-theory. The method gives new results for lacunary maximal functions and for multipliers in Triebel-Lizorkin-spaces.References
- A.-P. Calderón and A. Torchinsky, Parabolic maximal functions associated with a distribution. II, Advances in Math. 24 (1977), no. 2, 101–171. MR 450888, DOI 10.1016/S0001-8708(77)80016-9
- Anthony Carbery, Variants of the Calderón-Zygmund theory for $L^p$-spaces, Rev. Mat. Iberoamericana 2 (1986), no. 4, 381–396. MR 913694, DOI 10.4171/RMI/40
- Michael Christ, On almost everywhere convergence of Bochner-Riesz means in higher dimensions, Proc. Amer. Math. Soc. 95 (1985), no. 1, 16–20. MR 796439, DOI 10.1090/S0002-9939-1985-0796439-7
- Michael Christ, Javier Duoandikoetxea, and José L. Rubio de Francia, Maximal operators related to the Radon transform and the Calderón-Zygmund method of rotations, Duke Math. J. 53 (1986), no. 1, 189–209. MR 835805, DOI 10.1215/S0012-7094-86-05313-5
- Michael Christ and Elias M. Stein, A remark on singular Calderón-Zygmund theory, Proc. Amer. Math. Soc. 99 (1987), no. 1, 71–75. MR 866432, DOI 10.1090/S0002-9939-1987-0866432-6
- Javier Duoandikoetxea and José L. Rubio de Francia, Maximal and singular integral operators via Fourier transform estimates, Invent. Math. 84 (1986), no. 3, 541–561. MR 837527, DOI 10.1007/BF01388746
- Charles Fefferman, A note on spherical summation multipliers, Israel J. Math. 15 (1973), 44–52. MR 320624, DOI 10.1007/BF02771772
- C. Fefferman and E. M. Stein, $H^{p}$ spaces of several variables, Acta Math. 129 (1972), no. 3-4, 137–193. MR 447953, DOI 10.1007/BF02392215 L. Hörmander, Estimates for translation invariant operators in ${L_p}$-spaces, Acta Math. 104 (1960), 93-139.
- R. Johnson, Maximal subspaces of Besov spaces invariant under multiplication by characters, Trans. Amer. Math. Soc. 249 (1979), no. 2, 387–407. MR 525680, DOI 10.1090/S0002-9947-1979-0525680-5
- Robert H. Latter and Akihito Uchiyama, The atomic decomposition for parabolic $H^{p}$ spaces, Trans. Amer. Math. Soc. 253 (1979), 391–398. MR 536954, DOI 10.1090/S0002-9947-1979-0536954-6
- W. Littman, C. McCarthy, and N. Rivière, $L^{p}$-multiplier theorems, Studia Math. 30 (1968), 193–217. MR 231126, DOI 10.4064/sm-30-2-193-217
- Wolodymyr R. Madych, On Littlewood-Paley functions, Studia Math. 50 (1974), 43–63. MR 344797, DOI 10.4064/sm-50-1-43-63
- Jaak Peetre, New thoughts on Besov spaces, Duke University Mathematics Series, No. 1, Duke University, Mathematics Department, Durham, N.C., 1976. MR 0461123
- N. M. Rivière, Singular integrals and multiplier operators, Ark. Mat. 9 (1971), 243–278. MR 440268, DOI 10.1007/BF02383650
- Andreas Seeger, On quasiradial Fourier multipliers and their maximal functions, J. Reine Angew. Math. 370 (1986), 61–73. MR 852510, DOI 10.1515/crll.1986.370.61
- Andreas Seeger, Necessary conditions for quasiradial Fourier multipliers, Tohoku Math. J. (2) 39 (1987), no. 2, 249–257. MR 887941, DOI 10.2748/tmj/1178228328
- Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR 0290095
- Hans Triebel, Theory of function spaces, Monographs in Mathematics, vol. 78, Birkhäuser Verlag, Basel, 1983. MR 781540, DOI 10.1007/978-3-0346-0416-1
Additional Information
- © Copyright 1988 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 308 (1988), 259-272
- MSC: Primary 42B15; Secondary 46E35, 47B38
- DOI: https://doi.org/10.1090/S0002-9947-1988-0955772-3
- MathSciNet review: 955772