Transference for radial multipliers and dimension free estimates

Auscher, P.; Carro, M. J.

doi:10.1090/S0002-9947-1994-1152319-9

Transference for radial multipliers and dimension free estimates
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by P. Auscher and M. J. Carro

Trans. Amer. Math. Soc. 342 (1994), 575-593

DOI: https://doi.org/10.1090/S0002-9947-1994-1152319-9

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Abstract:

For a large class of radial multipliers on ${L^p}({{\mathbf {R}}^{\mathbf {n}}})$, we obtain bounds that do not depend on the dimension n. These estimates apply to well-known multiplier operators and also give another proof of the boundedness of the Hardy-Littlewood maximal function over Euclidean balls on ${L^p}({{\mathbf {R}}^{\mathbf {n}}})$, $p \geq 2$, with constant independent of the dimension. The proof is based on the corresponding result for the Riesz transforms and the method of rotations.

References

J. Bourgain, Averages in the plane over convex curves and maximal operators, J. Analyse Math. 47 (1986), 69–85. MR 874045, DOI 10.1007/BF02792533
Ronald R. Coifman and Guido Weiss, Transference methods in analysis, Conference Board of the Mathematical Sciences Regional Conference Series in Mathematics, No. 31, American Mathematical Society, Providence, R.I., 1976. MR 0481928
A.-P. Calderón and A. Zygmund, On higher gradients of harmonic functions, Studia Math. 24 (1964), 211–226. MR 167631, DOI 10.4064/sm-24-2-211-226
Peter L. Duren, Theory of $H^{p}$ spaces, Pure and Applied Mathematics, Vol. 38, Academic Press, New York-London, 1970. MR 0268655
Javier Duoandikoetxea and José L. Rubio de Francia, Estimations indépendantes de la dimension pour les transformées de Riesz, C. R. Acad. Sci. Paris Sér. I Math. 300 (1985), no. 7, 193–196 (French, with English summary). MR 780616
Gilles Pisier, Riesz transforms: a simpler analytic proof of P.-A. Meyer’s inequality, Séminaire de Probabilités, XXII, Lecture Notes in Math., vol. 1321, Springer, Berlin, 1988, pp. 485–501. MR 960544, DOI 10.1007/BFb0084154
E. M. Stein, Some results in harmonic analysis in $\textbf {R}^{n}$, for $n\rightarrow \infty$, Bull. Amer. Math. Soc. (N.S.) 9 (1983), no. 1, 71–73. MR 699317, DOI 10.1090/S0273-0979-1983-15157-1
Elias M. Stein, Maximal functions. I. Spherical means, Proc. Nat. Acad. Sci. U.S.A. 73 (1976), no. 7, 2174–2175. MR 420116, DOI 10.1073/pnas.73.7.2174
Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR 0290095
E. M. Stein and J.-O. Strömberg, Behavior of maximal functions in $\textbf {R}^{n}$ for large $n$, Ark. Mat. 21 (1983), no. 2, 259–269. MR 727348, DOI 10.1007/BF02384314
Elias M. Stein and Guido Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton Mathematical Series, No. 32, Princeton University Press, Princeton, N.J., 1971. MR 0304972
Elias M. Stein and Stephen Wainger, Problems in harmonic analysis related to curvature, Bull. Amer. Math. Soc. 84 (1978), no. 6, 1239–1295. MR 508453, DOI 10.1090/S0002-9904-1978-14554-6
N. Ja. Vilenkin, Special functions and the theory of group representations, Translations of Mathematical Monographs, Vol. 22, American Mathematical Society, Providence, R.I., 1968. Translated from the Russian by V. N. Singh. MR 0229863

Theory of Bessel functions

Modern analysis

A. Zygmund, Trigonometric series. 2nd ed. Vols. I, II, Cambridge University Press, New York, 1959. MR 0107776