Mixed norm estimates for certain means
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- by Lennart Börjeson PDF
- Trans. Amer. Math. Soc. 309 (1988), 517-541 Request permission
Abstract:
We obtain estimates of the mean \[ F_x^\gamma (t) = {C_\gamma }\int _{|y| < 1} {{{(1 - |y{|^2})}^\gamma }f(x - ty) dy} \] in mixed Lebesgue and Sobolev spaces. They generalize earlier estimates of the spherical mean $F_x^{ - 1}(t) = C\;\int _{{S^{n - 1}}} {f(x - ty) dS(y)}$ and of solutions of the wave equation ${\Delta _x}u = {\partial ^2}u/\partial {t^2}$.References
- A. Benedek and R. Panzone, The space $L^{p}$, with mixed norm, Duke Math. J. 28 (1961), 301–324. MR 126155, DOI 10.1215/S0012-7094-61-02828-9 J. Bergh and J. Löfström, Interpolation spaces, Springer-Verlag, Berlin and New York, 1976. L. Börjeson, Estimates for the Bochner-Riesz operator with negative index, Reports, Dept. of Math., Univ. of Stockholm, No. 5, 1984.
- Jean Bourgain, Estimations de certaines fonctions maximales, C. R. Acad. Sci. Paris Sér. I Math. 301 (1985), no. 10, 499–502 (French, with English summary). MR 812567 —, Averages in the plane over convex curves and maximal operators, preprint. —, On the spherical maximal function in the plane, preprint.
- Michael Cwikel and Svante Janson, Interpolation of analytic families of operators, Studia Math. 79 (1984), no. 1, 61–71. MR 772005, DOI 10.4064/sm-79-1-61-71 A. Erdélyi (Editor), Higher transcendental functions, Vols. 1, 2, McGraw-Hill, New York, 1955.
- 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 J. L. Lions and E. Magenes, Non-homogeneous boundary value problems and applications, Vol. 1, Springer-Verlag, Berlin and New York, 1972.
- Bernard Marshall, $L^{p}$-$L^{q}$ multipliers of anisotropic wave equations, Indiana Univ. Math. J. 33 (1984), no. 3, 435–457. MR 740959, DOI 10.1512/iumj.1984.33.33024
- Akihiko Miyachi, On some estimates for the wave equation in $L^{p}$ and $H^{p}$, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 27 (1980), no. 2, 331–354. MR 586454
- D. M. Oberlin and E. M. Stein, Mapping properties of the Radon transform, Indiana Univ. Math. J. 31 (1982), no. 5, 641–650. MR 667786, DOI 10.1512/iumj.1982.31.31046 J. Peetre, ${H^p}$ spaces, Lecture notes, Lund, 1974.
- Juan C. Peral, $L^{p}$ estimates for the wave equation, J. Functional Analysis 36 (1980), no. 1, 114–145. MR 568979, DOI 10.1016/0022-1236(80)90110-X
- J. Peyrière and P. Sjölin, Regularity of spherical means, Ark. Mat. 16 (1978), no. 1, 117–126. MR 499142, DOI 10.1007/BF02385987
- Per Sjölin, Lipschitz continuity of spherical means, Linear spaces and approximation (Proc. Conf., Math. Res. Inst., Oberwolfach, 1977) Lecture Notes in Biomath., vol. 21, Springer, Berlin-New York, 1978, pp. 229–234. MR 501480
- Per Sjölin, Regularity and integrability of spherical means, Monatsh. Math. 96 (1983), no. 4, 277–291. MR 729040, DOI 10.1007/BF01471211
- Per Sjölin, Regularity and integrability properties of spherical means, Proceedings of the nineteenth Nordic congress of mathematicians (Reykjavík, 1984) Vísindafél. Ísl., XLIV, Icel. Math. Soc., Reykjavík, 1985, pp. 215–218. MR 828037 —, Regularity properties of spherical means, Proc. A. Haar Memorial Conference (Budapest, 1985) (to appear).
- Per Sjölin, Norm inequalities for spherical means, Monatsh. Math. 100 (1985), no. 2, 153–161. MR 809120, DOI 10.1007/BF01295672
- Sigrid Sjöstrand, On the Riesz means of the solutions of the Schrödinger equation, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 24 (1970), 331–348. MR 270219
- Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR 0290095
- 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, Mitchell H. Taibleson, and Guido Weiss, Weak type estimates for maximal operators on certain $H^{p}$ classes, Proceedings of the Seminar on Harmonic Analysis (Pisa, 1980), 1981, pp. 81–97. MR 639468
- 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
- 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
- Robert S. Strichartz, Convolutions with kernels having singularities on a sphere, Trans. Amer. Math. Soc. 148 (1970), 461–471. MR 256219, DOI 10.1090/S0002-9947-1970-0256219-1
- H. Triebel, Interpolation theory, function spaces, differential operators, VEB Deutscher Verlag der Wissenschaften, Berlin, 1978. MR 500580
- G. N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge University Press, Cambridge, England; The Macmillan Company, New York, 1944. MR 0010746
Additional Information
- © Copyright 1988 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 309 (1988), 517-541
- MSC: Primary 42B15
- DOI: https://doi.org/10.1090/S0002-9947-1988-0929662-6
- MathSciNet review: 929662