Best constants for two nonconvolution inequalities
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- by Michael Christ and Loukas Grafakos PDF
- Proc. Amer. Math. Soc. 123 (1995), 1687-1693 Request permission
Abstract:
The norm of the operator which averages $|f|$ in ${L^p}({\mathbb {R}^n})$ over balls of radius $\delta |x|$ centered at either 0 or x is obtained as a function of n , p and $\delta$. Both inequalities proved are n-dimensional analogues of a classical inequality of Hardy in ${\mathbb {R}^1}$. Finally, a lower bound for the operator norm of the Hardy-Littlewood maximal function on ${L^p}({\mathbb {R}^n})$ is given.References
- Albert Baernstein II and B. A. Taylor, Spherical rearrangements, subharmonic functions, and $^*$-functions in $n$-space, Duke Math. J. 43 (1976), no. 2, 245–268. MR 402083 G. Hardy, J. Littlewood, and G. Pólya, Inequalities, The University Press, Cambridge, 1959.
- William G. Faris, Weak Lebesgue spaces and quantum mechanical binding, Duke Math. J. 43 (1976), no. 2, 365–373. MR 425598 G. Pólya and G. Szegö, Isoperimatric inequalities in mathematical physics, Princeton Univ. Press, Princeton, NJ, 1951.
- Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR 0290095 S. L. Sobolev, On a theorem of functional analysis, Mat. Sb. (N.S.) 4 (1938), 471-497; English transl., Amer. Math. Soc. Transl. Ser. 2, vol. 34, Amer. Math. Soc., Providence, RI, 1963, pp. 39-68.
- 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
Additional Information
- © Copyright 1995 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 123 (1995), 1687-1693
- MSC: Primary 42B25; Secondary 26D15, 47B38
- DOI: https://doi.org/10.1090/S0002-9939-1995-1239796-6
- MathSciNet review: 1239796