Best constants for two nonconvolution inequalities

Authors:
Michael Christ and Loukas Grafakos

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

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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.

- 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, - 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, - 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, - E. M. Stein and J.-O. Strömberg,
*Behavior of maximal functions in ${\bf R}^{n}$ for large $n$*, Ark. Mat.**21**(1983), no. 2, 259–269. MR**727348**, DOI https://doi.org/10.1007/BF02384314 - Elias M. Stein and Guido Weiss,
*Introduction to Fourier analysis on Euclidean spaces*, Princeton University Press, Princeton, N.J., 1971. Princeton Mathematical Series, No. 32. MR**0304972**

*Inequalities*, The University Press, Cambridge, 1959.

*Isoperimatric inequalities in mathematical physics*, Princeton Univ. Press, Princeton, NJ, 1951.

*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.

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Article copyright:
© Copyright 1995
American Mathematical Society