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Zbl 0830.42009
Christ, Michael; Grafakos, Loukas
Best constants for two nonconvolution inequalities. (English)
[J] Proc. Am. Math. Soc. 123, No.6, 1687-1693 (1995). ISSN 0002-9939; ISSN 1088-6826/e

The authors consider higher-dimensional versions of Hardy's inequality. Moreover, they also consider the problem of finding the best possible constants for these inequalities. A typical result is the following inequality $$\Biggl( \int_{\bbfR^n} \biggl(|D( x, |x|)|^{- 1} \int_{D(x, |x|)} |f(y)|dy \biggr)^p dx\Biggr)^{1/p}\le c_{p, n} |f|_p,$$ where $1< p< \infty$, $D(x, |x|)$ is the ball of radius $|x|$ centered at $x$, and where $$c_{p, n}= p'(w_{n- 1}/ w_{n- 2}) 2^{n/p'- 1} B(2^{- 1}(n/p'- 1),(n- 3)/2),$$ is the best possible constant, $w_{m- 1}$ denotes the area of the unit sphere $S^{m- 1}$, and $B(s, t)$ is the usual beta-function.
[M.Milman (Boca Raton)]

MSC 2000:: *42B25 Maximal functions

Keywords: nonconvolution inequalities; higher-dimensional versions of Hardy's inequality; beta-function

Cited in: Zbl 1040.42018

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