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Bochner-Riesz means of functions in weak-L p

Authors:	Leonardo Colzani Giancarlo Travaglini Marco Vignati

Institution:	(1) Dipartimento di Matematica, Università degli Studi di Milano, Via C. Saldini 50, I-20 133 Milano, Italia;(2) Dipartimento di Matematica ed Applicazioni, Università di Palermo, Via Archrafi 34, I-90 123 Palermo, Italia

Abstract:	The Bochner-Riesz means of order 0 for suitable test functions on ^N are defined via the Fourier transform by $(S_R^\delta f)(\xi ) = (1 - \|\xi \|^2 /R^2 )^\delta + \hat f(\xi )$ . We show that the means of the critical index $\delta = \frac{N}{p} - \frac{{N + 1}}{2},1< p< \frac{{2N}}{{N + 1}}$ , do not mapL ^p,(^N) intoL ^p,(^N), but they map radial functions ofL ^p,(^N) intoL ^p,(^N). Moreover, iff is radial and in theL ^p,(^N) closure of test functions,S _R f(x) converges, asR+, tof(x) in norm and for almost everyx in ^N. We also observe that the means of the function\|x\| ^–N/p, which belongs toL ^p,(^N) but not to the closure of test functions, converge for nox.

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