ISSN:
1436-5081
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
Abstract The Bochner-Riesz means of order δ≥0 for suitable test functions on ℝ 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,∞(ℝ N ) intoL p,∞(ℝ N ), but they map radial functions ofL p,∞(ℝ N ) intoL p,∞(ℝ N ). Moreover, iff is radial and in theL p,∞(ℝ N ) closure of test functions,S R δ f(x) converges, asR→+∞, tof(x) in norm and for almost everyx in ℝ N . We also observe that the means of the function|x| −N/p, which belongs toL p,∞(ℝ N ) but not to the closure of test functions, converge for nox.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF01311209
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