The Bochner–Riesz means are defined by the Fourier multiplier operators(formula presented). Here we prove that if f has β derivatives in Lp(Rd), then (formula presented)converges pointwise to f (x) as R → + ∞ with a possible exception of a set of points with Hausdorff dimension at most d – βp if one of the following conditions holds: either α > (d – 1)|1/p – 1/2|, or α > d(1/2 – 1/p) – 1/2 and α + β ≥ (d – 1)/2. If β > d/p, then pointwise convergence holds everywhere.
Colzani, L., Volpi, S. (2013). Pointwise convergence of Bochner-Riesz means in Sobolev spaces. In M.A. Piccardello (a cura di), Trends in harmonic analysis (pp. 135-146). Springer [10.1007/978-88-470-2853-1_7].
Pointwise convergence of Bochner-Riesz means in Sobolev spaces
Colzani, L;
2013
Abstract
The Bochner–Riesz means are defined by the Fourier multiplier operators(formula presented). Here we prove that if f has β derivatives in Lp(Rd), then (formula presented)converges pointwise to f (x) as R → + ∞ with a possible exception of a set of points with Hausdorff dimension at most d – βp if one of the following conditions holds: either α > (d – 1)|1/p – 1/2|, or α > d(1/2 – 1/p) – 1/2 and α + β ≥ (d – 1)/2. If β > d/p, then pointwise convergence holds everywhere.
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