Abstract
The Bochner–Riesz means are defined by the Fourier multiplier operators \((S_{R}^{\alpha}\ast f)\hat{\ }(\xi)=( 1-|R^{-1} \xi|^{2})^{\alpha}_{+}\hat{f}(\xi)\). Here we prove that if f has β derivatives in L p(R d), then \(S_{R}^{\alpha}\ast f(x)\) 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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Colzani, L., Volpi, S. (2013). Pointwise Convergence of Bochner–Riesz Means in Sobolev Spaces. In: Picardello, M. (eds) Trends in Harmonic Analysis. Springer INdAM Series, vol 3. Springer, Milano. https://doi.org/10.1007/978-88-470-2853-1_7
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DOI: https://doi.org/10.1007/978-88-470-2853-1_7
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