Abstract
A general summability method of more-dimensional Fourier transforms is given with the help of a continuous function θ. Under some weak conditions on θ we show that the maximal operator of the ℓ 1–θ-means of a tempered distribution is bounded from H p (ℝd) to L p (ℝd) for all d/(d+α)<p≤∞ and, consequently, is of weak type (1,1), where 0<α≤1 depends only on θ. As a consequence we obtain a generalization of the one-dimensional summability result due to Lebesgue, more exactly, the ℓ 1–θ-means of a function f∈L 1(ℝd) converge a.e. to f. Moreover, we prove that the ℓ 1–θ-means are uniformly bounded on the spaces H p (ℝd), and so they converge in norm (d/(d+α)<p<∞). Similar results are shown for conjugate functions. Some special cases of the ℓ 1–θ-summation are considered, such as the Weierstrass, Picar, Bessel, Fejér, de La Vallée-Poussin, Rogosinski, and Riesz summations.
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Communicated by Karlheinz Groechenig.
This research was supported by the Hungarian Scientific Research Funds (OTKA) No K67642.
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Weisz, F. ℓ1-Summability of d-Dimensional Fourier Transforms. Constr Approx 34, 421–452 (2011). https://doi.org/10.1007/s00365-011-9128-9
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DOI: https://doi.org/10.1007/s00365-011-9128-9