This paper provides a fairly general approach to summability questions for multi-dimensional Fourier transforms. It is based on the use of Wiener amalgam spaces W(Lp, lq)(ℝd), Herz spaces and weighted versions of Feichtinger 's algebra and covers a wide range of concrete special cases (20 of them are listed at the end of the paper). It is proved that under some conditions the maximal operator of the θ-means σTθf can be estimated pointwise by the Hardy-Littlewood maximal function. From this it follows that σTθf → f a.e. for all ∈ W(L 1,l∞)(ℝd), hence f ∈ L p(ℝd) for any 1 ≤ p ≤ ∞. Moreover, σTθf(x) converges to f(x) at each Lebesgue point of f ∈ L1(ℝd) (resp. f ∈ W(L 1,l∞)(ℝd)) if and only if the Fourier transform of θ is in a suitable Herz space. In case θ is in a Besov space or in a weighted Feichtinger's algebra or in a Sobolev-type space then the a.e. convergence is obtained. Some sufficient conditions are given for θ to be in the weighted Feichtinger's algebra. The same results are presented for multi-dimensional Fourier series.
|Number of pages||28|
|Journal||Mathematical Proceedings of the Cambridge Philosophical Society|
|Publication status||Published - May 1 2006|
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