Pointwise convergence in Pringsheim’s sense of the summability of Fourier transforms onWiener amalgam spaces

摘要

New multi-dimensional Wiener amalgam spaces W_c(L p, l_∞)(R~d) are introduced by taking the usual one-dimensional spaces coordinatewise in each dimension. The strong Hardy-Littlewood maximal function is investigated on these spaces. The pointwise convergence in Pringsheim’s sense of the θ-summability of multidimensional Fourier transforms is studied. It is proved that if the Fourier transform of θ is in a suitable Herz space, then the θ-means σ_T~θ f converge to f a.e. for all f ∈ W_c(L_1(log L)~(d?1), l_∞)(R~d). Note that W_c(L_1(log L)~(d?1), l_∞)(R~d) ? W_c(L_r, l_∞)(R~d) ? L_r (R~d) and W_c(L_1(log L)~(d?1), l_∞)(R~d) ? L_1(log L)~(d?1)(R~d), where 1 < r ≤ ∞. Moreover, σ_T~θ f (x) converges to f (x) at each Lebesgue point of f ∈ W_c(L1(log L)~(d?1), l_∞)(R~d).