Abstract
We consider sequences of compact bounded linear operators \(U_n:L^p(0,1)\rightarrow ~L^p(0,1)\) with certain convergence properties. Several divergence theorems for multiple sequences of tensor products of these operators are proved. These theorems in particular imply that \(L\log ^{d-1} L\) is the optimal Orlicz space guaranteeing almost everywhere summability of rectangular partial sums of multiple Fourier series in general orthogonal systems.
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Gát, G., Karagulyan, G. On Convergence Properties of Tensor Products of Some Operator Sequences. J Geom Anal 26, 3066–3089 (2016). https://doi.org/10.1007/s12220-015-9662-y
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DOI: https://doi.org/10.1007/s12220-015-9662-y