Abstract
For double Walsh-Fourier series and with f ∈ L2([0, 1) × [0, 1)) we prove two almost orthogonality results relative to the linearized maximal square partial sums operator SN(x,y)f (x, y). Assumptions are N(x, y) non-decreasing as a function of x and of y and, roughly speaking, partial derivatives with approximately constant ratio \({{N_y^\prime \left({x,y} \right)} \over {N_x^\prime \left({x,y} \right)}} \cong {2^{{n_0}}}\) for all x and y, where n0 is any fixed non-negative integer. Estimates, independent of N(x, y) and n0, are then extended to Lr, 1 < r < 2. We give an application to the family N(x, y) = λxy on [0, 1) × [0, 1), any λ > 10.
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Supported by MIUR Excellence Department Project awarded to the Department of Mathematics, University of Rome Tor Vergata (Grant No. CUP E83C1800010 0006)
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Prestini, E. New Exceptional Sets and Convergence of the Square Partial Sums of Walsh-Fourier Series. Acta. Math. Sin.-English Ser. 36, 733–748 (2020). https://doi.org/10.1007/s10114-020-9353-x
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DOI: https://doi.org/10.1007/s10114-020-9353-x