For every symmetric quasi-Banach sequence space contained in ell-log, the paper constructs and proves the minimal symmetric quasi-Banach space that contains the discrete Hilbert transform image, and computes it concretely for weak ell-1.
Boyd, The Hilbert transform on rearrangement-invariant spaces , Can
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The optimal symmetric quasi-Banach range of the discrete Hilbert transform
For every symmetric quasi-Banach sequence space contained in ell-log, the paper constructs and proves the minimal symmetric quasi-Banach space that contains the discrete Hilbert transform image, and computes it concretely for weak ell-1.