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Optimal rearrangement-invariant Banach function range for the Hilbert transform. (English) Zbl 1488.46065

Summary: We describe the optimal rearrangement-invariant Banach function range of the classical Hilbert transform acting on a rearrangement-invariant Banach function space. We also show the existence of such optimal range for the Lorentz and Marcinkiewicz spaces.

MSC:

46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
44A15 Special integral transforms (Legendre, Hilbert, etc.)
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