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Non-extendability of the finite Hilbert transform. (English) Zbl 1477.44003

Summary: The finite Hilbert transform \(T:X\rightarrow X\) acts continuously on every rearrangement invariant space \(X\) on \((-1,1)\) having non-trivial Boyd indices. It is proved that \(T\) cannot be further extended, whilst still taking its values in \(X\), to any larger domain space. That is, \(T:X\rightarrow X\) is already optimally defined.

MSC:

44A15 Special integral transforms (Legendre, Hilbert, etc.)
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
47A53 (Semi-) Fredholm operators; index theories
47B34 Kernel operators
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References:

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