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  • Author or Editor: J. Betancor x
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We present new properties of the Besov--Hankel spaces introduced in [10]. We prove a Hankel version of a result of Bui, Paluszyński and Taibleson obtaining new norms that define the topology of the Besov--Hankel spaces. Also we get atomic representations for the distributions in the spaces under consideration.

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We study the translation and the convolution associated to the discrete Jacobi transformation on complex sequences of slow and rapid growth. Also we establish new topological properties for these spaces of sequences.

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We consider some aspects of harmonic analysis of the differential operator C v = d 2 / d x 2 + { v 2 1 / 4 ) / a ? 2 , v > 1 . Spectral decomposition of its self-adjoint extension is given in terms of the Hankel transform H ν . We present a fairly detailed analysis of the corresponding Poisson semigroup {P t } t > 0: this is given in a weighted setting with A p -weights involved. Then, we consider conjugate Poisson integrals of functions from L p (w), wA p , 1 ≦ p < ∞. Boundary values of the conjugate Poisson integrals exist both in L p (w) and a.e., and the resulting mapping is called the generalized Hilbert transform. Mapping properties of that transform are then proved. All this complements, in some sense, the analysis of conjugacy for the modified Hankel transform H ν which was initiated in the classic paper of Muckenhoupt and Stein [15], then continued in a series of papers by Andersen, Kerman, Rooney and others.

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