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Hilbert transform and singular integrals on the spaces of tempered ultradistributions

Andrzej Kamiński, Dušanka Perišić, Stevan Pilipović (2000)

Banach Center Publications

The Hilbert transform on the spaces S ' * ( R d ) of tempered ultradistributions is defined, uniquely in the sense of hyperfunctions, as the composition of the classical Hilbert transform with the operators of multiplying and dividing a function by a certain elliptic ultrapolynomial. We show that the Hilbert transform of tempered ultradistributions defined in this way preserves important properties of the classical Hilbert transform. We also give definitions and prove properties of singular integral operators...

Hilbert transform, Toeplitz operators and Hankel operators and invariant A∞ weights.

Sergei Treil, Alexander Volberg, Dechao Zheng (1997)

Revista Matemática Iberoamericana

In this paper, several sufficient conditions for boundedness of the Hilbert transform between two weighted Lp-spaces are obtained. Invariant A∞ weights are obtained. Several characterizations of invariant A∞ weights are given. We also obtain some sufficient conditions for products of two Toeplitz operators of Hankel operators to be bounded on the Hardy space of the unit circle using Orlicz spaces and Lorentz spaces.

Hilbert transforms and the Cauchy integral in euclidean space

Andreas Axelsson, Kit Ian Kou, Tao Qian (2009)

Studia Mathematica

We generalize the notions of harmonic conjugate functions and Hilbert transforms to higher-dimensional euclidean spaces, in the setting of differential forms and the Hodge-Dirac system. These harmonic conjugates are in general far from being unique, but under suitable boundary conditions we prove existence and uniqueness of conjugates. The proof also yields invertibility results for a new class of generalized double layer potential operators on Lipschitz surfaces and boundedness of related Hilbert...

Hyperbolic singular integral operators.

Andrea R. Nahmod (1995)

Revista Matemática Iberoamericana

We define a class of integral operators which are singular relative to the hyperbolic metric in simply connected domains of the plane. We study the necessary and sufficient conditions for such operators to be bounded on L2 of the upper half plane relative to the hyperbolic metric.

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