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A new of looking at distributional estimates; applications for the bilinear Hilbert transform.

Dimitriy Bilyk, Loukas Grafakos (2006)

Collectanea Mathematica

Distributional estimates for the Carleson operator acting on characteristic functions of measurable sets of finite measure were obtained by Hunt. In this article we describe a simple method that yields such estimates for general operators acting on one or more functions. As an application we discuss how distributional estimates are obtained for the linear and bilinear Hilbert transform. These distributional estimates show that the square root of the bilinear Hilbert transform is exponentially lntegrable...

A note on a problem arising from risk theory

Ulrich Abel, Ovidiu Furdui, Ioan Gavrea, Mircea Ivan (2010)

Czechoslovak Mathematical Journal

In this note we give an answer to a problem of Gheorghiță Zbăganu that arose from the study of the properties of the moments of the iterates of the integrated tail operator.

A Parseval equation and a generalized finite Hankel transformation

Jorge J. Betancor, Manuel T. Flores (1991)

Commentationes Mathematicae Universitatis Carolinae

In this paper, we study the finite Hankel transformation on spaces of generalized functions by developing a new procedure. We consider two Hankel type integral transformations h μ and h μ * connected by the Parseval equation ∑ n = 0 ∞ ( h μ f ) ( n ) ( h μ * ϕ ) ( n ) = ∫ 0 1 f ( x ) ϕ ( x ) d x . A space S μ of functions and a space L μ of complex sequences are introduced. h μ * is an isomorphism from S μ onto L μ when μ ≥ - 1 2 . We propose to define the generalized finite Hankel transform h μ ' f of f ∈ S μ ' by 〈 ( h μ ' f ) , ( ( h μ * ϕ ) ( n ) ) n = 0 ∞ 〉 = 〈 f , ϕ 〉 , for ϕ ∈ S μ .

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