For a wide family of multivariate Hausdorff operators, the boundedness of an operator from this family is proved on the real Hardy space. By this we extend and strengthen previous results due to Andersen and Moricz.2000 Mathematics subject classification: primary 47B38, 42B10; secondary 46E30.
Abstract. We prove that the Hausdorff operator generated by a function ϕ ∈ L 1 (R) is bounded on the real Hardy space H 1 (R). The proof is based on the closed graph theorem and on the fact that if a function f in L 1 (R) is such that its Fourier transform f (t) equals 0 for t < 0 (or for t > 0), then f ∈ H 1 (R).
In this survey, results on the representation of a function as an absolutely convergent Fourier integral are collected, classified and discussed. Certain applications are also given.
In this paper we investigate properties of the general monotone sequences and functions, a generalization of monotone sequences and functions as well as of those of bounded variation. Some applications to various problems of analysis are given.
For a wide family of multivariate Hausdorff operators, a new stronger condition for the boundedness of an operator from this family on the real Hardy space H 1 by means of atomic decomposition.
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