Hausdorff operators on Hardy type spaces
https://doi.org/10.22405/2226-8383-2021-22-3-133-142
Abstract
During last 20 years, an essential part of the theory of Hausdorff operators is concentrated on their boundedness on the real Hardy space 𝐻1(R𝑑). The spaces introduced by Sweezy are,
in many respects, natural extensions of this space. They are nested in full between 𝐻1(R𝑑) and 𝐿10 (R𝑑). Contrary to 𝐻1(R𝑑), they are subject only to atomic characterization. For the estimates of Hausdorff operators on 𝐻1(R𝑑), other characterizations have always been applied.
Since this option is excluded in the case of Sweezy spaces, in this paper an approach to the estimates of Hausdorff operators is elaborated, where only atomic decompositions are used.
While on 𝐻1(R𝑑) this approach is applicable to the atoms of the same type, on the Sweezy spaces the same approach is not less effective for the sums of atoms of various types. For a
single Hausdorff operator, the boundedness condition does not depend on the space but only on the parameters of the operator itself. The space on which this operator acts is characterized
by the choice of atoms. An example is given (two-dimensional, for simplicity), where a matrix dilates the argument only in one variable.
About the Authors
Elijah Rafailovich Liflyand
Bar-Ilan University
Israel
Israel
department of mathematics
Maria Alexandrovna Skopina
St. Petersburg State University, Regional Mathematical Center of Southern Federal University.
Russian Federation
Russian Federation
doctor of physical and mathematical sciences,
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Review
For citations:
Liflyand E.R., Skopina M.A. Hausdorff operators on Hardy type spaces. Chebyshevskii Sbornik. 2021;22(3):133-142. (In Russ.) https://doi.org/10.22405/2226-8383-2021-22-3-133-142
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