Boundedness of Convolution Operators on Hardy Spaces
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Boundedness of Convolution Operators on Hardy Spaces Eduard Belinsky1 · Elijah Liflyand2,3 Received: 3 February 2017 / Revised: 7 May 2018 / Accepted: 8 May 2018 / Published online: 29 April 2019 © Springer-Verlag GmbH Germany, part of Springer Nature 2019
Abstract Establishing conditions for the boundedness of an operator taking H p (Rn ) into L p (Rn ), with 0 < p ≤ 1, is a classical subject. A standard approach to such problems is using the atomic characterization of H p (Rn ), 0 < p ≤ 1, and working with atoms. Unlike in certain earlier work on the subject we apply this machinery not to specific operators but to a wide general family of multivariate linear means generated by a multiplier. We illustrate the use of these new conditions applying them to some methods known from before. Keywords Fourier transform · Hardy space · Atomic decomposition Mathematics Subject Classification Primary 42B30; Secondary 42B08 · 42B15 · 42B35 · 42B10
Communicated by Doron Lubinsky. The idea of this work was suggested by Eduard Belinsky long ago. For the second named author, Eduard Belinsky was an elder and more experienced friend and colleague, numerous discussions with whom forged largely his outlook on mathematics. Belinsky’s untimely death brought to a halt this project as well as many other joined plans. It took much time for the second author to return to this project, to start working on the problems and examples suggested by Eduard, and to clarify certain issues. Belinsky’s initial contribution is more than enough for him to qualify to be a coauthor of this work. However, the second author takes full responsibility for possible inaccuracies.
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Elijah Liflyand [email protected]; [email protected]
1
Cave Hill, Barbados
2
Department of Mathematics, Bar-Ilan University, 52900 Ramat Gan, Israel
3
S.M. Nikolskii Institute of Mathematics, RUDN University, 6 Miklukho-Maklay St, Moscow, Russia 117198
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184
E. Belinsky Z”L, E. Liflyand
1 Introduction We consider the linear means, generated by an appropriate multiplier function ϕ (on the Fourier transform side), or by its Fourier transform as a kernel of the convolution operator n f (u) ϕ N (x − u) du, (1) (H f ) N (x) = (Hϕ f ) N (x) = N Rn
where
ϕ (x) =
ϕ(u)e−i xu du
Rn
is the Fourier transform of ϕ, with xu = x1 u 1 + · · · + xn u n . As is common in harmonic analysis, in order to study summability and convergence properties of these means, we associate with (H f ) N (x) their maximal operator (H∗ f )(x) = (Hϕ∗ f )(x) = sup N n f (u) ϕ N (x − u) du . N >0 n
(2)
R
Functions f from a Hardy space H p (Rn ), 0 < p ≤ 1, are of special interest since one can study the behavior of (2) on much simpler functions, the so-called atoms. In one of the first papers in this area [6] two special cases of ϕ were studied. Later on, some other cases were treated in the same manner, see, e.g., [3] and [8,9]. A survey of these and related results can also be found in [10]. In [2], such results were related to those on discrete Hardy
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