Real Hardy Space, Multidimensional Variations, and Integrability of the Fourier Transform
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Complex Analysis and Operator Theory
Real Hardy Space, Multidimensional Variations, and Integrability of the Fourier Transform L. Angeloni1 · E. Liflyand2,3 · G. Vinti1 Received: 29 October 2018 / Accepted: 18 September 2019 © Springer Nature Switzerland AG 2020
Abstract A new class of functions is introduced closely related to that of functions with bounded Tonelli variation and to the real Hardy space. For this class, conditions for integrability of the Fourier transform are established. Keywords Fourier transform · Hilbert transform · Riesz transform · Bounded variation · Absolute continuity · Tonelli variation · Hardy space Mathematics Subject Classification Primary 42B10; Secondary 42B20 · 42B30 · 42B35 · 42A38 · 26B30
1 Introduction In dimension one, there is a nice and quite strong condition (see e.g., [9, Ch.3] or [11]) that guarantees the integrability of the Fourier transform
Communicated by Irene Sabadini. This article is part of the topical collection “Higher Dimensional Geometric Function Theory and Hypercomplex Analysis” edited by Irene Sabadini, Michael Shapiro and Daniele Struppa.
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E. Liflyand [email protected] L. Angeloni [email protected] G. Vinti [email protected]
1
Dipartimento di Matematica e Informatica, Universita’ degli Studi di Perugia, Via Vanvitelli 1, 06123 Perugia, Italy
2
Department of Mathematics, Bar-Ilan University, 52900 Ramat Gan, Israel
3
Regional Mathematical Center of Southern Federal University, Bolshaya Sadovaya Str. 105/42, Rostov-on-Don, Russia 344006 0123456789().: V,-vol
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f (x) =
f (t)e−i xt dt
R
(1)
of a function of bounded variation f : R → C. The function f is also assumed to be (locally) absolutely continuous and vanishing at infinity but these assumptions can be considered as minimal and almost necessary. The condition itself is f ∈ H 1 (R), where the latter is the real Hardy space (recall that the derivative of a function of bounded variation exists a.e. and is Lebesgue integrable). Indeed, we just integrate by parts in (1) and apply Hardy’s inequality (see e.g., [6, (7.24)]) R
| g (x)| d x g H 1 (R) . |x|
(2)
We shall use the notation “ ” as abbreviation for “ ≤ C ”, with C being an absolute positive constant, maybe different in different occurrences. The norm in H 1 (R) is understood as the sum of the L 1 (R) norms of the function and its Hilbert transform. To be precise, under the assumption f ∈ H 1 (R) there is no need in assuming the boundedness of variation, since, on the one side, this assumption itself provides f ∈ L 1 (R) and, on the other side, the integrability of the derivative along with the absolute continuity of the function makes it to be of bounded variation (at least, to be equal almost everywhere to such a function; see e.g., [5, Ch. 1] or [9, 3.5.1]). This will allow us to discuss the topic in terms of bounded variations even if any concrete variation is not explicitly involved. In fact, within this scope there is a possibility to arrive at the necessary a
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