Fourier Multipliers in Hardy Spaces in Tubes over Open Cones
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Fourier Multipliers in Hardy Spaces in Tubes over Open Cones Alexander V. Tovstolis
Received: 8 August 2013 / Revised: 19 March 2014 / Accepted: 26 March 2014 © Springer-Verlag Berlin Heidelberg 2014
Abstract We obtain effective sufficient conditions for multipliers of Fourier integrals acting from H p (T ) to H q (T ), 0 < p ≤ q ≤ 1. We also show that they are sharp in some cases. Special attention is paid to the means of Fourier integrals with compactly supported radial kernels. As an application, the critical index for the Bochner–Riesz means to define a bounded linear operator from H p to H q is found. Surprisingly, it does not depend on p. Keywords Fourier multiplier · Hardy spaces in tubes over open cones · Fourier integral · Multiplier defined by a radial function · Bochner–Riesz means · Nikol’skij type inequality · Non-increasing rearrangement Mathematics Subject Classification 2010
42B15 · 42B30 · 41A17
Multipliers of Fourier series and integrals have been investigated and widely used since 1923, when they were introduced by Fekete [4]. It is well-known that the Fourier series of a 2π -periodic function may not converge, or it may converge not to its generating function. However, it is possible to introduce some multiplicative factors λn into the Fourier series, i.e., to consider a modified Fourier series f ∼
cn λn e2πinx
n∈Z
Communicated by Alexander Yu. Solynin. A. V. Tovstolis (B) Department of Mathematics, Oklahoma State University, 401 Mathematical Sciences, Stillwater, OK 74078, USA e-mail: [email protected]
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that has better properties. This tool has been successfully applied to problems of approximation theory, differential equations, numerical analysis, etc., in particular if defines a bounded linear operator on the corresponding function space. The first effective sufficient condition for boundedness of in L p (T), p ∈ (1, ∞), and its applications were found by Marcinkiewicz [14]. Later, for the non-periodic case of multipliers of Fourier integrals, these conditions were obtained by Michlin [15,16] and Hörmander [9] (see also [22, Ch. IV]). The cases most often investigated are p = 1, 2, ∞, which is not a surprise. Employing the Riesz–Thorin Theorem, it is easy to transfer such results to the case p ∈ (1, ∞). These results and techniques became classical and are well described, e.g., in [23]. For p ∈ (0, 1), L p -spaces are pre-normed, and there are no linear continuous functionals, and no Fourier series in these spaces. This is the reason for considering the H p (D) spaces of functions analytic in the unit disk D and having their boundary values in L p (T). For any f ∈ H p (D), p > 0, one can consider the Taylor series of f that coincides with the Fourier series of the limit values of f on the unit circle when p ≥ 1. Since the functions under consideration are holomorphic, such investigation for H p (D), p ∈ (0, 1), yields many interesting results. Several efficient conditions for multipliers in H p spaces in polydisk Dm , and their applications t
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