Summability of Fourier series in periodic Hardy spaces with variable exponent
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SUMMABILITY OF FOURIER SERIES IN PERIODIC HARDY SPACES WITH VARIABLE EXPONENT F. WEISZ Department of Numerical Analysis, E¨ otv¨ os L. University, P´ azm´ any P. s´ et´ any 1/C, H-1117 Budapest, Hungary e-mail: [email protected] (Received December 17, 2019; accepted March 15, 2020)
Abstract. Let p(·) : Tn → (0, ∞) be a variable exponent function satisfying the globally log-H¨ older condition and 0 < q ≤ ∞. We introduce the periodic variable Hardy and Hardy–Lorentz spaces Hp(·) (Td ) and Hp(·),q (Td ) and prove their atomic decompositions. A general summability method, the so called θ-summability is considered for multi-dimensional Fourier series. Under some conditions on θ, it is proved that the maximal operator of the θ-means is bounded from Hp(·) (Td ) to Lp(·) (Td ) and from Hp(·),q (Td ) to Lp(·),q (Td ). This implies some norm and almost everywhere convergence results for the summability means. The Riesz, Bochner–Riesz, Weierstrass, Picard and Bessel summations are investigated as special cases.
1. Introduction It was proved by Lebesgue [24] that the Fej´er means [11] of the trigonometric Fourier series of a one-dimensional integrable function f ∈ L1 (T) converge almost everywhere to the function. This result was generalized for several summability methods, such as for the Riesz, Weierstrass, Abel, etc. summations in Zygmund [46], Butzer and Nessel [2], Stein and Weiss [38] or Trigub and Belinsky [39]. A general method of summation, the so called θ-summation method, which is generated by a single function θ and which includes all summations just mentioned, is studied intensively in the literature (see e.g. Butzer and Nessel [2], Trigub and Belinsky [39], G´ at [12–14], Goginava [15–17], Persson, Tephnadze and Wall [31], Simon [33–35] and Feichtinger and Weisz [9,10, This research was supported by the Hungarian National Research, Development and Innovation Office – NKFIH, K115804 and KH130426. Key words and phrases: variable Hardy space, variable Hardy–Lorentz space, atomic decomposition, θ-summability, maximal operator. Mathematics Subject Classification: primary 42B08, secondary 42B30, 42A24, 42B25. c 2020 0236-5294/$ 20.00 © 0 Akad´ emiai Kiad´ o, Budapest 0236-5294/$20.00 Akade ´miai Kiado ´, Budapest, Hungary
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F. F. WEISZ WEISZ
41–43]). For multi-dimensional Fourier series, the summability means are defined by |k| f (k)e2πık·x, ··· θ σnθ f (x) := n k1 ∈Z
kd ∈Z
where | · | denotes the Euclidean norm and f (k) is the kth Fourier coefficient of f . The choice θ(u) = max(1 − |u|, 0) yields the Fej´er summation. Stein, Taibleson and Weiss [37] proved for the Bochner–Riesz summability that the maximal operator σ∗θ of the θ-means is bounded from the Hardy space Hp (Td ) to Lp (Td ) if p > p0 (see also Grafakos [18] and Lu [28]). Recently, the author [44] generalized this result and verified for multi-dimensional Fourier transforms that σ∗θ is bounded from Hp(·) (Rd ) to Lp(·) (Rd ) and from Hp(·),q (Rd ) to Lp(·),q (Rd ) for all p(·) > p0 , where p(·): Rd → (0, ∞) is a variable exponen
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