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The boundedness of Weyl multiplier on Hardy spaces associated with twisted convolution. (English) Zbl 1174.42026

On \(\mathbb{C}^n\), consider the \(2n\) linear differential operators \[ Z_j = \frac{\partial}{ \partial_{z_j}}+ \bar{z}_j ,\quad \bar{Z}_j = \frac{\partial}{\partial_{\bar{z}_j}} -{z_j}, \quad j= 1,2, \ldots ,n. \] They generate a nilpotent Lie algebra isomorphic with the Heisenberg algebra of dimension \(2n+1\). These operators generate a family of “twsited translation” \(\tau_w\) on \(C^n\) defined by \[ (\tau_w f)(z)= f(z+w) \exp ( i \;\text{Im} (z \cdot \bar{w}) ). \] The “twisted convolution” of two functions \(f\) and \(g\) can be defined as \[ f \times g (z) = \int_{C^n} f(w) \tau_{-w}g(z)dw. \] Twisted convolution has been investigated in connection with the Weyl functional calculus. S. Thangavelu [Rev. Mat. Iberoam. 6, No. 1–2, 75–90 (1990; Zbl 0734.42010)] proved \(L^p\) boundedness of Weyl multiplier of the form \(\phi (H)\) where \(H = -\Delta + | x |^2\). On the other hand G. Mauceri, M. A. Picardello and F. Ricci [Adv. Math. 39, 270–288 (1981; Zbl 0503.46037)] introduced Hardy space \(H^1\) associated with twisted convolution operators.
The author proves the boundedness of Weyl multiplier on this Hardy space \(H^1\). For the proof, the author gives a characterization of \(H^1\) by Littlewood-Paley \(g\) function.

MSC:

42B30 \(H^p\)-spaces
42B25 Maximal functions, Littlewood-Paley theory
42B35 Function spaces arising in harmonic analysis
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[1] Duong, X. T.; Ouhabaz, E. M.; Yan, L. X., Endpoint estimates for Riesz transforms of magnetic Schrödinger operators, Ark. Mat., 44, 261-275 (2006) · Zbl 1172.35370
[2] Dziubański, J.; Zienkiewicz, J., \(H^p\) spaces associated with Schrödinger operators with potentials from reverse Hölder classes, Colloq. Math., 98, 5-38 (2003) · Zbl 1083.42015
[3] Folland, G. B.; Stein, E. M., Hardy Spaces on Homogeneous Groups (1982), Princeton Univ. Press: Princeton Univ. Press Princeton · Zbl 0508.42025
[4] Mauceri, G.; Picardello, M.; Ricci, F., A Hardy space associated with twisted convolution, Adv. Math., 39, 270-288 (1981) · Zbl 0503.46037
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[6] Thangavelu, S., Lectures on Hermite and Laguerre Expansions, Math. Notes, vol. 42 (1993), Princeton Univ. Press: Princeton Univ. Press Princeton, NJ · Zbl 0791.41030
[7] Thangavelu, S., Littlewood-Paley-Stein theory on \(C^n\) and Weyl multipliers, Rev. Mat. Iberoamericana, 6, 75-90 (1990) · Zbl 0734.42010
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