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Multipliers of Hardy spaces associated with Laguerre expansions. (English) Zbl 1342.42028

Let \[ H^p([0,\infty))=\{ f\in H^p(\mathbb R): \mathrm{supp}(f) \subset [0,\infty) \}, \quad 0<p\leq 1, \] where \(H^p(\mathbb R)\) is the Hardy space on \(\mathbb R\). The authors consider Laguerre expansions for \(f \in H^p([0,\infty))\): \[ f \sim \sum_{n=0}^\infty c_n^{(\alpha)}(f)\mathcal L_n^{(\alpha)}(x), \] where \(\{\mathcal L_n^{(\alpha)}\}_{n=0}^\infty\) is a complete \(\alpha\)-Laguerre orthonormal system on \([0, \infty)\) with respect to the Lebesgue measure and the coefficients \(c_n^{(\alpha)}(f)\) are suitably defined for \(f \in H^p([0,\infty))\). Let \(\alpha\geq 0\). Define \(\alpha^*=+\infty\) if \(\alpha\) is even and \(\alpha^*=\alpha/2+1\) otherwise. Suppose that \((\alpha^*)^{-1}<p<1\leq q<\infty\). Then, it is shown that \[ \left(\sum_{n=0}^\infty |\lambda_n c_n^{(\alpha)}(f)|^q \right)^{1/q} \leq c\| f\|_{H^p([0,\infty))}, \] where \(\{\lambda_n\}_{n=0}^\infty\) is a sequence of complex numbers satisfying \[ \sum_{n=1}^N n^{q/p}|\lambda_n|^q=O(N^q). \]

MSC:

42C10 Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.)
42B30 \(H^p\)-spaces
42A45 Multipliers in one variable harmonic analysis
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References:

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