## 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

### Keywords:

Hardy space; Laguerre expansion
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### References:

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