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Hardy spaces for Laguerre expansions. (English) Zbl 1152.42301

Summary: Let \({\mathcal L}_n^a(x)\) be the standard Laguerre functions of type \(a\). We denote \(\varphi_n^a(x)={\mathcal L}_n^a(x^2)(2x)^{1/2}\). Let \[ {\mathcal T}_tf(x)=\sum_{n}e^{-(n+(a+1)/ 2)t}\langle f,{\mathcal L}_n^a\rangle {\mathcal L}_n^a(x) \text{ and } T_tf(x)=\sum_{n}e^{-(4n+2a+2)t} \langle f,\varphi_n^a\rangle \varphi_n^a(x) \] be the semigroups associated with the orthonormal systems \({\mathcal L}^a_n\) and \(\varphi_n^a\). We say that a function \(f\) belongs to the Hardy space \(H^1\) associated with one of the semigroups if the corresponding maximal function belongs to \(L^1((0,\infty), dx)\). We prove special atomic decompositions of the elements of the Hardy spaces.

MSC:

42A99 Harmonic analysis in one variable
35J10 Schrödinger operator, Schrödinger equation
42B25 Maximal functions, Littlewood-Paley theory
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