Constructive Approximation

, Volume 27, Issue 3, pp 269–287 | Cite as

Hardy Spaces for Laguerre Expansions

  • Jacek DziubanskiEmail author


Let \({\cal L}_n^a(x)\) be the standard Laguerre functions of type a. We denote \(\varphi_n^a(x)={\cal L}_n^a(x^2)(2x)^{1\slash 2}\). Let \({\cal T}_tf(x)=\sum_{n}e^{-(n+(a+1)\slash 2)t} \langle f,{\cal L}_n^a\rangle {\cal L}_n^a(x)\) 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 \({\cal L}^a_n\) 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.


Hardy Space Maximal Function Integral Kernel Weak Type Orthonormal System 
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Copyright information

© Springer 2007

Authors and Affiliations

  1. 1.Institute of Mathematics, University of Wroclaw50-384 Wroclaw, pl. Grunwaldzki 2/4Poland

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