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Atomic Decomposition of Hardy Spaces Associated with Certain Laguerre Expansions

  • Jacek DziubańskiEmail author
Article

Abstract

Let L n a (x), n=0,1,…, be the Laguerre polynomials of order a>−1. Denote n a (x)=(n!/Γ(n+a+1))1/2 L n a (x)e x/2. Let
$$T_{t}(x,y)=\sum_{n}e^{-(n+(a+1)/2)t}\ell_{n}^{a}(x)\ell_{n}^{a}(y)$$
be the kernel of the semigroup {T t } t>0 associated with the system n a considered on ((0,∞),x a dx). We say that a function f belongs to the Hardy space H 1 associated with the semigroup if the maximal function
$$\mathcal{M}f(x)=\sup_{t>0}\biggl|\int_{0}^{\infty}T_{t}(x,y)f(y)y^{a}\,dy\biggr|$$
belongs to L 1((0,∞),x a dx). We prove a special atomic decomposition of the elements of the Hardy space.

Keywords

Hardy spaces Maximal functions Laguerre expansions 

Mathematics Subject Classification (2000)

42B30 33C45 42B25 

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Copyright information

© Birkhäuser Boston 2008

Authors and Affiliations

  1. 1.Institute of MathematicsUniversity of WrocławWrocławPoland

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