Atomic Decomposition of Hardy Spaces Associated with Certain Laguerre Expansions

Let L _n ^a (x), n=0,1,…, be the Laguerre polynomials of order a>−1. Denote ℓ _n ^a (x)=(n!/Γ(n+a+1))^1/2 L _n ^a (x)e ^−x/2. Let be the kernel of the semigroup {T _t } _t>0 associated with the system ℓ _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 belongs to L ¹((0,∞),x ^a   dx). We prove a special atomic decomposition of the elements of the Hardy space. Research supported by the European Commission Marie Curie Host Fellowship for the Transfer of Knowledge “Harmonic Analysis, Nonlinear Analysis and Probability” MTKD-CT-2004-013389, and by Polish funds for science in the years 2005–2008 (research project 1P03A03029).