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!/Gamma(n + a + 1))L-1/2(n)a(x)e(-x/2). Let T-t(x, y) = Sigma(n)e(-(n+(a+1)/2)t)l(n)(a)(x)l(n)(a)(y) be the kernel of the semigroup {T-t}(t>0) associated with the system l(n)(a) considered on ((0, infinity), x(a) dx). We say that a function f belongs to the Hardy space H-1 associated with the semigroup if the maximal function M f(x) = sup(t>0)vertical bar integral(infinity)(0) T-t(x, y) f(y)y(a) dy vertical bar belongs to L-1((0, infinity), x(a) dx). We prove a special atomic decomposition of the elements of the Hardy space.