Molecular decompositions of homogeneous Besov type spaces for Laguerre function expansions and applications

In this paper we consider the Laguerre operator \(L=-\frac{d^2}{dx^2}-\frac{\alpha }{x}\frac{d}{dx}+x^2\) $L=-\frac{d^2}{d{x}^2}-\frac{\alpha }{x}\frac{d}{\mathit{dx}}+x^2$ on the Euclidean space \(\mathbb R_{+}\) ${\mathbb{R}}_{+}$ . The main aim of this article is to develop a theory of homogeneous Besov type spaces associated to the Laguerre operator. To achieve our expected goals, Schwartz type spaces on \(\mathbb R_{+}\) ${\mathbb{R}}_{+}$ are introduced and then tempered type distributions are constructed. Using a suitable distribution of the Laguerre operator, the Calderón reproducing formula and the Harnack type inequality for subharmonic functions are established. With these tools in hand, we define the Besov type spaces \(\dot{B}_{p,q}^{s,L,m}\) ${\dot{B}}_{p,q}^{s,L,m}$ and obtain the molecular decompositions of \(\dot{B}_{p,q}^{s,L,m}\) ${\dot{B}}_{p,q}^{s,L,m}$ . As applications, the embedding theorem and square functions characterization of Besov type spaces \(\dot{B}_{p,q}^{s,L,m}\) ${\dot{B}}_{p,q}^{s,L,m}$ are also investigated.