Tent spaces and solutions of Weinstein type equations with CMO(R_+,dm_λ) boundary values

Abstract

Let $\{P_t^{[\lambda ]}{\}}_{t>0}$ be the Poisson semigroup associated with the Bessel operator ${\mathrm{\Delta }}_{\lambda }$ on ${\mathbb{R}}_{+}:=(0,\mathrm{\infty })$, where $\lambda >0$ and   ${\mathrm{\Delta }}_{\lambda }:=-x^{-2\lambda }\frac{d}{dx}{x}^{2\lambda }\frac{d}{dx}$.   In this paper, the authors show that a function $u(y,t)$ on ${\mathbb{R}}_{+}\times {\mathbb{R}}_{+}$, has the form $u(y,t)=P_t^{[\lambda ]}f(y)$ with $f\in$ CMO$({\mathbb{R}}_{+},d{m}_{\lambda })$, where $d{m}_{\lambda }(x):=x^{2\lambda }dx$, if and only if $u$ satisfies the Weinstein type equation   ${\mathbb{L}}_{\lambda }u(x,t):=\frac{{\partial }^{2}u(x,t)}{\partial t^2}-{\mathrm{\Delta }}_{\lambda }u(x,t)=0$, $(x,t)\in {\mathbb{R}}_{+}\times {\mathbb{R}}_{+}$,   a Carleson type condition and certain limiting conditions. For this purpose, the authors first introduce the tent spaces $T_2^p$ with $p\in [1,\mathrm{\infty }]$ and $T_{2,C}^{\mathrm{\infty }}$ in the Bessel setting and then show that CMO$({\mathbb{R}}_{+},d{m}_{\lambda })$ has a connection with $T_{2,C}^{\mathrm{\infty }}$ via $\{P_t^{[\lambda ]}{\}}_{t>0}$. In addition, the authors obtain some boundedness results on the operator ${\pi }_{\lambda }$ from tent spaces to some "ordinary" function spaces.