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 \(\Delta_{\lambda}\) on \(\mathbb{R}_+:=(0,\infty)\), where \(\lambda>0\) and   \(\Delta_{\lambda}:=-x^{-2\lambda}\frac{d}{dx}x^{2\lambda}\frac{d}{dx}\).   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\((\mathbb{R}_{+},dm_{\lambda})\), where \(dm_{\lambda}(x):=x^{2\lambda}\,dx\), if and only if \(u\) satisfies the Weinstein type equation   \(\mathbb{L}_{\lambda}u(x,t):=\frac{\partial^{2}u(x,t)}{\partial t^{2}}-\Delta_{\lambda}u(x,t)=0\), \((x,t)\in{\mathbb{R}_{+}\times\mathbb{R}_{+}}\),   a Carleson type condition and certain limiting conditions. For this purpose, the authors first introduce the tent spaces \(T_{2}^{p}\) with \(p\in[1,\infty]\) and \(T_{2,C}^{\infty}\) in the Bessel setting and then show that CMO\((\mathbb{R}_{+},dm_{\lambda})\) has a connection with \(T_{2,C}^{\infty}\) 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.