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

Jorge J. Betancor¹ ; Qingdong Guo² ; Dongyong Yang²

¹Universidad de La Laguna, Departamento de Análisis Matemático
²Xiamen University, School of Mathematical Sciences

Annales Fennici Mathematici, Tome 50 (2025) no. 1, p. 29–48

Résumé

  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.  

Détail

DOI : 10.54330/afm.155908

Keywords: Bessel operator, CMO(R_, dm_λ), Weinstein type equation, tent space, boundedness