Boundedness and Nuclearity of pseudo-differential operators on Homogeneous Trees

Let $\text{x}$ be the homogeneous tree of degree $q+1(q\ge 2)$. In this article, we present symbolic criteria for boundedness of pseudo-differential operators on homogeneous tree $\text{x}$. A sufficient condition for weak type $(p,q)$ boundedness is also given. We present necessary and sufficient conditions on the symbols $\sigma$ such that the corresponding pseudo-differential operators $T_{\sigma }$ from $L^{p_1}(\text{x})$ into $L^{p_2}(\text{x})$ to be nuclear for $1\leq p_1, p_2<\infty$\mathrm{.}Weshowthattheadjointofthenuclearoperatorfrom$L^{p_2’}(\x)$into$L^{p_1’}(\x)$ is again a nuclear operator. Explicit formulas for the symbol of the adjoint and product of the nuclear pseudo differential operators on $\text{x}$ are given.