Boundedness and Nuclearity of pseudo-differential operators on Homogeneous Trees

Let $\x$ be the homogeneous tree of degree $q+1~(q\geq2)$. In this article, we present symbolic criteria for boundedness of pseudo-differential operators on homogeneous tree $\x$. 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 $L^{p_1} (\x)$ into $L^{p_2} (\x)$ to be nuclear for $1\leq p_1, p_2<\infty$. We show that the adjoint of the nuclear operator from $L^{p_2'}(\x)$ 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 $\x$ are given.