Boundedness and nuclearity of pseudo-differential operators on homogeneous trees

Abstract

Let \(\mathfrak X\) 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 \(\mathfrak X\). 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 \(L^{p_1} (\mathfrak X)\) into \(L^{p_2} (\mathfrak X)\) to be nuclear for \(1\le p_1, p_2<\infty \). Explicit formulas for the symbol of the adjoint and product of the nuclear pseudo differential operators on \(\mathfrak X\) are given.