Functional calculus of Laplace transform type on non-doubling parabolic manifolds with ends

摘要

Let M be a non-doubling parabolic manifold with ends and L a non-negative self-adjoint operator on L~2(M) which satisfies a suitable heat kernel upper bound named the upper bound of Gaussian type. These operators include the Schrodinger operators L = △+V where A is the Laplace-Beltrami operator and V is an arbitrary non-negative potential. This paper will investigate the behaviour of the Poisson semi-group kernels of L together with its time derivatives and then apply them to obtain the weak type (1,1) estimate of the functional calculus of Laplace transform type of L~(1/2) which is denned by ♍(L~(1/2))f(ⅹ) := ∫_0~∞ [L~(1/2)e~(-tL~(1/2))f(ⅹ)] m(t)dt where m(t) is a bounded function on [0,∞). In the setting of our study, both doubling condition of the measure on M and the smoothness of the operators' kernels are missing. The purely imaginary power L~(is),s∈ R, is a special case of our result and an example of weak type (1,1) estimates of a singular integral with non-smooth kernels on non-doubling spaces.