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

Let M be a non-doubling parabolic manifold with ends and L a non-negative self-adjoint operator on L2(M) which satisfies a suitable heat kernel upper bound named the upper bound of Gaussian type. These operators include the Schrödinger operators L = ∆ + V where ∆ 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 which is defined by M( √ L)f(x) := ́∞ 0 [√ Le−t √ Lf(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 Lis, 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.