Speaker：Ji Li（Macquarie University）

Time：2019-01-09 14:30

Location：Conference Room 108 at Experiment Building at Haiyun Campus

Abstract: Let Δ be the Laplace-Beltrami operator acting on a non-doubling manifold with two ends Rm#Rn with m > n ≥3. Let ht(x; y) be the kernels of the semigroup e-tΔgenerated by Δ. We say that a non-negative self-adjoint operator L on L2( Rm#Rn ) has a heat kernel with upper bound of Gaussian type if the kernel ht(x; y) of the semigroup e-tL satisfies ht(x; y)≤Chαt(x; y) for some constants C α. This class of operators includes the Schrödinger operator L = Δ + V where V is an arbitrary non-negative potential. We then obtain upper bounds of the Poisson semigroup kernel of L together with its time derivatives use them to show the weak type (1; 1) estimate for the holomorphic functional calculus M(L1/2) where M(z) is a function of Laplace transform type. Our result covers the purely imaginary powers Lis; s∈R, as a special case serves as a model case for weak type (1; 1) estimates of singular integrals with non-smooth kernels on non-doubling spaces. The results we provide here are based on recent result with The Anh Bui, Xuan Thinh Duong Brett D. Wick.