Functional calculus of operators with heat kernel bounds on non-doubling manifolds with ends

Abstract

Let ∆ be the Laplace-Beltrami operator acting on a non-doubling manifold with two ends R m ♯R n with m > n ≥ 3. Let h t (x, y) be the kernels of the semigroup e -t∆ generated by ∆.We say that a non-negative self-adjoint operator L on L 2 (R m ♯R n ) has a heat kernel with upper bound of Gaussian type if the kernel h t (x, y) of the semigroup e -tL satisfies h t (x, y) ≤ Ch αt (x, y) for some constants C and α.This class of operators includes the Schrödinger operator L = ∆ + 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 and use them to show the weak type (1, 1) estimate for the holomorphic functional calculus M( √ L) where M(z) is a function of Laplace transform type.Our result covers the purely imaginary powers L is , s ∈ R, as a special case and serves as a model case for weak type (1, 1) estimates of singular integrals with non-smooth kernels on non-doubling spaces.If the measure µ satisfies the condition (1.4), then the space (R d , µ) is called a nonhomogeneous space.Calderón-Zygmund theory has been developed on such non-homogeneous spaces; see for example [22,23,24,29].For the BMO and H 1 function space, the Littlewood-Paley theory, and weighted norm inequalities on such non-homogeneous spaces, see [2,25,28,30]; for Morrey spaces, Besov spaces and Triebel-Lizorkin spaces in this setting, see [9,15,26].See also [2,18,19,20] for recent work in this direction which studies a more general setting for non-homogeneous analysis on metric spaces (X, d, µ), where (X, d) is said to be geometrically doubling.However, to obtain boundedness of singular integrals in this setting, one needs certain strong regularity on the associated kernels in terms of the upper doubling measure, i.e., r n as in (1.4) rather than µ(B(x, r)).For example Hölder continuity on the space variables of the kernels is needed for weak type (1, 1) estimate.• Singular integrals with non-smooth kernels: A lot of work has been carried out to study singular integrals whose associated kernels are not smooth enough to satisfy the Hörmander condition.Substantial progress has been made by X.

Locations

• Indiana University Mathematics Journal - View
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