Weak Type (1,1) Bounds for Schrodinger Groups

Let L be a nonnegative self-adjoint operator acting on L-2(X), where X is a space of homogeneous type of dimension n. Suppose that the heat kernel of L satisfies a Gaussian upper bound. It is known that the operator (I + L)(-s)eitL is bounded on L-p(X) for s > n|1/2 - 1/p| and p is an element of (1, infinity) (see, e.g., [7; 22; 33]). The index s = n|1/2-1/p| was only obtained recently in [9; 10], and this range of s is sharp since it is precisely the range known in the case where L is the Laplace operator Delta on X = R-n[30]. In this paper, we establish that for p = 1, the operator (1 + L)(-n/2)eitL is of weak type (1, 1), that is, there is a constant C, independent oft and f, such that mu({x:|(I +L)(-n/2)eitL f (x)| > lambda}) <= C lambda-1(1 + |t|)(n/2) ||f vertical bar vertical bar(L)1(X), t is an element of R (for lambda > 0 when mu(X) = infinity and lambda > mu(X)(-1) ||f || (L)1(X) when mu(X) < infinity). Moreover, we also show that the index n/2 is sharp when L is the Laplacian on R-n by providing an example. Our results are applicable to Schrodinger groups for large classes of operators including elliptic operators on compact manifolds, Schrodinger operators with nonnegative potentials and Laplace operators acting on Lie groups of polynomial growth or irregular non-doubling domains of Euclidean spaces.