Estimates for functions of the Laplace operator on homogeneous trees

In this paper, we study the heat equation on a homogeneous graph, relative to the natural (nearest–neighbour) Laplacian. We find pointwise estimates for the heat and resolvent kernels, and the L p − L q L^{p}-L^{q} mapping properties of the corresponding operators

On the H^1-L^1 boundedness of operators

We prove that if q is in (1,\infty), Y is a Banach space and T is a linear operator defined on the space of finite linear combinations of (1,q)-atoms in R^n which is uniformly bounded on (1,q)-atoms, then T admits a unique continuous extension to a bounded linear operator from H^1(R^n) to Y. We show that the same is true if we replace (1,q)-atoms with continuous (1,\infty)-atoms. This is known to be false for (1,\infty)-atoms.Comment: This paper will appear in Proceedings of the American Mathematical Societ

Endpoint results for spherical multipliers on noncompact symmetric spaces

In this paper we prove boundedness results on atomic Hardy type spaces for multipliers of the spherical transform on noncompact symmetric spaces of arbitrary rank. The multipliers we consider satisfy either inhomogeneous or homogeneous Mihlin-H\uf6rmander type conditions. In particular, we are able to treat the case of {strongly singular multipliers} whose convolution kernels are not integrable at infinity. Thus our results apply also to negative and imaginary powers of the Laplacian

Maximal characterisation of local Hardy spaces on locally doubling manifolds

We prove a radial maximal function characterisation of the local atomic Hardy space h^1(M) on a Riemannian manifold M with positive injectivity radius and Ricci curvature bounded from below. As a consequence, we show that an integrable function belongs to h^1(M) if and only if either its local heat maximal function or its local Poisson maximal function are integrable. A key ingredient is a decomposition of H\"older cut-offs in terms of an appropriate class of approximations of the identity, which we obtain on arbitrary Ahlfors-regular metric measure spaces and generalises a previous result of A. Uchiyama.Comment: 31 page