No abstract

We modify Hörmander's well-known weak type (1,1) condition for integral operators (in a weakened version due to Duong and McIntosh) and present a weak type (p,p) condition for arbitrary operators. Given an operator A on L_2 with a bounded H^\infty calculus, we show as an application the L_r -boundedness of the H^\infty calculus for all r\in(p,q) , provided the semigroup (e^{-tA}) satisfies suitable weighted L_p\to L_q -norm estimates with 2\in(p,q) . This generalizes results due to Duong, McIntosh and Robinson for the special case (p,q)=(1,\infty) where these weighted norm estimates are equivalent to Poisson-type heat kernel bounds for the semigroup (e^{-tA}) . Their results fail to apply in many situations where our improvement is still applicable, e.g. if A is a Schrödinger operator with a singular potential, an elliptic higher order operator with bounded measurable coefficients or an elliptic second order operator with singular lower order terms.

We prove comparison theorems for the H ∞ -calculus that allow to transfer the property of having a bounded H ∞ -calculus from one sectorial operator to another. The basic technical ingredient are suitable square function estimates. These comparison results provide a new approach to perturbation theorems for the H ∞ -calculus in a variety of situations suitable for applications. Our square function estimates also give rise to a new interpolation method, the Rademacher interpolation. We show that a bounded H ∞ -calculus is characterized by interpolation of the domains of fractional powers with respect to Rademacher interpolation. This leads to comparison and perturbation results for operators defined in interpolation scales such as the L pscale. We apply the results to give new proofs on the H ∞ -calculus for elliptic differential operators, including Schrödinger operators and perturbed boundary conditions. As new results we prove that elliptic boundary value problems with bounded uniformly coefficients have a bounded H ∞ -calculus in certain Sobolev spaces and that the Stokes operator on bounded domains with ∂ ∈ C 1,1 has a bounded H ∞ -calculus in the Helmholtz scale L p,σ ( ), p ∈ (1, ∞).

We consider the problem of L p -boundedness of higher order Riesz transforms ∇ m L −1/2 associated to elliptic operators L of order 2m on R D . As an application of the recently solved Kato conjecture, we show ∇ m L −1/2 ∈ L(L p (R D )) for all p ∈ ( 2D 2m+D ∨1 , 2]. This generalizes the result of Auscher and Tchamitchian restricted to the case D ≤ 2m.

Abstract. Let X be a space of homogeneous type and let L be an injective, non-negative, selfadjoint operator on L 2 (X) such that the semigroup generated by −L fulfills Davies-Gaffney estimates of arbitrary order. We prove that the operator, acts as a bounded linear operator on the Hardy space H 1 L (X) associated with L whenever F is a bounded, sufficiently smooth function. Based on this result, together with interpolation, we establish Hörmander type spectral multiplier theorems on Lebesgue spaces for non-negative, self-adjoint operators satisfying generalized Gaussian estimates in which the required differentiability order is relaxed compared to all known spectral multiplier results.