From refined estimates for spherical harmonics to a sharp multiplier theorem on the Grushin sphere

We prove a sharp multiplier theorem of Mihlin–Hörmander type for the Grushin operator on the unit sphere in R 3 , and a corresponding boundedness result for the associated Bochner–Riesz means. The proof hinges on precise pointwise bounds for spherical harmonics

On the maximal operator of a general Ornstein--Uhlenbeck semigroup

If Q is a real, symmetric and positive definite nxn matrix, and B a real nxn matrix whose eigenvalues have negative real parts, we consider the Ornstein--Uhlenbeck semigroup on R^n with covariance Q and drift matrix B. Our main result is that the associated maximal operator is of weak type (1,1) with respect to the invariant measure. The proof has a geometric gist and hinges on the ``forbidden zones method'' previously introduced by the third author. For large values of the time parameter, we also prove a refinement of this result, in the spirit of a conjecture due to Talagrand

Spectral multipliers for the Kohn Laplacian on forms on the sphere in Cn

The unit sphere S in Cn is equipped with the tangential Cauchy–Riemann complex and the associated Laplacian □ b. We prove a Hörmander spectral multiplier theorem for □ b with critical index n- 1 / 2 , that is, half the topological dimension of S. Our proof is mainly based on representation theory and on a detailed analysis of the spaces of differential forms on S

Stability and Convergence of Product Formulas for Operator Matrices

We present easy to verify conditions implying stability estimates for operator matrix splittings which ensure convergence of the associated Trotter, Strang and weighted product formulas. The results are applied to inhomogeneous abstract Cauchy problems and to boundary feedback systems.Comment: to appear in Integral Equations and Operator Theory (ISSN: 1420-8989