El Maati Ouhabaz

Plancherel-type estimates and sharp spectral multipliers

Abstract We study general spectral multiplier theorems for self-adjoint positive definite operators on L2(X,μ), where X is any open subset of a space of homogeneous type. We show that the sharp Hormander-type spectral multiplier theorems follow from the appropriate estimates of the L2 norm of the kernel of spectral multipliers and the Gaussian bounds for the corresponding heat kernel. The sharp Hormander-type spectral multiplier theorems are motivated and connected with sharp estimates for the critical exponent for the Riesz means summability, which we also study here. We discuss several examples, which include sharp spectral multiplier theorems for a class of scattering operators on R3 and new spectral multiplier theorems for the Laguerre and Hermite expansions.

Invariance of closed convex sets and domination criteria for semigroups

Let a and b be two positive continuous and closed sesquilinear forms on the Hilbert space H=L2(Ω, μ). Denote by T=T(t)t≧0and S=S(t)t≧0the semigroups generated by a and b on H. We give criteria in terms of a and b guaranteeing that the semigroup T is dominated by S, i.e. |T(t)f|≦S(t)|f| for all t≧0 and f∈H. The method proposed uses ideas on invariance of closed convex sets of H under semigroups. Applications to elliptic operators and concrete examples are given.

Maximal regularity for non-autonomous evolution equations

AbstractWe consider the maximal regularity problem for non-autonomous evolution equations 1Each operator

Sharp Gaussian bounds and ^{}-growth of semigroups associated with elliptic and Schrödinger operators

A(t)

The heat kernel of a Schrödinger operator with inverse square potential

A(t) is associated with a sesquilinear form

A spectral multiplier theorem for non-self-adjoint operators

{\mathop {{\varvec{\mathfrak {a}}}}(t; \cdot , \cdot )}