Spectral multiplier theorems for abstract harmonic oscillators on UMD lattices

We consider operators acting on a UMD Banach lattice X that have the same algebraic structure as the position and momentum operators associated with the harmonic oscillator (Formula Presented) acting on L²(R^d). More precisely, we consider abstract harmonic oscillators of the form (Formula Presented) for tuples of...

Maximal inequalities for stochastic convolutions and pathwise uniform convergence of time discretisation schemes

We prove a new Burkholder–Rosenthal type inequality for discrete-time processes taking values in a 2-smooth Banach space. As a first application we prove that if (S(t, s)) _⩽_s_≤_t_⩽_T is a C-evolution family of contractions on a 2-smooth Banach space X and (Wt)t∈[0,T] is a...

Maximal inequalities for stochastic convolutions in 2-smooth Banach spaces and applications to stochastic evolution equations

This paper presents a survey of maximal inequalities for stochastic convolutions in 2-smooth Banach spaces and their applications to stochastic evolution equations. This article is part of the theme issue 'Semigroup applications everywhere'.

The weyl calculus for group generators satisfying the canonical commutation relations

We generalise the classical Weyl pseudo-differential calculus on R^d to the setting of two d-tuples of operators A = (A₁,..., A_d) and B = (B₁,..., B_d) acting on a Banach space generating bounded C0-groups satisfying the Weyl canonical commutation relations. We show that the resulting Weyl...