Spectral multiplier theorems for abstract harmonic oscillators on UMD lattices

TL;DR

This paper establishes spectral multiplier theorems for abstract harmonic oscillators on UMD Banach lattices, extending classical results to non-doubling spaces and non-Hilbert settings, with applications to Bargmann-Fock spaces.

Contribution

It introduces a functional calculus for abstract harmonic oscillators on UMD lattices, generalizing classical spectral multiplier estimates to broader, non-Hilbert, non-doubling contexts.

Findings

Proves spectral multiplier theorems for abstract harmonic oscillators on UMD lattices.

Extends classical Hörmander multiplier results to non-doubling metric measure spaces.

Applies results to harmonic oscillators on mixed norm Bargmann-Fock spaces.

Abstract

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 $-\frac12\Delta + \frac12|x|^{2} $ acting on $L^{2}(\mathbb{R}^{d})$. More precisely, we consider abstract harmonic oscillators of the form $\frac12 \sum _{j=1} ^{d}(A_{j}^{2}+B_{j}^{2})$ for tuples of operators $A=(A_{j})_{j=1} ^{d}$ and $B=(B_{k})_{k=1} ^{d}$, where $iA_j$ and $iB_k$ are assumed to generate $C_{0}$ groups and to satisfy the canonical commutator relations. We prove functional calculus results for these abstract harmonic oscillators that match classical H\"ormander spectral multiplier estimates for the harmonic oscillator $-\frac12\Delta + \frac12|x|^{2}$ on $L^{p}(\mathbb{R}^{d})$. This covers situations where the underlying metric measure space is not doubling and the use of function spaces that are not particularly well suited to extrapolation arguments. For instance, as an application we treat the harmonic oscillator on mixed norm Bargmann-Fock spaces. Our approach is based on a transference principle for the Schr\"odinger representation of the Heisenberg group that allows us to reduce the problem to the study of the twisted Laplacian on the Bochner spaces $L^{2}(\mathbb{R}^{2d};X)$. This can be seen as a generalisation of the Stone-von Neumann theorem to UMD lattices $X$ that are not Hilbert spaces.