Campanato spaces via quantum Markov semigroups on finite von Neumann algebras

TL;DR

This paper investigates noncommutative Campanato spaces linked to quantum Markov semigroups on finite von Neumann algebras, revealing surprising equivalences and properties that extend classical results to the quantum setting.

Contribution

It establishes the equivalence of column and row Campanato spaces for certain parameters, introduces higher order cancellation properties, and connects these spaces with Lipschitz and BMO spaces in the quantum context.

Findings

Column and row spaces coincide for 0<α<2.

Higher order cancellation property holds without extra conditions.

Connections with Lipschitz and BMO spaces are established.

Abstract

We study the Campanato spaces associated with quantum Markov semigroups on a finite von Neumann algebra $\mathcal M$. Let $\mathcal T=(T_{t})_{t\geq0}$ be a Markov semigroup, $\mathcal P=(P_{t})_{t\geq0}$ the subordinated Poisson semigroup and $\alpha>0$. The column Campanato space ${\mathcal{L}^{c}_{\alpha}(\mathcal{P})}$ associated to $\mathcal P$ is defined to be the subset of $\mathcal M$ with finite norm which is given by \begin{align*} \|f\|_{\mathcal{L}^{c}_{\alpha}(\mathcal{P})}=\left\|f\right\|_{\infty}+\sup_{t>0}\frac{1}{t^{\alpha}}\left\|P_{t}|(I-P_{t})^{[\alpha]+1}f|^{2}\right\|^{\frac{1}{2}}_{\infty}. \end{align*} The row space ${\mathcal{L}^{r}_{\alpha}(\mathcal{P})}$ is defined in a canonical way. In this article, we will first show the surprising coincidence of these two spaces ${\mathcal{L}^{c}_{\alpha}(\mathcal{P})}$ and ${\mathcal{L}^{r}_{\alpha}(\mathcal{P})}$ for $0<\alpha<2$. This equivalence of column and row norms is generally unexpected in the noncommutative setting. The approach is to identify both of them as the Lipschitz space ${\Lambda_{\alpha}(\mathcal{P})}$. This coincidence passes to the little Campanato spaces $\ell^{c}_{\alpha}(\mathcal{P})$ and $\ell^{r}_{\alpha}(\mathcal{P})$ for $0<\alpha<\frac{1}{2}$ under the condition $\Gamma^{2}\geq0$. We also show that any element in ${\mathcal{L}^{c}_{\alpha}(\mathcal{P})}$ enjoys the higher order cancellation property, that is, the index $[\alpha]+1$ in the definition of the Campanato norm can be replaced by any integer greater than $\alpha$. It is a surprise that this property holds without further condition on the semigroup. Lastly, following Mei's work on BMO, we also introduce the spaces ${\mathcal{L}^{c}_{\alpha}(\mathcal{T})}$ and explore their connection with ${\mathcal{L}^{c}_{\alpha}(\mathcal{P})}$. All the above-mentioned results seem new even in the (semi-)commutative case.