Campanato spaces via quantum Markov semigroups on finite von Neumann algebras

Abstract

We study the Campanato spaces associated with quantum Markov semigroups on a finite von Neumann algebra $\mathcal{M}$. Let $\mathcal{T}=(T_t{)}_{t\ge 0}$ be a Markov semigroup, $\mathcal{P}=(P_t{)}_{t\ge 0}$ the subordinated Poisson semigroup and $\alpha >0$. The column Campanato space ${\mathcal{L}}_{\alpha }^c(\mathcal{P})$ 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}_α(\mathcal{P})}=\left\|f\right\|_{\infty}+\sup_{t>0}\frac{1}{t^α}\left\|P_{t}|(I-P_{t})^{[α]+1}f|^{2}\right\|^{\frac{1}{2}}_{\infty}. \end{align*} The row space ${\mathcal{L}}_{\alpha }^r(\mathcal{P})$ is defined in a canonical way. In this article, we will first show the surprising coincidence of these two spaces ${\mathcal{L}}_{\alpha }^c(\mathcal{P})$ and ${\mathcal{L}}_{\alpha }^r(\mathcal{P})$ 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 ${\ell }_{\alpha }^c(\mathcal{P})$ and ${\ell }_{\alpha }^r(\mathcal{P})$ for $0<\alpha <\frac{1}{2}$ under the condition ${\Gamma }^2\ge 0$. We also show that any element in ${\mathcal{L}}_{\alpha }^c(\mathcal{P})$ 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 ${\mathcal{L}}_{\alpha }^c(\mathcal{T})$ and explore their connection with ${\mathcal{L}}_{\alpha }^c(\mathcal{P})$. All the above-mentioned results seem new even in the (semi-)commutative case.