Campanato Spaces via Quantum Markov Semigroups on Finite von Neumann Algebras

We study the Campanato spaces associated with quantum Markov semigroups on a finite von Neumann algebra M. Let T=(Tₜ)t≥ 0 be a Markov semigroup, P=(Pₜ)t≥ 0 the subordinated Poisson semigroup and α>0. The column Campanato space Lᶜα(P) associated to P is defined to be the subset of M with finite norm which is given by $$ {align*} \|f\|_{Lᶜα(P)}=\|f\|∞+t>0{1}{tα}\|Pₜ|(I-Pₜ)[α]+1f|²\|1/2∞. {align*} $$The row space ${Lʳα(P)}$ is defined in a canonical way. In this article, we will first show the surprising coincidence of these two spaces ${Lᶜα(P)}$ and ${Lʳα(P)}$ for $0<α <2$. 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 ${Λ α(P)}$. This coincidence passes to the little Campanato spaces $ ᶜα(P)$ and $ ʳα(P)$ for $0<α <1/2$ under the condition $Γ ²≥ 0$. We also show that any element in ${Lᶜα(P)}$ enjoys the higher-order cancellation property, that is, the index $[α ]+1$ in the definition of the Campanato norm can be replaced by any integer greater than $α $. 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 ${Lᶜα(T)}$ and explore their connection with ${Lᶜα(P)}$. All the above-mentioned results seem new even in the (semi-)commutative case.