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 M. Let T= (Tₓ) ₓ ₀ be a Markov semigroup, P= (Pₓ) ₓ ₀ the subordinated Poisson semigroup and 0. The column Campanato space L^{c (P) } associated to P is defined to be the subset of M with finite norm which is given by align* \|f\|₋^₂_{ (P) }=\|f\|_+ₓ₀1t^{}\|Pₓ| (I-Pₓ) ^+1f|^2\|^1{2}_. align* The row space L^{r (P) } is defined in a canonical way. In this article, we will first show the surprising coincidence of these two spaces L^{c (P) } and L^{r (P) } for 0 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 (P). This coincidence passes to the little Campanato spaces ^c (P) and ^r (P) for 0 12 under the condition ^2 0. We also show that any element in L^{c (P) } enjoys the higher-order cancellation property, that is, the index +1 in the definition of the Campanato norm can be replaced by any integer greater than. 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 L^{c (T) } and explore their connection with L^{c (P) }. All the above-mentioned results seem new even in the (semi-) commutative case.