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=(T-t)(t >= 0) be a Markov semigroup, P=(P-t)(t >= 0) the subordinated Poisson semigroup and alpha>0. The column Campanato space L alpha c(P) associated to P is defined to be the subset of M with finite norm which is given by & Vert;f & Vert;L-alpha (c)(P)=& Vert;f & Vert;(infinity)+sup(t>0)(1)/(t)alpha & Vert;P-t|(I-P-t)([alpha]+1)f|(2)& Vert;(1/2)(infinity). The row space L-alpha (R)(P) is defined in a canonical way. In this article, we will first show the surprising coincidence of these two spaces L-alpha (c)(P) and L-alpha (R)(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(P). This coincidence passes to the little Campanato spaces & ell;(alpha)(c)(P) and & ell;(alpha)(R)(P) for 0<alpha<(1)/(2 )under the condition Gamma(2)>= 0. We also show that any element in L-alpha (c)(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 L-alpha (c)(T) and explore their connection with L-alpha (c)(P). All the above-mentioned results seem new even in the (semi-)commutative case.