Analysis Seminar: Approximation in Hölder spaces with applications to the compactness of the bi-commutator

Abstract: We discuss commutators of Calderón-Zygmund operators. It is classical that the Lp-to-Lp bounded commutators are those with symbols in BMO; that Lp-to-Lp compact commutators are those with symbols in VMO, a subspace of BMO; and that VMO coincides with those BMO symbols approximable by N := compactly supported smooth functions.

 More recently, it was shown that all non-trivial Lp-to-Lq compact commutators, when p<q, are exactly those bounded commutators with symbols in Hölder classes approximable by N. A parallel holds in the case p>q. Thus, the high-level structure of the Lp-to-Lp case was fully recovered in the Lp-to-Lq cases, when p is not q. It was later noted that in fact all the sufficient conditions for compactness in the cases “p<q”, “p=q” and “p>q” follow from each other by a simple abstract principle coupled with the technically difficult fact that N constitutes a common shared dense subspace of all the symbol classes for boundedness.

The following is morally true: to understand the Lp-to-Lq compactness of commutators, with p,q ordered anyhow, it is enough to completely understand:

      - approximability by nice functions N in ALL the different symbol classes that characterize boundedness,

      - compactness only in ONE of the ranges “p<q”, “p=q” and “p>q”.

In this talk, we discuss how the above two-step scheme can be realized in the study of the mixed Lp(Lq)-to-Ls(Lt) compactness of the bi-commutator [T2,[b,T1]]. Among other novel results, we have obtained a neat proof of the sufficiency of product VMO for bi-commutator compactness.

Joint work with Carlos Mudarra and Henri Martikainen.

Host: Walton Green