2024/12/03 by Kangwei Li, Li, Kangwei, Henri Martikainen +1
Mathematics · #42B20 #Advanced Topics in Algebra #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Mathematical and Theoretical Analysis #advanced mathematical theories
paper · pdf · doi:10.48550/arxiv.2412.02497
openalex publication_date 2024/12/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We complement the recent theory of general singular integrals T invariant under the Zygmund dilations (x1, x2, x3) ↦ (s x1, tx2, st x3) by proving necessary and sufficient conditions for the boundedness and compactness of commutators [b,T] from Lp → Lq. Previously, only the p=q upper bound in terms of a Zygmund type little BMO space was known for general operators, and it appears that there has been some confusion about the corresponding lower bound in recent literature. We give complete characterizations whenever p ≤ q for a general class of non-degenerate Zygmund type singular integrals. Some of the results are somewhat surprising in view of existing papers - for instance, compactness always forces b to be constant. Even in the simpler situation of bi-parameter singular integrals it appears that this has not been observed previously.