2024/10/14 by Cao, Mingming, Yabuta, Kôzô
#42B20 #42B35 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics
paper · doi:10.48550/arxiv.2410.10304
We develop the compactness theory of multilinear singular integrals on product spaces using a modern point of view. The first main result is a compact T1 theorem for multilinear Calderón--Zygmund operators on product spaces. More specifically, we prove that a multilinear singular integral operator T on product spaces can be extended to a compact multilinear operator from Lp1(w1p1) × ⋯ × Lpm(wmpm) to Lp(wp) for all exponents \frac1p = ∑j=1m (1)/(pj)>0 with p1, …, pm ∈ (1, ∞] and for all weights w ∈ A_p(ℝn1 × ℝn2) if the following hypotheses are satisfied: (H1) T admits a compact full kernel representation, (H2) T admits a compact partial kernel representation, (H3) T satisfies the weak compactness property, (H4) T satisfies the diagonal CMO condition, and (H5) T satisfies the product CMO condition. This is a multilinear compact extension of Journé's T1 theorem on product spaces. The second main result establishes the mean continuity of commutators [\boldsymbolb, T]\boldsymbolα on weighted Lebesgue spaces as above, which can be viewed as a substitution of compactness because the compactness of [\boldsymbolb, T]\boldsymbolα is equivalent to \boldsymbolb ≡ constant when T is a non-degenerate bi-parameter singular integral. Our main tools include multilinear bi-parameter dyadic representation, multilinear extrapolation, multilinear interpolation, and Kolmogorov--Riesz compactness criterion.