Off-Diagonal Estimates for Bi-Commutators

Abstract

We study the bi-commutators $[T_1, [b, T_2]]$ of pointwise multiplication and Calderón–Zygmund operators and characterize their $L^{p_1}L^{p_2} \to L^{q_1}L^{q_2}$ boundedness for several off-diagonal regimes of the mixed-norm integrability exponents $(p_1,p_2)\neq (q_1,q_2)$. The strategy is based on a bi-parameter version of the recent approximate weak factorization method.

Lower bound of the parabolic Hilbert commutator

“…Notice that b BMO β (R d ) depends only on the vector β and not on ε, δ ∈ {−1, 1} d . With this notation and γ(t) = (t, t 2 ) we have BMO γ = BMO (1,2) . For Theorem 1.3 the extension is the following.…”

“…by a finite number of dyadic operators over some anisotropic dyadic grids D i β . Such grids are constructed at least in [3], and these grids have all the important properties that we would expect of a dyadic grid, hence, a standard argument shows that M D i β is of weak type (1,1). With this detail in the clear, we estimate the second of the martingale differences as…”

“…multi-parameter and multilinear extensions. For the multiparameter variants of the awf argument see Airta, Hytönen, Li, Martikainen, Oikari [1] and Oikari [15], where, respectfully, the commutators…”

“…The settings considered in all the aforementioned research articles, apart from [1] which treats an iterated commutator of product nature, are such that the dimension of the ambient space R d is the same as that of the singular integral. Detaching from this, we consider singular integrals in the plane that are lower dimensional compared to the functions they hit, i.e., the kernel is localized to a curve.…”

Off-diagonal estimates for bilinear commutators

“…The awf approach for off-diagonal estimates has been generalized to the bi-parameter setting in Airta, Hytönen, Li, Martikainen and Oikari [1] and Oikari [19]. In [1] and [19], respectfully, the commutators T 2 , [b, T 1 ] , b, T : L p 1 (R d 1 ; L p 2 (R d 2 )) → L q 1 (R d 1 ; L q 2 (R d 2 )), (1.6) where 1 < p 1 , p 2 , q 1 , q 2 < ∞, and T 1 , T 2 are non-degenerate one-parameter CZOs and T is a non-degenerate bi-parameter CZO, were treated. The adaptability of the awf method to the bi-parameter settings was not effortless and for both of the commutator on the line (1.6) the characterization of some cases is still open.…”