Weighted Endpoint Estimates for Singular Integral Operators Associated with Zygmund Dilations

Multiresolution analysis and Zygmund dilations

“…This limitation was recently addressed by Han et al [10] who introduced a new set of (1.6)-invariant conditions on convolution kernels, involving only minimal Hölder-continuity type smoothness assumptions analogous to those in the classical theory, and proved that the corresponding class of SIOs is well-defined and bounded on (the unweighted) L p . (See also [11] for related end-point estimates, although these go to a somewhat different direction than our presents goals.) In particular, the kernel size bound of [10] takes the form…”

“…In other words, for the first time in the Zygmund dilation setting, we deal with kernels of the general form K(x, y) instead of K(x − y), and we obtain a (special) T (1)-type theorem in this framework. The assumptions on the kernel consists of a natural generalisation of the conditions of [10], and indeed we will check (see Proposition 4.19) that the operators considered in [6,10,11] are a special case of those that we deal with. (For kernel size and smoothness, this will be fairly obvious, but the connections between the cancellation assumptions in the two settings are not as straightforward.…”

“…Besides describing rather precise limits for the weighted theory of the class of singular convolution operators introduced by [10], we actually go a step further by dropping the translation-invariance assumption on the operators present in all previous related contributions [6,10,11]. In other words, for the first time in the Zygmund dilation setting, we deal with kernels of the general form K(x, y) instead of K(x − y), and we obtain a (special) T (1)-type theorem in this framework.…”

“…Achieving this level of generality depends on a methodological change of paradigm compared to the previous contributions [6,7,10,11] in the Zygmund dilation setting. In short, our approach consists of developing a multiresolution analysis, and more specifically a dyadic representation theorem, in this framework.…”

Exotic Calderón-Zygmund operators

“…Zygmund dilations are a group of dilations lying in between the standard product theory and the one-parameter setting -in R 3 = R × R 2 they are the dilations (x 1 , x 2 , x 3 ) → (δ 1 x 1 , δ 2 x 2 , δ 1 δ 2 x 3 ). Recently, in [7] and subsequently in [3,8] general convolution form singular integrals invariant under Zygmund dilations were studied. In these papers the decay factor…”