Hilbert transform and Littlewood–Paley theory

Variational estimates for martingale paraproducts

“…. It also has a weighted version, formulated for instance as a part of Theorem 3.2.3 in the book [HvNVW16]:…”

“…The proof of (2.7) is based on (2.5) and it already exists as the proof of [HvNVW16, Theorem 3.2.7]. However, the working assumption in [HvNVW16] is that f (k) are arbitrary martingales with respect to the same filtration, which is not the case here. For this reason and for the sake of completeness we prefer to repeat the short argument, rather than just invoke the result from [HvNVW16].…”

“…By the homogeneity of (3.4) we can normalize: assume f L p = 1 and g L 1 = 1. Fix a number ν > 0 and perform Gundy's decomposition [Gun68] of the martingale g at height α = ν r ; see its formulation as Theorem 3.4.1 in the book [HvNVW16]. It splits g as g n = g good n + g bad n + g harmless n ,…”

On conditional expectations in $$L^p(\mu ;L^q(\nu ;X))$$ L p ( μ ; L q

“…The reader is referred to [2,6] for the basic theory of the Lebesgue-Bochner spaces and conditional expectations in these spaces. The same reference contains some standard results concerning the Radon-Nikodým property that will be needed later on.…”

$$L^p$$-Analysis of the Hodge–Dirac Operator Associated with Witten Laplacians on Complete Riemannian Manifolds

“…Motivated by certain square function estimates in harmonic analysis, the theory of R-boundedness was initiated in [18] and has found widespread use in various areas of analysis, among them parabolic PDE, harmonic analysis and stochastic analysis. We refer the reader to [21,35,36,40] for detailed accounts.…”