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As a tool for solving the Neumann problem for divergence-form equations, Kenig and Pipher introduced the space ${\mathcal{X}}$ of functions on the half-space, such that the non-tangential maximal function of their$L$_{2}Whitney averages belongs to$L$_{2}on the boundary. In this paper, answering questions which arose from recent studies of boundary value problems by Auscher and the second author, we find the pre-dual of ${\mathcal{X}}$ , and characterize the pointwise multipliers from ${\mathcal{X}}$ to$L$_{2}on the half-space as the well-known Carleson-type space of functions introduced by Dahlberg. We also extend these results to$L$_{$p$}generalizations of the space ${\mathcal{X}}$ . Our results elaborate on the well-known duality between Carleson measures and non-tangential maximal functions.

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The global existence and uniqueness of classical solutions to the Boltzmann equation without angular cut-off is proved under the condition that the initial data is in some Sobolev space. In addition, the solutions thus obtained are shown to be positive and C in all variables. One of the key observations is the introduction of a new important norm related to the singularity in the cross section of the collision operator. This norm captures the essential property of the singularity and yields precisely the dissipation description of the linearized collision operator through the celebrated H-theorem.

We prove generalized Fefferman-Stein type theorems on sharp functions with Ap weights in spaces of homogeneous type with either finite or infinite underlying measure. We then apply these results to establish mixed-norm weighted Lp-estimates for elliptic and parabolic equations/systems with (partially) BMO coefficients in regular or irregular domains.