A counterexample to a multilinear endpoint question of Christ and Kiselev

Christ-Kiselev极大定理的多参数推广：强变差界

Multiparameter extensions of the Christ-Kiselev maximal theorem: strong variational bounds

本文证明了Christ-Kiselev极大定理的多参数强变差界，并推广到多线性算子情形。

Multilinear singular integral forms of Christ-Journé type

We prove $L^{p_1}(\mathbb R^d)\times \dots \times L^{p_{n+2}}(\mathbb R^{d})$ polynomial growth estimates for the Christ-Journé multilinear singular integral forms and suitable generalizations.

Multilinear Oscillatory Integrals Via Reduction of Dimension

Christ, Li, Tao, and Thiele have established multilinear oscillatory integral operator inequalities under severe dimensional restrictions. These restrictions are relaxed substantially in the present paper, but are by no means eliminated. The simple method used here has the advantage of yielding power type bounds, but cannot be expected to extend to the general case.

A sharp nonlinear Hausdorff-Young inequality for small potentials

The nonlinear Hausdorff-Young inequality follows from the work of Christ and Kiselev. Later Muscalu, Tao, and Thiele asked if the constants can be chosen independently of the exponent. We show that the nonlinear Hausdorff-Young quotient admits an even better upper bound than the linear one, provided that the function is sufficiently small in the $L^1$ norm. The proof combines perturbative techniques with the sharpened version of the linear Hausdorff-Young inequality due to Christ.

Endpoint estimates for the commutators of multilinear Calderón-Zygmund operators with Dini type kernels

Let $T_{\vec{b}}$ and $T_{Πb}$ be the commutators in the $j$-th entry and iterated commutators of the multilinear Calderón-Zygmund operators, respectively. It was well-known that $T_{\vec{b}}$ and $T_{Πb}$ were not of weak type $(1,1)$ and $(H^1, L^1)$, but they did satisfy certain endpoint $L\log L$ type estimates. In this paper, our aim is to give more natural sharp endpoint results. We show that $T_{\vec{b}}$ and $T_{Πb}$ are bounded from product Hardy space $H^1\times\cdot\cdot\cdot\times H^1$ to weak $L^{\frac{1}{m},\infty}$ space, whenever the kernel satisfies a class of Dini type condit…

On Holder-Brascamp-Lieb inequalities for torsion-free discrete Abelian groups

Hölder-Brascamp-Lieb inequalities provide upper bounds for a class of multilinear expressions, in terms of $L^p$ norms of the functions involved. They have been extensively studied for functions defined on Euclidean spaces. Bennett-Carbery-Christ-Tao have initiated the study of these inequalities for discrete Abelian groups and, in terms of suitable data, have characterized the set of all tuples of exponents for which such an inequality holds for specified data, as the convex polyhedron defined by a particular finite set of affine inequalities. In this paper we advance the theory of such inequ…