方向平方函数

Square functions for bi-Lipschitz maps and directional operators

First we prove a Littlewood-Paley diagonalization result for bi-Lipschitz perturbations of the identity map on the real line. This result entails a number of corollaries for the Hilbert transform along lines and monomial curves in the plane. Second, we prove a square function bound for a single scale directional operator. As a corollary we give a new proof of part of a theorem of Katz on direction fields with finitely many directions.

Directional square functions and a sharp Meyer lemma

Quantitative formulations of Fefferman's counterexample for the ball multiplier are naturally linked to square function and vector-valued estimates for directional singular integrals. The latter are usually referred to as Meyer-type lemmas and are traditionally attacked by combining weighted inequalities with sharp estimates for maximal directional averaging operators. This classical approach fails to give sharp bounds. In this article we develop a novel framework for square function estimates, based on directional Carleson embedding theorems and multi-parameter time-frequency analysis, which …

某些方向极大函数的几乎正交性原理

Almost-orthogonality principles for certain directional maximal functions

本文为线段平均和方向奇异积分相关的极大函数建立几乎正交性原理，并由此在方向集均匀分布时得到尖锐的$L^2$界。

Directional maximal function along the primes

We study a two-dimensional discrete directional maximal operator along the set of the prime numbers. We show existence of a set of vectors, which are lattice points in a sufficiently large annulus, for which the $\ell^2$ norm of the associated maximal operator with supremum taken over all large scales grows with an epsilon power in the number of vectors. This paper is a follow-up to a prior work on the discrete directional maximal operator along the integers by the first and third author.

Directional convexity and characterizations of Beta and Gamma functions

The logarithmic convexity of restrictions of the Beta functions to rays parallel to the main diagonal and the functional equation \[ ϕ\left( x+1\right) =\frac{x\left( x+k\right) }{\left( 2x+k+1\right) \left( 2x+k\right) }ϕ\left( x\right) ,\ \ \ \ \ \ x>0, \] for $k>0$ allow to get a characterizations of the Beta function. This fact and a notion of the beta-type function lead to a new characterization of the Gamma function.

Discrete Analogues in Harmonic Analysis: Directional Maximal Functions in $\mathbb{Z}^2$

Let $V = \{ v_1,\dots,v_N\}$ be a collection of $N$ vectors that live near a discrete sphere. We consider discrete directional maximal functions on $\mathbb{Z}^2$ where the set of directions lies in $V$, given by \[ \sup_{v \in V, k \geq C \log N} \left| \sum_{n \in \mathbb{Z}} f(x-v\cdot n ) \cdot ϕ_k(n) \right|, \ f:\mathbb{Z}^2 \to \mathbb{C}, \] where and $ϕ_k(t) := 2^{-k} ϕ(2^{-k} t)$ for some bump function $ϕ$. Interestingly, the study of these operators leads one to consider an "arithmetic version" of a Kakeya-type problem in the plane, which we approach using a combination of geometric…