关于differential forms上Beurling-Ahlfors算子的新估计

格点上Beurling--Ahlfors算子虚部的sharp $L^p$ 界

The sharp $L^p$ bound for the imaginary part of the Beurling-Ahlfors operator on the lattice

本文证明了二维整数格上离散Beurling--Ahlfors算子虚部的sharp $\ell^p$ 范数界为 $p^\star-1$，通过鞅表示但需克服强微分从属缺失的困难。

尖锐的高阶可积性理论，第一部分：Beurling--Ahlfors 变换的 $L^p$ 范数

Sharp Higher Integrability Theory, Part I: The $L^p$-norm of the Beurling--Ahlfors transform

本文证实了 Iwaniec 1982 年关于 Beurling--Ahlfors 变换 $L^p$ 范数的猜想，给出了 $p\geq 2$ 与 $1<p\leq 2$ 时的精确值。

Sharp Lower bound estimates for vector-valued and matrix-valued multipliers in $L^p$

We generalize the idea of a multiplier in two different ways and generalize a recent result of Geiss, Montomery-Smith and Saksman. First of all, we consider multipliers in the form of a vector acting on a scalar function. Using this technique we compute the sharp lower bound estimate for $L^p$ operator norm of a quadratic perturbation of the real part of the Ahlfors-Beurling operator. Secondly, we consider matrix-valued multipliers to obtain a new proof showing that the $L^p$ operator norm of the Ahlfors-Beurling operator is bounded below by p^*-1.

Sharp weighted estimates for approximating dyadic operators

We give a new proof of the sharp weighted $L^2$ inequality ||T||_{L^2(w)} \leq c [w]_{A_2} where $T$ is the Hilbert transform, a Riesz transform, the Beurling-Ahlfors operator or any operator that can be approximated by Haar shift operators. Our proof avoids the Bellman function technique and two weight norm inequalities. We use instead a recent result due to A. Lerner to estimate the oscillation of dyadic operators.

Bellman VS Beurling: sharp estimates of uniform convexity for $L^p$ spaces

We obtain the classical Hanner inequalities by the Bellman function method. These inequalities give sharp estimates for the moduli of convexity of Lebesgue spaces. Easy ideas from differential geometry help us to find the Bellman function using neither "magic guesses" nor calculations.

Sharp weighted estimates for classical operators

We give a new proof of the sharp one weight $L^p$ inequality for any operator $T$ that can be approximated by Haar shift operators such as the Hilbert transform, any Riesz transform, the Beurling-Ahlfors operator. Our proof avoids the Bellman function technique and two weight norm inequalities. We use instead a recent result due to A. Lerner to estimate the oscillation of dyadic operators. Our method is flexible enough to prove the corresponding sharp one-weight norm inequalities for some operators of harmonic analysis: the maximal singular integrals associated to $T$, Dyadic square functions …

关于奇异积分与鞅变换

On singular integral and martingale transforms

研究了向量值奇异积分算子范数与向量值鞅变换范数的线性等价性。特别地，证明了巴拿赫空间X的UMD（p）常数等于Beurling-Ahlfors奇异积分算子作用在平面上X值L^p空间上的实部（或虚部）的范数.此外，用线性等价代替等式，发现这是偶乘子的典型性质。对奇乘子和希尔伯特变换给出了相应的结果。

Laminates Meet Burkholder Functions

We will explain how to compute the exact $L^p$ operator norm of a "quadratic perturbation" of the real part of the Ahlfors--Beurling operator. For the lower bound estimate we use a new approach of constructing a sequence of laminates (probability measures for which Jensen's inequality holds, but for rank one concave functions) to give an almost extremal sequence to approximate the operator. The upper bound estimate is given by extending the estimates of the quadratic perturbation of the martingale transform to continuous martingales. The use of "heat martingales" then allow us to connect the R…

多线性分数次有界平均振荡算子理论 I：稀疏控制、稀疏 $T1$ 定理、非对角外推、定量加权估计——针对广义交换子

The multilinear fractional bounded mean oscillation operator theory I: sparse domination, sparse $T1$ theorem, off-diagonal extrapolation, quantitative weighted estimate -- for generalized commutators

本文在球基测度空间上引入并研究了多线性分数次有界平均振荡算子及其广义交换子，建立了稀疏控制与稀疏 $T1$ 定理，并在多线性分数次权下获得了多种加权不等式与非对角外推框架。

拟正则映射的定量Sobolev正则性

Quantitative Sobolev regularity of quasiregular maps

该研究量化了Beltrami预解式的Sobolev范数，推导了拟正则分布的定量自提升不等式，还得到了区域上Beltrami方程预解式的定量估计，通过加权Sobolev估计将Astala-Iwaniec-Saksman技巧推广至更高正则性。

Regularizations of general singular integral operators

In the theory of singular integral operators significant effort is often required to rigorously define such an operator. This is due to the fact that the kernels of such operators are not locally integrable on the diagonal, so the integral formally defining the operator or its bilinear form is not well defined (the integrand is not in L^1) even for nice functions. However, since the kernel only has singularities on the diagonal, the bilinear form is well defined say for bounded compactly supported functions with separated supports. One of the standard ways to interpret the boundedness of a sin…

第二阶里斯茨变换的离散类比

Discrete analogues of second-order Riesz transforms

本文研究了第二阶里斯茨变换的离散类比的\ell^p范数，构造了新的离散里斯茨变换并探讨其与经典算子的关系。

Cotlar鞅变换与相关的奇异积分

Cotlar martingale transforms and related singular integrals

本文探讨Cotlar鞅变换及其相关奇异积分的结构，揭示其与Hilbert变换的相似性，并研究Riesz变换的范数渐进行为，同时解决两个长期未解的问题。

Fourier integral operators, fractal sets and the regular value theorem

We prove that if ${\mathcal E} \subset {\Bbb R}^{2d}$, $d \ge 2$, is an Ahlfors-David regular product set of sufficiently large Hausdorff dimension, denoted by $dim_{\mathcal H}({\mathcal E})$, and $ϕ$ is a sufficiently regular function, then the upper Minkowski dimension of the set $$ \{w \in {\mathcal E}: ϕ_l(w)=t_l; 1 \leq l \leq m \}$$ does not exceed $dim_{\mathcal H}({\mathcal E})-m$, in line with the regular value theorem from the elementary differential geometry. Our arguments are based on the mapping properties of the underlying Fourier Integral Operators and are intimately connected …