Sharp estimates of martingale transforms in higher dimensions and applications to the Ahlfors-Beurling operator

New estimates for the Beurling-Ahlfors operator on differential forms

We establish new $p$-estimates for the norm of the generalized Beurling--Ahlfors transform $\mathcal{S}$ acting on form-valued functions. Namely, we prove that $\norm{\mathcal{S}}_{L^p(\R^n;Λ)\to L^p(\R^n;Λ)}\leq n(p^{*}-1)$ where $p^*=\max\{p, p/(p-1)\},$ thus extending the recent Nazarov--Volberg estimates to higher dimensions. The even-dimensional case has important implications for quasiconformal mappings. Some promising prospects for further improvement are discussed at the end.

Sharp estimates for the commutators of the Hilbert, Riesz transforms and the Beurling-Ahlfors operator on weighted Lebesgue spaces

We prove that the operator norm on weighted Lebesgue space L2(w) of the commutators of the Hilbert, Riesz and Beurling transforms with a BMO function b depends quadratically on the A2-characteristic of the weight, as opposed to the linear dependence known to hold for the operators themselves. It is known that the operator norms of these commutators can be controlled by the norm of the commutator with appropriate Haar shift operators, and we prove the estimate for these commutators. For the shift operator corresponding to the Hilbert transform we use Bellman function methods, however there is n…

无迹Beurling--Ahlfors变换与Hodge系统的Bourgain--Brezis问题

The trace-free Beurling--Ahlfors transform and the Bourgain--Brezis problem for Hodge systems

该研究引入无迹Beurling--Ahlfors变换，利用其符号的反射奇性得到端点抵消估计，证明了任意维度与形式次数下Bourgain--Brezis猜想的希尔伯特情形，还得到多类新的临界估计及端点Hodge分解，并证明多数情形下有界选择算子非线性。

The essential spectrum of the Neumann--Poincare operator on a domain with corners

Exploiting the homogeneous structure of a wedge in the complex plane, we compute the spectrum of the anti-linear Ahlfors-Beurling transform acting on the associated Bergman space. Consequently, the similarity equivalence between the Ahlfors-Beurling transform and the Neumann-Poincare operator provides the spectrum of the latter integral operator on a wedge. A localization technique and conformal mapping lead to the first complete description of the essential spectrum of the Neumann-Poincare operator on a planar domain with corners, with respect to the energy norm of the associated harmonic fie…

Stability in Burkholder's differentially subordinate martingales inequalities and applications to Fourier multipliers

We study stability estimates for the almost extremal functions associated with the $L^p$-bound for the real and imaginary parts of the Beurling-Ahlfors operator. The proof exploits probabilistic methods and rests on analogous results for differentially subordinate martingales which are of independent interest. This allows us to obtain stability inequalities for a larger class of Fourier multipliers.

The foundational inequalities of D.L. Burkholder and some of their ramifications

This paper present an overview of some of the applications of the martingale inequalities of D.L. Burkholder to $L^p$-bounds for singular integral operators, concentrating on the Hilbert transform, first and second order Riesz transforms, the Beurling-Ahlfors operator and other multipliers obtained by projections (conditional expectations) of transformations of stochastic integrals. While martingale inequalities can be used to prove the boundedness of a wider class of Calderón-Zygmund singular integrals, the aim of this paper is to show results which give optimal or near optimal bounds in the …