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

Abstract

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.

Best constants in some estimates for the harmonic maximal operator on the real line

“…It might seem quite unexpected that the exponents behave differently for p ≤ 2 and p ≥ 2. However, a similar phenomenon occurs in the context of martingale transforms and wide classes of Fourier multipliers [1]. For various examples of other stability results in geometry and spectral theory, we refer the reader to the work by Brasco and De Philippis [3], as well as to Bianchi and Egnell [2], Chen, Frank and Weth [5], Christ [6], Dolbeault and Toscani [10], Fathi, Indrei and Ledoux [12], and the very recent paper of Carlen [4].…”

Sharpening Hölder's inequality

“…Throughout the paper we assume the summability exponents are in the interval (1, +∞), in particular, 1 < θ < +∞. We reserve the symbol p for the range [2, ∞) and q for (1,2] (we also usually assume that p and q are dual in the sense 1 p + 1 q = 1). Also, to simplify the presentation, we assume µ has no atoms.…”

“…Such sharpenings may be viewed as stability results: the new inequality says that if the equality almost holds, then the functions are close to the optimizers. See [5] for the Hausdorff-Young and Young's convolutional inequalities, [6] for the Riesz-Sobolev inequality, [2] for various martingale inequalities, and [4] for Sobolev-type embedding theorems. The latter paper also suggests a theoretical approach to the stability phenomenon.…”