A sharp Bernstein-type inequality and application to the Carleson embedding theorem with matrix weights

TL;DR

This paper establishes a sharp Bernstein-type inequality for positive complex polynomials with polynomial growth, leading to improved bounds in matrix-weighted Carleson embedding theorems, with optimal results in the scalar case.

Contribution

It introduces a new sharp Bernstein-type inequality for positive polynomials and applies it to enhance bounds in matrix-weighted Carleson embedding theorems.

Findings

Derived a sharp Bernstein-type inequality for positive polynomials.

Improved the upper estimate in matrix-weighted Carleson embedding theorem.

Achieved optimal bounds in the scalar case.

Abstract

We prove a sharp Bernstein-type inequality for complex polynomials which are positive and satisfy a polynomial growth condition on the positive real axis. This leads to an improved upper estimate in the recent work of Culiuc and Treil on the weighted martingale Carleson embedding theorem with matrix weights. In the scalar case this new upper bound is optimal.