# Failure of the matrix weighted bilinear Carleson embedding theorem

## TL;DR

This paper demonstrates the failure of a natural matrix weighted bilinear Carleson embedding theorem under various conditions, highlighting the necessity of specific matrix weight conditions for the theorem to hold.

## Contribution

It establishes that a uniform bound on the conditioning number of the matrix weight is necessary and sufficient for the bilinear embedding, and shows the failure of several natural formulations.

## Key findings

- Failure of natural matrix weighted bilinear Carleson embedding formulations
- Necessity and sufficiency of conditioning number bounds for embedding
- Proof of matrix weighted redundancy condition

## Abstract

We prove failure of the natural formulation of a matrix weighted bilinear Carleson embedding theorem, featuring a matrix valued Carleson sequence as well as products of norms for the embedding. We show that assuming an A2 weight is also not sufficient. Indeed, a uniform bound on the conditioning number of the matrix weight is necessary and sufficient to get the bilinear embedding. We show that any improvement of a recent matrix weighted bilinear embedding, featuring a scalar Carleson sequence and inner products instead of norms must fail. In particular, replacing the scalar sequence by a matrix sequence results in failure even when maintaining the formulation using inner products. Any formulation using norms, even in the presence of a scalar Carleson sequence must fail. As a positive result, we prove the so called matrix weighted redundancy condition in full generality.