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arXiv · 1906.08715

Failure of the matrix weighted bilinear Carleson embedding theorem

Komla Domelevo·Stefanie Petermichl·Kristina Ana Škreb
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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.

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Komla Domelevo, Stefanie Petermichl, Kristina Ana Škreb. 2019-06-20. Failure of the matrix weighted bilinear Carleson embedding theorem. https://arxiv.org/abs/1906.08715

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