Spectral perturbation theory and the two weights problem

2013/06/08 by Alexander Pushnitski, Alexander Volberg, Pushnitski, Alexander +1
Mathematics · #47A40 #47G10 #Advanced Banach Space Theory #Advanced Harmonic Analysis Research #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Holomorphic and Operator Theory #Spectral Theory (math.SP) #math.CA #math.SP #msc:47A40 #msc:47G10

paper · pdf · doi:10.48550/arxiv.1306.1912

14 pages, Latex

arxiv created 2013/06/08 · openalex publication_date 2013/06/08 · arxiv updated 2013/06/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The famous two weights problem consists in characterising all possible pairs of weights such that the Hardy projection is bounded between the corresponding weighted L² spaces. Koosis' theorem of 1980 gives a way to construct a certain class of pairs of weights. We show that Koosis' theorem is closely related to (in fact, is a direct consequence of) a spectral perturbation model suggested by de Branges in 1962. Further, we show that de Branges' model provides an operator-valued version of Koosis' theorem.

Citations

• Two weight estimate for the Hilbert transform and corona decomposition for non-doubling measures
• Scattering theory and Banach space valued singular integrals

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