Bilinear singular integral operators with kernels in weighted spaces

2024/12/09 by Petr Honzík, Honzík, Petr, Stefanos Lappas +3 · 2 citations
Mathematics · #Advanced Harmonic Analysis Research #Differential Equations and Boundary Problems #Numerical methods in inverse problems

paper · pdf · doi:10.48550/arxiv.2412.07014

Abstract

We establish the full quasi-Banach range of L^p₁(\mathbb R) × L^p₂(\mathbb R) → L^p(\mathbb R) bounds for one-dimensional bilinear singular integral operators with homogeneous kernels whose restriction Ω to the unit sphere \mathbb S¹ is supported away from the degenerate line θ₁=θ₂, belongs to L^q(\mathbb S¹) for some q>1 and has vanishing integral. In fact, a more general result is obtained by dropping the support condition on Ω and requiring that Ω∈ L^q(\mathbb S¹,u^q), where u(θ₁,θ₂)=|θ₁-θ₂|^-1 for (θ₁,θ₂)∈ \mathbb S¹. In addition, we provide counterexamples that show the failure of the n-dimensional version of the previous result when n≥ 2, as well as the failure of its m-linear variant in dimension one when m≥ 3. The relationship of these results to (un)boundedness properties of higher-dimensional multilinear Hilbert transforms is also discussed.

Cited by

• Sharp bilinear estimates for maximal singular integrals with kernels in weighted L^q spaces
• An alternate approach to bilinear rough singular integrals

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• An alternate approach to bilinear rough singular integrals