Bilinear singular integral operators with kernels in weighted spaces
Petr Honzík, Stefanos Lappas, Lenka Slavíková
TL;DR
This work analyzes a one-dimensional bilinear singular integral with a rough kernel Ω, obtained by averaging directional bilinear Hilbert transforms over the unit circle. It proves full quasi-Banach L^p boundedness for the operator T_Ω in the range 1<p_1,p_2<∞, 1/2<p<∞ with 1/p=1/p_1+1/p_2, under Ω ∈ L^q(\mathbb{S}^1) with q>1 and vanishing integral, provided Ω is away from the degenerate direction; it also extends to Ω ∈ L^q(\mathbb{S}^1,u^q) with a degeneracy weight. The paper develops a dyadic/kernel decomposition, a Calderón-Zygmund decomposition, and a two-stage interpolation (bilinear Marcinkiewicz and weighted multilinear interpolation) to obtain strong-type bounds, and it establishes sharp counterexamples in higher dimensions and for multilinear variants. These results elucidate the (un)boundedness phenomena of averaged multilinear Hilbert transforms and indicate the limits of simple averaging when degeneracies are present.
Abstract
We establish the full quasi-Banach range of $L^{p_1}(\mathbb R) \times L^{p_2}(\mathbb R) \rightarrow L^p(\mathbb R)$ bounds for one-dimensional bilinear singular integral operators with homogeneous kernels whose restriction $Ω$ to the unit sphere $\mathbb S^1$ is supported away from the degenerate line $θ_1=θ_2$, belongs to $L^q(\mathbb S^1)$ 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 $Ω\in L^q(\mathbb S^1,u^q)$, where $u(θ_1,θ_2)=|θ_1-θ_2|^{-1}$ for $(θ_1,θ_2)\in \mathbb S^1$. In addition, we provide counterexamples that show the failure of the $n$-dimensional version of the previous result when $n\geq 2$, as well as the failure of its $m$-linear variant in dimension one when $m\geq 3$. The relationship of these results to (un)boundedness properties of higher-dimensional multilinear Hilbert transforms is also discussed.