Sparse bounds for maximal rough singular integrals via the Fourier transform
Francesco Di Plinio, Tuomas Hytönen, Kangwei Li
Abstract
We prove a quantified sparse bound for the maximal truncations of convolution-type singular integrals with suitable Fourier decay of the kernel. Our result extends the sparse domination principle by Conde-Alonso, Culiuc, Ou and the first author to the maximally truncated case, and covers the rough homogeneous singular integrals <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>T</mml:mi> <mml:mi>Ω</mml:mi> </mml:msub> </mml:math> on <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>ℝ</mml:mi> <mml:mi>d</mml:mi> </mml:msup> </mml:math> with bounded angular part <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Ω</mml:mi> </mml:math> having vanishing integral on the sphere. Among several consequences, we obtain new quantitative weighted norm inequalities for the maximal truncation of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>T</mml:mi> <mml:mi>Ω</mml:mi> </mml:msub> </mml:math> , extending a result by Roncal, Tapiola and the second author. A convex-body valued version of the sparse bound is also deduced and employed towards novel matrix-weighted norm inequalities for the maximal truncations of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>T</mml:mi> <mml:mi>Ω</mml:mi> </mml:msub> </mml:math> . Our result is quantitative, but even the qualitative statement is new, and the present approach via sparse domination is the only one currently known for the matrix weighted bounds of this class of operators.