2026/07/19 by Binwei Dan, Qingying Xue
#math.CA #math.AP
We prove endpoint theorems for one-dimensional bilinear rough singular integrals. Our starting point is a sharp structural characterization of the associated angular multiplier. For every mean-zero Ω∈ L1(\mathbbS1), the finite-part angular multiplier associated with TΩ has bounded variation if and only if the antipodal even part of Ω belongs to H1(\mathbbS1). This characterization identifies the precise rotational regularity required in the one-dimensional bilinear setting. It also yields a Stieltjes decomposition compatible with uniform estimates for the bilinear Hilbert transform. We then establish two boundedness criteria under critical kernel assumptions. First, if Ω∈ Llog L(\mathbbS1), then TΩ is bounded from Lp1(ℝ)× Lp2(ℝ)to Lp(ℝ) whenever 1<p1,p2,p<∞ and (1)/(p)=(1)/(p1)+(1)/(p2). Moreover, the logarithmic exponent 1 is optimal within the scale L(log L)A. Second, at the critical directional index, the same boundedness holds for Ω\inK1/2,β(\mathbbS1), provided that β>(3)/(2)max\p1,p1',p2,p2'\-1.The two critical kernel classes are incomparable. The Llog L result is obtained by reducing the multiplier to a finite-part angular profile of bounded variation. The directional result follows from endpoint Fourier decay, product wavelet decompositions, and interpolation.