Bilinear operators along variable curves
Abstract
Motivated by Zygmund's differentiation conjecture and Stein's conjecture for Lipschitz vector fields, we study bilinear singular integral and maximal operators along variable curves given by BHaα(f,g)(x) &= p.v.∫ℝ f(x−t) g(x + a(x)tα) dt/t, BMaα(f,g)(x) &= supr>01/2r∫−rr |f(x−t)g(x+a(x)tα)| dt, BSHcα(f,g)(x) &= p.v. ∫ℝ f(x−t) g(x + tα) dt/(t−c(x)), where α> 0 and a,c:ℝ→ℝ are measurable. For every α>0 and 0<A<B<∞, we construct a smooth, globally Lipschitz coefficient a:ℝ→(A,B) such that, for every Hölder conjugate pair 1<p,q<∞, no strong--type Lp(ℝ)× Lq(ℝ)→ L1(ℝ) estimate holds for BHaα or BMaα on Cc∞(ℝ)× Cc∞(ℝ); moreover, there exist nonnegative smooth inputs in ⋂s>1Ls(ℝ), one of which is compactly supported, whose outputs under BHaα and BMaα do not belong to L1(ℝ). For any measurable a:ℝ→ℝ, BMaα maps Lp(ℝ) × Lq(ℝ) boundedly into Lr(ℝ) whenever 1<p,q≤∞, 1/r = 1/p + 1/q, and r > 1, and maps Lp(ℝ) × Lq(ℝ) into weak L1(ℝ) when 1/p+1/q=1, with constants independent of a. Finally, we investigate the shifted operator BSHcα, including dyadic-integer-valued c, and prove that there is a measurable choice of c such that the operator admits no bounded Lp(ℝ) × Lq(ℝ) → Lr,∞(ℝ) estimates in the range 1 ≤ p,q ≤ ∞ and 1/2<r<∞, resolving an open problem posed by Gaitan and Lie.
Related
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BibTeX
@misc{gismondi2026bilinear,
title = {{Bilinear operators along variable curves}},
author = {Nicholas Gismondi and Mandon Pathak},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.40472},
eprint = {2610.40472},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.40472},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Bilinear operators along variable curves
AU - Nicholas Gismondi
AU - Mandon Pathak
PY - 2026
DA - 2026/10/29
PB - arXiv
DO - 10.48550/arxiv.2610.40472
UR - https://arxiv.org/abs/2610.40472
AB - Motivated by Zygmund's differentiation conjecture and Stein's conjecture for Lipschitz vector fields, we study bilinear singular integral and maximal operators along variable curves given by BH_a^α(f,g)(x) &= p.v.∫_ℝ f(x−t) g(x + a(x)t^α) dt/t, BM_a^α(f,g)(x) &= sup_r>01/2r∫_−r^r |f(x−t)g(x+a(x)t^α)| dt, BSH_c^α(f,g)(x) &= p.v. ∫_ℝ f(x−t) g(x + t^α) dt/(t−c(x)), where α> 0 and a,c:ℝ→ℝ are measurable. For every α>0 and 0<A<B<∞, we construct a smooth, globally Lipschitz coefficient a:ℝ→(A,B) such that, for every Hölder conjugate pair 1<p,q<∞, no strong--type L^p(ℝ)× L^q(ℝ)→ L^1(ℝ) estimate holds for BH_a^α or BM_a^α on C_c^∞(ℝ)× C_c^∞(ℝ); moreover, there exist nonnegative smooth inputs in ⋂_s>1L^s(ℝ), one of which is compactly supported, whose outputs under BH_a^α and BM_a^α do not belong to L^1(ℝ). For any measurable a:ℝ→ℝ, BM_a^α maps L^p(ℝ) × L^q(ℝ) boundedly into L^r(ℝ) whenever 1<p,q≤∞, 1/r = 1/p + 1/q, and r > 1, and maps L^p(ℝ) × L^q(ℝ) into weak L^1(ℝ) when 1/p+1/q=1, with constants independent of a. Finally, we investigate the shifted operator BSH_c^α, including dyadic-integer-valued c, and prove that there is a measurable choice of c such that the operator admits no bounded L^p(ℝ) × L^q(ℝ) → L^r,∞(ℝ) estimates in the range 1 ≤ p,q ≤ ∞ and 1/2<r<∞, resolving an open problem posed by Gaitan and Lie.
ER -