ON MULTILINEAR MAXIMAL OPERATORS ALONG HOMOGENEOUS CURVES

Suppose that (formula prseneted) yᵗ (t) := (γ₁ (t), …, γₙ (t)) = (a₁ tᵈ₁, …, aₙ tᵈₙ), 1 ≤ d₁ < … < dₙ ∈ ℕ, aᵢ ≠ 0 is a homogeneous polynomial curve. We prove that whenever p₁, …, pₙ > 1 and 1/p = ∑_{j=1}ⁿ 1/pⱼ ≤ 1, there exists an absolute constant 0 < C = C_{p₁,…,pₙ} < ∞ such that sup_{r > 0} (1/r ∫₀ʳ ∏_{i=1}ⁿ |fᵢ(x − γᵢ(t))| dt) ∈ Lᵖ(ℝ) ≤ C · ∏_{i=1}ⁿ ||fⱼ||_{Lᵖⱼ(ℝ)} (formula prseneted). Our main tool is a smoothing estimate, adapted from work of Kosz, Mirek, Peluse, Wan, and Wright.