Mixed norm estimates for dilated averages over planar curves

2025/03/07 by Li, Junfeng, Lou, Zengjian, Yu, Haixia
#Analysis of PDEs (math.AP) #Classical Analysis and ODEs (math.CA) #FOS: Mathematics

paper · doi:10.48550/arxiv.2503.05140

Abstract

In this paper, we investigate the mixed norm estimates for the operator T associated with a dilated plane curve (ut, uγ(t)), defined by Tf(x, u) := ∫₀¹ f(x₁ - ut, x₂ - uγ(t)) dt, where x := (x₁, x₂) and γ is a general plane curve satisfying appropriate smoothness and curvature conditions. More precisely, we establish the L_x^p(ℝ²) → L_x^q L_u^r(ℝ² × [1, 2]) (space-time) estimates for T , whenever ((1)/(p),(1)/(q)) satisfy max\0, (1)/(2p) - (1)/(2r), (3)/(p) - (r+2)/(r)\ lt; (1)/(q) ≤ (1)/(p) lt; (r+1)/(2r) and 1 + (1 + ω)((1)/(q) - (1)/(p)) gt; 0, where r ∈ [1, ∞] and ω:= \limsup_{t → 0⁺} (ln|γ(t)|)/(ln t) . These results are sharp, except for certain borderline cases. Additionally, we examine the L_x^p(ℝ²) → L_u^r L_x^q(ℝ² × [1, 2]) (time-space) estimates for T , which are especially almost sharp when p=2 or p∈ [1, (3)/(2)]∪ [4, ∞].

Related

• L_x^p→ L^q_x,u estimates for dilated averages over planar curves
• Sharp variational inequalities for average operators over finite type curves in the plane
• Uniform L_x^p - L^q_x,r Improving for Dilated Averages over Polynomial Curves
• Sharp smoothing properties of averages over curves
• L^p-improving bounds of maximal functions along planar curves