2024/01/29 by Junfeng Li, Li, Junfeng, Naijia Liu +5
Computer Science · Mathematics · #Advanced Harmonic Analysis Research #Advanced Mathematical Modeling in Engineering #Classical Analysis and ODEs (math.CA) #FOS: Mathematics
paper · pdf · doi:10.48550/arxiv.2401.16040
openalex publication_date 2024/01/29 · openalex created_date 2024/01/31 · openalex updated_date 2026/07/28
In this paper, we consider the Lxp(ℝ2)→ Lx,uq(ℝ2× [1,2]) estimate for the operator T along a dilated plane curve (ut,uγ(t)), where Tf(x,u):=∫01f(x1-ut,x2-u γ(t)) \textrmdt, x:=(x1,x2) and γ is a general plane curve satisfying some suitable smoothness and curvature conditions. We show that T is Lxp(ℝ2) to Lx,uq(ℝ2× [1,2]) bounded whenever ((1)/(p),(1)/(q))∈ \square ∪ \(0,0)\∪ \((2)/(3),(1)/(3))\ and 1+(1 +ω)((1)/(q)-(1)/(p))>0, where the trapezium \square:=\((1)/(p),(1)/(q)): (2)/(p)-1≤(1)/(q)≤ (1)/(p), (1)/(q)>(1)/(3p), (1)/(q)>(1)/(p)-(1)/(3)\ and ω:=\limsupt→ 0+(ln|γ(t)|)/(ln t). This result is sharp except for some borderline cases. On the other hand, in a smaller ((1)/(p),(1)/(q)) region, we also obtain the almost sharp estimate T : Lxp(ℝ2)→ Lxq(ℝ2) uniformly for u∈ [1,2]. These results imply that the operator T has the so called local smoothing phenomenon, i.e., the Lq integral about u on [1,2] extends the region of ((1)/(p),(1)/(q)) in uniform estimate T : Lxp(ℝ2)→ Lxq(ℝ2).