Mixed norm estimates for dilated averages over planar curves
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909942218227712 |
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| author | Li, Junfeng Lou, Zengjian Yu, Haixia |
| author_facet | Li, Junfeng Lou, Zengjian Yu, Haixia |
| contents | In this paper, we investigate the mixed norm estimates for the operator $ T $associated with a dilated plane curve $(ut, uγ(t))$, defined by \[ Tf(x, u) := \int_{0}^{1} f(x_1 - ut, x_2 - uγ(t)) \, dt, \] where $ x := (x_1, x_2) $ and $γ$ is a general plane curve satisfying appropriate smoothness and curvature conditions. More precisely, we establish the $ L_x^p(\mathbb{R}^2) \rightarrow L_x^q L_u^r(\mathbb{R}^2 \times [1, 2]) $ (space-time) estimates for $ T $, whenever $(\frac{1}{p},\frac{1}{q})$ satisfy \[ \max\left\{0, \frac{1}{2p} - \frac{1}{2r}, \frac{3}{p} - \frac{r+2}{r}\right\} < \frac{1}{q} \leq \frac{1}{p} < \frac{r+1}{2r} \] and $$1 + (1 + ω)\left(\frac{1}{q} - \frac{1}{p}\right) > 0,$$ where $ r \in [1, \infty] $ and $ ω:= \limsup_{t \rightarrow 0^+} \frac{\ln|γ(t)|}{\ln t} $. These results are sharp, except for certain borderline cases. Additionally, we examine the $ L_x^p(\mathbb{R}^2) \rightarrow L_u^r L_x^q(\mathbb{R}^2 \times [1, 2]) $ (time-space) estimates for $T $, which are especially almost sharp when $p=2$ or $p\in [1, \frac{3}{2}]\cup [4, \infty]$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_05140 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Mixed norm estimates for dilated averages over planar curves Li, Junfeng Lou, Zengjian Yu, Haixia Analysis of PDEs Classical Analysis and ODEs In this paper, we investigate the mixed norm estimates for the operator $ T $associated with a dilated plane curve $(ut, uγ(t))$, defined by \[ Tf(x, u) := \int_{0}^{1} f(x_1 - ut, x_2 - uγ(t)) \, dt, \] where $ x := (x_1, x_2) $ and $γ$ is a general plane curve satisfying appropriate smoothness and curvature conditions. More precisely, we establish the $ L_x^p(\mathbb{R}^2) \rightarrow L_x^q L_u^r(\mathbb{R}^2 \times [1, 2]) $ (space-time) estimates for $ T $, whenever $(\frac{1}{p},\frac{1}{q})$ satisfy \[ \max\left\{0, \frac{1}{2p} - \frac{1}{2r}, \frac{3}{p} - \frac{r+2}{r}\right\} < \frac{1}{q} \leq \frac{1}{p} < \frac{r+1}{2r} \] and $$1 + (1 + ω)\left(\frac{1}{q} - \frac{1}{p}\right) > 0,$$ where $ r \in [1, \infty] $ and $ ω:= \limsup_{t \rightarrow 0^+} \frac{\ln|γ(t)|}{\ln t} $. These results are sharp, except for certain borderline cases. Additionally, we examine the $ L_x^p(\mathbb{R}^2) \rightarrow L_u^r L_x^q(\mathbb{R}^2 \times [1, 2]) $ (time-space) estimates for $T $, which are especially almost sharp when $p=2$ or $p\in [1, \frac{3}{2}]\cup [4, \infty]$. |
| title | Mixed norm estimates for dilated averages over planar curves |
| topic | Analysis of PDEs Classical Analysis and ODEs |
| url | https://arxiv.org/abs/2503.05140 |