$L_x^p\rightarrow L^q_{x,u}$ estimates for dilated averages over planar curves
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| Format: | Preprint |
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2024
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| _version_ | 1866916109056212992 |
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| author | Li, Junfeng Liu, Naijia Lou, Zengjian Yu, Haixia |
| author_facet | Li, Junfeng Liu, Naijia Lou, Zengjian Yu, Haixia |
| contents | In this paper, we consider the $L_x^p(\mathbb{R}^2)\rightarrow L_{x,u}^q(\mathbb{R}^2\times [1,2])$ estimate for the operator $T$ along a dilated plane curve $(ut,uγ(t))$, where
$$Tf(x,u):=\int_{0}^{1}f(x_1-ut,x_2-u γ(t))\,\textrm{d}t,$$
$x:=(x_1,x_2)$ and $γ$ is a general plane curve satisfying some suitable smoothness and curvature conditions. We show that $T$ is $L_x^p(\mathbb{R}^2)$ to $L_{x,u}^q(\mathbb{R}^2\times [1,2])$ bounded whenever $(\frac{1}{p},\frac{1}{q})\in \square \cup \{(0,0)\}\cup \{(\frac{2}{3},\frac{1}{3})\}$ and $1+(1 +ω)(\frac{1}{q}-\frac{1}{p})>0$, where the trapezium $\square:=\{(\frac{1}{p},\frac{1}{q}):\ \frac{2}{p}-1\leq\frac{1}{q}\leq \frac{1}{p}, \frac{1}{q}>\frac{1}{3p}, \frac{1}{q}>\frac{1}{p}-\frac{1}{3}\}$ and $ω:=\limsup_{t\rightarrow 0^{+}}\frac{\ln|γ(t)|}{\ln t}$. This result is sharp except for some borderline cases. On the other hand, in a smaller $(\frac{1}{p},\frac{1}{q})$ region, we also obtain the almost sharp estimate $T : L_x^p(\mathbb{R}^2)\rightarrow L_{x}^q(\mathbb{R}^2)$ uniformly for $u\in [1,2]$. These results imply that the operator $T$ has the so called local smoothing phenomenon, i.e., the $L^q$ integral about $u$ on $[1,2]$ extends the region of $(\frac{1}{p},\frac{1}{q})$ in uniform estimate $T : L_x^p(\mathbb{R}^2)\rightarrow L_{x}^q(\mathbb{R}^2)$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2401_16040 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | $L_x^p\rightarrow L^q_{x,u}$ estimates for dilated averages over planar curves Li, Junfeng Liu, Naijia Lou, Zengjian Yu, Haixia Classical Analysis and ODEs In this paper, we consider the $L_x^p(\mathbb{R}^2)\rightarrow L_{x,u}^q(\mathbb{R}^2\times [1,2])$ estimate for the operator $T$ along a dilated plane curve $(ut,uγ(t))$, where $$Tf(x,u):=\int_{0}^{1}f(x_1-ut,x_2-u γ(t))\,\textrm{d}t,$$ $x:=(x_1,x_2)$ and $γ$ is a general plane curve satisfying some suitable smoothness and curvature conditions. We show that $T$ is $L_x^p(\mathbb{R}^2)$ to $L_{x,u}^q(\mathbb{R}^2\times [1,2])$ bounded whenever $(\frac{1}{p},\frac{1}{q})\in \square \cup \{(0,0)\}\cup \{(\frac{2}{3},\frac{1}{3})\}$ and $1+(1 +ω)(\frac{1}{q}-\frac{1}{p})>0$, where the trapezium $\square:=\{(\frac{1}{p},\frac{1}{q}):\ \frac{2}{p}-1\leq\frac{1}{q}\leq \frac{1}{p}, \frac{1}{q}>\frac{1}{3p}, \frac{1}{q}>\frac{1}{p}-\frac{1}{3}\}$ and $ω:=\limsup_{t\rightarrow 0^{+}}\frac{\ln|γ(t)|}{\ln t}$. This result is sharp except for some borderline cases. On the other hand, in a smaller $(\frac{1}{p},\frac{1}{q})$ region, we also obtain the almost sharp estimate $T : L_x^p(\mathbb{R}^2)\rightarrow L_{x}^q(\mathbb{R}^2)$ uniformly for $u\in [1,2]$. These results imply that the operator $T$ has the so called local smoothing phenomenon, i.e., the $L^q$ integral about $u$ on $[1,2]$ extends the region of $(\frac{1}{p},\frac{1}{q})$ in uniform estimate $T : L_x^p(\mathbb{R}^2)\rightarrow L_{x}^q(\mathbb{R}^2)$. |
| title | $L_x^p\rightarrow L^q_{x,u}$ estimates for dilated averages over planar curves |
| topic | Classical Analysis and ODEs |
| url | https://arxiv.org/abs/2401.16040 |