沿平面曲线的极大函数的$L^p$改进界
$L^p$-improving bounds of maximal functions along planar curves
AI总结：
本文研究沿平面曲线的极大函数的$L^p$改进界，在一般光滑性和曲率条件下给出了$L^p\rightarrow L^q$估计的充分条件，并证明结果在除边界情形外是尖锐的。
摘要：
本文研究了沿平面曲线$(t,\gamma(t))$的极大函数$M_\gamma$的$L^p(\mathbb{R}^2)$改进界，即$L^p(\mathbb{R}^2)\rightarrow L^q(\mathbb{R}^2)$估计，其中$$M_\gamma f(x_1,x_2):=\sup_{u\in [1,2]}\left|\int_{0}^{1}f(x_1-ut,x_2-u \gamma(t))\\,\textrm{d}t\right|,$$且$\gamma$是满足适当光滑性和曲率条件的一般平面曲线。我们得到当$(\frac{1}{p},\frac{1}{q})\in \Delta\cup \{(0,0)\}$且$(\frac{1}{p},\frac{1}{q})$满足$1+(1+\omega)(\frac{1}{q}-\frac{1}{p})>0$时，$M_\gamma : L^p(\mathbb{R}^2)\rightarrow L^q(\mathbb{R}^2)$成立，其中$\Delta:=\{(\frac{1}{p},\frac{1}{q}):\\ \frac{1}{2p}<\frac{1}{q}\leq \frac{1}{p}, \frac{1}{q}>\frac{3}{p}-1 \}$且$\omega:=\limsup_{t\rightarrow 0^{+}}\frac{\ln|\gamma(t)|}{\ln t}$。除某些边界情形外，该结果是尖锐的。正如Hickman在[J. Funct. Anal. 270 (2016), pp. 560--608]中所指出的，这是一个非常不同的情形。
英文摘要：
In this paper, we study the $L^p(\mathbb{R}^2)$-improving bounds, i.e., $L^p(\mathbb{R}^2)\rightarrow L^q(\mathbb{R}^2)$ estimates, of the maximal function $M_γ$ along a plane curve $(t,γ(t))$, where $$M_γf(x_1,x_2):=\sup_{u\in [1,2]}\left|\int_{0}^{1}f(x_1-ut,x_2-u γ(t))\,\textrm{d}t\right|,$$ and $γ$ is a general plane curve satisfying some suitable smoothness and curvature conditions. We obtain $M_γ : L^p(\mathbb{R}^2)\rightarrow L^q(\mathbb{R}^2)$ if $(\frac{1}{p},\frac{1}{q})\in Δ\cup \{(0,0)\}$ and $(\frac{1}{p},\frac{1}{q})$ satisfying $1+(1 +ω)(\frac{1}{q}-\frac{1}{p})>0$, where $Δ:=\{(\frac{1}{p},\frac{1}{q}):\ \frac{1}{2p}<\frac{1}{q}\leq \frac{1}{p}, \frac{1}{q}>\frac{3}{p}-1 \}$ and $ω:=\limsup_{t\rightarrow 0^{+}}\frac{\ln|γ(t)|}{\ln t}$. This result is sharp except for some borderline cases. As Hickman stated in [J. Funct. Anal. 270 (2016), pp. 560--608], this is a very different situation.
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