Endpoint estimate on Rubio de Francia's maximal function
数学专题报告
报告题目(Title)：Endpoint estimate on Rubio de Francia's maximal function
报告人(Speaker)：Ji Li教授 (Macquarie University, Australia)
地点(Place)：后主楼1124
时间(Time)：2026年4月6日（周一）15:30-16:30
邀请人(Inviter)：杨大春
报告摘要
Let $\mu$ be a compactly supported Borel measure in $\mathbb R^d$, $d\ge 1$. For $t>0$, let us denote by $\mu_t$ its dilation given by
\[ \langle \mu_t,\phi\rangle = \int \phi(t y )\, \md\mu(y) \]
for $\phi\in C_c(\mathbb R^d)$. Consider the maximal function
\[ M_\mu f(x) = \sup_{t>0} | f\ast\mu_t |. \]
The following theorem concerning $L^p$ boundedness for maximal operator was proved by Rubio de Francia in his influential paper
Theorem. [Duke Math. J. 1986].
Suppose that $\mu$ is a compactly supported Borel measure and
\[ |\widehat\mu(\xi)|\le C |\xi|^{-a} \]
with $a >\frac12$. Then, for all $ p >p_a := \frac{ 2a+1}{2a}$, we have
\[ \| \mm f \|_{L^p(\mathbb R^d)} \le C \|f\|_{L^p(\mathbb R^d)}. \]
We establish the missing endpoint estimate:
Theorem. [HKLL].
Suppose that $\mu$ is a compactly supported Borel measure and satisfying \eqref{decay} with $a >\frac12$. Then, we have
\[ \| \mm f\|_{L^{p_a, \infty}} \le C \| f\|_{L^{p_a,1}}. \]
This talk is mainly based on the recent progress:
Seheon Ham, Jiwon Kah, Sanghyuk Lee and Ji Li, On Rubio de Francia's maximal theorem, arXiv:2602.03465.
