球面极大函数与伸缩集的分形维数

球形极大函数问题

Problems on spherical maximal functions

本文综述了球形极大函数的各种猜想与结果，重点讨论了具有分形扩张集的问题。

球极大函数的幂权重不等式

Power weight inequalities for spherical maximal functions

本文通过Legendre-Assouad函数确定了球极大函数在加权$ L^p(|x|^α) $空间中的可允许范围，解决了Duoandikoetxea和Seijo的开放问题。

Upper bounds for Fourier decay rates of fractal measures

For spherical and parabolic averages of the Fourier transform of fractal measures, we obtain new upper bounds on rates of decay by an "intermediate dimension" trick.

关于平面中有限型曲线相关极大算子的 $L^{p}$-改进界

On $L^{p}$-improving bounds for maximal operators associated with curves of finite type in the plane

本文研究平面中有限型曲线极大算子的 $L^{p}$-改进性质，揭示其改进区域由曲线有限型阶数与伸缩集分形维数共同决定，并进一步分析非各向同性伸缩的影响。

Endpoint mapping properties of spherical maximal operators

For a function $f\in L^p(\Bbb R^d)$, $d\ge 2$, let $A_t f(x)$ be the mean of $f$ over the sphere of radius $t$ centered at $x$. Given a set $E\subset (0,\infty)$ of dilations we prove endpoint bounds for the maximal operator $M_E$ defined by $M_E f(x)=\sup_{t\in E} |A_t f(x)|$.

Remarks on dimension of unions of curves

We study an analogue of Marstrand's circle packing problem for curves in higher dimensions. We consider collections of curves which are generated by translation and dilation of a curve $γ$ in $\mathbb R^d$, i.e., $ x + t γ$, $(x,t) \in \mathbb R^d \times (0,\infty)$. For a Borel set $F \subset \mathbb R^d\times (0,\infty)$, we show the unions of curves $\bigcup_{(x,t) \in F} ( x+tγ)$ has Hausdorff dimension at least $α+1$ whenever $F$ has Hausdorff dimension bigger than $α$, $α\in (0, d-1)$. We also obtain results for unions of curves generated by multi-parameter dilation of $γ$. One of the ma…

Spherical means on the Heisenberg group: Stability of a maximal function estimate

Consider the surface measure $μ$ on a sphere in a nonvertical hyperplane on the Heisenberg group $\mathbb{H}^n$, $n\ge 2$, and the convolution $f*μ$. Form the associated maximal function $Mf=\sup_{t>0}|f*μ_t|$ generated by the automorphic dilations. We use decoupling inequalities due to Wolff and Bourgain-Demeter to prove $L^p$-boundedness of $M$ in an optimal range.

球形极大函数的端点幂权不等式

Endpoint power weight inequalities for spherical maximal functions

该研究针对球形极大函数，证明了精确 $L^p(|x|^\alpha)$ 估计，解决了特定参数下全极大算子的开放问题，还得到了含局部 Assouad 维数的一般结果并开展了模型案例研究。

强 $\ell^r$ 球面极大函数

The strong $\ell^r$ spherical maximal function

本文研究 $\ell^r$ 单位球面强极大算子的 $L^p$ 有界性，给出充要条件，并证明局部极大算子的最优 $L^p\to L^q$ 界，方法结合二进分解、驻相法、$TT^*$ 论证及多参数局部光滑。

两步幂零李群上的球面极大函数

Spherical maximal functions on two step nilpotent Lie groups

该研究聚焦两步幂零李群上的球面极大函数，去掉Métivier群相关结果的非退化条件，证明了所有$d\ge3$的两步幂零李群上该极大函数的精确$L^p$有界性。

Nikodym集与球面相关的极大函数

Nikodym sets and maximal functions associated with spheres

本文研究球面相关的Nikodym集与极大函数，证明尖锐L^p估计，得出Nikodym集须有完整Hausdorff维数，并利用波动方程局部光滑估计推导一般极大函数的L^p界。

Average decay of the Fourier transform of measures with applications

We consider spherical averages of the Fourier transform of fractal measures and improve both the upper and lower bounds on the rate of decay. Maximal estimates with respect to fractal measures are deduced for the Schrödinger and wave equations. This refines the almost everywhere convergence of the solution to its initial datum as time tends to zero. A consequence is that the solution to the wave equation cannot diverge on a $(d-1)$-dimensional manifold if the data belongs to the energy space $\dot{H}^1(\mathbb{R}^d)\times L^2(\mathbb{R}^d)$.

Sharp $L^2$ estimate of Schrödinger maximal function in higher dimensions

We show that, for $n\geq 3$, $\lim_{t \to 0} e^{itΔ}f(x) = f(x)$ holds almost everywhere for all $f \in H^s (\mathbb{R}^n)$ provided that $s>\frac{n}{2(n+1)}$. Due to a counterexample by Bourgain, up to the endpoint, this result is sharp and fully resolves a problem raised by Carleson. Our main theorem is a fractal $L^2$ restriction estimate, which also gives improved results on the size of divergence set of Schrödinger solutions, the Falconer distance set problem and the spherical average Fourier decay rates of fractal measures. The key ingredients of the proof include multilinear Kakeya esti…

A generalized polar-coordinate integration formula with applications to the study of convolution powers of complex-valued functions on $\mathbb{Z}^d$

In this article, we consider a class of functions on $\mathbb{R}^d$, called positive homogeneous functions, which interact well with certain continuous one-parameter groups of (generally anisotropic) dilations. Generalizing the Euclidean norm, positive homogeneous functions appear naturally in the study of convolution powers of complex-valued functions on $\mathbb{Z}^d$. As the spherical measure is a Radon measure on the unit sphere which is invariant under the symmetry group of the Euclidean norm, to each positive homogeneous function $P$, we construct a Radon measure $σ_P$ on $S=\{η\in \math…

Annulus Maximal Averages on Variable Hyperplanes

By giving a thin width of $0<δ\ll 1$ to both a unit circle and a unit line, we set an annulus and a tube on the Euclidean plane $\mathbb{R}^2$. Consider the maximal means $M_δ$ over dilations of the annulus, and $N_δ$ over rotations of the tube. It is known that their operator norms on $L^2(\mathbb{R}^2)$ are $O(|\log 1/δ|^{1/2})$. In this paper, we study the maximal averages $\mathcal{M}^A_δ$ and $\mathcal{N}^A_δ$ over those annuli and tubes now imbedded on the variable hyperplanes $(x,x_3)+\left\{\left(y, \langle A(x), y\rangle\right): y\in\mathbb{R}^2\right\}\subset \mathbb{R}^3$ where $A$ …