Maximal averages over flat radial hypersurfaces

L^p boundedness of maximal averages over hypersurfaces in R^3

Extending the methods developed in the author's previous paper and using adapted coordinate systems in two variables, an L^p boundedness theorem is proven for maximal operators over hypersurfaces in R^3 when p > 2. When the best possible p is greater than 2, the theorem typically provides sharp estimates. This gives another approach to the subject of recent work of Ikromov, Kempe, and Muller on this subject

Estimates for maximal functions associated to hypersurfaces in $\Bbb R^3$ with height $h<2:$ Part II -- A geometric conjecture and its proof for generic 2-surfaces

In this article, we continue the study of $L^p$-boundedness of the maximal operator $\mathcal M_S$ associated to averages along isotropic dilates of a given, smooth hypersurface $S$ in 3-dimensional Euclidean space. We focus here on small surface-patches near a given point $x^0$ exhibiting singularities of type $\mathcal A$ in the sense of Arnol'd at this point; this is the situation which had yet been left open. Denoting by $p_c$ the minimal Lebesgue exponent such that $\mathcal M_S$ is $L^p$-bounded for $p>p_c,$ we are able to identify $p_c$ for all analytic surfaces of type $\mathcal A$ (wi…

$L^p$估计：$\boldsymbol{\text{R}}^3$中沿混合齐次超曲面的极大平均算子

$L^p$-Estimates for maximal averages along mixed homogeneous hypersurfaces in $\mathbb{R}^{3}$

本文研究$\boldsymbol{\text{R}}^3$中沿混合齐次超曲面的极大平均算子的$L^p$估计，确定临界勒贝格指数$p_c$，其有界性与黑塞行列式实根重数相关，且关联了FIO锥乘子理论的近期成果。

Sharp $L^p$-estimates for maximal operators associated to hypersurfaces in $\bR^3$ for $p>2.$

We study the boundedness problem for maximal operators $\M$ associated to smooth hypersurfaces $S$ in 3-dimensional Euclidean space. For $p>2,$ we prove that if no affine tangent plane to $S$ passes through the origin and $S$ is analytic, then the associated maximal operator is bounded on $L^p(\RR^3)$ if and only if $p>h(S),$ where $h(S)$ denotes the so-called height of the surface $S.$ For non-analytic finite type $S$ we obtain the same statement with the exception of the exponent $p=h(S).$ Our notion of height $h(S)$ is closely related to A. N. Varchenko's notion of height $h(ϕ)$ for functio…

Estimates for maximal functions associated to hypersurfaces in $\Bbb R^3$ with height $h<2:$ Part I

In this article, we continue the study of the problem of $L^p$-boundedness of the maximal operator $M$ associated to averages along isotropic dilates of a given, smooth hypersurface $S$ of finite type in 3-dimensional Euclidean space. An essentially complete answer to this problem had been given about seven years ago by the last named two authors in joint work with M. Kempe for the case where the height h of the given surface is at least two. In the present article, we turn to the case $h<2.$ More precisely, in this Part I, we study the case where $h<2,$ assuming that $S$ is contained in a suf…

The variation of the maximal function of a radial function

We study the problem concerning the variation of the Hardy-Littlewood maximal function in higher dimensions. As the main result, we prove that the variation of the non-centered Hardy-Littlewood maximal function of a radial function is comparable to the variation of the function itself.

Global Range Estimates for Maximal Oscillatory Integrals with Radial Testfunctions

We consider the maximal function of oscillatory integrals and prove a global estimate for radial test functions which is almost sharp with respect to the Sobolev regularity.

The best constant for the centered maximal operator on radial decreasing functions

We show that the lowest constant appearing in the weak type (1,1) inequality satisfied by the centered Hardy-Littlewood maximal operator on radial integrable functions is 1.

Maximal averages associated to families of finite type surfaces

We study the boundedness problem for maximal operators $\mathcal{M}$ associated to averages along families of hypersurfaces $S$ of finite type in $\mathbb{R}^n.$ In this paper, we prove that if $S$ is a finite type hypersurface which is of finite type $k$ at $x_0 \in \mathbb{R}^n$, then the associated maximal operator is bounded on $L^p(\mathbb{R}^n)$ for $p>k.$ We shall also consider a variable coefficient version of maximal theorem and we obtain the same $L^p-$ boundedness result for $p>k.$ We also discuss the consequence of this result. In particular, we verify a conjecture by E. M. Stein a…

Square functions and maximal operators associated with radial Fourier multipliers

We begin with an overview on square functions for spherical and Bochner-Riesz means which were introduced by Eli Stein, and discuss their implications for radial multipliers and associated maximal functions. We then prove new endpoint estimates for these square functions, for the maximal Bochner-Riesz operator, and for more general classes of radial Fourier multipliers.

Bounds for Lacunary maximal functions given by Birch--Magyar averages

We obtain positive and negative results concerning lacunary discrete maximal operators defined by dilations of sufficiently nonsingular hypersurfaces arising from Diophantine equations in many variables. Our negative results show that this problem differs substantially from that of lacunary discrete maximal operators defined along a nonsingular hypersurface. Our positive results are improvements over bounds for the corresponding full maximal functions which were initially studied by Magyar. In order to obtain positive results, we use an interpolation technique of the second author to reduce pr…

Regularity of the centered fractional maximal function on radial functions

We study the regularity properties of the centered fractional maximal function $M_β$. More precisely, we prove that the map $f \mapsto |\nabla M_βf|$ is bounded and continuous from $W^{1,1}(\mathbb{R}^d)$ to $L^q(\mathbb{R}^d)$ in the endpoint case $q=d/(d-β)$ if $f$ is radial function. For $d=1$, the radiality assumption can be removed. This corresponds to the counterparts of known results for the non-centered fractional maximal function. The main new idea consists in relating the centered and non-centered fractional maximal function at the derivative level.

Dimension dependency of the weak type $(1,1)$ bounds for maximal functions associated to finite radial measures

We show that the best constants appearing in the weak type (1,1) inequalities satisfied by the centered Hardy-Littlewood maximal function associated to some finite radial measures, such as the standard gaussian measure, grow exponentially fast with the dimension.

A characterization of maximal operators associated with radial Fourier multipliers

We give a simple necessary and sufficient condition for maximal operators associated with radial Fourier multipliers to be bounded on $L^p_{rad}$ and $L^p$ for certain $p$ greater than $2$. The range of exponents obtained for the $L^p_{rad}$ characterization is optimal for the given condition. The $L^p$ characterization is derived from an inequality of Heo, Nazarov, and Seeger regarding a characterization of radial Fourier multipliers.

The circular maximal operator on Heisenberg radial functions

Lebesgue space estimates are obtained for the circular maximal function on the Heisenberg group $\mathbb{H}^1$ restricted to a class of Heisenberg radial functions. Under this assumption, the problem reduces to studying a maximal operator on the Euclidean plane. This operator has a number of interesting features: it is associated to a non-smooth curve distribution and, furthermore, fails both the usual rotational curvature and cinematic curvature conditions.

On the lack of dimension free estimates in Lp for maximal functions associated to radial measures

In a recent article J. Aldaz proved that the weak L1 bounds for the centered maximal operator associated to finite radial measures cannot be taken independently with respect to the dimension. We show that at least for small p near to 1 the same result holds for the Lp bounds of such measures with decreasing densities. We also give some concrete examples, that include Gaussian measure, where better estimates with respect to the general case are obtained.

径向函数上广义球面极大算子的端点估计与最优性

Endpoint estimates and optimality for the generalized spherical maximal operator on radial functions

本文研究径向函数上广义球面平均 Radon 变换极大算子 $M^{\alpha,\beta}_t$，给出其在幂加权 Lebesgue 空间有界的尖锐条件及最优端点弱型估计。

$L^p-L^q$ estimates for the circular maximal operators on Heisenberg radial functions

$L^p$ boundedness of the circular maximal function $\mathcal M_{\mathbb{H}^1}$ on the Heisenberg group $\mathbb{H}^1$ has received considerable attentions. While the problem still remains open, $L^p$ boundedness of $\mathcal M_{\mathbb{H}^1}$ on Heisenberg radial functions was recently shown for $p>2$ by Beltran, Guo, Hickman, and Seeger [2]. In this paper we extend their result considering the local maximal operator $M_{\mathbb{H}^1}$ which is defined by taking supremum over $1<t<2$. We prove $L^p-L^q$ estimates for $M_{\mathbb{H}^1}$ on Heisenberg radial functions on the optimal range of $p,…

Behavior of weak type bounds for high dimensional maximal operators defined by certain radial measures

As shown in [A1], the lowest constants appearing in the weak type (1,1) inequalities satisfied by the centered Hardy-Littlewood maximal operator associated to certain finite radial measures, grow exponentially fast with the dimension. Here we extend this result to a wider class of radial measures and to some values of p>1. Furthermore, we improve the previously known bounds for p=1. Roughly speaking, whenever p\in (1, 1.03], if μis defined by a radial, radially decreasing density satisfying some mild growth conditions, then the best constants c_{p,d,μ} in the weak type (p,p) inequalities satis…

Maximal estimates for a generalized spherical mean Radon transform acting on radial functions

We study a generalized spherical means operator, viz. generalized spherical mean Radon transform, acting on radial functions. As the main results, we find conditions for the associated maximal operator and its local variant to be bounded on power weighted Lebesgue spaces. This translates, in particular, into almost everywhere convergence to radial initial data results for solutions to certain Cauchy problems for classical Euler-Poisson-Darboux and wave equations. Moreover, our results shed some new light to the interesting and important question of optimality of the yet known $L^p$ boundedness…

Maximal Functions for Lacunary Dilation Structures

If mu is a smooth density on a hypersurface in R^d whose curvature never vanishes to infinite order, and A is a d-by-d matrix whose eigenvalues all have absolute value greater than 1, then the maximal function given by convolving f with dilates of mu by powers of A, and taking the maximum, is bounded from a corresponding version of H^1 to weak L^1.

Sparse domination and $L^{p} \rightarrow L^{q}$ estimates for maximal functions associated with curvature

In this paper, we study maximal functions along some finite type curves and hypersurfaces. In particular, various impacts of non-isotropic dilations are considered. Firstly, we provide a generic scheme that allows us to deduce the sparse domination bounds for global maximal functions under the assumption that the corresponding localized maximal functions satisfy the $L^{p}$ improving properties. Secondly, for the localized maximal functions with non-isotropic dilations of curves and hypersurfaces whose curvatures vanish to finite order at some points, we establish the $L^{p}\rightarrow L^{q}$ …

The Variation of the Fractional Maximal Function of a Radial Function

In this paper we study the regularity of the non-centered fractional maximal operator $M_β$. As the main result, we prove that there exists $C(n,β)$ such that if $q=n/(n-β)$ and $f$ is a radial function, then $\|DM_βf\|_{L^{q}(\mathbb{R}^n)}\leq C(n,β)\|Df\|_{L^{1}(\mathbb{R}^n)}$. The corresponding result was previously known only if $n=1$ or $β=0$. Our proofs are almost free from one-dimensional arguments. Therefore, we believe that the new approach may be very useful when trying to extend the result for all $f\in W^{1,1}(\mathbb{R}^n)$.

具有分形伸缩集的双线性极大算子的Sobolev光滑性估计

Sobolev smoothing estimates for bilinear maximal operators with fractal dilation sets

本文研究超曲面上双线性平均算子及其极大版本，利用Fourier分析方法在分形伸缩集情形下建立L^2×L^2→L^2的Sobolev光滑性估计，并由此推导L^p改进界、稀疏界及加权推论，同时推广到三角形平均与双参数平均算子。

On high dimensional maximal operators

In this note we describe some recent advances in the area of maximal function inequalities. We also study the behaviour of the centered Hardy-Littlewood maximal operator associated to certain families of doubling, radial decreasing measures, and acting on radial functions. In fact, we precisely determine when the weak type $(1,1)$ bounds are uniform in the dimension.

The Best constant for the almost uncentered maximal operator on radial decreasing functions

The purpose of this note is to find the least weak type $(1,1)$ bound for the almost uncentered maximal operator on radial decreasing functions.

Smooth and singular maximal averages over 2D hypersurfaces and associated Radon transforms

We prove $L^p$ boundedness results, $p > 2$, for local maximal averaging operators over a smooth 2D hypersurface $S$ with either a $C^1$ density function or a density function with a singularity that grows as $|(x,y)|^{-β}$ for $β< 2$. Suppose one is in coordinates such that the surface is localized near some $(x_0,y_0,z_0)$ at which $(0,0,1)$ is normal to the surface, and suppose the surface is represented as the graph of $z_0 + s(x - x_0, y - y_0)$ near $(x_0,y_0)$, with $s(0,0) = 0$. It is shown that as long as the Taylor series of the Hessian determinant of $s(x,y)$ at $(0,0)$ is not ident…

带有分数集缩放的球形最大算子

Spherical maximal operators with fractal sets of dilations on radial functions

研究了在径向函数上，具有分数集缩放的球形最大算子在勒贝格空间中的映射性质，确定了高维情况下类型集仅依赖于E的上闵可夫斯基维度，并在二维情况下通过与E的上阿索夫谱相关的维度谱确定了L^p→L^q类型集的闭包。

Gradient bounds for radial maximal functions

In this paper we study the regularity properties of certain maximal operators of convolution type at the endpoint $p=1$, when acting on radial data. In particular, for the heat flow maximal operator and the Poisson maximal operator, when the initial datum $u_0 \in W^{1,1}( \mathbb{R}^d)$ is a radial function, we show that the associated maximal function $u^*$ is weakly differentiable and $$\|\nabla u^*\|_{L^1(\mathbb{R}^d)} \lesssim_d \|\nabla u_0\|_{L^1(\mathbb{R}^d)}.$$ This establishes the analogue of a recent result of H. Luiro for the uncentered Hardy-Littlewood maximal operator, now in a…

Problems and Results related to Waring's problem: Maximal functions and ergodic averages

We study the arithmetic analogue of maximal functions on diagonal hypersurfaces. This paper is a natural step following the papers of \cite{Magyar_dyadic}, \cite{Magyar_ergodic} and \cite{MSW}. We combine more precise knowledge of oscillatory integrals and exponential sums to generalize the asymptotic formula in Waring's problem to an approximation formula for the fourier transform of the solution set of lattice points on hypersurfaces arising in Waring's problem and apply this result to arithmetic maximal functions and ergodic averages. In sufficiently large dimensions, the approximation form…

球极最大函数与傅里叶积分算子的Hardy空间

Spherical maximal functions and Hardy spaces for Fourier integral operators

本文利用傅里叶积分算子的Hardy空间，研究球极最大函数的界及其在不同几何结构上的扩展，同时改进了波动方程的收敛性结果。

Fourier restriction implies maximal and variational Fourier restriction

We give an abstract argument that an a priori Fourier restriction estimate for a certain choice of exponents automatically implies maximal and variational Fourier restriction estimates. These, in turn, provide pointwise and quantitative interpretations of restriction of the Fourier transform to the hypersurface in question.