Let $H^n\cong \Bbb R^{2n}\ltimes \Bbb R$ be the Heisenberg group and let $μ_t$ be the normalized surface measure for the sphere of radius $t$ in $\Bbb R^{2n}$. Consider the maximal function defined by $Mf=\sup_{t>0} |f*μ_t|$. We prove for $n\ge 2$ that $M$ defines an operator bounded on $L^p(H^n)$ provided that $p>2n/(2n-1)$. This improves an earlier result by Nevo and Thangavelu, and the range for $L^p$ boundedness is optimal. We also extend the result to a more general setting of surfaces and to groups satisfying a nondegeneracy condition; these include the groups of Heisenberg type.
Let $G\cong\mathbb{R}^{d} \ltimes \mathbb{R}$ be a finite-dimensional two-step nilpotent group with the group multiplication $(x,u)\cdot(y,v)\rightarrow(x+y,u+v+x^{T}Jy)$ where $J$ is a skew-symmetric matrix satisfying a degeneracy condition with $2\leq {\rm rank}\, J <d$. Consider the maximal function defined by $$ {\frak M}f(x, u)=\sup_{t>0}\big|\int_Σ f(x-ty, u- t x^{T}Jy) dμ(y)\big|, $$ where $Σ$ is a smooth convex hypersurface and $dμ$ is a compactly supported smooth density on $Σ$ such that the Gaussian curvature of $Σ$ is nonvanishing on supp $dμ$. In this paper we prove that when $d\ge…
In this paper, we give the expressions for the bounded spherical functions, or equivalently the spherical functions of positive type, for the free two-step nilpotent Lie groups endowed with the actions of orthogonal groups or their special subgroups. Next we deduce some results about the (Kohn) sub-Laplacian, and we compute the radial Plancherel measure.
We discuss some of our work on averages along polynomial sequences in nilpotent groups of step 2. Our main results include boundedness of associated maximal functions and singular integrals operators, an almost everywhere pointwise convergence theorem for ergodic averages along polynomial sequences, and a nilpotent Waring theorem. Our proofs are based on analytical tools, such as a nilpotent Weyl inequality, and on complex almost-orthogonality arguments that are designed to replace Fourier transform tools, which are not available in the non-commutative nilpotent setting. In particular, we pres…
We study a family of maximal operators that provides a continuous link connecting the Hardy-Littlewood maximal function to the spherical maximal function. Our theorems are proved in the multilinear setting but may contain new results even in the linear case. For this family of operators we obtain bounds between Lebesgue spaces in the optimal range of exponents.
We prove an expanded range of $\ell ^{p}(\mathbb{Z}^d)$-improving properties and sparse bounds for discrete spherical maximal means in every dimension $d\geq 6$. Essential elements of the proofs are bounds for high exponent averages of Ramanujan and restricted Kloosterman sums.
We obtain boundedness for the bilinear spherical maximal function in a range of exponents that includes the Banach triangle and a range of $L^p$ with $p<1$. We also obtain counterexamples that are asymptotically optimal with our positive results on certain indices as the dimension tends to infinity.
In this paper we introduce and study a bilinear spherical maximal function of product type in the spirit of bilinear Calderón-Zygmund theory. This operator is different from the bilinear spherical maximal function considered by Geba et al. We deal with lacunary and full versions of this operator, and we prove weighted estimates with respect to bilinear weights. Our approach involves sparse forms following ideas by Lacey, but we also use other techniques to handle the particular triplet $(2,2,1)$.
In this paper we deal with lacunary and full versions of the spherical maximal function on the Heisenberg group $\mathbb{H}^n$, for $n\ge 2$. By suitable adaptation of an approach developed by M. Lacey in the Euclidean case, we obtain sparse bounds for these maximal functions, which lead to new unweighted and weighted estimates. In particular, we deduce the $L^p$ boundedness, for $1<p<\infty$, of the lacunary maximal function associated to the spherical means on the Heisenberg group. In order to prove the sparse bounds, we establish $ L^p-L^q $ estimates for local (single scale) variants of th…
In dimensions $n\ge 2$ we obtain $L^{p_1}(\mathbb R^n) \times\dots\times L^{p_m}(\mathbb R^n)$ to $L^p(\mathbb R^n)$ boundedness for the multilinear spherical maximal function in the largest possible open set of indices and we provide counterexamples that indicate the optimality of our results. Moreover, we obtain weak type and Lorentz space estimates as well as counterexamples in the endpoint cases.
We exhibit a range of $\ell ^{p}(\mathbb{Z}^d)$-improving properties for the discrete spherical maximal average in every dimension $d\geq 5$. The strategy used to show these improving properties is then adapted to establish sparse bounds, which extend the discrete maximal theorem of Magyar, Stein, and Wainger to weighted spaces. In particular, the sparse bounds imply that the discrete spherical maximal average is a bounded map from $\ell^2(w)$ into $\ell^2(w)$ provided $w^{\frac{d}{d-4}+δ}$ belongs to the Muckenhoupt class $A_2$ for some $δ>0.$
We study the maximal estimates for the bilinear spherical average and the bilinear Bochner-Riesz operator. Firstly, we obtain $L^p\times L^q \to L^r$ estimates for the bilinear spherical maximal function on the optimal range. Thus, we settle the problem which was previously considered by Geba, Greenleaf, Iosevich, Palsson and Sawyer, later Barrionevo, Grafakos, D. He, Honzík and Oliveira, and recently Heo, Hong and Yang. Secondly, we consider $L^p\times L^q \to L^r$ estimates for the maximal bilinear Bochner-Riesz operators and improve the previous known results. For the purpose we draw a conn…
We derive sparse bounds for the bilinear spherical maximal function in any dimension $d\geq 1$. When $d\geq 2$, this immediately recovers the sharp $L^p\times L^q\to L^r$ bound of the operator and implies quantitative weighted norm inequalities with respect to bilinear Muckenhoupt weights, which seems to be the first of their kind for the operator. The key innovation is a group of newly developed continuity $L^p$ improving estimates for the single scale bilinear spherical averaging operator.
In this paper we investigate the $L^p$ boundedness of the lacunary maximal function $ M_{\Ha}^{lac} $ associated to the spherical means $ A_r f$ taken over Koranyi spheres on the Heisenberg group. Closely following an approach used by M. Lacey in the Euclidean case, we obtain sparse bounds for these maximal functions leading to new unweighted and weighted estimates. The key ingredients in the proof are the $L^p$ improving property of the operator $A_rf$ and a continuity property of the difference $A_rf-τ_y A_rf$, where $τ_yf(x)=f(xy^{-1})$ is the right translation operator.
We prove that the discrete spherical maximal functions (in the spirit of Magyar, Stein and Wainger) corresponding to the Euclidean spheres in $\mathbb Z^d$ with dyadic radii have $\ell^p(\mathbb Z^d)$ bounds for all $p\in[2, \infty]$ independent of the dimensions $d\ge 5$. An important part of our argument is the asymptotic formula in the Waring problem for the squares with a dimension-free multiplicative error term. By considering new approximating multipliers we will show how to absorb an exponential in dimension (like $C^d$ for some $C>1$) growth in norms arising from the sampling principle…
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 $M_E f(x)=\sup_{t\in E} |A_t f(x)|$.
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.
Let $f\in L^p(\mathbb{R}^d)$, $d\ge 3$, and let $A_t f(x)$ the average of $f$ over the sphere with radius $t$ centered at $x$. For a subset $E$ of $[1,2]$ we prove close to sharp $L^p\to L^q$ estimates for the maximal function $\sup_{t\in E} |A_t f|$. A new feature is the dependence of the results on both the upper Minkowski dimension of $E$ and the Assouad dimension of $E$. The result can be applied to prove sparse domination bounds for a related global spherical maximal function.
We consider the averages of a function $ f$ on $ \mathbb R ^{n}$ over spheres of radius $ 0< r< \infty $ given by $ A_{r} f (x) = \int_{\mathbb S ^{n-1}} f (x-r y) \; d σ(y)$, where $ σ$ is the normalized rotation invariant measure on $ \mathbb S ^{n-1}$. We prove a sharp range of sparse bounds for two maximal functions, the first the lacunary spherical maximal function, and the second the full maximal function. $$ M_{lac} f = \sup_{j\in \mathbb Z } A_{2^j} f , \qquad M_{full} f = \sup_{ r>0 } A_{r} f . $$ The sparse bounds are very precise variants of the known $L^p$ bounds for these maximal …
We provide a new direct proof of the $\ell^2$-boundedness of the Discrete Spherical Maximal Function that neither relies on abstract transference theorems (and hence Stein's Spherical Maximal Function Theorem) nor on delicate asymptotics for the Fourier transform of discrete spheres.
We prove $L^p\to L^q$ estimates for the local maximal operator associated with dilates of the Kóranyi sphere in Heisenberg groups. These estimates are sharp up to endpoints and imply new bounds on sparse domination for the corresponding global maximal operator. We also prove sharp $L^p\to L^q$ estimates for spherical means over the Korányi sphere, which can be used to improve the sparse domination bounds for the associated lacunary maximal operator.
In this paper, we show that there exists a positive density subsequence of orthonormal spherical harmonics which achieves the maximal Lp norm growth for 2<p<=6, therefore giving an example of a Riemannian surface supporting such subsequence of eigenfunctions. This answers the question proposed by Sogge and Zelditch (arXiv:1011.0215). Furthermore, we provide an explicit lower bound on the density in this example.
We establish pointwise almost everywhere convergence for ergodic averages along polynomial sequences in nilpotent groups of step two of measure-preserving transformations on $σ$-finite measure spaces. We also establish corresponding maximal inequalities on $L^p$ for $1<p\leq \infty$ and $ρ$-variational inequalities on $L^2$ for $2<ρ<\infty$. This gives an affirmative answer to the Furstenberg-Bergelson-Leibman conjecture in the linear case for all polynomial ergodic averages in discrete nilpotent groups of step two. Our proof is based on almost-orthogonality techniques that go far beyond Fouri…
We extend the main result of Harrow, Kolla, and Schulman -- the existence of dimension-free $L^2$-bounds for the spherical maximal function in the hypercube -- to all $L^p, p > 1$. Our approach is motivated by the spectral technique developed by Stein and Nevo and Stein in the context of pointwise ergodic theorems on general groups. We provide an example which demonstrates that no dimension-free weak-type (1-1) bound exists at the endpoint.
We improve the range of $\ell^p(\mathbb Z^d)$-boundedness of the integral $k$-spherical maximal functions introduced by Magyar. The previously best known bounds for the full $k$-spherical maximal function require the dimension $d$ to grow at least cubicly with the degree $k$. Combining ideas from our prior work with recent advances in the theory of Weyl sums by Bourgain, Demeter, and Guth and by Wooley, we reduce this cubic bound to a quadratic one. As an application, we deduce improved bounds in the ergodic Waring--Goldbach problem.
In this paper, we use the Approximation Formula for the Fourier transform of the solution set of lattice points on k-spheres and methods of Bourgain and Ionescu to refine the l^p(Z^d)-boundedness results for discrete k- spherical maximal functions to a restricted weak-type result at the endpoint. Moreover we introduce a novel Approximation Formula for a single average; this allows us to improve our bounds for discrete k-spherical maximal functions along sparse subsequences of radii by exploiting recent progress of Wooley on the Vinogradov mean value conjectures. In particular we have improved …
Sharp power-weight estimates for the fractional spherical maximal operator
National Taiwan Normal University（台湾师范大学）；National Center for Theoretical Sciences, NTU（台湾理论科学中心）；National Taiwan University（台湾大学）；University of Pittsburgh（匹兹堡大学）
We prove sparse bounds for the spherical maximal operator of Magyar, Stein and Wainger. The bounds are conjecturally sharp, and contain an endpoint estimate. The new method of proof is inspired by ones by Bourgain and Ionescu, is very efficient, and has not been used in the proof of sparse bounds before. The Hardy-Littlewood Circle method is used to decompose the multiplier into major and minor arc components. The efficiency arises as one only needs a single estimate on each element of the decomposition.
