Maximal Function Estimates Research Articles

Pointwise convergence problem of the one‐dimensional Boussinesq‐type equation

In this paper, we consider the initial data problem of the one‐dimensional Boussinesq‐type equation. Inspired by the work of Compaan et al. (Int. Math. Res. Not. 2021, 599–650), by using the Fourier restriction norm method, we investigate the pointwise convergence of the one‐dimensional Boussinesq equation and cubic Boussinesq equation. We also only use the Strichartz estimates instead of Fourier restriction norm method to establish convergence conclusions of the one‐dimensional cubic Boussinesq equation. The key ingredients are some Strichartz estimates and high‐low frequency decomposition related to the nonlinear part of the integral form solution in the Duhamel form and the maximal function estimate related to inhomogeneous estimate.

Uniform convergence of the one-dimensional cubic Schrödinger equation

In this article, we investigate the uniform convergence problem of the one-dimensional cubic Schrödinger equation. By using the Kato smoothing estimate, the maximal function estimate and the dyadic mixed Lebesgue spaces, we establish the uniform convergence of the one-dimensional cubic Schrödinger equation in H s ( R ) ( s > 1 6 ) which is an alternative proof of Theorem 1.1 ( n = 1 , p = 3 ) of Compaan et al. [Pointwise convergence of the Schrödinger flow. Int Math Res Not. 2021;1:596–647].

Pointwise convergence and nonlinear smoothing of the generalized Zakharov–Kuznetsov equation

Abstract This paper is devoted to studying the pointwise convergence problem and nonlinear smoothing of the generalized Zakharov–Kuznetsov equation. Firstly, we present an alternative proof of Theorem 1.5 of Linares and Ramos [Maximal function estimates and local well-posedness for the generalized Zakharov–Kuznetsov equation, SIAM J. Math. Anal. 53 (2021), 1, 914–936] and Theorem 1.8 of Linares and Ramos [The Cauchy problem for the L 2 L^{2} -critical generalized Zakharov–Kuznetsov equation in dimension 3, Comm. Partial Differential Equations 46 (2021), 9, 1601–1627]. Secondly, we give an alternative proof of Theorem 1.1 of Ribaud and Vento [A note on the Cauchy problem for the 2D generalized Zakharov–Kuznetsov equations, C. R. Math. Acad. Sci. Paris 350 (2012), 9–10, 499–503] and present the nonlinear smoothing and uniform convergence of two-dimensional generalized Zakharov–Kuznetsov equation. Thirdly, we give an alternative proof of Theorem 1.4 of Linares and Ramos [Maximal function estimates and local well-posedness for the generalized Zakharov–Kuznetsov equation, SIAM J. Math. Anal. 53 (2021), 1, 914–936]. Finally, we study the nonlinear smoothing and uniform convergence of 𝑛-dimensional generalized Zakharov–Kuznetsov equation with n ≥ 3 n\geq 3 .

Capacitary maximal inequalities and applications

In this paper we introduce capacitary analogues of the Hardy-Littlewood maximal function,MCf(x):=supr>0⁡1C(B(x,r))∫B(x,r)|f|dC, for C= the Hausdorff content or a Riesz capacity. For these maximal functions, we prove a strong-type (p,p) bound for 1<p≤+∞ on the capacitary integration spaces Lp(C) and a weak-type (1,1) bound on the capacitary integration space L1(C). We show how these estimates clarify and improve the existing literature concerning maximal function estimates on capacitary integration spaces. As a consequence, we deduce correspondingly stronger differentiation theorems of Lebesgue-type, which in turn, by classical capacitary inequalities, yield more precise estimates concerning Lebesgue points for functions in Sobolev spaces.

On the persistence of spatial analyticity for generalized KdV equation with higher order dispersion

AbstractPersistence of spatial analyticity is studied for solutions of the generalized Korteweg‐de Vries (KdV) equation with higher order dispersion where , are integers. For a class of analytic initial data with a fixed radius of analyticity , we show that the uniform radius of spatial analyticity of solutions at time cannot decay faster than as . In particular, this improves a recent result due to Petronilho and Silva [Math. Nachr. 292 (2019), no. 9, 2032–2047] for the modified Kawahara equation (, ), where they obtained a decay rate of order . Our proof relies on an approximate conservation law in a modified Gevrey spaces, local smoothing, and maximal function estimates.

Estimates for maximal functions associated to hypersurfaces in $\mathbbm{R}^3$ with height h &lt; 2: part II : A geometric conjecture and its proof for generic 2-surfaces

Estimates for maximal functions associated to hypersurfaces in $\mathbbm{R}^3$ with height h &lt; 2: part II : A geometric conjecture and its proof for generic 2-surfaces

On the Global Well-Posedness and Orbital Stability of Standing Waves for the Schrödinger Equation with Fractional Dissipation

In this paper, we are concerned with the nonlinear fractional Schrödinger equation. We extend the result of Guo and Huo and prove that the Cauchy problem of the nonlinear fractional Schrödinger equation is global well-posed in H32−γ(R) with 12≤γ&lt;1. In view of the complexity of the nonlinear fractional Schrödinger equation itself, the local smoothing effect and maximal function estimates are not enough for presenting the global well-posedness for the nonlinear fractional Schrödinger equation. In this paper, we use a suitably iterative scheme and complete the global well-posed result for Equation (R). Moreover, we obtain the orbital stability of standing waves for the above equations via establishing the profile decomposition of bounded sequences in Hs(RN) (0&lt;s&lt;1) with N≥2.

Local well-posedness and spatial analyticity for the generalized Zakharov–Kuznetsov equation

The initial value problem for the generalized Zakharov–Kuznetsov equation on R2 is shown to be local well-posed in spaces of functions which are analytic on a strip without shrinking the width of the strip in time. The proof mainly relies upon the local smoothing effect, a multi-dimensional maximal function estimate and Bourgain-type spaces. These techniques are significant for derivative nonlinear dispersive equation in low regularity spaces. In addition, under the boundedness assumption of a suitable Sobolev norm the generalized Zakharov–Kuznetsov equation is proved to be Gevrey-class analysis. Especially, we obtain an explicit lower bound on the possible decreasing rate of the uniform radius of analyticity of a solution starting from analytic initial data.

$$L^{p}$$-Improving Bounds and Weighted Estimates for Maximal Functions Associated with Curvature

$$L^{p}$$-Improving Bounds and Weighted Estimates for Maximal Functions Associated with Curvature

Dimension-Free Estimates for the Discrete Spherical Maximal Functions

Abstract 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 $C&amp;gt;1$) growth in norms arising from the sampling principle of Magyar, Stein, and Wainger and ultimately deduce dimension-free estimates for the discrete spherical maximal functions.

Two-weight estimates for Hardy-Littlewood maximal functions and Hausdorff operators on $p$-adic Herz spaces

This paper is concerned with some sufficient conditions for the boundedness of Hardy-Littlewood maximal functions, rough Hausdorff and matrix Hausdorff operators on two-weighted Herz spaces on $p$-adic fields through its atomic decomposition. Bibliography: 34 titles.

Two-weight estimates for Hardy-Littlewood maximal functions and Hausdorff operators on $p$-adic Herz spaces

This paper is concerned with some sufficient conditions for the boundedness of Hardy-Littlewood maximal functions, rough Hausdorff and matrix Hausdorff operators on two-weighted Herz spaces on $p$-adic fields through its atomic decomposition.

Square function and non-tangential maximal function estimates for elliptic operators in 1-sided NTA domains satisfying the capacity density condition

Abstract Let Ω ⊂ ℝ n + 1 {\Omega\subset\mathbb{R}^{n+1}} , n ≥ 2 {n\geq 2} , be a 1-sided non-tangentially accessible domain (also known as uniform domain), that is, Ω satisfies the interior Corkscrew and Harnack chain conditions, which are respectively scale-invariant/quantitative versions of openness and path-connectedness. Let us assume also that Ω satisfies the so-called capacity density condition, a quantitative version of the fact that all boundary points are Wiener regular. Consider two real-valued (non-necessarily symmetric) uniformly elliptic operators L 0 ⁢ u = - div ⁡ ( A 0 ⁢ ∇ ⁡ u ) and L ⁢ u = - div ⁡ ( A ⁢ ∇ ⁡ u ) L_{0}u=-\operatorname{div}(A_{0}\nabla u)\quad\text{and}\quad Lu=-% \operatorname{div}(A\nabla u) in Ω, and write ω L 0 {\omega_{L_{0}}} and ω L {\omega_{L}} for the respective associated elliptic measures. The goal of this article and its companion [M. Akman, S. Hofmann, J. M. Martell and T. Toro, Perturbation of elliptic operators in 1-sided NTA domains satisfying the capacity density condition, preprint 2021, https://arxiv.org/abs/1901.08261v3] is to find sufficient conditions guaranteeing that ω L {\omega_{L}} satisfies an A ∞ {A_{\infty}} -condition or a RH q {\operatorname{RH}_{q}} -condition with respect to ω L 0 {\omega_{L_{0}}} . In this paper, we are interested in obtaining a square function and non-tangential estimates for solutions of operators as before. We establish that bounded weak null-solutions satisfy Carleson measure estimates, with respect to the associated elliptic measure. We also show that for every weak null-solution, the associated square function can be controlled by the non-tangential maximal function in any Lebesgue space with respect to the associated elliptic measure. These results extend previous work of Dahlberg, Jerison and Kenig and are fundamental for the proof of the perturbation results in the paper cited above.

Convergence problem of reduced Ostrovsky equation in Fourier–Lebesgue spaces with rough data and random data

This paper is devoted to studying the convergence problem of free reduced Ostrovsky equation in Fourier–Lebesgue spaces with rough data and the stochastic continuity of free reduced Ostrovsky equation in Fourier–Lebesgue spaces with random data. On the one hand, we establish the pointwise convergence related to the free reduced Ostrovsky equation in Fourier–Lebesgue spaces [Formula: see text] with rough data. In particular, we show that [Formula: see text] is the necessary condition for the maximal function estimate in [Formula: see text], which means that [Formula: see text] is optimal for rough data. On the other hand, we present the stochastic continuity of free reduced Ostrovsky equation at [Formula: see text] in Fourier–Lebesgue spaces [Formula: see text] with random data.

Convergence problem of Schrödinger equation in Fourier-Lebesgue spaces with rough data and random data

In this paper, we consider the convergence problem of Schrödinger equation. Firstly, we show the almost everywhere pointwise convergence of Schrödinger equation in Fourier-Lebesgue spaces H ^ 1 p , p 2 ( R ) ( 4 ≤ p &gt; ∞ ) , \hat {H}^{\frac {1}{p},\frac {p}{2}}(\mathbf {R})(4\leq p&gt;\infty ), H ^ 3 s 1 p , 2 p 3 ( R 2 ) ( s 1 &gt; 1 3 , 3 ≤ p &gt; ∞ ) , \hat {H}^{\frac {3 s_{1}}{p},\frac {2p}{3}}(\mathbf {R}^{2})(s_{1}&gt;\frac {1}{3},3\leq p&gt;\infty ), H ^ 2 s 2 p , p ( R n ) ( s 2 &gt; n 2 ( n + 1 ) , 2 ≤ p &gt; ∞ , n ≥ 3 ) \hat {H}^{\frac {2 s_{2}}{p},p}(\mathbf {R}^{n})(s_{2}&gt;\frac {n}{2(n+1)},2\leq p&gt;\infty ,n\geq 3) with rough data. Secondly, we show that the maximal function estimate related to one dimensional Schrödinger equation can fail with data in H ^ s , p 2 ( R ) ( s &gt; 1 p ) \hat {H}^{s,\frac {p}{2}}(\mathbf {R})(s&gt;\frac {1}{p}) . Finally, we show the stochastic continuity of Schrödinger equation with random data in L ^ r ( R n ) ( 2 ≤ r &gt; ∞ ) \hat {L}^{r}(\mathbf {R}^{n})(2\leq r&gt;\infty ) almost surely. The main ingredients are maximal function estimates and density theorem in Fourier-Lebesgue spaces as well as some large deviation estimates.

Sharp function estimates and boundedness for Toeplitz-type operators associated with general fractional integral operators

Abstract In this paper, we establish some sharp maximal function estimates for certain Toeplitz-type operators associated with some fractional integral operators with general kernel. As an application, we obtain the boundedness of the Toeplitz-type operators on the Lebesgue, Morrey and Triebel-Lizorkin spaces. The operators include the Littlewood-Paley operator, Marcinkiewicz operator and Bochner-Riesz operator.

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 ℍn, n ≥ 2, and the convolution f * μ. Form the associated maximal function Mf = supt>0 ∣f * μt∣ generated by the automorphic dilations. We use decoupling inequalities due to Wolff and Bourgain—Demeter to prove Lp-boundedness of M in an optimal range.

On the nonlinear Dysthe equation

This work is dedicated to putting on a solid analytic ground the theory of local well-posedness for the two dimensional Dysthe equation. This equation can be derived from the incompressible Navier–Stokes equation after performing an asymptotic expansion of a wavetrain modulation to the fourth order. Recently, this equation has been used to numerically study rare phenomena on large water bodies such as rogue waves. In order to study well-posedness, we use Strichartz, and improved smoothing and maximal function estimates. We follow ideas from the pioneering work of Kenig, Ponce and Vega, but since the equation is highly anisotropic, several technical challenges had to be resolved. We conclude our work by also presenting an ill-posedness result.

New Kakeya estimates using Gromov's algebraic lemma

This paper presents several new results related to the Kakeya problem. First, we establish a geometric inequality which says that collections of direction-separated tubes (thin neighborhoods of line segments that point in different directions) cannot cluster inside thin neighborhoods of low degree algebraic varieties. We use this geometric inequality to obtain a new family of multilinear Kakeya estimates for direction-separated tubes. Using the linear / multilinear theory of Bourgain and Guth, these multilinear Kakeya estimates are converted into Kakeya maximal function estimates. Specifically, we obtain a Kakeya maximal function estimate in Rn at dimension d(n)=(2−2)n+c(n) for some c(n)>0. Our bounds are new in all dimensions except n=2,3,4, and 6.

Weak type $(1, 1)$ estimates for maximal functions along $1$-regular sequences of integers

We show the pointwise convergence of the averages \[ \mathcal A _N f(x) = \frac {1}{\# {\bf B} _N} \sum _{n \in {\bf B} _N} f(x + n) \] for $f \in \ell ^1(\mathbb Z )$ where ${\bf B} _N = {\bf B} \cap [1, N]$, and ${\bf B} $ is a $1$-regular sequence of i