A sharp square function estimate for the cone in ℝ3

  title={A sharp square function estimate for the cone in
  author={Larry Guth and Houjie Wang and Ruixiang Zhang},
  journal={arXiv: Classical Analysis and ODEs},
We prove a sharp square function estimate for the cone in $\mathbb{R}^3$ and consequently the local smoothing conjecture for the wave equation in $2+1$ dimensions. 
Square function estimates and Local smoothing for Fourier Integral Operators
We prove a variable coefficient version of the square function estimate of Guth--Wang--Zhang. By a classical argument of Mockenhaupt--Seeger--Sogge, it implies the full range of sharp local smoothingExpand
Sharp smoothing properties of averages over curves
  • Hyerim Ko, Sanghyuk Lee, Sewook Oh
  • Mathematics
  • 2021
We study smoothing properties of the averaging operator defined by convolution with a measure supported on a smooth nondegenerate curve γ in R, d ≥ 3. Despite the simple geometric structure of suchExpand
Improved local smoothing estimate for the wave equation in higher dimensions
In this paper, we consider the local smoothing estimate associated with half-wave operator $e^{it\sqrt{-\Delta}}$ and the related fractional Schrodinger operator $e^{it(-\Delta)^{\alpha/2}}$ withExpand
$L^p-L^q$ estimates for the circular maximal operators on Heisenberg radial functions
L boundedness of the circular maximal function M H1 on the Heisenberg group H has received considerable attentions. While the problem still remains open, L boundedness of M H1 on Heisenberg radialExpand
$L^{p}$-estimate of Schr\"odinger maximal function in higher dimensions
Almost everywhere convergence on the solution of Schrödinger equation is an important problem raised by Carleson in harmonic analysis. In recent years, this problem was essentially solved by buildingExpand
$\ell^2$ decoupling for certain surfaces of finite type in $\mathbb{R}^3$
In this article, we establish an l decoupling inequality for the surface F 2 4 := { (ξ1, ξ2, ξ 4 1 + ξ 4 2) : (ξ1, ξ2) ∈ [0, 1] 2 } associated with the decomposition adapted to finite type geometryExpand
Local smoothing and Hardy spaces for Fourier integral operators
We show that the Hardy spaces for Fourier integral operators form a natural space of initial data when applying l decoupling inequalities to local smoothing for the wave equation. This yields newExpand
Maximal estimate for average over space curve
Let M be the maximal operator associated to a smooth curve in R which has nonvanishing curvature and torsion. We prove that M is bounded on Lp if and only if p > 3.
Variation bounds for spherical averages
We consider $r$-variation operators for the family of spherical means, with special emphasis on $L^p\to L^q$ estimates.
A sharp $L^{10}$ decoupling for the twisted cubic
We prove a sharp $l^{10}(L^{10})$ decoupling for the moment curve in $\mathbb{R}^3$. The proof involves a two-step decoupling combined with new incidence estimates for planks, tubes and plates.


The proof of the l2 decoupling conjecture
  • Ann. of Math. (2)
  • 2015
Wave front sets, local smoothing and Bourgain's circular maximal theorem
The purpose of this paper is to improve certain known regularity results for the wave equation and to give a simple proof of Bourgain's circular maximal theorem [1]. We use easy wave front analysisExpand
Mean square of zeta function, circle problem and divisor problem revisited
This paper is closely related to the recent work [BW17] of the same authors and our purpose is to elaborate more on some of the results and methods from [BW17]. More specifically our goal isExpand
A trilinear approach to square function and local smoothing estimates for the wave operator
The purpose of this paper is to improve Mockenhaupt's square function estimate and Sogge's local smoothing estimate in $\mathbb R^3$. For this we use the trilinear approach of S. Lee and A. VargasExpand
On the cone multiplier in R3
Abstract We prove the sharp L 3 bounds for the cone multiplier in R 3 and the associated square function, which is known as Mockenhauptʼs square function.
Homogenous Fourier multipliers of Marcinkiewicz type
This 1995 paper contains a sharp version of the classical Marcinkiewicz multiplier theorem for the class of homogeneous Fourier multipliers in two dimensions; here a one-dimensional MarcinkiewiczExpand
Propagation of singularities and maximal functions in the plane
SummaryIn this work we mainly generalize Bourgain's circular maximal function to include variable coefficient averages. Our techniques involve a combination of Bourgain's basic ideas plus microlocalExpand
Small cap decouplings
We develop a toolbox for proving decouplings into boxes with diameter smaller than the canonical scale. As an application of this new technique, we solve three problems for which earlier methods haveExpand
Incidence Estimates for Well Spaced Tubes
We prove analogues of the Szemeredi-Trotter theorem and other incidence theorems using $\delta$-tubes in place of straight lines, assuming that the $\delta$-tubes are well-spaced in a strong sense.