Related papers: A sharp square function estimate for the moment cu…

We use high-low frequency methods developed in the context of decoupling to prove sharp (up to $C_\epsilon R^\epsilon$) square function estimates for the moment curve $(t,t^2,\ldots,t^n)$ in $\mathbb{R}^n$. Our inductive scheme incorporates…

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.

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.

We prove sharp small cap decoupling estimates for the moment curve in $\mathbb{R}^3$. Our formulation of the small caps is motivated by a conjecture about $L^p$ estimates for exponential sums from the small cap decoupling paper of Demeter,…

We consider a type of maximal operators associated to moment curves in $\mathbb R^d, d\geq 3.$ We derive $L^p$ mapping properties for these operators. In a special case, the estimate is sharp.

This paper proves sharp small cap decoupling estimates for the moment curve $\mathcal{M}^n=\{(t,t^2,\ldots,t^n):0\leq t\leq 1\}$ in the remaining small cap parameter ranges for $\mathbb{R}^2$ and $\mathbb{R}^3$.

Building on the classical work of C\'{o}rdoba--Fefferman and the recent work of Schippa, we establish $L^4$ reverse square function estimates for functions whose Fourier support is contained in a $\delta$-neighborhood of the curve…

We provide sharp bounds for the exponential moments and $p$-moments, $1\leqslant p \leqslant 2$, of the terminate distribution of a martingale whose square function is uniformly bounded by one. We introduce a Bellman function for the…

We give a sharp convexity estimate for L-functions which have a functional equation and an Euler product.

We calculate the the sharp constant and characterise the extremal initial data in $\dot{H}^{\frac{3}{4}}\times\dot{H}^{-\frac{1}{4}}$ for the $L^4$ Sobolev--Strichartz estimate for the wave equation in four space dimensions.

We prove sharp local smoothing estimates for curve averages in all dimensions. As a corollary, we prove the sharp $L^p$ boundedness of the helical maximal operator in $\mathbb{R}^4$, which was previously known only for $\mathbb{R}^2$ and…

We prove a sharp estimate for the inscribed radius under certain fully nonlinear curvature flows. This estimate is asymptotically sharp on cylinders.

We introduce small cap square function estimates for parabola and cone, and prove the sharp estimates. More precisely, we study the inequalities of form \[ \|f\|_p\le C_{\alpha,p}(R)…

We prove the sharp local L^1 - L^\infty smoothing estimate for the logarithmic fast diffusion equation, or equivalently, for the Ricci flow on surfaces. Our estimate almost instantly implies an improvement of the known L^p - L^\infty…

A representation of the sharp coefficient in a pointwise estimate for the gradient of the generalized Poisson integral of a function $f$ on ${\mathbb R}^n$ is obtained under the assumption that $f$ belongs to $L^p$. The explicit value of…

We calculate the first and second moments of L-functions in the family of quadratic twists of a fixed elliptic curve E over F_q[x], asymptotically in the limit as the degree of the twists tends to infinity. We also compute moments involving…

We give sharp sectional curvature estimates for complete immersed cylindrically bounded $m$-submanifolds $\phi:M\to N\times\mathbb{R}^{\ell}$, $n+\ell\leq 2m-1$ provided that either $\phi$ is proper with the second fundamental form with…

We obtain a priori estimates for solutions to the prescribed scalar curvature equation on $S^3$. The usual non-degeneracy assumption on the curvature function is replaced by a new condition, which is necessary and sufficient for the…

In this note we give a sharp weighted estimate for square function from $L^2(w)$ to $L^2(w)$, $w\in A_2$. This has been known. But we also give a sharpening of this weighted estimate in the spirit of $T1$-type testing conditions. Finally we…

We establish the $L^p(\mathbb{R}^3)$ boundedness of the helical maximal function for the sharp range $p>3$. Our results improve the previous known bounds for $p>4$. The key ingredient is a new microlocal smoothing estimate for averages…