Related papers: Square function estimates for cones over quadratic…

We show sharp square function estimates for curves in the plane whose curvature degenerates at a point and estimates sharp up to endpoints for cones over these curves. To this end, for curves of finite type we extend the classical…

In this paper, we extend the C\'ordoba-Fefferman square function estimate for the parabola to a weighted setting. Our weighted square function estimate is derived from a weighted wave envelope estimate for the parabola. The bounds are…

In this paper we establish square-function estimates on the double and single layer potentials for divergence-form elliptic operators, of arbitrary even order 2m, with variable t-independent coefficients in the upper half-space. This…

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 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)…

Weighted cone-volume functionals are introduced for the convex polytopes in $\mathbb{R}^n$. For these functionals, geometric inequalities are proved and the equality conditions are characterized. A variety of corollaries are derived,…

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 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 square function estimates for certain conical regions. Specifically, let $\{\Delta_j\}$ be regions of the unit sphere $\mathbb{S}^{n-1}$ and let $S_j f$ be the smooth Fourier restriction of $f$ to the conical region…

The goal of this paper is to prove bilinear $L^p$ estimates for rough dispersive evolutions satisfying non-degeneracy and transversality assumptions. The estimates generalize the sharp Fourier extension estimates for the cone and the…

We present explicit constructions of orthogonal polynomials inside quadratic bodies of revolution, including cones, hyperboloids, and paraboloids. We also construct orthogonal polynomials on the surface of quadratic surfaces of revolution,…

Weighted quadratic estimates are proved for certain bisectorial firstorder differential operators with bounded measurable coefficients which are (not necessarily pointwise) accretive, on complete manifolds with positive injectivity radius.…

A formula for calculating Extensions of (mainly integral) Polynomial Functors is established, based upon projective resolutions. Sample computations are performed, which, in particular, exhibit a surprising non-trivial extension of Divided…

In this note, we provide explicit expressions for the projections onto the graph of a quadratic polynomial. The projections are obtained by examining the critical points of the associated quartic polynomial, that is, the roots of the cubic…

In this paper we study the spherical convexity of quadratic functions on spherically convex sets. In particular, conditions characterizing the spherical convexity of quadratic functions on spherical convex sets associated to the positive…

We prove sharp local smoothing estimates for wave equations on compact Riemannian manifolds in $n+1$ dimensions for odd $n$ and obtain improved estimates in even dimensions. This is achieved by deriving local smoothing estimates for certain…

Second derivative pinching estimates are proved for a class of elliptic and parabolic equations, including motion of hypersurfaces by curvature functions such as quotients of elementary symmetric functions of curvature. The estimates imply…

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…

In this paper, we prove small cap square function and decoupling estimates for the parabola, where the small caps are essentially axis-parallel rectangles of dimensions $\delta\times \delta^\beta$ for $0\leq \beta\leq 1$. Our estimates…

Given a quaternionic slice regular function $f$, we give a direct and effective way to compute the coefficients of its spherical expansion at any point. Such coefficients are obtained in terms of spherical and slice derivatives of the…