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We prove mixed-norm estimates for circular averages with respect to $\alpha$-dimensional fractal measures on $\mathbb{R}^2$, using circle tangency bounds when $\alpha \in (0,1]$ and a $\delta$-discretized slicing lemma for fractals when…
We study interior curvature estimates for convex graphs which satisfy the quotient equation $\frac{\sigma_{n}}{\sigma_{n-2}}(\lambda)=f(X)>0$ in this paper.
Recently, two of the authors obtained estimates for the adjoint restriction operator to finite type curves with respect to general measures. Strikingly, it turns out that some of such estimates are sharp, especially when the measures are…
The purpose of this paper is to prove an essentially sharp L^2 Fourier restriction estimate for light cones, of the type which is called bilinear in the recent literature.
We address Gromov's Quantification of $C^0$ Convergence Conjecture in dimension three. Let $B$ be the unit ball in $\mathbb R^3$. Let $g$ and $g_0$ be smooth metrics on $B$. We prove there are constants $C$ and $\epsilon_0$ depending only…
We present new sharp results concerning multipliers and distance estimates in various spaces of harmonic functions in the unit ball of $R^n$.
In this short note, we prove a decay estimate for non-linear solutions of 3D cubic defocusing non-linear Schr\"odinger equation.
The equations for waves on the surface of an irrotational incompressible fluid are derived in the coordinates of the velocity potential/stream function. The low frequency shallow water approximation for these waves is derived for a varying…
We have derived an analytical formulation for estimating the volume of geometries enclosed by implicitly defined surfaces. The novelty of this work is due to two aspects. First we provide a general analytical formulation for all…
We prove that finite area isolated singularities of surfaces with constant positive curvature in R^3 are removable singularities, branch points or immersed conical singularities. We describe the space of immersed conical singularities of…
We prove L^1 --> L^\infty estimates for the linear Schroedinger equation in three dimensions. The potential is assumed to belong to certain L^p spaces, but no pointwise decay estimates and no additional regularity is required.
We prove local unique solvability of the wave equation for a large class of weakly singular, locally bounded space-time metrics in a suitable space of generalised functions.
We prove a smooth compactness theorem for the space of embedded self-shrinkers in $\RR^3$. Since self-shrinkers model singularities in mean curvature flow, this theorem can be thought of as a compactness result for the space of all…
The 3-dimensional wave function for a sharply cut-off Coulomb potential is analytically derived. The asymptotic form of the related scattering amplitude reveals a failure of the standard renormalization factor which is believed to be…
We discuss the validity of Minkowski integral identities for hypersurfaces inside a cone, intersecting the boundary of the cone orthogonally. In doing so we correct a formula provided in [3]. Then we study rigidity results for constant mean…
The Fourier restriction conjecture is a fundamental problem in harmonic analysis. In this paper, we investigate restriction estimates for degenerate higher codimensional quadratic surfaces and obtain sharp results for some types of…
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…
Weakly stable constant mean curvature (CMC) hypersurfaces are stable critical points of the area functional with respect to volume preserving deformations. We establish a pointwise curvature estimate (in the non-singular dimensions) and a…
There has been substantial recent work on methods for estimating the slope function in linear regression for functional data analysis. However, as in the case of more conventional finite-dimensional regression, much of the practical…
We show that on the $d$-dimensional cube $I^d\equiv [0,1]^d$ the discrete moduli of smoothness which use only the values of the function on a diadic mesh are sufficient to determine the moduli of smoothness of that function. As an important…