arXiv research

Optimizes coordinate charts for smooth elliptic structures.

Sharp Hölder regularity found for complex Frobenius theorem coordinates.

Smooth functions preserve Zygmund class on curves.

New stability estimate for metric rigidity in hyperbolic dynamics.

Establishes Calderón-Zygmund inequalities on evolving Riemannian manifolds.

The paper improves the smoothness of vector fields on manifolds.

Survey of global Calderón-Zygmund inequalities on Riemannian manifolds.

Sharp Sobolev theory for scalar elliptic equations on minimal regular manifolds.

The Calderón--Zygmund estimate fails on certain open manifolds.

Density of smooth functions in Sobolev space on manifolds with curvature bounds.

The paper addresses geometric analysis on non-compact Riemannian manifolds, proving Calderón-Zygmund inequalities.

We parametrize the space $\mathcal{Z}$ of Zygmund vector fields on the unit circle in terms of infinitesimal shear functions on the Farey tesselation. Then we express the Hilbert transform and the Fourier coefficients of the Zygmund vector fields in terms of the above parametrization by infinitesimal shear functions. F…

New example shows non-compact manifolds can lack $L^p$-Calderón-Zygmund inequalities.

The study shows that close hypersurfaces have uniformly bounded inequalities.

Being motivated by the problem of deducing $L^p$-bounds on the second fundamental form of an isometric immersion from $L^p$-bounds on its mean curvature vector field, we prove a (nonlinear) Calderón-Zygmund inequality for maps between complete (possibly noncompact) Riemannian manifolds.

Paper proves a quantitative estimate for transforming almost complex structures into standard ones.

An essentially unique homeomorphic solution to the Beltrami equation was found in the 1960s using the theory of Calderón-Zygmund and singular integral operators in $L^p(\mathbb C)$. We will present an alternative method to solve the Beltrami equation using the Hodge star operator and standard elliptic PDE theory. We wi…

Study interior estimates for solutions of Poisson equation on Riemann surfaces.

Study on smoothness of solutions to nonlinear equations on Riemannian manifolds.

We introduce the concept of Calderón-Zygmund inequalities on Riemannian manifolds. For $1<p<\infty$, these are inequalities of the form $\left\Vert \mathrm{Hess}\left( u\right) \right\Vert _{L^p}\leq C_{1}\left\Vert u\right\Vert _{L^p}+C_{2}\left\Vert Δu\right\Vert _{L^p},$ valid a priori for all smooth functions $…

In the category of metrics with conical singularities along a smooth divisor with angle in $(0, 2π)$, we show that locally defined weak solutions ($C^{1,1}-$solutions) to the Kähler-Einstein equations actually possess maximum regularity, which means the metrics are actually Hölder continuous in the singular polar coord…

A new subdivision scheme for Heisenberg group values with central smoothness loss.

Study surfaces in Half-Pipe space and vector fields on hyperbolic plane.

The study bounds Riesz transforms on manifolds with controlled curvature.

We give dimension-free regularity conditions for a class of possibly degenerate sub-elliptic equations in the Heisenberg group exhibiting super-quadratic growth in the horizontal gradient; this solves an issue raised by Manfredi & Mingione (Math. Ann. 2007) where only dimension dependent bounds for the growth exponent …

Study $L^p$ boundedness of Riesz transform on differential forms for certain manifolds.

Paper generalizes paracomposition and change of variables for paradifferential operators.

The paper examines $L^p$ gradient and Riesz transform estimates under Ricci lower bounds.

The paper proves a theorem about earthquake extensions of vector fields on circles.

The paper establishes conditions for complex structures on manifolds with given vector fields.

The paper proves boundedness of a Riesz transform on weighted manifolds.

We prove a version of Smirnov type theorem and Charatheodory type theorem for a harmonic homeomorphism of the unit disk onto a Jordan surface with rectifiable boundary. Further we establish the classical isoperimetric inequality and Riesz--Zygmund inequality for Jordan harmonic surfaces without any smoothness assumptio…

In this work the Isoperimetric Inequality for integral varifolds is used to obtain sharp estimates for the size of the set where the density quotient is small and to generalise Calderón's and Zygmund's theory of first order differentiability for functions in Lebesgue spaces from Lebesgue measure to integral varifolds.

The paper examines functional properties on manifolds with very negative curvature.

The Global Newlander-Nirenberg theorem is proven for domains with finite smooth boundary in complex manifolds.

We prove existence and uniqueness of solutions to the Minkowski problem in any domain of dependence $D$ in $(2+1)$-dimensional Minkowski space, provided $D$ is contained in the future cone over a point. Namely, it is possible to find a smooth convex Cauchy surface with prescribed curvature function on the image of the …

Let M be a globally Riemannian symmetric space. We prove a duality estimate between pairings of vector fields with divergence zero and and in L^1 with vector fields in a critical Sobolev space on M. As a consequence we get a sharp Calderon-Zygmund estimate for solutions to Poisson's equation on M, where the right side …

Given a finite collection of $C^1$ vector fields on a $C^2$ manifold which span the tangent space at every point, we consider the question of when there is locally a coordinate system in which these vector fields have a higher level of smoothness. For example, when is there a coordinate system in which the vector field…

We give an overview of the generalized Calderón-Zygmund theory for "non-integral" singular operators, that is, operators without kernels bounds but appropriate off-diagonal estimates. This theory is powerful enough to obtain weighted estimates for such operators and their commutators with $\BMO$ functions. $L^p-L^q$ of…

Constructs distance-like functions on manifolds with controlled curvature to study Sobolev spaces.

Given a finite collection of $C^1$ vector fields on a $C^2$ manifold which span the tangent space at every point, we consider the question of when there is locally a coordinate system in which these vector fields are real analytic. We give necessary and sufficient, coordinate-free conditions for the existence of such a…

New algorithm for batch list-decodable linear regression with stronger guarantees.

Study on harmonic functions in RCD spaces, focusing on singular points and vanishing gradients.

The paper proves density of smooth functions in Sobolev spaces on certain manifolds.

Given an arbitrary closed set A of $\mathbf{R}^{n}$, we establish the relation between the eigenvalues of the approximate differential of the spherical image map of A and the principal curvatures of A introduced by Hug-Last-Weil, thus extending a well known relation for sets of positive reach by Federer and Zaehle. The…

Paper proves spectral rigidity of hyperbolic cusped manifolds for compact deformations.

Given a finite collection of $C^1$ vector fields on a $C^2$ manifold which span the tangent space at every point, we consider the question of when there is locally a coordinate system in which these vector fields are $\mathscr{C}^{s+1}$ for $s\in (1,\infty]$, where $\mathscr{C}^s$ denotes the Zygmund space of order $s$…

Deep networks can efficiently approximate functions on curved manifolds.