Related papers: Averages over curves with torsion

By extending the classical quantitative approximation results for positive and linear operators in $L^{p}([0, 1]), 1\le p \le +\infty$ of Berens and DeVore in 1978 and of Swetits and Wood in 1983 to the more general case of sublinear,…

It is proved the generalization of Toponogov theorem about the length of the curve in two-dimensional Riemannian manifolds in the case of two-dimensional Alexandrov spaces.

In this paper we investigate the approximation properties of kernel interpolants on manifolds. The kernels we consider will be obtained by the restriction of positive definite kernels on $\R^d$, such as radial basis functions (RBFs), to a…

This paper considers the problem of establishing $L^p$-improving inequalities for Radon-like operators in intermediate dimensions (i.e., for averages overs submanifolds which are neither curves nor hypersurfaces). Due to limitations in…

We obtain a local Sobolev constant estimate for integral Ricci curvature, which enables us to extend several important tools such as the maximal principle, the gradient estimate, the heat kernel estimate and the $L^2$ Hessian estimate to…

We introduce a new notion of generalized log twisted curves, which are marked nodal curves with additional data at the marked points. In the case when the markings are distinct this notion agrees with the notion of twisted curve introduced…

We show that every $\mathbb{R}^d$-valued Sobolev path with regularity $\alpha$ and integrability $p$ can be lifted to a Sobolev rough path provided $\alpha < 1/p<1/3$. The novelty of our approach is its use of ideas underlying Hairer's…

We prove logarithmic Sobolev inequalities on higher-dimensional bounded smooth domains based on novel Gagliardo-Nirenberg type interpolation inequalities. Moreover, we use them to address the long-time dynamics of some nonlinear nonlocal…

We prove $L^p$ quantitative differentiability estimates for functions defined on uniformly rectifiable subsets of the Euclidean space. More precisely, we show that a Dorronsoro-type theorem holds in this context: the $L^p$ norm of the…

The minimal surfaces meeting in triples with equal angles along a common boundary naturally arise from soap films and other physical phenomenon. They are also the natural extension of the usual minimal surface. In this paper, we consider…

We obtain a coarse relationship between geometric intersection numbers of curves and the sum of their subsurface projection distances with explicit quasi-constants. By using this relationship, we give applications in the studies of the…

We analyze the distribution of unitarized L-polynomials Lp(T) (as p varies) obtained from a hyperelliptic curve of genus g <= 3 defined over Q. In the generic case, we find experimental agreement with a predicted correspondence (based on…

The paper is devoted to proving Allard-Michael-Simon-type $L^p$-Sobolev inequalities $(p>1)$ with explicit constants in the setting of Euclidean minimal submanifolds of arbitrary codimension. Our results require separate discussions for the…

We obtain some weighted $L^{p}$-Sobolev estimates with gain on $p$ and the weight for solutions of the $\overline{\partial}$-equation in lineally convex domains of finite type in $\mathbb{C}^{n}$ and apply them to obtain weighted…

In this paper we compute the Hausdorff distance between sets of continuous curves and sets of piecewise constant or linear discretizations. These sets are Sobolev balls given by the continuous or discrete $L^p$-norm of the derivatives. We…

For a general open set, we characterize the compactness of the embedding $W^{1,p}_0\hookrightarrow L^q$ in terms of the summability of its torsion function. In particular, for $1\le q<p$ we obtain that the embedding is continuous if and…

We follow the method of ABP estimate in \cite{brendle2021} and apply it to spacelike submanifolds in $\mathbb R^{n,1}$. We then obtain Michael-Simon type inequalities. Surprisingly, our investigation leads to a Sobolev inequality without a…

In this paper, we establish the curvature estimates for $p$-convex hypersurfaces in $\mathbb{R}^{n+1}$ of prescribed curvature with $p\geq \frac{n}{2}$. The existence of a star-shaped hypersurface of prescribed curvature is obtained. We…

The conformal nature of smooth curves in $\mathbb{R}^3$ is characterised by conformal length, curvature and torsion. We present a derivation of these conformal parameters via a limiting process using inscribed polygons with circular edges .…

We prove $l^p$-improving estimates for the averaging operator along the discrete paraboloid in the sharp range of $p$ in all dimensions $n\ge 2$.