Related papers: Characterization of function spaces via low regula…

We study minimization of a structured objective function, being the sum of a smooth function and a composition of a weakly convex function with a linear operator. Applications include image reconstruction problems with regularizers that…

A $C^0$-Finsler structure is a continuous function $F:TM \rightarrow [0,\infty)$ defined on the tangent bundle of a differentiable manifold $M$ such that its restriction to each tangent space is an asymmetric norm. We use the convolution of…

In a doubling metric measure space $(X,\rho,\mu)$ supporting a Poincar\'e inequality, we give a new characterisation of first-order Sobolev spaces by mean oscillations, and extend previous characterisations of constant functions in terms of…

We study minimax density estimation on the product space $\mathbb{R}^{d_1}\times\mathbb{R}^{d_2}$. We consider $L^p$-risk for probability density functions defined over regularity spaces that allow for different level of smoothness in each…

We prove that smooth $C^\infty$ functions are dense in weighted fractional Sobolev spaces on an arbitrary open set, under some mild conditions on the weight. We also obtain a~similar result in non-weighted spaces defined by some kernel…

We consider the problem of estimating an unknown function f* and its partial derivatives from a noisy data set of n observations, where we make no assumptions about f* except that it is smooth in the sense that it has square integrable…

The resolvent Krylov subspace method builds approximations to operator functions $f(A)$ times a vector $v$. For the semigroup and related operator functions, this method is proved to possess the favorable property that the convergence is…

A basilar property and a useful tool in the theory of Sobolev spaces is the density of smooth compactly supported functions in the space $W^{k,p}(\R^n)$ (i.e. the functions with weak derivatives of orders $0$ to $k$ in $L^p$). On Riemannian…

Newtonian spaces generalize first-order Sobolev spaces to abstract metric measure spaces. In this paper, we study regularity of Newtonian functions based on quasi-Banach function lattices. Their (weak) quasi-continuity is established,…

We provide a comprehensive study of interrelations between different measures of smoothness of functions on various domains and smoothness properties of approximation processes. Two general approaches to this problem have been developed:…

Sobolev quantities (norms, inner products, and distances) of probability density functions are important in the theory of nonparametric statistics, but have rarely been used in practice, partly due to a lack of practical estimators. They…

For any connected component $H_0$ of the space of real meromorphic functions we build a compactification $N(H_0)$ of the space $H_0$. Then we express the Euler characteristics of the spaces $H_0$ and $N(H_0)$ in terms of topological…

In this paper we present a new characterization of Sobolev spaces on Euclidian spaces ($\mathbb{R}^n$). Our characterizing condition is obtained via a quadratic multiscale expression which exploits the particular symmetry properties of…

We give an intrinsic characterization of the restrictions of Sobolev, Triebel-Lizorkin and Besov spaces to regular subsets of $R^n$ via sharp maximal functions and local approximations.

A generalization of the classical Sard theorem in the plane is the following. Let $f$ be a function defined on a subset $A\subset{\mathbb R}^2$. If $f$ has modulus of continuity $\omega(r)\lesssim r^2$, then $f(A)\subset{\mathbb R}$ has…

In 2013, Masson and Siljander determined a method to prove that the $p$-minimal upper gradient $g_{f_\varepsilon}$ for the time mollification $f_\varepsilon$, $\varepsilon>0$, of a parabolic Newton-Sobolev function $f\in…

For certain weighted locally convex spaces $X$ and $Y$ of one real variable smooth functions, we characterize the smooth functions $\varphi: \mathbb{R} \to \mathbb{R}$ for which the composition operator $C_\varphi: X \to Y, \, f \mapsto f…

We survey a (nonlinear) Fredholm theory for a new class of ambient spaces called polyfolds, and develop the analytical foundations for some of the applications of the theory. The basic feature of these new spaces, which can be finite and…

The Sobolev regularity of invariant measures for diffusion processes is proved on non-smooth metric measure spaces with synthetic lower Ricci curvature bounds. As an application, the symmetrizability of semigroups is characterized, and the…

Suppose $X$ is an $\rm{RCD}(K,N)$ space with $K \in \mathbb{R}$ and $N \in (1,\infty)$. We obtain a characterisation of the Newtonian-Sobolev space $N^{1,2}(X)$ in terms of a quantity which measures to what extent a function is locally…