Related papers: Sections of functions and Sobolev type inequalitie…

In this paper we give a geometric condition which ensures that $(q,p)$-Poincar\'e-Sobolev inequalities are implied from generalized $(1,1)$-Poincar\'e inequalities related to $L^1$ norms in the context of product spaces. The concept of…

We provide some necessary and sufficient conditions for a proper lower semicontinuous convex function, defined on a real Banach space, to be locally or globally Lipschitz continuous. Our criteria rely on the existence of a bounded selection…

In this paper we consider an abstract Wiener space $(X,\gamma,H)$ and an open subset $O\subseteq X$ which satisfies suitable assumptions. For every $p\in(1,+\infty)$ we define the Sobolev space $W_{0}^{1,p}(O,\gamma)$ as the closure of…

The purpose of this survey is a comprehensive study of operator Lip\-schitz functions. A continuous function $f$ on the real line ${\Bbb R}$ os called operator Lipschitz if $\|f(A)-f(B)\|\le\operatorname{const}\|A-B\|$ for arbitrary…

On metric spaces equipped with doubling measures, we prove that a differentiability theorem holds for Lipschitz functions if and only if the space supports nontrivial (metric) derivations in the sense of Weaver that satisfy an additional…

We introduce the generalized notion of semicontinuity of a function defined on a topological space and derive the useful classification of the so-called Lipschitz derivatives of functions defined on a metric space. Secondly, we investigate…

We investigate variable coefficient analogs of a recent work of Bortz, Hofmann, Martell and Nystr\"om [BHMN25]. In particular, we show that if $\Omega$ is the region above the graph of a Lip(1,1/2) (parabolic Lipschitz) function and $L$ is…

We prove the continuity of Sobolev functions $\varphi \in W^{1,n}_{\mathrm{loc}}(\Omega)$, $\Omega \subset \mathbb{R}^n$, that satisfy \[ \lvert\nabla \varphi(x)\rvert^n \le K(x)\bigl(\langle \nabla \varphi(x), \xi(x)\rangle + A(x)\bigr),…

We study $p$-harmonic functions, $ 1 < p\neq 2 < \infty$, in $ \mathbb{R}^{2}_+ = \{ z = x + i y : y > 0, - \infty < x < \infty \} $ and $B( 0, 1 ) = \{ z : |z| < 1 \}$. We first show for fixed $ p$, $1 < p\neq 2 < \infty$, and for all…

Our aim is to characterize the homogeneous fractional Sobolev-Slobodecki\u{\i} spaces $\mathcal{D}^{s,p} (\mathbb{R}^n)$ and their embeddings, for $s \in (0,1]$ and $p\ge 1$. They are defined as the completion of the set of smooth and…

The paper deals with the interplay between boundedness, order and ring structures in function lattices on the line and related metric spaces. It is shown that the lattice of all Lipschitz functions on a normed space $E$ is isomorphic to its…

We study local regularity properties of local minimizer of scalar integral functionals with controlled $(p,q)$-growth in the two-dimensional plane. We establish Lipschitz continuity for local minimizer under the condition $1<p\leq q<\infty$…

In this paper, we show that the density in energy of Lipschitz functions in a Sobolev space $N^{1,p}(X)$ holds for all $p\in [1,\infty)$ whenever the space $X$ is complete and separable and the measure is Radon and finite on balls.…

The purpose of this survey article is a comprehensive study of operator Lipschitz functions. A continuous function $f$ on the real line ${\Bbb R}$ is called operator Lipschitz if $\|f(A)-f(B)\|\le{\rm const}\|A-B\|$ for arbitrary…

It was already known that a p-adic, locally Lipschitz continuous semi-algebraic function is piecewise Lipschitz continuous, where the pieces can be taken semi-algebraic. We prove that if the function has locally Lipschitz constant 1, then…

This note corrects a gap and improves results in an earlier paper by the first named author. More precisely, it is shown that on weakly compactly generated Banach spaces X which admit a C^{p} smooth norm, one can uniformly approximate…

Sufficient and necessary results have been proven on Lipschitz type integral conditions and bounds of its Fourier transform for an $L^2$ function, in the setting of Riemannian symmetric spaces of rank $1$ whose growth depends on a…

We prove local Lipschitz regularity for bounded minimizers of functionals with nonstandard $p,q$-growth with the source term in the Lorentz space $L(N,1)$ under the restriction $q<p+1+p\,\min\left\{\frac 1N,\frac{2(p-1)}{Np-2p+2}\right\}$.…

We show that if $p_-\geq 2$, then a sufficient condition for the density of smooth functions with compact support, in the variable exponent Sobolev space $W^{1,p(\cdot)}(\mathbb R^n)$, is that the Riesz potentials of compactly supported…

We prove an uniform boundedness principle for the Lipschitz seminorm of continuous, monotone, positively homogeneous and subadditive mappings on suitable cones of functions. The result is applicable to several classes of classically…