Related papers: The sharp Whitney extension theorem for convex $C^…

We prove an extension of Eells and Sampson's rigidity theorem for harmonic maps from a closed manifold of non-negative Ricci curvature to a manifold of non-positive sectional curvature. We give an application of our result in the setting of…

The Lewy extension theorem asserts the holomorphic extendability of CR functions defined in a neighborhood of a point on a hypersurface in C^{n+1}. The edge-of-the-wedge theorem asserts the extendability of holomorphic functions defined in…

Let $E$ be a real vector space with dual space $E^*$ and let $C\subset E$ be a convex subset with more than one point. Let $f : C\to\mathbb{R}$ be a function satisfying a mild stability property at 'flat' points of the (relative) boundary…

For $-1\leq B<A\leq 1$, let $\mathcal{C}(A,B)$ denote the class of normalized Janowski convex functions defined in the unit disk $\mathbb{D}:=\{z\in\mathbb{C}:|z|<1\}$ that satisfy the subordination relation $1+zf''(z)/f'(z)\prec…

In this paper, we prove some rigidity theorems for compact Bach-flat $n$-manifold with the positive constant scalar curvature. In particular, our conditions in Theorem 1.4 have the additional properties of being sharp.

We consider a space of infinitely smooth functions on an unbounded closed convex set in ${\mathbb R}^n$. It is shown that each function of this space can be extended to an entire function in ${\mathbb C}^n$ satisfying some prescribed growth…

For a finite $E \subset \R^2$, $f:E \rightarrow \R$, and $p>2$, we produce a continuous $F:\R^2 \rightarrow \R$ depending linearly on $f$, taking the same values as $f$ on $E$, and with $L^{2,p}(\R^2)$ semi-norm minimal up to a factor…

Let G be a finitely generated group, let $\pi \colon G \to {\rm GL}(E)$ be a uniformly bounded $c_0$-representation on a superreflexive Banach space $E$, and let $b \colon G \to E$ be a $1$-cocycle for $\pi$. Then $b$ has sublinear growth…

Smooth functions $f:G\to E$ from a topological group $G$ to a locally convex space $E$ were considered by Riss (1953), Boseck, Czichowski and Rudolph (1981), Belti\c{t}\u{a} and Nicolae (2015), and others, in varying degrees of generality.…

We show that a Lipschitz domain can be expanded solely near a part of its boundary, assuming that the part is enclosed by a piecewise C1 curve. The expanded domain as well as the extended part are both Lipschitz. We apply this result to…

Assuming GRH and the Ramanujan-Petersson conjecture we prove explicit bounds for $L(1,f)$ for a large class of $L$-functions $L(s,f)$, which includes $L$-functions attached to automorphic cuspidal forms on $GL(n)$. The proof generalizes…

Let $n\geq 1$, $K>0$, and let $X=(X_1,X_2,\dots,X_n)$ be a random vector in $\mathbb{R}^n$ with independent $K$--subgaussian components. We show that for every $1$--Lipschitz convex function $f$ in $\mathbb{R}^n$ (the Lipschitzness with…

We study local regularity properties of local minimizer of scalar integral functionals of the form $$\mathcal F[u]:=\int_\Omega F(\nabla u)-f u\,dx$$ where the convex integrand $F$ satisfies controlled $(p,q)$-growth conditions. We…

We establish the first extension results for divergence-free (or solenoidal) elements of $\mathrm{L}^{1}$-based function spaces. Here, the key point is to preserve the solenoidality constraint while simultaneously keeping the underlying…

We study expansion/contraction properties of some common classes of mappings of the Euclidean space ${\mathbb R}^n, n\ge 2\,,$ with respect to the distance ratio metric. The first main case is the behavior of M\"obius transformations of the…

Let $D_j\subset\Bbb C^{k_j}$ be a pseudoconvex domain and let $A_j\subset D_j$ be a locally pluripolar set, $j=1,...,N$. Put$$X:=\bigcup_{j=1}^N A_1\times...\times A_{j-1}\times D_j\times A_{j+1}\times...\times A_N\subset\Bbb…

The Lipschitz extension modulus $e(M)$ of a metric space $M$ is the infimum over $L\ge 1$ such that for any Banach space $Z$ and any $C\subset M$, any 1-Lipschitz function $f:C\to Z$ can be extended to an $L$-Lipschitz function $F:M\to Z$.…

We show that Lipschitz solutions $u$ of $\mathrm{div}\, G(\nabla u)=0$ in $B_1\subset\mathbb R^2$ are $C^1$, for strictly monotone vector fields $G\in C^0(\mathbb R^2;\mathbb R^2)$ satisfying a mild ellipticity condition. If $G=\nabla F$…

We prove that, for any closed semialgebraic subset $W$ of $\mathbb{R}^n$ and for any positive integer $p$, there exists a Nash function $f:\mathbb{R}^n\setminus W\longrightarrow (0, \infty)$ which is equivalent to the distance function from…

We show that for every Lipschitz function $f$ defined on a separable Riemannian manifold $M$ (possibly of infinite dimension), for every continuous $\epsilon:M\to (0,+\infty)$, and for every positive number $r>0$, there exists a $C^\infty$…