A Lipschitz map f between the metric spaces X and Y is called a Lipschitz quotient map if there is a C > 0 (the smallest such C, the co-Lipschitz constant, is denoted coLip(f), while Lip(f) denotes the Lipschitz constant of f) so that for every x ∈ X and r > 0, fBX(x, r) ⊃ BY (f(x), r/C). Thus Lipschitz quotient maps are surjective maps which by definition have the property ensured by the open ...

This is the second and last paper of a series aimed at solving local Cauchy problem for polarized $${\mathbb {U}}(1)$$ symmetric solutions to Einstein vacuum equations featuring nonlinear interaction three small amplitude impulsive gravitational waves. Such are characterized by their singular “wave-fronts” across which curvature tensor allowed admit delta singularity. Under symmetry, reduce Ein...

In this note we investigate propagation of smallness properties for solutions to heat equations. We consider spectral projector estimates the Laplace operator with Dirichlet or Neumann boundary conditions on a Riemanian manifold without boundary. show that using new approach Logunov and Malinnikova (2018) allows one extend type Jerison Lebeau (1999) from localisation open sets localization arbi...

We study the regularity of viscosity solution u $u$ σ k $\sigma _k$ -Loewner–Nirenberg problem on a bounded smooth domain Ω ⊂ R n $\Omega \subset \mathbb {R}^n$ for ⩾ 2 $k \geqslant 2$ . It was known that is locally Lipschitz in $\Omega$ prove that, with d $d$ being distance function to ∂ $\partial \Omega$ and δ > 0 $\delta 0$ sufficiently small, { < ( x ) } $\lbrace d(x) \delta \rbrace$ first ...

We first show that a bounded linear operator $ T $ on a real Banach space $ E $ is quasicompact (Riesz, respectively) if and only if $T': E_{mathbb{C}}longrightarrow E_{mathbb{C}}$ is quasicompact  (Riesz, respectively), where the complex Banach space $E_{mathbb{C}}$ is a suitable complexification of $E$ and $T'$ is the complex linear operator on $E_{mathbb{C}}$ associated with $T$. Next, we pr...

Lipschitz domains. Our presentations here will almost exclusively be for bounded Lipschitz domains. Roughly speaking, a domain (a connected open set) Ω ⊂ R is called a Lipschitz domain if its boundary ∂Ω can be locally represented by Lipschitz continuous function; namely for any x ∈ ∂Ω, there exists a neighborhood of x, G ⊂ R, such that G ∩ ∂Ω is the graph of a Lipschitz continuous function und...

Let $mathfrak{A}$ be an algebra. A linear mapping $delta:mathfrak{A}tomathfrak{A}$ is called a textit{derivation} if $delta(ab)=delta(a)b+adelta(b)$ for each $a,binmathfrak{A}$. Given two derivations $delta$ and $delta'$ on a $C^*$-algebra $mathfrak A$, we prove that there exists a derivation $Delta$ on $mathfrak A$ such that $deltadelta'=Delta^2$ if and only if either $delta'=0$ or $delta=sdel...

We give conditions that ensure an operator satisfying a Piestch domination in given setting also satisfies different setting. From this we derive bounded multilinear T T </mml:semant...

In this paper we prove an analogue of Banach and Kannan fixed point theorems by generalizing the Lipschitz constat $k$, in generalized Lipschitz mapping on cone metric space over Banach algebra, which are answers for the open problems proposed by Sastry et al, [K. P. R. Sastry, G. A. Naidu, T. Bakeshie, Fixed point theorems in cone metric spaces with Banach algebra cones, Int. J. of Math. Sci. ...