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Given a closed orientable surface (\Sigma) of genus at least two, we establish an affine isomorphism between the convex compact set of isotopy-invariant topological measures on (\Sigma) and the convex compact set of additive functions on…

Let $P_{\alpha} f(x,t)$ be the Caffarelli-Silvestre extension of a smooth function $f(x): \mathbb{R}^n \rightarrow \mathbb{R}^{n+1}_+:=\mathbb{R}^n\times (0,\infty).$ The purpose of this article is twofold. Firstly, we want to characterize…

Let $G$ be a Polish locally compact group acting on a Polish space $X$ with a $G$-invariant probability measure $\mu$. We factorize the integral with respect to $\mu$ in terms of the integrals with respect to the ergodic measures on $X$,…

It is well-known that solvability of the $\mathrm{L}^{p}$-Dirichlet problem for elliptic equations $Lu:=-\mathrm{div}(A\nabla u)=0$ with real-valued, bounded and measurable coefficients $A$ on Lipschitz domains…

Let $\Omega$ be a strongly Lipschitz domain of $\mathbb{R}^n$, whose complement in $\mathbb{R}^n$ is unbounded. Let $L$ be a second order divergence form elliptic operator on $L^2 (\Omega)$ with the Dirichlet boundary condition, and the…

We study the entropy and Lyapunov exponents of invariant measures $\mu$ for smooth surface diffeomorphisms $f$, as functions of $(f,\mu)$. The main result is an inequality relating the discontinuities of these functions. One consequence is…

For a general measure space $(\Omega,\mu)$, it is shown that for every band $M$ in $L_p(\mu)$ there exists a decomposition $\mu=\mu'+\mu^{\prime\prime}$ such that $M=L_p(\mu')=\{f\in L_p(\mu);f=0\ \mu^{\prime\prime}\text{-a.e.}\}$. The…

Let $\sigma>0$. For $1\le p\le \infty$, the Bernstein space $B^p_{\sigma}$ is a Banach space of all $f\in L^p(R)$ such that $f$ is bandlimited to $\sigma$; that is, the distributional Fourier transform of $f$ is supported in $[-\sigma,…

An old question of A.V. Arhangel'skii asks if the Menger property of a Tychonoff space $X$ is preserved by homeomorphisms of its function space $C_p(X)$. We provide affirmative answer in the case of linear homeomorphisms. To this end, we…

The aim of this paper is twofold. First, we obtain a Schwarz-Pick type lemma for the $\alpha$-harmonic mapping $u=P_{\alpha}[\phi]$, where $\phi\in L^{p}(\mathbb{S}^{n-1},\mathbb{R} )$ and $p\in[1,\infty]$. We get an explicit form of the…

A topological space $G$ is said to be a {\it rectifiable space} provided that there are a surjective homeomorphism $\phi :G\times G\rightarrow G\times G$ and an element $e\in G$ such that $\pi_{1}\circ \phi =\pi_{1}$ and for every $x\in G$…

Let $H^p(L^2(M))$ be the space of all $L^2$-harmonic $p$-forms $(2\leq p\leq n-2)$ on complete submanifolds $M$ with flat normal bundle in spheres. In this paper, we first show that $H^p(L^2(M))$ is trivial if the total curvature of $M$ is…

The reliable recovery and uncertainty quantification of a fixed effect function $\mu$ in a functional mixed model, for modelling population- and object-level variability in noisily observed functional data, is a notoriously challenging…

Let $\mathcal{M}$ be a $\sigma$-finite von Neumann algebra equipped with a normal faithful state $\varphi$, and let $\Phi$ be a growth function. We consider Haagerup noncommutative Orlicz spaces $L^\Phi(\M,\varphi)$ associated with $\M$ and…

In this article, for $N \geq 2, s \in (1,2), p\in (1, \frac{N}{s}), \sigma=s-1 $ and $a \in [0, \frac{N-sp}{2})$, we establish an isometric isomorphism between the higher order fractional weighted Beppo-Levi space \begin{align*} {\mathcal…

Let $0<p<\infty$, $\Gamma$ be a Lipschitz curve on the complex plane~$\mathbb{C}$ and $\Omega_+$ is the domain above $\Gamma$, we define Hardy space $H^p(\Omega_+)$ as the set of analytic functions $F$ satisfying…

Let $(\varphi_i)_{i=1}^n$ be mutually orthogonal functions on a probability space such that $\|\varphi_i\|_\infty \leq 1 $ for all $i \in [n]$. Let $\alpha > 0$. Let $\Phi(u) = u^2 \log^{\alpha}(u)$ for $u \geq u_{0}$, and $\Phi(u) =…

Let $\Gamma \curvearrowright \Omega$ be a measure-preserving action and $\mathcal{L} \Gamma \hookrightarrow L^\infty(\Omega) \rtimes \Gamma$ the natural inclusion of the group von Neumann algebra into the crossed product. When $\mu(\Omega)…

Let $E$ be one of the spaces $C(K)$ and $L_1$, $F$ be an arbitrary Banach space, $p>1,$ and $(X,\sigma)$ be a space with a finite measure. We prove that $E$ is isometric to a subspace of the Lebesgue-Bochner space $L_p(X;F)$ only if $E$ is…

Given a Moebius homeomorphism $f : \partial X \to \partial Y$ between boundaries of proper, geodesically complete CAT(-1) spaces $X,Y$, and a family of probability measures $\{ \mu_x \}_{x \in X}$ on $\partial X$, we describe a continuous…