On the "Zero-Two" Law for Positive Contractions on Banach-Kantorovich Lattice $L^p(\nabla,μ)$

Improvements of some operator inequalities involving positive linear maps via the Kantorovich constant

We present some operator inequalities for positive linear maps that generalize and improve the derived results in some recent years. For instant, if $A$ and $B$ are positive operators and $m,m^{'},M,M^{'}$ are positive real numbers satisfying either one of the condition $ 0<m \leq B \leq m^{'} <M^{'} \leq A \leq M $ or $0<m \leq A \leq m^{'} <M^{'} \leq B \leq M$, then \begin{align*} Φ^{p} \big(A \nabla _{v} B+2 r Mm (A^{-1}\nabla B^{-1}- &A^{-1} \sharp B^{-1} )\big)\\ & \leq \left( \frac{K(h)}{ 4^{\frac{2}{p}-1} K^{r_{1}} \left( \sqrt {h^{'}}\right)} \right) ^{p} Φ^{p} (A \sharp_ν B) \end{ali…

Coincidence points of Mappings in Banach Spaces

In this article we prove an existence theorem for coincidence points of mappings in Banach spaces. This theorem generalizes the Kantorovich fixed point theorem.

一般巴拿赫空间中 Kantorovich 型算子的逼近

Approximation by Kantorovich-type operators in general Banach spaces

本文在一般巴拿赫格中研究线性 Kantorovich 型采样算子的逼近性质，在 Steklov 平均算子一致有界假设下建立相关估计，其框架无需空间平移不变，扩展了仅针对$L_p$空间的相关估计。

Narrow Orthogonally Additive Operators on Lattice-Normed Spaces

The aim of this article is to extend results of M.~Popov and second named author about orthogonally additive narrow operators on vector lattices. The main object of our investigations are an orthogonally additive narrow operators between lattice-normed spaces. We prove that every $C$-compact laterally-to-norm continuous orthogonally additive operator from a Banach-Kantorovich space $V$ to a Banach lattice $Y$ is narrow. We also show that every dominated Uryson operator from Banach-Kantorovich space over an atomless Dedekind complete vector lattice $E$ to a sequence Banach lattice $\ell_p(Γ)$ o…

Improved operator Kantorovich and Wielandt inequalities for positive linear maps

In this paper, we improve and generalize the operator versions of Kantorovich and Wielandt inequalities for positive linear maps on Hilbert space. Our results are more extensive and precise than many previous results due to Fu and He [Linear Multilinear Algebra, doi: 10. 1080/03081087. 2014. 880432.] and Zhang [Banach J. Math. Anal., 9 (2015), no. 1, 166-172.].

On a Kantorovich-Rubinstein inequality

An easy consequence of Kantorovich-Rubinstein duality is the following: if $f:[0,1]^d \rightarrow \infty$ is Lipschitz and $\left\{x_1, \dots, x_N \right\} \subset [0,1]^d$, then $$ \left| \int_{[0,1]^d} f(x) dx - \frac{1}{N} \sum_{k=1}^{N}{f(x_k)} \right| \leq \left\| \nabla f \right\|_{L^{\infty}} \cdot W_1\left( \frac{1}{N} \sum_{k=1}^{N}{δ_{x_k}} , dx\right),$$ where $W_1$ denotes the $1-$Wasserstein (or Earth Mover's) Distance. We prove another such inequality with a smaller norm on $\nabla f$ and a larger Wasserstein distance. Our inequality is sharp when the points are very regular, i.e…

Newton-Kantorovich-Vietoris method

In this short work we consider nonlinear equation in Banach space $x = A(x)$. We combine classical Newton method with Vietoris method to propose new more general method. Under rather general conditions about the noise we investigate local convergence of the method.

Dominated Uryson operators on lattice-normed spaces

The aim of this note is to introduce the space D_{U}(V,W) of the dominated Uryson operators on lattice-normed spaces. We prove an "Up-and-down" type theorem for a positive abstract Uryson operator defined on a vector lattice and taking values in a Dedekind complete vector lattice. This result we apply to prove the decomposability of the lattice valued norm of the space D_{U}(V,W) of all dominated Uryson operators. We obtain that, for a lattice-normed space V and a Banach-Kantorovich space W the space D_{U}(V,W) is also a Banach-Kantorovich space. We prove that under some mild conditions, a dom…

Narrow operators on lattice-normed spaces

The aim of this article is to extend results of Maslyuchenko O., Mykhaylyuk V. Popov M. about narrow operators on vector lattices. We give a new definition of a narrow operator where a vector lattice as the domain space of a narrow operator is replaced with a lattice-normed space. We prove that every GAM-compact (bo)-norm continuous linear operator from a Banach-Kantorovich space V to a Banach lattice Y is narrow. Then we show that, under some mild conditions, a continuous dominated operator is narrow if and only if its exact dominant is narrow.

Weak regularity of Gauss mass transport

Given two probability measures $μ$ and $ν$ we consider a mass transportation mapping $T$ satisfying 1) $T$ sends $μ$ to $ν$, 2) $T$ has the form $T = ϕ\frac{\nabla ϕ}{|\nabla ϕ|}$, where $ϕ$ is a function with convex sublevel sets. We prove a change of variables formula for $T$. We also establish Sobolev estimates for $ϕ$, and a new form of the parabolic maximum principle. In addition, we discuss relations to the Monge-Kantorovich problem, curvature flows theory, and parabolic nonlinear PDE's.

Maximal Sobolev regularity for solutions of elliptic equations in Banach spaces endowed with a weighted Gaussian measure: the convex subset case

Let $X$ be a separable Banach space endowed with a non-degenerate centered Gaussian measure $μ$. The associated Cameron--Martin space is denoted by $H$. Consider two sufficiently regular convex functions $U:X\rightarrow\mathbb{R}$ and $G:X\rightarrow \mathbb{R}$. We let $ν=e^{-U}μ$ and $Ω=G^{-1}(-\infty,0]$. In this paper we are interested in the $W^{2,2}$ regularity of the weak solutions of elliptic equations of the type \begin{align}\label{Probelma in abstract} λu-L_{ν,Ω} u=f, \end{align} where $λ>0$, $f\in L^2(Ω,ν)$ and $L_{ν,Ω}$ is the self-adjoint operator associated with the quadratic fo…

When each continuous operator is regular, II

The following theorem is essentially due to L.~Kantorovich and B. Vulikh and it describes one of the most important classes of Banach lattices between which each continuous operator is regular. {\bf Theorem 1.1.} {\sl Let $E$ be an arbitrary L-space and $F$ be an arbitrary Banach lattice with Levi norm. Then ${\cal L}(E,F)={\cal L}^r(E,F),\ (\star) $ that is, every continuous operator from $E$ to $F$ is regular.} In spite of the importance of this theorem it has not yet been determined to what extent the Levi condition is essential for the validity of equality $(\star)$. Our main aim in this w…

Optimal Gagliardo-Nirenberg interpolation inequality for rearrangement invariant spaces

We prove optimality of the Gagliardo-Nirenberg inequality $$ \|\nabla u\|_{X}\lesssim\|\nabla^2 u\|_Y^{1/2}\|u\|_Z^{1/2}, $$ where $Y, Z$ are rearrangement invariant Banach function spaces and $X=Y^{1/2}Z^{1/2}$ is the Calderón--Lozanovskii space. By optimality, we mean that for a certain pair of spaces on the right-hand side, one cannot reduce the space on the left-hand, remaining in the class of rearrangement invariant spaces. The optimality for the Lorentz and Orlicz spaces is given as a consequence, exceeding previous results. We also discuss pointwise inequalities, their importance and co…

Some convergent results for Backtracking Gradient Descent method on Banach spaces

Our main result concerns the following condition: {\bf Condition C.} Let $X$ be a Banach space. A $C^1$ function $f:X\rightarrow \mathbb{R}$ satisfies Condition C if whenever $\{x_n\}$ weakly converges to $x$ and $\lim _{n\rightarrow\infty}||\nabla f(x_n)||=0$, then $\nabla f(x)=0$. We assume that there is given a canonical isomorphism between $X$ and its dual $X^*$, for example when $X$ is a Hilbert space. {\bf Theorem.} Let $X$ be a reflexive, complete Banach space and $f:X\rightarrow \mathbb{R}$ be a $C^2$ function which satisfies Condition C. Moreover, we assume that for every bounded set …

关于Banach函数空间中的$H=W$

On $H=W$ in Banach function spaces

本文在Banach函数空间$X(\Omega)$中证明了$H=W$，即$W^1_X(\Omega)$等于$H^1_X(\Omega)$，并给出了两个推论。