Uniformly continuous superposition operators in the space of bounded variation functions

Let I, J ? ? be intervals. The main result says that if a superposition operator H generated by a function of two variables h: I × J → ?, H (φ)(x) ? h (x, φ (x)), maps the set BV (I, J) of all bounded variation functions, φ: I → J into the Banach space BV (I, ?) and is uniformly continuous with respect to the BV ‐norm, then h (x, y) = a (x)y + b (x), x ∈ I, y ∈ J, for some a, b ∈ BV (I, ?) (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A Semigroup Approach to Harmonic Maps

We present a semigroup approach to harmonic maps between metric spaces. Our basic assumption on the target space (N,d) is that it admits a barycenter contraction, i.e. a contracting map which assigns to each probability measure q on N a point b(q) in N. This includes all metric spaces with globally nonpositive curvature in the sense of Alexandrov as well as all metric spaces with globally nonpositive curvature in the sense of Busemann. It also includes all Banach spaces.The analytic input comes from the domain space (M,) where we assume that we are given a Markov semigroup (p_t)_t>0. Typical examples come from elliptic or parabolic second-order operators on Rⁿ, from Lévy type operators, from Laplacians on manifolds or on metric measure spaces and from convolution operators on groups. In contrast to the work of Korevaar and Schoen (1993, 1997), Jost (1994, 1997), Eells and Fuglede (2001) our semigroups are not required to be symmetric.The linear semigroup acting, e.g., on the space of bounded measurable functions u:MR gives rise to a nonlinear semigroup (P_t^*)_t acting on certain classes of measurable maps f:MN. We will show that contraction and smoothing properties of the linear semigroup (p_t)_t can be extended to the nonlinear semigroup (P_t^*)_t, for instance, L_p–L_q smoothing, hypercontractivity, and exponentially fast convergence to equilibrium. Among others, we state existence and uniqueness of the solution to the Dirichlet problem for harmonic maps between metric spaces. Moreover, for this solution we prove Lipschitz continuity in the interior and Hölder continuity at the boundary.Our approach also yields a new interpretation of curvature assumptions which are usually required to deduce regularity results for the harmonic map flow: lower Ricci curvature bounds on the domain space are equivalent to estimates of the L₁-Wasserstein distance between the distribution of two Brownian motions in terms of the distance of their starting points; nonpositive sectional curvature on the target space is equivalent to the fact that the L₁-Wasserstein distance of two distributions always dominates the distance of their barycenters.Dedicated to the memory of Professor Dr. Heinz Bauer     

Continuous homogeneous selections of set-valued metric generalized inverses of linear operators in Banach spaces

In this paper,continuous homogeneous selections for the set-valued metric generalized inverses T of linear operators T in Banach spaces are investigated by means of the methods of geometry of Banach spaces.Necessary and sufficient conditions for bounded linear operators T to have continuous homogeneous selections for the set-valued metric generalized inverses T are given.The results are an answer to the problem posed by Nashed and Votruba.     

An Extension of Set-Valued Contraction Principle for Mappings on a Metric Space with a Graph and Application

In this paper, we obtain an existence theorem for fixed points of contractive set-valued mappings on a metric space endowed with a graph. This theorem unifies and extends several fixed point theorems for mappings on metric spaces and for mappings on metric spaces endowed with a graph. As an application, we obtain a theorem on the convergence of successive approximations for some linear operators on an arbitrary Banach space. This result yields the well-known Kelisky–Rivlin theorem on iterates of the Bernstein operators on C[0,1].     

Continuity of the metric-projection operators

We consider the metric-projection operators in the space LP(), 1 < p < , where M^b, M^b being the space of bounded Randon measures. The metric-projection operators project a fixed element g C^b in LP() onto a closed convex subset K C_u ^b, where C_u ^b is the space of bounded continuous functions in the topology of uniform convergence. We obtain statements on the continuity of the metric-projection operators, considered as mappings from M^b into K.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 8, pp. 1141–1144, August, 1992.     

Maps of several variables of finite total variation. I. Mixed differences and the total variation

Given two points a=(a₁,…,an) and b=(b₁,…,bn) from Rn with a<b componentwise and a map f from the rectangle into a metric semigroup M=(M,d,+), we study properties of the total variation of f on introduced by the first author in [V.V. Chistyakov, A selection principle for mappings of bounded variation of several variables, in: Real Analysis Exchange 27th Summer Symposium, Opava, Czech Republic, 2003, pp. 217-222] such as the additivity, generalized triangle inequality and sequential lower semicontinuity. This extends the classical properties of C. Jordan's total variation (n=1) and the corresponding properties of the total variation in the sense of Hildebrandt [T.H. Hildebrandt, Introduction to the Theory of Integration, Academic Press, 1963] (n=2) and Leonov [A.S. Leonov, On the total variation for functions of several variables and a multidimensional analog of Helly's selection principle, Math. Notes 63 (1998) 61-71] (n∈N) for real-valued functions of n variables.     

Continuation of a linear operator to an involution operator

A bounded linear operatorA:X→X in a linear topological spaceX is called ap-involution operator,p≥2, ifA ^p=I, whereI is the identity operator. In this paper, we describe linearp-involution operators in a linear topological space over the field ℂ and prove that linear operators can be continued to involution operators. Translated fromMatematicheskie Zametki, Vol. 61, No. 5, pp. 671–676, May, 1997. Translated by M. A. Shishkova     

Composition operators between area-type Nevanlinna classes

This paper is concerned with the composition operators between the area-type Nevanlinna classes. Some sufficient and necessary conditions are given in terms of the concept of Carleson measure and the standard techniques of Montel Theorem for the composition operator C _φ: N _a ^p → N _a ^q to be bounded or compact, where 1 < p ⩽ q. Moreover, the inducing maps which induce invertible or Fredholm composition operators on N _a ^p are characterized. __________ Translated from Journal of Wuhan University (Natural Science Edition), 2004, 50(1): 1–5. This work was supported by the National Natural Science Foundation of China under grant number 19771063     

Bergman-Toeplitz operators: Radial component influence

We analyze the influence of the radial component of a symbol to spectral, compactness, and Fredholm properties of Toeplitz operators, acting on the Bergman space. We show that there existcompact Toeplitz operators whose (radial) symbols areunbounded near the unit circle . Studying this question we give several sufficient, and necessary conditions, as well as the corresponding examples. The essential spectra of Toeplitz operators with pure radial symbols have sufficiently rich structure, and even can be massive.TheC ^*-algebras generated by Toeplitz operators with radial symbols are commutative, but the semicommutators[T _a, T_b)=T_a·T_b–T_a·b are not compact in general. Moreover for bounded operatorsT _a andT _b the operatorT _a·b may not be bounded at all.This work was partially supported by CONACYT Project 27934-E, México.The first author acknowledges the RFFI Grant 98-01-01023, Russia.     

Some properties of the Bézier–Kantorovich type operators

The aim of this paper is to present estimates for the rate of pointwise convergence of the Bézier–Kantorovich modification of the discrete Feller operators in some classes of measurable functions bounded on an interval I, in particular, for functions of bounded pth power variation on I. Our theorems generalize and extend the recent results of Zeng and Piriou (J. Approx. Theory 95(1998) 369; 104(2000) 330) for the kantorovichians of the Bernstein–Bézier operators in the class of functions of bounded variation in the Jordan sense on [0,1].     

On tame operators between non-archimedean power series spaces

Let p ∈ {1, ∞}. We show that any continuous linear operator T from A1 (a) to Ap (b) is tame, i.e., there exists a positive integer c such that sup x||Tx||k/|x|ck ∞ for every k ∈ N. Next we prove that a similar result holds for operators from A∞(a) to Ap(b) if and only if the set Mb,a of all finite limit points of the double sequence (bi /aj ) i,j∈N is bounded. Finally we show that the range of every tame operator from A∞(a) to A∞(b) has a Schauder basis.     

Multi-dimensional coupled diffusion process

In this paper, the formal differential expression is considered, whereQ ₁,Q ₂ are general elliptic operators with bounded coefficientsa _ij (x), b _i (x) and bounded absorptionC ₁₁ (x),C ₂₂ (x),xR ⁿ . It is proved thatL generates semigroup the coupled diffusion semigroup acting on functions with zero limit at infinity. The associated Markov process is constructed and shown to have nice trajectory properties.     

具有粗糙核的多线性算子在Herz型空间的有界性

§ 1　Introduction and main resultsLet b∈BMO(Rn) and T be a standard Calderon-Zygmund singular integral operator.Define the commutator[b,T] as follows.[b,T] f(x) =b(x) Tf(x) -T(bf) (x) .In [3 ] ,the boundedness ofthe commutator[b,T] wasestablished on Lp(Rn) .There are thesimilar results in [1 ,2 ] when the commutator was substituted with the multilinearoperators generated by the singular integral operator T and a Taylor series A(see thedefinition below) .Recently,many mathematicians h…     

The Spectra of General Differential Operators in the Direct Sum Spaces

In this paper, the general ordinary quasi-differential expression M _pof n-th order with complex coefficients and its formal adjoint M _p ⁺ on any finite number of intervals I _p =(a _p ,b _p ),p= 1,...,N, are considered in the setting of the direct sums of L _wp ² (a _p ,b _p )-spaces of functions defined on each of the separate intervals, and a number of results concerning the location of the point spectra and the regularity fields of general differential operators generated by such expressions are obtained. Some of these are extensions or generalizations of those in a symmetric case in [1], [14], [15], [16], [17] and of a general case with one interval in [2], [11], [12], whilst others are new.     

Kolmogorov equations for measures

We consider a semigroup of operators in the Banach space C _b (H) of uniformly continuous and bounded functions on a separable Hilbert space H. We prove an existence and uniqueness result for a measure valued equation involving this class of semigroups. Then we apply the result to the transition semigroup and the Kolmogorov operator corresponding to a stochastic PDE in H. For this purpose, we characterize the generator of the transition semigroup on a core.      

Bicommutants and ranges of derivations

Let V be a vector space, V* its dual space and L(V) the algebra of all linear operators on V. For an operator a?∈?L(V) let a* be its adjoint acting on V*, and for a subset R of L(V) let R″ be its bicommutant. If R is a left noetherian subalgebra of L(V), then {a*: a?∈?R}″?=?{a*: a?∈?R″}. When R is singly generated R″ is described precisely. Further, for any two operators a, b?∈?L(V), b?∈?(a)″ if and only if the derivations d _a and d _b satisfy d _b (F(V))???d _a (F(V)), where F(V) is the set of all finite rank operators on V. In this case the inclusion d _b (L(V))???d _a (L(V)) also holds.     

Rigidity of Area-Minimizing Hyperbolic Surfaces in Three-Manifolds

We prove that if M is a three-manifold with scalar curvature greater than or equal to ?2 and Σ?M is a two-sided compact embedded Riemann surface of genus greater than 1 which is locally area-minimizing, then the area of Σ is greater than or equal to 4π(g(Σ)?1), where g(Σ) denotes the genus of Σ. In the equality case, we prove that the induced metric on Σ has constant Gauss curvature equal to ?1 and locally M splits along Σ. We also obtain a rigidity result for cylinders (I×Σ,dt ²+g _Σ), where I=[a,b]?? and g _Σ is a Riemannian metric on Σ with constant Gauss curvature equal to ?1.