On the continuity of superposition operators in the space of functions of bounded variation

In the paper we present results on the continuity of nonlinear superposition operators acting in the space of functions of bounded variation in the sense of Jordan. It is shown that the continuity of an autonomous superposition operator is automatically guaranteed if the acting condition is met. We also give a simple proof of the fact that a nonautonomous superposition operator generated by a continuously differentiable function is uniformly continuous on bounded sets. Moreover, we present necessary and sufficient conditions for the continuity of a superposition operator (autonomous or nonautonomous) in a general setting. Thus, we give the answers to two basic open problems mentioned in the monograph (Appell et al. in Bounded variation and around, series in nonlinear analysis and application, De Gruyter, Berlin, 2014).     

Locally Lipschitz composition operators in spaces of functions of bounded variation

We give a necessary and sufficient condition on a function \({f:\mathbb{R}\to\mathbb{R}}\) under which the composition operator (Nemytskij operator) F defined by \({Fh=f\circ h}\) acts in the spaces \({BV_\varphi[a,b], HBV[a,b], {\rm and} \,RV_\varphi[a,b]}\) and satisfies a local Lipschitz condition. While the proof of sufficiency consists in a straightforward calculation, the proof of necessity builds on nontrivial arguments like Helly’s selection principle or the Arzelà–Ascoli compactness criterion.     

Uniformly bounded Nemytskij operators between the Banach spaces of functions of bounded n-th variation

The main result says that the generator of any uniformly bounded composition operator acting between Banach algebras of functions of bounded n-th variation is an affine function with respect to the function variable.     

On the compactness of convolution-type operators in Morrey spaces

In a Morrey space, the product of the convolution operator with summable kernel and the operator of multiplication by an essentially bounded function is considered. Sufficient conditions for such a product to be compact are obtained. In addition, it is shown that the commutator of the convolution operator and the operator of multiplication by a function of weakly oscillating type is compact in a Morrey space.     

Smooth points in some spaces of bounded operators

We determine the smooth points of certain spaces of bounded operatorsL(X,Y), including the cases whereX andY arel ^p -orc ₀-direct sums of finite dimensional Banach spaces or subspaces of the latter enjoying the metric compact approximation property. We also remark that the operators not attaining their norm are nowhere dense inL(X,Y) wheneverK(X,Y) is anM-ideal inL(X,Y).     

Boundedness and unboundedness results for some maximal operators on functions of bounded variation

We characterize the space BV(I) of functions of bounded variation on an arbitrary interval I⊂R, in terms of a uniform boundedness condition satisfied by the local uncentered maximal operator MR from BV(I) into the Sobolev space W1,1(I). By restriction, the corresponding characterization holds for W1,1(I). We also show that if U is open in Rd, d>1, then boundedness from BV(U) into W1,1(U) fails for the local directional maximal operator , the local strong maximal operator , and the iterated local directional maximal operator . Nevertheless, if U satisfies a cone condition, then boundedly, and the same happens with , , and MR.     

On the fast approximation of some nonlinear operators in nonregular wavelet spaces

We focus our attention on the approximation of some nonlinear operators in adapted wavelet spaces. We show the interest of the construction of scaling functions with a large number of zero moments. We present the convergence estimate of an algorithm based on paraproducts for the approximation of nonlinear operators using wavelets connected to scaling functions with zero moments. Numerical tests are performed on univariate examples. This revised version was published online in June 2006 with corrections to the Cover Date.     

On weak compactness of bounded sets in Banach and locally convex spaces

We investigate the compactness of one class of bounded subsets in Banach and locally convex spaces. We obtain a generalization of the Banach-Alaoglu theorem to a class of subsets that are not polars of convex balanced neighborhoods of zero. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 52, No. 6, pp 731–739, June. 2000.     

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)     

Trace theorems for functions of bounded variation in metric spaces

In this paper we show existence of traces of functions of bounded variation on the boundary of a certain class of domains in metric measure spaces equipped with a doubling measure supporting a 1-Poincaré inequality, and obtain $L^1$ estimates of the trace functions. In contrast with the treatment of traces given in other papers on this subject, the traces we consider do not require knowledge of the function in the exterior of the domain. We also establish a Maz'ya-type inequality for functions of bounded variation that vanish on a set of positive capacity.     

Local compactness in right bounded asymmetric normed spaces

We characterize the finite dimensional asymmetric normed spaces which are right bounded and the relation of this property with the natural compactness properties of the unit ball, such as compactness and strong compactness. In contrast with some results found in the existing literature, we show that not all right bounded asymmetric norms have compact closed balls. We also prove that there are finite dimensional asymmetric normed spaces that satisfy that the closed unit ball is compact, but not strongly compact, closing in this way an open question on the topology of finite dimensional asymmetric normed spaces. In the positive direction, we will prove that a finite dimensional asymmetric normed space is strongly locally compact if and only if it is right bounded.     

Uniform continuity and compactness for resolvent families of operators

We characterize the uniform continuity and the compactness of a resolvent family of operators {R(t)_t0 for a Volterra equation of convolution type denned in a Banach spaceX. In particular, we extend similar results to those for semigroups of operators and cosine families of operators studied in other works.Work partially supported by DICYT 91-33 and FONDECYT 91-0471.     

2-Local isometries between spaces of functions of bounded variation

Positivity - Given two subsets X and Y of the real line with at least two points, we apply results on surjective linear isometries between Banach spaces of all functions of bounded variation BV(X)...