Controlled Hahn-Mazurkiewicz Theorem and some new dimension functions of Peano continua

Given a metric Peano continuum X we introduce and study the Hölder Dimension there is a -Hölder onto map of X as well as its topological counterpart is an admissible metric for X}. We show that for each convex metric continuum X the dimension Hö-dim(X) equals the fractal dimension of X. The topological Hölder dimension Hö-dim(Mn) of the n-dimensional universal Menger cube Mn equals n. On the other hand, there are 1-dimensional rim-finite Peano continua X with arbitrary prescribed Hö-dim(X)?1.     

Baire generic histograms of wavelet coefficients and large deviation formalism in Besov and Sobolev spaces

Histograms of wavelet coefficients are expressed in terms of the wavelet profile and the wavelet density. The large deviation multifractal formalism states that if a function f has a minimal uniform Hölder regularity then its Hölder spectrum is equal to the wavelet density. The purpose of this paper is twofold. Firstly, we compute generically (in the sense of Baire's categories) these histograms in Besov and Lp,s(T) spaces, where T is the torus Rd/Zd (resp. in the Baire's vector space where s:q?s(q) is a C¹ and concave function on R⁺ satisfying 0?s^′?d and s(0)>0). Secondly, as an application, we deduce some extra generic properties for the histograms in these spaces, and study the generic validity of the large deviation multifractal formalism in Besov and Lp,s spaces for s>d/p (resp. in the above space V).     

Operator Hölder-Zygmund functions

It is well known that a Lipschitz function on the real line does not have to be operator Lipschitz. We show that the situation changes dramatically if we pass to Hölder classes. Namely, we prove that if f belongs to the Hölder class Λα(R) with 0<α<1, then for arbitrary self-adjoint operators A and B. We prove a similar result for functions f in the Zygmund class Λ₁(R): for arbitrary self-adjoint operators A and K we have . We also obtain analogs of this result for all Hölder-Zygmund classes Λα(R), α>0. Then we find a sharp estimate for ‖f(A)−f(B)‖ for functions f of class for an arbitrary modulus of continuity ω. In particular, we study moduli of continuity, for which for self-adjoint A and B, and for an arbitrary function f in Λω. We obtain similar estimates for commutators f(A)Q−Qf(A) and quasicommutators f(A)Q−Qf(B). Finally, we estimate the norms of finite differences for f in the class Λω,m that is defined in terms of finite differences and a modulus continuity ω of order m. We also obtain similar results for unitary operators and for contractions.     

Blow-up criterion of strong solutions to the Navier-Stokes equations in Besov spaces with negative indices

A Hölder type inequality in Besov spaces is established and applied to show that every strong solution u(t,x) on (0,T) of the Navier-Stokes equations can be continued beyond t>T provided that the vorticity for 0<α<1.     

Estimates of majorizing sequences in the Newton-Kantorovich method: A further improvement

Let be an operator, with X and Y Banach spaces, and f^′ be Hölder continuous with exponent θ. The convergence of the sequence of Newton-Kantorovich approximations

     

Compact coverings for Baire locally convex spaces

Very recently Tkachuk has proved that for a completely regular Hausdorff space X the space Cp(X) of continuous real-valued functions on X with the pointwise topology is metrizable, complete and separable iff Cp(X) is Baire (i.e. of the second Baire category) and is covered by a family of compact sets such that Kα⊂Kβ if α?β. Our general result, which extends some results of De Wilde, Sunyach and Valdivia, states that a locally convex space E is separable metrizable and complete iff E is Baire and is covered by an ordered family of relatively countably compact sets. Consequently every Baire locally convex space which is quasi-Suslin is separable metrizable and complete.     

Mappings and Prohorov spaces

We consider completely regular Hausdorff spaces. In this paper we investigate the space of probability Radon measures P(X) on a space X and the property to be a Prohorov space. We prove that the space P(X) is sieve-complete if and only if X is sieve-complete. Every mapping generates the mapping . Some properties of the mapping P(φ) are studied. In particular, we investigate under which conditions the open continuous image of a Prohorov space is Prohorov.     

Operators of p-evolution with nonregular coefficients in the time variable

We study the Cauchy problem for a class of p-evolution operators P(t,x,D_t,D_x) in , with less than coefficients with respect to the time variable.According to Lipschitz, log-lipschitz or Hölder regularity we find well-posedness in Sobolev spaces or in Gevrey classes.     

Compact almost automorphic solutions to integral equations with infinite delay

Given a∈L¹(R) and the generator A of an L¹-integrable resolvent family of linear bounded operators defined on a Banach space X, we prove the existence of compact almost automorphic solutions of the semilinear integral equation for each f:R×X→X compact almost automorphic in t, for each x∈X, and satisfying Lipschitz and Hölder type conditions. In the scalar linear case, we prove that a∈L¹(R) positive, nonincreasing and log-convex is sufficient to obtain the existence of compact almost automorphic solutions.     

Invariant functionals and the uniqueness of invariant norms

Let τ be a representation of a compact group G on a Banach space (X,||·||). The question we address is whether X carries a unique invariant norm in the sense that ||·|| is the unique norm on X for which τ is a representation. We characterize the uniqueness of norm in terms of the automatic continuity of the invariant functionals in the case when X is a dual Banach space and τ is a -continuous representation of G on X such that τ(G) consists of -continuous operators. We illustrate the usefulness of this characterization by studying the uniqueness of the norm on the spaces Lp(Ω), where Ω is a locally compact Hausdorff space equipped with a positive Radon measure and G acts on Ω as a group of continuous invertible measure-preserving transformations.     

Absolutely summing operators and integration of vector-valued functions

Let (Ω,Σ,μ) be a complete probability space and an absolutely summing operator between Banach spaces. We prove that for each Dunford integrable (i.e., scalarly integrable) function the composition u○f is scalarly equivalent to a Bochner integrable function. Such a composition is shown to be Bochner integrable in several cases, for instance, when f is properly measurable, Birkhoff integrable or McShane integrable, as well as when X is a subspace of an Asplund generated space or a subspace of a weakly Lindelöf space of the form C(K). We also study the continuity of the composition operator f?u○f. Some other applications are given.     

On sequence space representations of Hörmander-Beurling spaces

It is shown that is isomorphic to (Ω open set in Rn, 1?p<∞, k Beurling-Björck weight) extending a Hörmander's result (the proof we give is valid in the vector-valued case, too). As a consequence, and using Vogt's representation theorems and weighted Lp-spaces of entire analytic functions, a number of results on sequence space representations of Hörmander-Beurling are given.     

Topological entropy for set valued maps

Any continuous map T on a compact metric space X induces in a natural way a continuous map on the space K(X) of all non-empty compact subsets of X. Let T be a homeomorphism on the interval or on the circle. It is proved that the topological entropy of the induced set valued map is zero or infinity. Moreover, the topological entropy of is zero, where C(X) denotes the space of all non-empty compact and connected subsets of X. For general continuous maps on compact metric spaces these results are not valid.     

Reduction theorems for Sobolev embeddings into the spaces of Hölder,Morrey and Campanato type

Let X be a rearrangement‐invariant Banach function space on Q where Q is a cube in and let be the Sobolev space of real‐valued weakly differentiable functions f satisfying . We establish a reduction theorem for an embedding of the Sobolev space into spaces of Campanato, Morrey and Hölder type. As a result we obtain a new characterization of such embeddings in terms of boundedness of a certain one‐dimensional integral operator on representation spaces.     

A bilinear version of Orlicz-Pettis theorem

Given three Banach spaces X, Y and Z and a bounded bilinear map , a sequence x=n(xn)⊆X is called B-absolutely summable if is finite for any y∈Y. Connections of this space with are presented. A sequence x=n(xn)⊆X is called B-unconditionally summable if is finite for any y∈Y and z^∗∈Z^∗ and for any M⊆N there exists xM∈X for which _∑n∈M〈B(xn,y),z^∗〉=〈B(xM,y),z^∗〉 for all y∈Y and z^∗∈Z^∗. A bilinear version of Orlicz-Pettis theorem is given in this setting and some applications are presented.     

An asymptotic formula for the ranks of homotopy groups

Let X be a finite simply connected CW complex of dimension n. The loop space homology H_∗(ΩX;Q) is the universal enveloping algebra of a graded Lie algebra LX isomorphic with π∗−1(X)⊗Q. Let QX⊂LX be a minimal generating subspace, and set .Theorem: If dimLX=∞ and , then

     

Bounded tightness for locally convex spaces and spaces C(X)

We show that metrizability and bounded tightness are actually equivalent for a large class of locally convex spaces including (LF)-spaces, (DF)-spaces, the space of distributions D′(Ω), etc. A consequence of this fact is that for the bounded tightness for the weak topology of X is equivalent to the following one: X is linearly homeomorphic to a subspace of . This nicely supplements very recent results of Cascales and Raja. Moreover, we show that a metric space X is separable if the space C_p(X) has bounded tightness.