Rearrangements of functions on totally disconnected groups

A result of B.B. Wells Jr. claims that every complex valued continuous function on the compact ring of p-adic integers has a rearrangement which belongs to a certain class (W) of functions having absolutely convergent Fourier series. We point out that this is not the case since every real valued function from (W) has Lebesgue null range. On the other hand we prove the existence of a rearrangement with absolutely convergent Fourier series for every continuous real valued function on and on some other compact metric totally disconnected Abelian groups. We leave open if the same holds for all continuous complex valued functions on .     

On the Clark Ocone formula for the abstract Wiener space

Fix an abstract Wiener space where is a separable Hilbert space densely embedded into a Banach space . A pathwise construction of the Itô integral as a continuous square integrable martingale is given, where the integrands are -valued processes and the integrator is a -valued Brownian motion. We use this approach to the vector integral to prove that each Malliavin differentiable functional ? defined on the space of continuous -valued functions on [0,1], endowed with the Wiener measure, can be decomposed into the sum of the expected value of ? and the Itô integral of the conditional expectation of the Malliavin derivative of ? with respect to the Brownian filtration. The Malliavin derivative of ? is an -valued stochastic process. In a second application, it is shown that the iterated Itô integral, defined as a process on , is a continuous square integrable martingale.     

Holomorphy of spectral multipliers of the Ornstein-Uhlenbeck operator

Let γ be the Gauss measure on and the Ornstein-Uhlenbeck operator. For every p in [1,∞)?{2}, set , and consider the sector . The main results of this paper are the following. If p is in (1,∞)?{2}, and , i.e., if M is an Lp(γ)uniform spectral multiplier of in our terminology, and M is continuous on , then M extends to a bounded holomorphic function on the sector . Furthermore, if p=1 a spectral multiplier M, continuous on , satisfies the condition if and only if M extends to a bounded holomorphic function on the right half-plane, and its boundary value M(i·) on the imaginary axis is the Euclidean Fourier transform of a finite Borel measure on the real line. We prove similar results for uniform spectral multipliers of second order elliptic differential operators in divergence form on belonging to a wide class, which contains . From these results we deduce that operators in this class do not admit an H^∞ functional calculus in sectors smaller than .     

Derivatives of the L-cosine transform

The L^p-cosine transform of an even, continuous function is defined by

     

Quantitative recurrence properties for beta-dynamical system

This note is concerned with the quantitative recurrence properties of beta dynamical system ([0,1],Tβ) for general β>1; the size of points with the prescribed recurrence rate is determined. More precisely, Hausdorff dimensions of the sets and are obtained completely, where ψ is a positive function defined on N and f is a positive continuous function on [0,1]. Besides, the pressure function P(f,Tβ) with a continuous potential f is proven to be continuous with respect to β.     

Fixed-point free maps on the reals and more

In our main result we prove that a continuous function f:R→R fixes a point iff does, where is the continuous extension of f over the ?ech-Stone compactification. In the second half of the paper the statement of this theorem with R replaced by an arbitrary X is taken for a definition of a property and a number of results about the property are proved.     

Limiting subdifferentials of indefinite integrals

We compute the limiting subdifferential of the indefinite integral of the form where f is an essentially bounded measurable function, or a function continuous on an interval containing (except for, possibly, ), or a step-function which has a countable number of steps around . The related problem of computing the Aumann integral of the limiting subdifferential mapping ∂f(⋅), where f is a Lipschitz real function defined on an open set U⊂Rn, is also investigated.     

The existence of global attractors for the norm-to-weak continuous semigroup and application to the nonlinear reaction-diffusion equations

In this paper, first, we introduce a new concept, called the norm-to-weak continuous semigroup in a Banach space, and give a technical theorem to verify this notion of continuity. Then we establish a general method which is necessary and sufficient to obtain the existence of the global attractor for this kind of semigroup. As an application, we obtain the existence of the global attractor for a nonlinear reaction-diffusion equation with a polynomial growth nonlinearity of arbitrary order and with some weak derivatives in the inhomogeneous term, the global attractors are obtained in L^p(Ω), and , respectively. A new a priori estimate method, called asymptotic a priori estimate, has been introduced. Since the solutions of the equation has no higher regularity and the semigroup associated the solutions is not continuous in L^p(Ω), and , the results in this part are new and appear to be optimal.     

Injections into function spaces over ordinals

We investigate when and how function spaces over subspaces of ordinals admit continuous injections into each other. To formulate our results let τ be an uncountable regular cardinal. We prove, in particular, that: (1) If A and B are disjoint stationary subsets of τ then Cp(A) does not admit a continuous injection into Cp(B); (2) For A⊂ω₁, admits a continuous injection into iff A is countable or ω₁ embeds into A (which, in its turn, is equivalent to the statement “ embeds into ”).     

Lifting Galois representations of number fields

Let F be a number field. Given a continuous representation with insoluble image we show, under moderate assumptions at primes dividing ?∞, that for some continuous representation which is unramified outside finitely many primes. We also establish level lowering when F is totally real, is the reduction of a nearly ordinary Hilbert modular form and is distinguished at ?.     

On the automorphisms of the spectral completion of the algebraic numbers field

Let be the absolute Galois group of Q and let A=C(G,C) be the Banach algebra of all continuous functions defined on G with values in C. Let be the conjugation automorphism of C and let B be the R-Banach subalgebra of A consisting of continuous functions f such that for all σ∈G. Let ‖x‖=sup{|σ(x)|:σ∈G} be the spectral norm on and let be the spectral completion of . Using a canonical isometry between and B we study the structure of the group of R-algebras automorphisms of and the structure of its subgroup of all automorphisms of which when restricted to give rise to elements of G. We introduce a topology on and prove that this last one is homeomorphic and group isomorphic to G.     

Relaxation and regularity in the calculus of variations

In this work we prove that, if L(t,u,ξ) is a continuous function in t and u, Borel measurable in ξ, with bounded non-convex pieces in ξ, then any absolutely continuous solution to the variational problem

     

Eventually norm continuous semigroups on Hilbert space and perturbations

The eventually norm continuous semigroups on Hilbert space and perturbation are studied in this paper. By resolvent of infinitesimal generator, the sufficient and necessary conditions for eventually norm continuous semigroups are given. Using the result obtained, it is proved that if is infinitesimal generator of an eventually norm continuous semigroup T(t), then there is a subspace Ξ_A of such that, for any , the semigroup S(t) generated by preserves the property of T(t).     

Inverse spectral theory for symmetric operators with several gaps: scalar-type Weyl functions

Let S be the orthogonal sum of infinitely many pairwise unitarily equivalent symmetric operators with non-zero deficiency indices. Let J be an open subset of R. If there exists a self-adjoint extension S₀ of S such that J is contained in the resolvent set of S₀ and the associated Weyl function of the pair {S,S₀} is monotone with respect to J, then for any self-adjoint operator R there exists a self-adjoint extension such that the spectral parts and R_J are unitarily equivalent. It is shown that for any extension of S the absolutely continuous spectrum of S₀ is contained in that one of . Moreover, for a wide class of extensions the absolutely continuous parts of and S are even unitarily equivalent.     

The differentiable chain functor is not homotopy equivalent to the continuous chain functor

Let S_∗ and be the functors of continuous and differentiable singular chains on the category of differentiable manifolds. We prove that the natural transformation , which induces homology equivalences over each manifold, is not a natural homotopy equivalence.     

Exposed sets in potential theory

Let U be a relatively compact open subset of a harmonic space, and H(U) be the function space of all continuous functions on which are harmonic on U. We give a complete characterization of the H(U)-exposed subsets of . This extends the results of [J. Lukeš, T. Mocek, M. Smr?ka, J. Spurný, Choquet like sets in function spaces, Bull. Sci. Math. 127 (2003) 397-437].     

Toeplitz operators on the space of analytic functions with logarithmic growth

Continuous and compact Toeplitz operators for positive symbols are characterized on the space of analytic functions with logarithmic growth on the open unit disc of the complex plane. The characterizations are in terms of the behaviour of the Berezin transform of the symbol. The space was introduced and studied by Taskinen. The Bergman projection is continuous on this space in a natural way, which permits to define Toeplitz operators. Sufficient conditions for general symbols are also presented.     

An analytic characterization of the eigenvalues of self-adjoint extensions

Let be a self-adjoint extension in of a fixed symmetric operator A in . An analytic characterization of the eigenvalues of is given in terms of the Q-function and the parameter function in the Krein-Naimark formula. Here K and are Krein spaces and it is assumed that locally has the same spectral properties as a self-adjoint operator in a Pontryagin space. The general results are applied to a class of boundary value problems with λ-dependent boundary conditions.