We show that a measurement μ on a continuous dcpo D extends to a measurement μ̄ on the convex powerdomain CD iff it is a Lebesgue measurement. In particular, kerμ must be metrizable in its relative Scott topology. Moreover, the space ker μ̄ in its relative Scott topology is homeomorphic to the Vietoris hyperspace of kerμ, i.e., the space of nonempty compact subsets of kerμ in its Vietoris topolog...

We consider substitutions on compact alphabets and provide sufficient conditions for the diffraction to be pure point, absolutely continuous singular continuous. This allows one construct examples which Koopman operator associated function space has specific spectral components. For abelian bijective substitutions, we a dichotomy result regarding type of diffraction. also first example substitu...

This paper is devoted to the study of four integral operators that are basic generalizations and modifications of fractional integrals of Hadamard, in the space X c of Lebesgue measurable functions f on R+ = (0,∞) such that ∞ ∫ 0 ∣∣ucf (u)∣∣p du u <∞ (1 p <∞), ess sup u>0 [ u ∣∣f (u)∣∣]<∞ (p =∞), for c ∈ R = (−∞,∞), in particular in the space Lp(0,∞) (1 p ∞). Formulas for the Mellin transforms ...

Let D be a bounded domain in C with C boundary. Many holomorphic function spaces over D have been introduced in last half century, such as Hardy, Bergman, Besov and Sobolev spaces. Properties (boundary behaviour ect.) of functions in those function spaces have been received a great deal of studies. Readers can see parts of them from the following books [26] [29], [66], [71], [75], [76] and refe...

An analogous theorem holds for quasi-conformal homeomorphisms between subdomains of S, as can easily be seen from the fact that all of the above statements are local in nature. Here a.e. refers to the Lebesgue measure on S. Throughout these notes, all measures will be Borel measures on the appropriate topological space, which will always be some Borel subset of R for some n. ”Almost everywhere”...

We revisit the notion of statistical attractor defined by Ilyashenko. We argue to show that they are optimally defined to exist and to describe the asymptotical statistics of Lebesgue-almost all the orbits. We contribute to the theory, defining a minimality concept of α-observable statistical attractors and proving that the space is always full Lebesgue decomposable into pairwise disjoint sets ...

We show that for quasi-greedy bases in real or complex Banach spaces the error of the thresholding greedy algorithm of order N is bounded by the best N term error of approximation times a function of N which depends on the democracy functions and the quasi-greedy constant of the basis. If the basis is democratic this function is bounded by C logN . We show with two examples that this bound is a...

Let [Formula: see text] be a generalized Calderón-Zygmund operator or [Formula: see text] ( the identity operator), let [Formula: see text] and [Formula: see text] be the linear operators, and let [Formula: see text]. Denote the Toeplitz type operator by [Formula: see text]where [Formula: see text] and [Formula: see text] is fractional integral operator. In this paper, we establish the sharp ma...

Let H be a complex Hilbert space. Denote by B(H)+ the set of all positive bounded linear operators on H. A bijective map φ : B(H)+ → B(H)+ is said to preserve Lebesgue decompositions in both directions if for any quadruple A,B,C,D of positive operators, B = C +D is an A-Lebesgue decomposition of B if and only if φ(B) = φ(C)+φ(D) is a φ(A)-Lebesgue decomposition of φ(B). It is proved that every ...