Characterizations of Besov-type and Triebel-Lizorkin-type spaces via maximal functions and local means

Let s∈R. In this paper, the authors first establish the maximal function characterizations of the Besov-type space with and τ∈[0,∞), the Triebel-Lizorkin-type space with p∈(0,∞), q∈(0,∞] and τ∈[0,∞), the Besov-Hausdorff space with p∈(1,∞), q∈[1,∞) and and the Triebel-Lizorkin-Hausdorff space with and , where t^′ denotes the conjugate index of t∈[1,∞]. Using this characterization, the authors further obtain the local mean characterizations of these function spaces via functions satisfying the Tauberian condition and establish a Fourier multiplier theorem on these spaces. All these results generalize the existing classical results on Besov and Triebel-Lizorkin spaces by taking τ=0 and are also new even for Q spaces and Hardy-Hausdorff spaces.     

An explicit method for finding common solutions of variational inequalities and systems of equilibrium problems and fixed points of an infinite family of nonexpansive mappings

Let H be a Hilbert space and C be a nonempty closed convex subset of H, {T_i}_i∈N be a family of nonexpansive mappings from C into H, G_i:C×C→R be a finite family of equilibrium functions (i∈{1,2,…,K}), A be a strongly positive bounded linear operator with a coefficient and -Lipschitzian, relaxed (μ,ν)-cocoercive map of C into H. Moreover, let , {α_n} satisfy appropriate conditions and ; we introduce an explicit scheme which defines a suitable sequence as follows:

     

Decompositions of Besov-Hausdorff and Triebel-Lizorkin-Hausdorff spaces and their applications

Let p∈(1,∞), q∈[1,∞), s∈R and . In this paper, the authors establish the φ-transform characterizations of Besov-Hausdorff spaces and Triebel-Lizorkin-Hausdorff spaces (q>1); as applications, the authors then establish their embedding properties (which on is also sharp), smooth atomic and molecular decomposition characterizations for suitable τ. Moreover, using their atomic and molecular decomposition characterizations, the authors investigate the trace properties and the boundedness of pseudo-differential operators with homogeneous symbols in and (q>1), which generalize the corresponding classical results on homogeneous Besov and Triebel-Lizorkin spaces when p∈(1,∞) and q∈[1,∞) by taking τ=0.     

Further results on iterative methods for computing generalized inverses

In this paper, we reconsider the iterative method X_k=X_k−1+βY(I−AX_k−1), k=1,2,…,β∈C?{0} for computing the generalized inverse over Banach spaces or the generalized Drazin inverse a^d of a Banach algebra element a, reveal the intrinsic relationship between the convergence of such iterations and the existence of or a^d, and present the error bounds of the iterative methods for approximating or a^d. Moreover, we deduce some necessary and sufficient conditions for iterative convergence to or a^d.     

Explicit formulas for Laplace transforms of certain functionals of some time inhomogeneous diffusions

We consider a process given by the SDE , t∈[0,T), with initial condition , where T∈(0,∞], α∈R, (Bt)_t∈[0,T) is a standard Wiener process, b:[0,T)→R?{0} and σ:[0,T)→(0,∞) are continuously differentiable functions. Assuming , t∈[0,T), with some K∈R, we derive an explicit formula for the joint Laplace transform of and for all t∈[0,T) and for all α∈R. Our motivation is that the maximum likelihood estimator (MLE) of α can be expressed in terms of these random variables. As an application, we show that in case of α=K, K≠0,

     

Positivity properties in noncommutative convolution algebras with applications in pseudo-differential calculus

We study , of all such that for every ?∈C^∞₀, where denotes the twisted convolution. We prove that certain boundedness for are completely determined of the behaviour for a at origin, for example that , and that if a(0)<∞, then a∈L²∩L^∞. We use the results in order to determine wether positive pseudo-differential operators belong to certain Schatten-casses or not.     

Singular values, norms, and commutators

Let and X_i, i=1,…,n, be bounded linear operators on a separable Hilbert space such that X_i is compact for i=1,…,n. It is shown that the singular values of are dominated by those of , where ‖·‖ is the usual operator norm. Among other applications of this inequality, we prove that if A and B are self-adjoint operators such that a₁?A?a₂ and b₁?B?b₂ for some real numbers and b₂, and if X is compact, then the singular values of the generalized commutator AX-XB are dominated by those of max(b₂-a₁,a₂-b₁)(X⊕X). This inequality proves a recent conjecture concerning the singular values of commutators. Several inequalities for norms of commutators are also given.     

The Dirichlet Markov Ensemble

We equip the polytope of n×n Markov matrices with the normalized trace of the Lebesgue measure of Rⁿ². This probability space provides random Markov matrices, with i.i.d. rows following the Dirichlet distribution of mean (1/n,…,1/n). We show that if is such a random matrix, then the empirical distribution built from the singular values of tends as n→∞ to a Wigner quarter-circle distribution. Some computer simulations reveal striking asymptotic spectral properties of such random matrices, still waiting for a rigorous mathematical analysis. In particular, we believe that with probability one, the empirical distribution of the complex spectrum of tends as n→∞ to the uniform distribution on the unit disc of the complex plane, and that moreover, the spectral gap of is of order when n is large.     

Carleson measure problems for parabolic Bergman spaces and homogeneous Sobolev spaces

Let be the space of solutions to the parabolic equation having finite norm. We characterize nonnegative Radon measures μ on having the property , 1≤p≤q<∞, whenever . Meanwhile, denoting by v(t,x) the solution of the above equation with Cauchy data v₀(x), we characterize nonnegative Radon measures μ on satisfying , β∈(0,n), p∈[1,n/β], q∈(0,∞). Moreover, we obtain the decay of v(t,x), an isocapacitary inequality and a trace inequality.     

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 .     

Extension of Rotfel’d Theorem

Let f(t) be a non-negative concave function on the positive half-line. Given an arbitrary partitioned positive semi-definite matrix, we show that

     

Unique expansions of real numbers

It was discovered some years ago that there exist non-integer real numbers q>1 for which only one sequence (ci) of integers ci∈[0,q) satisfies the equality . The set of such “univoque numbers” has a rich topological structure, and its study revealed a number of unexpected connections with measure theory, fractals, ergodic theory and Diophantine approximation.In this paper we consider for each fixed q>1 the set Uq of real numbers x having a unique representation of the form with integers ci belonging to [0,q). We carry out a detailed topological study of these sets. For instance, we characterize their closures, and we determine those bases q for which Uq is closed or even a Cantor set. We also study the set consisting of all sequences (ci) of integers ci∈[0,q) such that . We determine the numbers r>1 for which the map (defined on (1,∞)) is constant in a neighborhood of r and the numbers q>1 for which is a subshift or a subshift of finite type.     

Preservers of spectral radius, numerical radius, or spectral norm of the sum on nonnegative matrices

Let be the set of entrywise nonnegative n×n matrices. Denote by r(A) the spectral radius (Perron root) of . Characterization is obtained for maps such that r(f(A)+f(B))=r(A+B) for all . In particular, it is shown that such a map has the form

     

Haj?asz-Sobolev imbedding and extension

The author establishes some geometric criteria for a Haj?asz-Sobolev -extension (resp. -imbedding) domain of Rn with n?2, s∈(0,1] and p∈[n/s,∞] (resp. p∈(n/s,∞]). In particular, the author proves that a bounded finitely connected planar domain Ω is a weak α-cigar domain with α∈(0,1) if and only if for some/all s∈[α,1) and p=(2−α)/(s−α), where denotes the restriction of the Triebel-Lizorkin space on Ω.     

Heat flow, BMO, and the compactness of Toeplitz operators

In this paper, it is shown that the Berezin-Toeplitz operator Tg is compact or in the Schatten class Sp of the Segal-Bargmann space for 1?p<∞ whenever (vanishes at infinity) or , respectively, for some s with , where is the heat transform of g on Cn. Moreover, we show that compactness of Tg implies that is in C₀(Cn) for all and use this to show that, for g∈BMO¹(Cn), we have is in C₀(Cn) for some s>0 only if is in C₀(Cn) for alls>0. This “backwards heat flow” result seems to be unknown for g∈BMO¹ and even g∈L^∞. Finally, we show that our compactness and vanishing “backwards heat flow” results hold in the context of the weighted Bergman space , where the “heat flow” is replaced by the Berezin transform Bα(g) on for α>−1.     

Identities for sums of characteristic values of entire matrix pencils

The paper deals with an entire matrix-valued function of a complex argument (an entire matrix pencil) f of order ρ(f)<∞. Identities for the following sums of the characteristic values of f are established:

     

Semilinear ordinary differential equation coupled with distributed order fractional differential equation

The system , where D^γ,γ∈[0,2] are operators of fractional differentiation, is investigated and the existence of a mild and classical solution is proven. Also, a necessary and sufficient condition for the existence and uniqueness of a solution to a general linear fractional differential equation , in is given.     

Maps preserving the spectrum of generalized Jordan product of operators

Let A₁,A₂ be standard operator algebras on complex Banach spaces X₁,X₂, respectively. For k?2, let (i₁,…,i_m) be a sequence with terms chosen from {1,…,k}, and define the generalized Jordan product

     

A complement to monotonicity of generalized Furuta-type operator functions

Let A?B?0 with A>0, t∈[0,1] and p?1. Then we shall show that