New applications of Besov-type and Triebel-Lizorkin-type spaces

In this paper, the authors prove that Besov-Morrey spaces are proper subspaces of Besov-type spaces and that Triebel-Lizorkin-Morrey spaces are special cases of Triebel-Lizorkin-type spaces . The authors also establish an equivalent characterization of when τ∈[0,1/p). These Besov-type spaces and Triebel-Lizorkin-type spaces were recently introduced to connect Besov spaces and Triebel-Lizorkin spaces with Q spaces. Moreover, for the spaces and , the authors investigate their trace properties and the boundedness of the pseudo-differential operators with homogeneous symbols in these spaces, which generalize the corresponding classical results of Jawerth and Grafakos-Torres by taking τ=0.     

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.     

Fixed points of dual quantum operations

Let ?A be a normal completely positive map on B(H) with Kraus operators . Denote M the subset of normal completely positive maps by . In this note, the relations between the fixed points of ?A and are investigated. We obtain that , where K(H) is the set of all compact operators on H and is the dual of ?A∈M. In addition, we show that the map is a bijection on M.     

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,

     

A remark on Ricceri's conjecture for a class of nonlinear eigenvalue problems

Consider the eigenvalue problem : −Δu=λf(x,u) in Ω, u=0 on ∂Ω, where Ω is a bounded smooth domain in RN. Denote by the set of all Carathéodory functions f:Ω×R→R such that for a.e. x∈Ω, f(x,⋅) is Lipschitzian with Lipschitz constant L, f(x,0)=0 and , and denote by (resp. ) the set of λ>0 such that has at least one nonzero classical (resp. weak) solution. Let λ₁ be the first eigenvalue for the Laplacian-Dirichlet problem. We prove that and . Our result is a positive answer to Ricceri's conjecture if use f(x,u) instead of f(u) in the conjecture.     

Well-posedness for the Navier-Stokes-Nernst-Planck-Poisson system in Triebel-Lizorkin space and Besov space with negative indices

This paper is concerned with the well-posedness of the Navier-Stokes-Nerst-Planck-Poisson system (NSNPP). Let sp=−2+n/p. We prove that the NSNPP has a unique local solution for in a subspace, i.e., Vu1×Vv1×Vv1, of with . We also prove that there exists a unique small global solution for any small initial data with .     

Exact rates in log law for positively associated random variables

Let be a strictly stationary sequence of positively associated random variables with mean zero and finite variance. Set , Mn=maxk?n|Sk|, n?1. Suppose . In this paper, we study the exact convergence rates of a kind of weighted infinite series of , and as ε↘0, respectively.     

Selfadjoint operators in S-spaces

We study S-spaces and operators therein. An S-space is a Hilbert space with an additional inner product given by , where U is a unitary operator in . We investigate spectral properties of selfadjoint operators in S-spaces. We show that their spectrum is symmetric with respect to the real axis. As a main result we prove that for each selfadjoint operator A in an S-space we find an inner product which turns S into a Krein space and A into a selfadjoint operator therein. As a consequence we get a new simple condition for the existence of invariant subspaces of selfadjoint operators in Krein spaces, which provides a different insight into this well-know and in general unsolved problem.     

An almost sure central limit theorem for products of sums of partial sums under association

Let be a strictly stationary positively or negatively associated sequence of positive random variables with EX₁=μ>0, and VarX₁=σ²<∞. Denote , and γ=σ/μ the coefficient of variation. Under suitable conditions, we show that

     

Strong convergence of the composite Halpern iteration

Let C be a closed convex subset of a uniformly smooth Banach space E and let T:C→C be a nonexpansive mapping with a nonempty fixed points set. Given a point u∈C, the initial guess x₀∈C is chosen arbitrarily and given sequences , and in (0,1), the following conditions are satisfied:

(i)

;

(ii)

αn→0, βn→0 and 0<a?γn, for some a∈(0,1);

(iii)

, and . Let be a composite iteration process defined by

     

Approximation of common fixed points for finite families of nonself asymptotically nonexpansive mappings in Banach spaces

Let E be a real uniformly convex Banach space, K be a closed convex nonempty subset of E which is also a nonexpansive retract with retraction P. Let be asymptotically nonexpansive mappings of K into E with sequences (respectively) satisfying kin→1 as n→∞, i=1,2,…,m, and . Let be a sequence in [?,1−?],?∈(0,1), for each i∈{1,2,…,m} (respectively). Let {xn} be a sequence generated for m?2 by

     

Quasi-convex density and determining subgroups of compact abelian groups

For an abelian topological group G, let denote the dual group of all continuous characters endowed with the compact open topology. Given a closed subset X of an infinite compact abelian group G such that w(X)<w(G), and an open neighborhood U of 0 in T, we show that . (Here, w(G) denotes the weight of G.) A subgroup D of G determines G if the map defined by r(χ)=χ?D for , is an isomorphism between and . We prove that

     

Some estimates for Bochner-Riesz operators on the weighted Herz-type Hardy spaces

In this paper, by using the atomic decomposition and molecular characterization of the homogeneous and non-homogeneous weighted Herz-type Hardy spaces , we obtain some weighted boundedness properties of the Bochner-Riesz operator and the maximal Bochner-Riesz operator on these spaces for α=n(1/p−1/q), 0<p?1 and 1<q<∞.     

Well-posedness of the fourth-order perturbed Schrödinger type equation in non-isotropic Sobolev spaces

In this paper we study the Cauchy problem of the non-isotropically perturbed fourth-order nonlinear Schrödinger type equation: ((x₁,x₂,…,xn)∈Rn, t?0), where a is a real constant, 1?d<n is an integer, g(x,|u|)u is a nonlinear function which behaves like α|u|u for some constant α>0. By using Kato method, we prove that this perturbed fourth-order Schrödinger type equation is locally well-posed with initial data belonging to the non-isotropic Sobolev spaces provided that s₁,s₂ satisfy the conditions: s₁?0, s₂?0 for or for with some additional conditions. Furthermore, by using non-isotropic Sobolev inequality and energy method, we obtain some global well-posedness results for initial data belonging to non-isotropic Sobolev spaces .     

A Representation of Projection Lattices and Their States in Euclidean Space

We propose a representationr : ∪ Ω → ^ν, where is the collection of closed subspaces of ann-dimensional real, complex, or quaternionic Hilbert space , or equivalently, the projection lattice of this Hilbert space, where Ω is the set of all states ω : → [0, 1]. The value that ω ∈ Ω takes ina ∈ is given by the scalar product of the representative points (r(a) andr(ω)). The representationr(a ∨ b) of the join of two orthogonal elementsa, b ∈ is equal tor(a) + r(b). The convex closure of the representation of Σ, the set of atoms of , is equal to the representation of Ω.     

Multifractal spectrum and generic properties of functions monotone in several variables

We study the singularity (multifractal) spectrum of continuous functions monotone in several variables. We find an upper bound valid for all functions of this type, and we prove that this upper bound is reached for generic functions monotone in several variables. Let be the set of points at which f has a pointwise exponent equal to h. For generic monotone functions f:d[0,1]→R, we have that for all h∈[0,1], and in addition, we obtain that the set is empty as soon as h>1. We also investigate the level set structure of such functions.     

The oscillation of differential transforms - entire Jacobi expansions

Let L=(1−x²)D²−((β−α)−(α+β+2)x)D with , and . Let f∈C^∞[−1,1], , with normalized Jacobi polynomials and the Cn decrease sufficiently fast. Set Lk=L(Lk−1), k?2. Let ρ>1. If the number of sign changes of (Lkf)(x) in (−1,1) is O(k1/(ρ+1)), then f extends to be an entire function of logarithmic order . For Legendre expansions, the result holds with replaced with .     

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.     

Monotonicity of zeros of Laguerre-Sobolev-type orthogonal polynomials

Denote by , k=1,…,n, the zeros of the Laguerre-Sobolev-type polynomials orthogonal with respect to the inner product