Let X be a real normed space with unit closed ball B. We prove that X is an inner product space if and only if it is true that whenever x, y are points in ?B such that the line through x and y supports 22B then x?y in the sense of Birkhoff.

We give a topological description of the Stone space of C(K,E), Banach lattices of continuous functions from a compact Hausdorff space K into a Banach lattice E.

We generalize the theory of Qp spaces, introduced on the unit disc in 1995 by Aulaskari, Xiao and Zhao, to bounded symmetric domains in Cd, as well as to analogous Moebius-invariant function spaces and Bloch spaces defined using higher order derivatives; the latter generalization contains new results even in the original context of the unit disc.

In the first part of this paper we present a representation theorem for the directional derivative of the metric projection operator in an arbitrary Hilbert space. As a consequence of the representation theorem, we present in the second part the development of the theory of projected dynamical systems in infinite dimensional Hilbert space. We show that this development is...

We study local uniform convexity and Kadec-Klee type properties in K-interpolation spaces of Lorentz couples. We show that a wide class of Banach couples of (commutative and) non-commutative Lorentz spaces possess the (so-alled) (DGL)-property originally introduced by Davis, Ghoussoub and Lindenstrauss in the context of renorming order continuous Banach latties. This property is...

Let s≺0. The author obtains some new frames for Besov spaces Bpqs(ℝn) with 1≤p, q≤∞ and Triebel-Lizorkin spaces Fpqs(ℝn) with 1≺p≺∞ and 1≺q≤∞ by a dual method via the subatomic characterizations of these spaces when s≻0.

In this paper the author obtains equivalent norms of Herz-type Besov and Triebel-Lizorkin spaces, which are generalizations of well-known Herz-type spaces and inhomogeneous Besov and Triebel-Lizorkin spaces.

We consider a generalized version of the small Lebesgue spaces, introduced in [5] as the associate spaces of the grand Lebesgue spaces. We find a simplified expression for the norm, prove relevant properties, compute the fundamental function and discuss the comparison with the Orlicz spaces.

We prove that the pointwise multipliers acting in a pair of fractional Sobolev spaces form the space of boundary traces of multipliers in a pair of weighted Sobolev space of functions in a domain.

A question about comparing norms of difference operators that was raised in [1] and presented at the Fourth ISAAC Congress is answered in the affirmative.

Let Dkf mean the vector composed by all partial derivatives of order k of a function f(x), x∈Ω⊂ℝn. Given a Banach function space A, we look for a possibly small space B such that ‖f‖B≤c‖|Dkf|‖A for all f∈C0k(Ω). The estimates obtained are applied to ultrasymmetric spaces A=Lφ,E, B=Lψ,E, giving some optimal (or rather sharp) relations between parameter-functions φ(t) and ψ(t) and...

Using the Duhamel product for holomorphic functions we give a new proof of Nagnibida’s theorem on unicellularity of integration operator Jα,(Jαf)(z)=∫αzf(t)dt, acting in the space Hoℓ(Ω).

We obtain n-harmonic extensions to the Cartesian product of ncopies of the upper half-plane, of distributions in the weighted space w12?wn2DL1', which is known to be the optimal space of tempered distributions that are S'-convolvable with a natural product domain version of the Poisson kernel.

Necessary and sufficient conditions for the existence of integral variational principles for boundary value problems for given ordinary and partial functional differential equations are obtained. Examples are given illustrating the results.

We introduce the space Bw(ℓ2) of linear (unbounded) operators on ℓ2 which map decreasing sequences from ℓ2 into sequences from ℓ2 and we find some classes of operators belonging either to Bw(ℓ2) or to the space of all Schur multipliers on Bw(ℓ2). For instance we show that the space B(ℓ2) of all bounded operators on ℓ2 is contained in the space of all Schur multipliers on Bw(ℓ2).

We consider the dilation operators Tk:f→f(2k.) in the frame of Besov spaces Bpqs(ℝd) with 1 ≤p,q≤∞. If s > 0, Tk is a bounded linear operator from Bpqs(ℝd) into itself and there are optimal bounds for its norm, see [4, 2.3.1]. We study the situation in the case s = 0, an open problem mentioned also in [4]. It turns out, that new effects based on Littlewood-Paley theory appear. In...

Given a bounded linear operator T:LPO(ℝn)→Lp1(ℝn), we state conditions under which T defines a bounded operator between corresponding pairs of Lp(ℝn;ιq) spaces and Triebel-Lizorkin spaces Fp,qs(ℝn). Applications are given to linear parabolic equations and to Schrödinger semigroups.

An asymptotic formula for the essential norm of the composition operator Cφ(f):=f∘φ, induced by an analytic self-map φ of the unit disc, mapping from the α-Bloch space ℬα or the Dirichlet type space Dαp into Qk(p,q) is established in terms of an integral condition.

We give a criterion of weak compactness for the operators on the Morse-Transue space MΨ, the subspace of the Orlicz space LΨ generated by L∞.

Under mild conditions on the weight function K we characterize lacunary series in QK(p,q) spaces, where QK(p,q) spaces are QK type spaces of functions analytic in the unit disk.

Let µ be a nonnegative Radon measure on ℝd which satisfies the growth condition that there exist constants C0 > 0 and n ∈ (0, d] such that for all x ∈ ℝd and r > 0, μ(B(x,r))≤C0rn, where B(x, r) is the open ball centered at x and having radius r . In this paper, when ℝd is not an initial cube which implies µ(ℝd) = ∞, the authors prove that the homogeneous Littlewood-Paley g...

Let ? be a holomorphic of the unit ball B in the n-dimensional complex space, and denote by Tg the extended Cesáro operator with symbol g. Let 0 < p <

In this paper, we study the boundedness of commutator [b,T] of Riesz transform associated with Schrödinger operator and b is BMO type function, note that the kernel of T has no smoothness, and the boundedness from Hb1(Rn)→L1(Rn) is obtained.

This paper characterizes the boundedness and compactness of the generalized composition operator (Cφgf)(z)=∫0zf'(φ(ξ))g(ξ)dξ from Bloch type spaces to QK type spaces.

We consider non-standard Hölder spaces Hλ(⋅)(X) of functions f on a metric measure space (X, d, μ), whose Hölder exponent λ(x) is variable, depending on x ∈ X. We establish theorems on mapping properties of potential operators of variable order α(x), from such a variable exponent Hölder space with the exponent λ(x) to another one with a “better“ exponent λ(x)