Complex locally uniform rotundity of Musielak-Orlicz spaces

The concepts of complex locally uniform rotundity and complex locally uniformly rotund point are introduced. The sufficient and necessary conditions of them are given in complex MusielakOrlicz spaces.     

Uniform Gateaux differentiability and weak uniform rotundity in Musielak-Orlicz function spaces of Bochner type equipped with the Luxemburg norm

Criteria for weak uniform rotundity and reflexivity of Musielak-Orlicz function spaces of Bochner type equipped with the Luxemburg norm are given. Although, criteria for uniform Gateaux differentiability and weak uniform rotundity of Musielak-Orlicz function spaces of real functions equipped with the Luxemburg norm were known, they can be also easily deduced from our main results.     

Onl p -copies in Musielak-Orlicz sequence spaces

The authors would like to thank Francisco L. Hernández for some friendly and interesting discussions on this paper.     

Extreme points and rotundity of Orlicz-Musielak sequence spaces

A.Kaminska and H.Hudzik[1-10] present a series of work concerning geometry ofsequence Orlicz-Musielak spaces. This paper continues their work, to give a character of extremepoints of the unit balls of sequence Orlicz-Musielak spaces epuipped with Luxemburg norm. Fromwhich a criterion of rotundity is obtained immediately.     

Local rotundity structure of Cesàro-Orlicz sequence spaces

Some criteria for extreme points and strong U-points in Cesàro-Orlicz spaces are given. In consequence we find a Cesàro-Orlicz sequence space different from c₀ which has no extreme points. Some examples show that in these spaces the notion of the strong U-point is essentially stronger than the notion of the extreme point. Various examples presented in this paper show that there are some differences between criteria for extreme points and strong U-points in Orlicz spaces and in Cesàro-Orlicz spaces. We also show that the uniqueness of the local best approximation needs the notion of SU-point, that is, the notion of the extreme point is not strong enough here.     

Uniformly non-l n (1) Musielak-Orlicz sequence spaces

We give a necessary and sufficient condition for the uniformly non-l _n ⁽¹⁾ property of Musielak-Orlicz sequence spacesl ^Φ generated by a sequence Φ=(ϕ_n:n⩾l) of finite Orlicz functions such that for eachn∈ℕ. As a result, forn ₀⩾2, there exist spacesl ^Φ which are only uniformly non-l _n ⁽¹⁾ forn⩾n ₀. Moreover we obtain a characterization of uniformly non-l _n ⁽¹⁾ and reflexive Orlicz sequence spaces over a wide class of purely atomic measures and of uniformly non-l _n ⁽¹⁾ Nakano sequence spaces. This extends a result of Luxemburg in [19]. Submitted in memory of Professor W. Orlicz     

Some multiplier generalized difference sequence spaces over n-normed spaces defined by a Musielak-Orlicz function

In the present paper, we introduce some multiplier sequence spaces over n-normed spaces defined by a Musielak-Orlicz functionM = (M _k ). We also study some topological properties and some inclusion relations between these spaces.     

Complex rotundity of Orlicz sequence spaces equipped with the p-Amemiya norm

In this paper, two equivalent definitions of complex strongly extreme points in general complex Banach spaces are shown. It is proved that for any Orlicz sequence space equipped with the p-Amemiya norm (1?p<∞, p is odd), complex strongly extreme points of the unit ball coincide with complex extreme points of the unit ball. Moreover, criteria for them in Orlicz sequence spaces equipped with the p-Amemiya norm are given. Criteria for complex mid-point locally uniform rotundity and complex rotundity of Orlicz sequence spaces equipped with the p-Amemiya norm are also deduced.     

Uniform convexity in Lorentz sequence spaces

Necessary and sufficient conditions for Lorentz sequence spacesd(a,p) (1<p<∞), to be uniformly convexifiable are given. In casep≧2 the modulus of convexity is calculated.     

Martingale Musielak-Orlicz Hardy spaces

In this article,we introduce the martingale Musielak-Orlicz Hardy spaces H_φ~*(?),Pφ(?),H_φ~S(?),Qφ(?)and H_φ~s(?),respectively,via the maximal function,the quadratic variation and the conditional quadratic variation of martingales.We then establish the atomic characterizations of H_φ~s(?),Pφ(?)and Qφ(?).As applications,we obtain the dual space of H_φ~s(?)and several martingale inequalities which further clarify the relations among H_φ~*(?),Pφ(?),H_φ~S(?),Qφ(?)and H_φ~s(?).Especially,as special cases,the results on atomic characterizations of H_φ~s(?),Pφ(?)and Qφ(?)as well as on the dual space of H_φ~s(?)in the weighted case are also new.     

Maximal function on Musielak-Orlicz spaces and generalized Lebesgue spaces

We consider the Hardy-Littlewood maximal operator M on Musielak-Orlicz Spaces Lφ(Rd). We give a necessary condition for the continuity of M on Lφ(Rd) which generalizes the concept of Muckenhoupt classes. In the special case of generalized Lebesgue spaces Lp(⋅)(Rd) we show that this condition is also sufficient. Moreover, we show that the condition is “left-open” in the sense that not only M but also Mq is continuous for some q>1, where .     

Local rotundity structure of Calderón-Lozanovskii spaces

General results saying that a point x of the unit sphere S(E) of a Köthe space E is an extreme point (a strongly extreme point) [an SU-point] of B(E) if and only if ‖x‖ is an extreme point (a strongly extreme point) [an SU-point] of B(E⁺) and ‖x‖ is a UM-point (a ULUM-point) [nothing more] of E are proved. These results are applied to get criteria for extreme points and SU-points of the unit ball in Caldern-Lozanovski spaces which refer to problem XII from [5]. Strongly extreme points in these spaces are also discussed.     

Capacity for potentials of functions in Musielak-Orlicz spaces

We define a capacity for potentials of functions in Musielak-Orlicz spaces. Basic properties of such capacity are studied. We also estimate the capacity of balls and give some applications of the estimates.     

Real-variable characterizations of anisotropic product Musielak-Orlicz Hardy spaces

Let A :=(A_1, A_2) be a pair of expansive dilations and φ : R~n×R~m×[0, ∞) → [0, ∞) an anisotropic product Musielak-Orlicz function. In this article, we introduce the anisotropic product Musielak-Orlicz Hardy space H~φ_A(R~n× R~m) via the anisotropic Lusin-area function and establish its atomic characterization, the g-function characterization, the g_λ~*-function characterization and the discrete wavelet characterization via first giving out an anisotropic product Peetre inequality of Musielak-Orlicz type. Moreover, we prove that finite atomic decomposition norm on a dense subspace of H~φ_A(R~n× R~m) is equivalent to the standard infinite atomic decomposition norm. As an application, we show that, for a given admissible triplet(φ, q, s), if T is a sublinear operator and maps all(φ, q, s)-atoms into uniformly bounded elements of some quasi-Banach spaces B, then T uniquely extends to a bounded sublinear operator from H~φ_A(R~n× R~m) to B. Another application is that we obtain the boundedness of anisotropic product singular integral operators from H~φ_A(R~n× R~m) to L~φ(R~n× R~m)and from H~φ_A(R~n×R~m) to itself, whose kernels are adapted to the action of A. The results of this article essentially extend the existing results for weighted product Hardy spaces on R~n× R~m and are new even for classical product Orlicz-Hardy spaces.     

Local Hardy spaces of Musielak-Orlicz type and their applications

Letφ:R n × [0,∞) → [0,∞) be a function such that φ(x,·) is an Orlicz function and (·,t) ∈ A ∞loc (Rn) (the class of local weights introduced by Rychkov).In this paper,the authors introduce a local Musielak-Orlicz Hardy space hφ(Rn) by the local grand maximal function,and a local BMO-type space bmoφ(Rn) which is further proved to be the dual space of hφ(Rn).As an application,the authors prove that the class of pointwise multipliers for the local BMO-type space bmo φ (Rn),characterized by Nakai and Yabuta,is just the dual of L 1 (Rn) + h Φ 0 (Rn),where φ is an increasing function on (0,∞) satisfying some additional growth conditions and Φ 0 a Musielak-Orlicz function induced by φ.Characterizations of hφ(Rn),including the atoms,the local vertical and the local nontangential maximal functions,are presented.Using the atomic characterization,the authors prove the existence of finite atomic decompositions achieving the norm in some dense subspaces of hφ(Rn),from which,the authors further deduce some criterions for the boundedness on hφ(Rn) of some sublinear operators.Finally,the authors show that the local Riesz transforms and some pseudo-differential operators are bounded on hφ(Rn).