Weighted Lorentz Spaces And Sets In Orlicz-the Lorentz Space

The weighted Lorentz spaces and the Orlicz-Lorentz spaces are important generalized forms of Lp spaces and classical Lorentz space Lp,q spaces, and they are the major manifestation of the rearrangement spaces. Since being introduced, they have always been of great importance in harmonic analysis. This paper mainly researches the analytic properties:difference norm, moduli of continuity, Sobolev embedding, Riesz convergence property. They are the generalization of the properties of LP spaces and classical Lorentz space Lp,q.The paper contains four chapters.Chapter 1 gives the estimate of partial moduli of continuity of functions with respect to the weighted Lorentz spaces.The problem of estimating the moduli of continuity of a function in Lq in term of its moduli of continuity in Lp (1≤p≤q≤∞) has been researched for a long time (see [6],§16, [11], [21] and [61]).For a function f∈Lp(Rn) (1≤p≤∞) we write In [21], we have the following result.Theorem 1.1 Let 1< p< oo and n≥1, or p=1 and n≥2. Suppose that f∈Lp(Rn), p< q<∞, andγ=n(p/1-q/1)<1.Then for anyδ>0.Now by (0.1) we can get that if f has all first-order generalized derivatives Djf∈Lp(Rn)(j＝1,...,n),then For p> 1 and n≥1 the inequality was proved in, and for p=1 and n≥2 the inequality (0.2) was proved in (see also and).Suppose that a function f has one partial derivative Djf∈Lp(Rn) with only one variable xj. The problem is to estimate the partial modulus of continuity of the function f with respect to the same variable in Lq, q> p. In this circum-stance, we often need to assume that f belongs to a certain space Lγ. In, the author obtained multiplicative inequalities of Gagliardo-Nirenberg type by using these conditions. Also in, a more general problem, namely, the problem of estimating the partial modulus of continuity of a function f with respect to a variable xj in Lq in terms of its partial modulus of continuity with respect to the same variable in Lp and the norm of f in Lγwas solved. Considering the exact integrability exponents for function in Sobolev spaces can be expressed in terms of the Lorentz spaces Lq,p, the author in studied multiplicative inequalities for moduli of continuity in the scale of these spaces and obtained more general results (see,Theorem 3.1,3.5,4.1, and 4.3).In many cases, the Lorentz space Lq,p should be substituted by weighted Lorentz spaceΛq,p(ω), whereωis a weight in R+(nonnegative locally integrable functions in R+). We want to know whether the similar inequalities hold with respect to modulus of continuityωjγ(f;δ)Λq,p(ω). In this chapter, we shall study this problem and obtain positive answer.The main results in the chapter are:Theorem 1.3.1 Let (p0,s0) and (p1,s1) be adimissible pairs,0<θ<1 and let numbers p, s, and p be defined by (1.3.1). Let weightω∈Bp if p<∞and W∈Δ2 if p=∞. Suppose that a function f∈Λp0,s0(ω) has the generalized derivative Djγf∈Λp1,s1(ω) with respect to the variable xj,1≤j≤n (n,j∈N). Then (with the obvious modification if s=∞),where C=Cγ[θ(1-θ)]-1/s (?)Theorem 1.3.3 Let(p0,s0)and(p1,s1)be adimissible pairs and p1>1. Let 0<θ<1,p=min(p0,p1),and numbers p and s be defined by the equalities (1.3.1).Suppose that w is given by and f∈Λp0,s0(w)∩Λp1,s1(w).Letγ∈N and 1≤j≤n.Then for anyδ>0,and For any 0<α<γ(with the obvious modification if s=∞);where and K1=2Kθ1/s.Chapter 2 researches the Sobolev embedding of the weighted Lorentz spaces and rearrangement spaces.Let 1≤p<∞andγ∈N.We denote by Wpr the isotropic Sobolev spaces for functions f∈LP(Rn)which have all generalized derivatives Dsf(s=(s1,...,sn)) of order |s|=s1+…+sn≤r, which belong to Lp(Rn).It is well known that for 1≤p0 Wp,sγ1,...,γn (?)Lq,θ.(*)Letωbe a weight in R+. We denote WΛp,s(ω)γ1,...,γn(1≤p<∞,0αi, ki∈N. We denote Besov space BΛq,pα1,...,αn the space of functions f∈Λq,p(w) for which We denote that definition of BΛq,pα1,...,αn does not depend on the choice of ki>αi (see [6], [31]).In this chapter, we prove in Theorem 3.1 that if w satisfies some special conditions, and 1/q*=1/p-r/n, then if 1≤p< n/r,0< s<∞we get WΛp,sr1,...,rn(?)Λq*,s(w), and if 1≤p< q< q*,0< s<∞, then for any qs/q*<θ<∞,we get WΛp,sr1,...,rn(?)Λq,θ(w), If we put p> 1, w=1, then the range of 9 may be enlarged to 0<θ<∞, and the above embedding is the result in [22]. The other result, Theorem 3.4, extends Theorem 4 in [22] by virtue of replacing the Lorentz space Lp,q by the weighted Lorentz spaceΛp,q(w) where w satisfies some mild conditions, which assumes as WΛp(w)r1,...,rn(?)BΛq,pα1,...,αn, where 1< p< q<∞,1/p-1/q< r/n, r=n(∑i=1 n ri-1)-1, andOn the other hand, we think it is meaningful if we can generalize (*) into the rearrangement invariant spaces. Indeed, there are many mathematicians, e.g., Bastero, Milman, Ruiz, Martin, Pustylnik, who have researched this kind of question and found many meaningful and important results. Readers may refer to their papers:[7,34,35,36,37,38,39,47]. The first result we get, Theorem 3.5, can be regarded as an extension of Theorem 1.2 and Corollary 1.3 in [39]. The secod result is Theorem 3.7,which can be regarded as a generalization of Theorem 3 in[22]in the background of the rearrangement invariant spaces.The main results in present chapter are:Theorem 2.3.1 Let 1≤p<∞,00 such that W(t)≥at,(?)t>0, (ⅲ)there exists a constantβwithβ<1 such that Then for every f∈WΛp,s T1,...,Tn,there holds that if pr/n, andαx<1.Then for any f∈WXr1,...,rn.Theorem 2.3.7 Letβx> 0,βX<1. Suppose f∈WxT1,...,rn.If E≠(?) where then for everyα∈E there holds where C is independent of f.Chapter 3 discusses the estimate of difference norm of functions with respect to the weighted Lorentz spaces.In this paper we study functions f on Rn which possess the generalized partial derivatives Our main goal is to obtain some norm estimates for the differences (ek being the unit coordinate vector).The sharp estimates of the norms of differences for the functions in Sobolev spaces have firstly been proved by V. P. Il'in [6]. For the space Wp1(Rn)(1≤p< n) Il'in's result reads as follows:If n∈N,1 0, then The generalization of the inequality(0.3)to the spaces Wp T1,...,rn was given in [22].That is where 0<1/p-1/q1,n≥1 or p=1,n≥2.In[28],there was the sharp estimates of the type(0.4)when the derivatives Dk Tk f belong to different Lorentz spaces LPk,skTheorem 3.1.1 Let n≥2,rk∈N,1≤pk,sk<∞and sk=1 if Pk=1. Letr,p and s be the numbers defined by(3.1.7),(3.1.8).For every pj(1≤j≤n) satisfying the condition take arbitrary qj>pj such that and denote then for any function f∈S0(Rn)which has the weak derivatives Dk Tk f∈Lpk,sk(Rn)(K= 1,...,n)there holds the inequality where C is a constant that does not depend on f.The main result of this chapter is:Theorem 3.3.1 Let n≥2,rk∈N,10; (ii)there exist two constantsη,βwithβ<1 such that and there holds q≡sup{η;(?)β<1,(0.5)holds)>max{pi；i=1,...,n}. For every pj(1≤j≤n)satisfying the condition take arbitrary qj such that pj