非交换加权Lorentz空间的对偶空间

设μ是一个半有限von Neumann代数.对于0P∞,0q≤∞,定义了非交换加权Lorentz空间Λ_ω~(p,q)(μ)及其associate空间Λ_ω~(p,q)(μ)',给出了空间Λ_ω~(p,q)(μ)'和Λ_ω~(p,q)(μ)'的一些基本性质.应用这些性质,还给出了非交换加权Lorentz空间Λ_ω~p(μ),0P∞的对偶空间.     

The Hardy-Littlewood Maximal Function and Weighted Lorentz Spaces

We study the boundedness of the Hardy–Littlewood maximalfunction in weighted Lorentz spaces, using some new techniqueson distribution and rearrangement functions. Our goal is togive a unified version of some well-known weighted inequalitiesin Rⁿ and R⁺, showing the relation between the theories.     

Fourier Inequalities and Moment Subspaces in Weighted Lebesgue Spaces

For define where Pointwise estimates and weighted inequalities describing the local Lipschitz continuity of are established. Sufficient conditions are found for the boundedness of from into and a spherical restriction property is proved. A study of the moment subspaces of is next developed in the one-variable case, for locally integrable, a.e. It includes a decomposition theorem and a complete classification of all possible sequences of moment subspaces in Characterizations are also given for each class. Applications related to the approximation and decomposition of are discussed.     

On Double Trigonometric Series with Multiply Monotone Coefficients in Lorentz Spaces

The sums of double series in terms of sines and cosines with multiply monotone coefficients in Lorentz spaces are estimated in this paper. The two-dimensional Lorentz spaces defined in this paper are the geneneralizations of Lebesgue spaces with mixed norm.     

Characterization of the Weak-Type Boundedness of the Hilbert Transform on Weighted Lorentz Spaces

We characterize the weak-type boundedness of the Hilbert transform H on weighted Lorentz spaces $\varLambda^{p}_{u}(w)$ , with p>0, in terms of some geometric conditions on the weights u and w and the weak-type boundedness of the Hardy–Littlewood maximal operator on the same spaces. Our results recover simultaneously the theory of the boundedness of H on weighted Lebesgue spaces L ^p (u) and Muckenhoupt weights A _p , and the theory on classical Lorentz spaces Λ ^p (w) and Ariño-Muckenhoupt weights B _p .     

Convolution Inequalities in Lorentz Spaces

In this paper we study boundedness of the convolution operator in different Lorentz spaces. We obtain the limit case of the Young-O’Neil inequality in the classical Lorentz spaces. We also investigate the convolution operator in the weighted Lorentz spaces.     

Lorentz序列空间的装球问题

Ｂａｎａｃｈ空间中装球问题的研究，近四十年来已取得了令人瞩目的发展。Ｂａｎａｃｈ空间的装球值的范围已经确定，Ｌ＿ｐ空间及Ｏｒｌｉｃｚ序列空间Ｉ＿Ｍ等许多经典Ｂａｎａｃｈ空间装球值已经找到．本文研究又一类经典Ｂａｎａｃｈ空间──Ｌｏｒｅｎｔｚ序列空间的装球问题，给出了Ｌｏｒｅｎｔｚ序列空间的装球值。     

Independent Random Variables in Lorentz Spaces

We investigate random variables in Lorentz spaces L_p,q. Conditionson the characteristic function are obtained which imply thata random variable belongs to the Lorentz space. Using them,we prove some estimates for the L_p,q-norm of sums of independentrandom variables. Some of these estimates are new for the spacesL_p.     

Refined Sobolev Inequalities in Lorentz Spaces

We establish refined Sobolev inequalities between the Lorentz spaces and homogeneous Besov spaces. The sharpness of these inequalities is illustrated on several examples, in particular based on non-uniformly oscillating functions known as chirps. These results are also used to derive refined Hardy inequalities.     

Lorentz Spaces and Lie Groups

This paper is motivated by the behavior of the heat diffusion kernelp_t(x) on a general unimodular Lie group. Indeed, contrary to what happens in ^n:, theP_t(x) on a general Lie group is behaving liket^−δ(t)/2for two possibly distinct integersδ(t), one forttending to 0 and another forttending to ∞, namelydandD. This forces us to consider a natural generalization of Lorentz spaces with different indices at “zero” and at “infinity.”     

Lorentz Spaces Of Vector-Valued Measures

Given a non-atomic, finite and complete measure space (,,µ)and a Banach space X, the modulus of continuity for a vectormeasure F is defined as the function _F(t) = sup_µ(E)t |F|(E)and the space V^p,q(X) of vector measures such that t^–1/p'_F(t) L^q((0,µ()],dt/t) is introduced. It is shown thatV^p,q(X) contains isometrically L^p,q(X) and that L^p,q(X) = V^p,q(X)if and only if X has the Radon–Nikodym property. It isalso proved that V^p,q(X) coincides with the space of cone absolutelysumming operators from L^p',q^' into X and the duality V^p,q(X^*)=(Lp^',q^'(X))^*where 1/p+1/p^'= 1/q+1/q^' = 1. Finally, V^p,q(X) is identifiedwith the interpolation space obtained by the real method (V¹(X),V(X))_1/p^',q. Spaces where the variation of F is replaced bythe semivariation are also considered.     

Lacunary Series in Weighted Hyperholomorphic B p,q (G) Spaces

The goal of this article is two-fold. First, we consider a class of hyperholomorphic functions, the so called B ^{p, q} (G) space in ?³. Then, we use the B ^{p, q} (G) space to characterize the hyperholomorphic α-Bloch space. Second, we obtain characterizations of the weighted hyperholomorphic B ^{p, q} (G)-functions by the coefficients of certain lacunary series expansions in Clifford Analysis.     

Weighted Tent Spaces

We generalize the theory of tent spaces introduced in [9] and [10], to consider weighted norms related to some function parameters (see [11]). We study their atomic decomposition, from which we obtain a weighted inequality for a certain fractional maximal operator. We also find the dual spaces, and get a new class of CARLESON measures and we identify the intermediate spaces when using several methods of interpolation.     

Adaptive Decomposition by Weighted Inner Functions: A Generalization of Fourier Series

In recent study adaptive decomposition of functions into basic functions of analytic instantaneous frequencies has been sought. Fourier series is a particular case of such decomposition. Adaptivity addresses certain optimal property of the decomposition. The present paper presents a fast decomposition of functions in the $\mathcal {L}^{2}(\partial {\mathbb{D}})$ spaces into a series of inner and weighted inner functions of increasing frequencies.     

Optimal Sobolev Type Inequalities in Lorentz Spaces

It is well known that the classical Sobolev embeddings may be improved within the framework of Lorentz spaces L ^p,q : the space $\mathcal{D}^{1,p}(\mathbb R^n)$ , 1?<?p?<?n, embeds into $L^{p^*,q}(\mathbb R^n)$ , p?≤?q?≤?∞. However, the value of the best possible embedding constants in the corresponding inequalities is known just in the case $L^{p^*,p}(\mathbb R^n)$ . Here, we determine optimal constants for the embedding of the space $\mathcal{D}^{1,p}(\mathbb R^n)$ , 1?<?p?<?n, into the whole Lorentz space scale $L^{p^{\ast}, q}(\mathbb R^n)$ , p?≤?q?≤?∞, including the limiting case q?=?p of which we give a new proof. We also exhibit extremal functions for these embedding inequalities by solving related elliptic problems.     

RUC-Bases in Orlicz and Lorentz Operator Spaces

If is an RUC-basis in somecouple of non-commutative L _p-spaces, then is an RUC-basic sequence in any non-commutative Orlicz or Lorentz space which is an interpolation space for this couple.     

Mixed Norm and Multidimensional Lorentz Spaces

In the last decade, the problem of characterizing the normability of the weighted Lorentz spaces has been completely solved ([16], [7]). However, the question for multidimensional Lorentz spaces is still open. In this paper, we consider weights of product type, and give necessary and sufficient conditions for the Lorentz spaces, defined with respect to the two-dimensional decreasing rearrangement, to be normable. To this end, it is also useful to study the mixed norm Lorentz spaces. Finally, we prove embeddings between all the classical, multidimensional, and mixed norm Lorentz spaces. Research partially supported by KAW 2000.0048 and STINT KU 2002-4025. Research partially supported by Grants MTM2004-02299, 2005SGR00556 and The Swedish Research Council no. 624-2003-571.