Space Λp Research Articles

On Discrete Weighted Lorentz Spaces and Equivalent Relations between Discrete ℓp-Classes

In this paper, we study some relations between different weights in the classes Bp,Bp*,Mp and Mp* that characterize the boundedness of the Hardy operator and the adjoint Hardy operator. We also prove that these classes generate the same weighted Lorentz space Λp. These results will be proven by using the properties of classes Bp,Bp*,Mp and Mp*, including the self-improving properties and also the properties of the generalized Hardy operator Hp, the adjoint operator Sq and some fundamental relations between them connecting their composition to their sum.

The problem of trigonometric Fourier series multipliers of classes in λp,q spaces

In this article, we consider weighted spaces of numerical sequences λp,q, which are defined as sets of sequences a = {ak}^∞_k=1, for which the norm ||a||λp,q :=\sum^∞_k=1|ak|^q k^(q/p −1)^1/q<∞ is finite. In the case of non-increasing sequences, the norm of the space λp,q coincides with the norm of the classical Lorentz space lp,q. Necessary and sufficient conditions are obtained for embeddings of the space λp,q into the space λp1,q1. The interpolation properties of these spaces with respect to the real interpolation method are studied. It is shown that the scale of spaces λp,q is closed in the relative real interpolation method, as well as in relative to the complex interpolation method. A description of the dual space to the weighted space λp,q is obtained. Specifically, it is shown that the space is reflective, where p', q' are conjugate to the parameters p and q. The paper also studies the properties of the convolution operator in these spaces. The main result of this work is an O’Neil type inequality. The resulting inequality generalizes the classical Young-O’Neil inequality. The research methods are based on the interpolation theorems proved in this paper for the spaces λp,q.

Abstract Lorentz spaces and Köthe duality

Given a symmetric Banach function space E and a decreasing positive weight w on I=(0,a), 0<a≤∞, the generalized Lorentz space ΛE,w is defined as the symmetrization of the canonical copy Ew of E on the measure space associated with the weight. A class of functions ME,w is similarly defined in the spirit of Marcinkiewicz spaces as the symmetrization of the space wEw. Differently from the Lorentz space, which is a Banach function space, the class ME,w does not need to be even a linear space; but we show that if the weight w is regular then this class is normable. Let also QE,w be the smallest fully symmetric Banach function space containing ME,w. The Köthe duality of these classes is developed here. The Köthe dual of the class ME,w is identified as the Lorentz space ΛE′,w, while the Köthe dual of ΛE,w is QE′,w. Several characterizations of QE,w are obtained, one of them states that a function belongs to QE,w if and only if its level function in Halperin’s sense with respect to w, belongs to ME,w. The other characterizations are by optimization with respect to the Hardy–Littlewood submajorization order. These results are applied to a number of concrete Banach function spaces. In particular a new description of the Köthe dual space is provided for the classical Lorentz space Λp,w and for the Orlicz–Lorentz space Λφ,w, which correspond respectively to the cases E=Lp and E=Lφ.

Weighted backward shift operators with invariant distributionally scrambled subsets

We obtain a sufficient condition to ensure that weighted backward shift operators on Köthe sequence spaces λp(A) admit an invariant distributionally ε-scrambled subset for any 0<ε<diamλp(A). In particular, every Devaney chaotic weighted backward shift operator on λp(A) supports such a subset.

Maximal distributional chaos of weighted shift operators on Köthe sequence spaces

During the last ten some years, many research works were devoted to the chaotic behavior of the weighted shift operator on the Kothe sequence space. In this note, a sufficient condition ensuring that the weighted shift operator Bwn: λp(A) → λp(A) defined on the Kothe sequence space λp(A) exhibits distributional ɛ-chaos for any 0 < ɛ < diamλp(A) and any n ∈ ℕ is obtained. Under this assumption, the principal measure of Bwn is equal to 1. In particular, every Devaney chaotic shift operator exhibits distributional ɛ-chaos for any 0 < ɛ < diam λp(A).

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 Lp(u) and Muckenhoupt weights Ap, and the theory on classical Lorentz spaces Λp(w) and Arino-Muckenhoupt weights Bp.

DOUBLING CONDITION AND LINEARITY OF THE SEQUENCE SPACE Λp(ƒ)

For 1 ≤ p < +∞,every ƒ (≠ 0) ∈Lp(R,dx) defines a sequence space Λp(ƒ) (Honda et al. Proc. Japan Acad. Ser. A 84 (2008), 39-41) which is an additive group but not necessarily a linear space. The main purpose of this paper is to discuss the linearity of Λp(ƒ). First we show that if ƒ is a piecewise monotone function, then Λp(ƒ) is a linear space. Next, specializing the case to p = 2, we characterize Λ2(ƒ) as a set, and discuss the linearity of it. With this aim, we extend the definition of the doubling condition and define the doubling dimension H(φ) of a non-negative function on [0, +∞).Let ƒ be the Fourier transform of ƒ and define a function φƒ associated with ƒ. Then we show that H(|ƒ|)< ∞ implies the linearity of Λ2(ƒ). In addition, we show that if H(φƒ)< 2, then Λ2(ƒ) is linear and give several examples.

SHARP CONSTANTS BETWEEN EQUIVALENT NORMS IN WEIGHTED LORENTZ SPACES

AbstractFor an increasing weight w in Bp (or equivalently in Ap), we find the best constants for the inequalities relating the standard norm in the weighted Lorentz space Λp(w) and the dual norm.

On embeddings between classical Lorentz spaces

Let p ∈ (0, ∞) , let v be a weight on (0, ∞) and let Λp(v) be the classical Lorentz space, determiined by the norm ∥f∥Λp(v) := (∫∞0(f*(t))pv(t)dt) 1/p. When p ∈ (1, ∞), this space is known to be a Banach space if and only if v is non-increasing, while it is only equivalent to a Banach space if and only if Λp(v) = Γp(v), where ∥f∥Γp(v) := (∫∞0(f**(t))pv(t) t/p. We may thus conclude that, for p ∈ (1, ∞), the space Λp(v) is equivalent to a Banach space if and only if the norm of a function f in it can be expressed in terms of f**. We study the question whether an analogous assertion holds when p = 1. Motivated by this problem, we consider general embeddings between four types of classical and weak Lorentz spaces, namely, Λp(v), Λp,∞(v), Γp(v), Γp,∞(v), where Λp,∞(v) and Γp∞(v) are certain weak analogues of the spaces Λp(v) and Γp(v), respectively. We present a unified approach to these embeddings, based on rearrangement techniques. We survey all the known results and prove new ones. Our main results concern the embedding Γp,∞(v) → Λq(w) which had not been characterized so far. We apply our results to the characterization of associate spaces of classical and weak Lorentz spaces and we give a characterization of fundamental functions for which the endpoint Lorentz space and the endpoint Marcinkiewicz space coincide.

Weighted Lorentz Spaces and the Hardy Operator

We find a new expression for the norm of a function in the weighted Loreritz space, with respect to the distribution funetion, and obtain as a simple consequence a generalization of the classical embeddings Lp, 1 ⊂ · · · ⊂ Lp ⊂ · · · ⊂ Lp, ∞ and a new definition of the weak space Λp, ∞u (w). We also give some applications to the boundedness of the Hardy operator Sƒ = ∫x0ƒ from Λp0u0(w0) into Λp1u1 (w1) with 0 < p0 ≤ p1.