We present new estimates in the setting of weighted Lorentz spaces of operators satisfying a limited Rubio de Francia condition; namely $T$ is bounded on $L^{p}(v)$ for every $v$ in an strictly smaller class of weights than the Muckenhoupt class $A_p$. Important examples will be the Bochner–Riesz operators $BR_\lambda$ with $0<\lambda <{(n-1)}/2$, sparse operators, Hörmander multipliers with a limited regularity condition and rough operators with $\Omega \in L^{r}(\Sigma )$, $1 < r < \infty$.

We continue the study of the boundedness of the operator

on the set of decreasing functions in Lp(w). This operator was first introduced by Braverman and Lai and also studied by Andersen, and although the weighted weak-type estimate was completely solved, the characterization of the weights w such that is bounded is still open for the case in which p > 1. The solution of this problem will have applications in the study of the boundedness on weighted Lorentz spaces of important operators in harmonic analysis.

We give necessary and sufficient conditions on a general cone of positive functions to satisfy the Decomposition Property (DP) introduced in [5] and connect the results with the theory of interpolation of cones introduced by Sagher [9]. One of our main result states that if Q satisfies DP or equivalently is divisible, then for the quasi-normed spaces E0 and E1,

According to this formula, it yields that the interpolation theory for divisible cones can be easily obtained from the classical theory.

We present some transference results for a convolution operator with kernel $K$ which is bounded from $L^{p_0}(w_0)$ into $L^{p_1}(w_1)$. Our results are a natural extension of the classical results of Coifman and Weiss and their further extensions. We shall give several applications in the setting of restriction of Fourier multipliers.

Let $T$ be a sublinear operator such that $(T\chi_E)^*(t)\,{\le}\, h(t, |E|)$ for some positive function $h(t,s)$ and every measurable set $E$. Then it is shown that under some conditions on the operator $T$, this restricted weak type estimate can be extended to the set of all functions $f\,{\in}\, L^1$ such that $\n f.\infty\,{\le}\, 1$, in the sense that $(Tf)^*(t)\,{\le}\, h(t, \n f.1)$. This inequality allows strong type estimates for $T$ to be obtained on several classes of spaces, such as logarithmic spaces and Lorentz spaces. A similar problem for operators $T$ acting on radial functions is also studied.

Let $M$ be the Hardy–Littlewood maximal function and let $ M_{(\alpha)}$ be the $\ell^\alpha$-maximal operator defined by $$M_{(\alpha)} \bar{f}(x)=\|{M\bar {f}\|_{\ell^\alpha}=\Bigg(\sum_{i=1}^\infty Mf_i(x)^\alpha\Bigg)^{1/\alpha},$$ where $\bar f=(f_i)_i$, $M\bar f=(Mf_i)_i$ and $\alpha >1$. The purpose of this work is to study modular inequalities of the form $$\int_{{\mathbb R}^n} P \big(\big|\overline M_{(\alpha)}f(x)\big| \big)\,dx \le \sum_j\int_{{\mathbb R}^n } Q(|f_j(x)|)\, dx,$$ where $P$ and $Q$ are modular functions. Our results apply to operators of the form $$\overline T_{(\alpha)} \bar f(x)= \Bigg(\sum_{i=1}^\infty |T_i f_i(x)|^\alpha\Bigg)^{1/\alpha},$$ where $T_i$ satisfies similar properties to $M$.

The work of Coifman and Weiss concerning Hardy spaces on spaces of homogeneous type gives, as a particular case, a definition of Hp(ZN) in terms of an atomic decomposition.

Other characterizations of these spaces have been studied by other authors, but it was an open question to see if they can be defined, as it happens in the classical case, in terms of a maximal function or via the discrete Riesz transforms.

In this paper, we give a positive answer to this question.

This work connects the theory of commutators with analytic families of operators in abstract interpolation theory. Our main result asserts that if {Lξ}0≤Reξ≤1 is an analytic family of operators satisfying some conditions, then [Lθ,Ω] +(Lξ)′(θ): Āθ→ Bθ is bounded. From this, we can deduce the boundedness of the commutator .

The paper shows that, if the operator T[ratio ]A(γ)→B(γ) is compact for almost every γ∈Γ, then T[ratio ]F(Ā)→F([Bmacron]) is compact when F([Xmacron])=(X)SK or F([Xmacron])=(X)SJ is the interpolation functor constructed for infinite families of Banach spaces and S satisfies certain conditions.

We study the boundedness of the Hardy–Littlewood maximal function in weighted Lorentz spaces, using some new techniques on distribution and rearrangement functions. Our goal is to give a unified version of some well-known weighted inequalities in ℝn and ℝ+, showing the relation between the theories.

We apply the expression for the norm of a function in the weighted Lorentz space, with respect to the distribution function, to obtain as a simple consequence some weighted inequalities for integral operators.