arXiv subjects
Publications and source records attributed to Javier Soria.
Our main goal in this work is to further improve the mixed norm estimates due to Fournier, and also Algervik and Kolyada, to more general rearrangement invariant (r.i.) spaces. In particular we find the optimal domains and the optimal ranges for these embeddings between mixed norm spaces and r.i. spaces.
We study functorial properties of the spaces $R(X)$, introduced in [Studia Math. 197 (2010), 69-79] as a central tool in the analysis of the Hardy operator minus the identity on decreasing functions. In particular, we provide conditions on a minimal Lorentz space $Λ_φ$ so that the equation $R(X)=Λ_φ$ has a solution within the category of rearrangement invariant (r.i.) spaces. Moreover, we show that if $R(X)=Λ_φ$, then we can always take $X$ to be the minimal r.i. Banach range space for the Hardy operator defined in $Λ_φ$.
We characterize, in the context of rearrangement invariant spaces, the optimal range space for a class of monotone operators related to the Hardy operator. The connection between optimal range and optimal domain for these operators is carefully analyzed.
Let $T$ be an arbitrary operator bounded from $L^{p_0}(w)$ into $L^{p_0, \infty}(w)$ for every weight $w$ in the Muckenhoupt class $A_{p_0}$. It is proved in this article that the distribution function of $Tf$ with respect to any weight $u$ can be essentially majorized by the distribution function of $Mf$ with respect to $u$ (plus an integral term easy to control). As a consequence, well-known extrapolation results, including results in a multilinear setting, can be obtained with very simple proofs. New applications in extrapolation for two-weight problems and estimates on rearrangement invariant spaces are established too.
We study the Lorentz spaces $L^{p,s}(R,μ)$ in the range $1 1$.
For a general set transformation ${\cal R}$ between two measure spaces, we define the rearrangement of a measurable function by means of the Layer's cake formula. We study some functional properties of the Lorentz spaces defined in terms of ${\cal R}$, giving a unified approach to the classical rearrangement, Steiner's symmetrization, the multidimensional case, and the discrete setting of trees.
We study the optimal domain for the Hardy operator considered with values in a rearrangement invariant space. In particular, this domain can be represented as the space of integrable functions with respect to a vector measure defined on a $δ$-ring. A precise description is given for the case of the minimal Lorentz spaces.
We characterize the weighted Hardy's inequalities for monotone functions in ${\mathbb R^n_+}.$ In dimension $n=1$, this recovers the classical theory of $B_p$ weights. For $n>1$, the result was only known for the case $p=1$. In fact, our main theorem is proved in the more general setting of partially ordered measure spaces.
We characterize the rearrangement invariant spaces for which there exists a non-constant fixed point, for the Hardy-Littlewood maximal operator (the case for the spaces $L^p(\mathbb{R}^{n})$ was first considered by Korry in \cite{Ko}). The main result that we prove is that the space $L^{\frac{n}{n-2},\infty}(\mathbb{R}^{n})\cap L^{\infty}(\mathbb{R}^{n})$ is minimal among those having this property
In the last decade, the problem of characterizing the normability of the weighted Lorentz spaces has been completely solved (\cite{Sa}, \cite{CaSoA}). 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.
We show that for every nontrivial potential space $L^{1}_{K}(\mathbb{R}^{n})$, there exists an approach region for which the associated maximal function is of weak-type, but the boundedness for the completed region is false, which is in contrast with the nontangential case.
We define a multidimensional rearrangement, which is related to classical inequalities for functions that are monotone in each variable. We prove the main measure theoretical results of the new theory and characterize the functional properties of the associated weighted Lorentz spaces.
The main objective of this work is to bring together two well known and, a priori, unrelated theories dealing with weighted inequalities for the Hardy-Littlewood maximal operator $M$, and thus, we consider the boundedness of $M$ in the weighted Lorentz space $Λ^p_u(w)$. Two examples are historically relevant as a motivation: If $w=1$, this corresponds to the study of the boundedness $M:L^p(u)\longrightarrow L^p(u),$ which was characterized by B. Muckenhoupt, giving rise to the so called $A_p$ weights. The second case is when we take $u=1$. This is a more recent theory, and was completely solved by M.A. Ariño and B. Muckenhoupt in 1991. It turns out that the boundedness $M:\llo\longrightarrow\llo,$ can be seen to be equivalent to the boundedness of the Hardy operator $A$ restricted to decreasing functions of $L^p(w)$. The class of weights satisfying this boundedness is known as $B_p$. Even though the $A_p$ and $B_p$ classes enjoy some similar features, they come from very different theories, and so are the techniques used on each case: Calderón--Zygmund decompositions and covering lemmas for $A_p$, rearrangement invariant properties and positive integral operators for $B_p$. It is our aim to give a unified version of these two theories. Contrary to what one could expect, the solution is not given in terms of the limiting cases above considered (i.e., $u=1$ and $w=1$), but in a rather more complicated condition, which reflects the difficulty of estimating the distribution function of the Hardy-Littlewood maximal operator with respect to general measures.
Associated to the class of restricted-weak type weights for the Hardy operator, we find a new class of Lorentz spaces for which the normability property holds. This result is analogous to the characterization given by Sawyer for the classical Lorentz spaces. We also show that these new spaces are very natural to study the existence of equivalent norms described in terms of the maximal function