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We prove endpoint and sparse-like bounds for Bergman projectors on nonhomogeneous, radial trees $X$ that model manifolds with possibly unbounded geometry. The natural Bergman measures on $X$ may fail to be doubling, and even locally doubling, with respect to the right metric in our setting. Weighted consequences of our sparse domination results are also considered, and are in line with the known results in the disk. Our endpoint results are partly a consequence of a new Calder\'on-Zygmund theory for discrete, non-locally doubling metric spaces., Comment: 38 pages

We study $L^p(\mu)$ estimates for the commutator $[H,b]$, where the operator $H$ is a dyadic model of the classical Hilbert transform introduced in \cite{arXiv:2012.10201,arXiv:2212.00090} and is adapted to a non-doubling Borel measure $\mu$ satisfying a dyadic regularity condition which is necessary for $H$ to be bounded on $L^p(\mu)$. We show that $\|[H, b]\|_{L^p(\mu) \rightarrow L^p(\mu)} \lesssim \|b\|_{\mathrm{BMO}(\mu)}$, but to {\it characterize} martingale BMO requires additional commutator information. We prove weighted inequalities for $[H, b]$ together with a version of the John-Nirenberg inequality adapted to appropriate weight classes $\widehat{A}_p$ that we define for our non-homogeneous setting. This requires establishing reverse H\"{o}lder inequalities for these new weight classes. Finally, we revisit the appropriate class of nonhomogeneous measures $\mu$ for the study of different types of Haar shift operators.

We study sparse domination for operators defined with respect to an atomic filtration on a space equipped with a general measure $\mu$. In the case of Haar shifts, $L^p$-boundedness is known to require a weak regularity condition, which we prove to be sufficient to have a sparse domination-like theorem. Our result allows us to characterize the class of weights where Haar shifts are bounded. A surprising novelty is that said class depends on the complexity of the Haar shift operator under consideration. Our results are qualitatively sharp.

In the line of previous work by Naor, we establish new forms of metric $\mathrm{X}_p$ inequalities in group algebras under very general assumptions. Our results' applicability goes beyond the previously known setting in two directions. In first place, we find continuous forms of the $\mathrm{X}_p$ inequality in the $n$-dimensional torus. Second, we consider transferred forms of the sharp scalar valued metric $\mathrm{X}_p$ inequality in the von Neumann algebra $\mathcal{L}(\mathrm{G})$ of a discrete group $\mathrm{G}$. As a byproduct of our results, some metric consequences and their relation with bi-Lipschitz nonembeddability of Banach spaces are explored in the context of noncommutative $L_p$ spaces., Comment: 16 pages

We investigate $L_p$-estimates for balanced averages of Fourier truncations in group algebras, in terms of differential operators acting on them. Our results extend a fundamental inequality of Naor for the hypercube (with profound consequences in metric geometry) to discrete groups. Different inequalities are established in terms of directional derivatives which are constructed via affine representations determined by the Fourier truncations. Our proofs rely on the Banach $\mathrm{X}_p$ nature of noncommutative $L_p$-spaces and dimension-free estimates for noncommutative Riesz transforms. In the particular case of free groups we use an alternative approach based on free Hilbert transforms., Comment: 22 pages

We establish regularity conditions for $L_p$-boundedness of Fourier multipliers on the group von Neumann algebras of higher rank simple Lie groups. This provides a natural H\"ormander-Mikhlin criterion in terms of Lie derivatives of the symbol and a metric given by the adjoint representation. In line with Lafforgue/de la Salle's rigidity theorem, our condition imposes certain decay of the symbol at infinity. It refines and vastly generalizes a recent result by Parcet, Ricard and de la Salle for $\SL$. Our approach is partly based on a sharp local H\"ormander-Mikhlin theorem for arbitrary Lie groups, which follows in turn from recent estimates by the authors on singular nonToeplitz Schur multipliers. We generalize the latter to arbitrary locally compact groups and refine the cocycle-based approach to Fourier multipliers in group algebras by Junge, Mei and Parcet. A few related open problems are also discussed., Comment: Small clarifications in the proof of Theorem B

We establish a rather unexpected and simple criterion for the boundedness of Schur multipliers $S_M$ on Schatten $p$-classes which solves a conjecture proposed by Mikael de la Salle. Given $1 < p < \infty$, a simple form our main result reads for $\mathbf{R}^n \times \mathbf{R}^n$ matrices as follows $$\big\| S_M: S_p \to S_p \big\|_{\mathrm{cb}} \lesssim \frac{p^2}{p-1} \sum_{|\gamma| \le [\frac{n}{2}] +1} \Big\| |x-y|^{|\gamma|} \Big\{ \big| \partial_x^\gamma M(x,y) \big| + \big| \partial_y^\gamma M(x,y) \big| \Big\} \Big\|_\infty.$$ In this form, it is a full matrix (nonToeplitz/nontrigonometric) amplification of the H\"ormander-Mikhlin multiplier theorem, which admits lower fractional differentiability orders $\sigma > \frac{n}{2}$ as well. It trivially includes Arazy's conjecture for $S_p$-multipliers and extends it to $\alpha$-divided differences. It also leads to new Littlewood-Paley characterizations of $S_p$-norms and strong applications in harmonic analysis for nilpotent and high rank simple Lie group algebras., Comment: To appear in Annals of Mathematics. Affiliations corrected

In this paper we solve three problems in noncommutative harmonic analysis which are related to endpoint inequalities for singular integrals. In first place, we prove that an $L_2$-form of H\"ormander's kernel condition suffices for the weak type (1,1) of Calder\'on-Zygmund operators acting on matrix-valued functions. To that end, we introduce an improved CZ decomposition for martingale filtrations in von Neumann algebras, and apply a very simple unconventional argument which notably avoids pseudolocalization. In second place, we establish as well the weak $L_1$ endpoint for matrix-valued CZ operators over nondoubling measures of polynomial growth, in the line of the work of Tolsa and Nazarov/Treil/Volberg. The above results are valid for other von Neumann algebras and solve in the positive two open problems formulated in 2009. An even more interesting problem is the lack of $L_1$ endpoint inequalities for singular Fourier and Schur multipliers over nonabelian groups. Given a locally compact group G equipped with a conditionally negative length $\psi: \mathrm{G} \to \mathbb{R}_+$, we prove that Herz-Schur multipliers with symbol $m \circ \psi$ satisfying a Mikhlin condition in terms of the $\psi$-cocycle dimension are of weak type $(1,1)$. Our result extends to Fourier multipliers for amenable groups and impose sharp regularity conditions on the symbol. The proof crucially combines our new CZ methods with novel forms of recent transference techniques. This $L_1$ endpoint gives a very much expected inequality which complements the $L_\infty \to \mathrm{BMO}$ estimates proved in 2014 by Junge, Mei and Parcet.

We present a general approach to sparse domination based on single-scale $L^p$-improving as a key property. The results are formulated in the setting of metric spaces of homogeneous type and avoid completely the use of dyadic-probabilistic techniques as well as of Christ-Hyt\"onen-Kairema cubes. Among the applications of our general principle, we recover sparse domination of Dini-continuous Calder\'on-Zygmund kernels on spaces of homogeneous type, we prove a family of sparse bounds for maximal functions associated to convolutions with measures exhibiting Fourier decay, and we deduce sparse estimates for Radon transforms along polynomial submanifolds of $\mathbb R^n$., Comment: 36 pages, submitted for publication. V2: improvement of the main results Theorems A and B; uniform bound on the truncations is now required for some L^p unrelated to the sparse exponents; upgraded to Dini modulus of continuity; L^{p_j} improving assumption simplified. Applications are unchanged but verification is simplified

Our first result is a noncommutative form of Jessen/Marcinkiewicz/Zygmund theorem for the maximal limit of multiparametric martingales or ergodic means. It implies bilateral almost uniform convergence with initial data in the expected Orlicz spaces. A key ingredient is the introduction of the $L_p$-norm of the $\limsup$ of a sequence of operators as a localized version of a $\ell_\infty/c_0$-valued $L_p$-space. In particular, our main result gives a strong $L_1$-estimate for the $\limsup$, as opposed to the usual weak $L_{1,\infty}$-estimate for the $\sup$. Let $\mathcal{L} \mathbf{F}_2$ denote the free group algebra and consider the free Poisson semigroup generated by the usual length function. It is an open problem to determine the largest class inside $L_1(\mathcal{L} \mathbf{F}_2)$ for which this semigroup converges to the initial data. Currently, the best known result is $L \log^2 L(\mathcal{L} \mathbf{F}_2)$. We improve this by adding to it the operators in $L_1(\mathcal{L} \mathbf{F}_2)$ spanned by words without signs changes. Contrary to other related results in the literature, this set has exponential growth. The proof relies on our estimates for the noncommutative $\limsup$ together with new transference techniques. We also establish a noncommutative form of C\'ordoba/Feffermann/Guzm\'an inequality for the strong maximal. More precisely, a weak $(\Phi,\Phi)$ inequality for noncommutative multiparametric martingales and $\Phi(s) = s (1 + \log_+ s)^{2 + \varepsilon}$. This logarithmic power is an $\varepsilon$-perturbation of the expected optimal one. The proof combines a refinement of Cuculescu's construction with a quantum probabilistic interpretation of de Guzm\'an's argument. The commutative form of our argument gives the simplest known proof of this classical inequality. A few interesting consequences are derived for Cuculescu's projections., Comment: 38 pages

We study the dual space of the variable Lebesgue space $\Lp$ with unbounded exponent function $\pp$ and provide an answer to a question posed in~[fiorenza-cruzuribe2013]. Our approach is to decompose the dual into a topological direct sum of Banach spaces. The first component corresponds to the dual in the bounded exponent case, and the second is, intuitively, the dual of functions that live where the exponent is unbounded (in a heuristic sense). The second space is extremely complicated, and we illustrate this with a series of examples. In the special case of the variable sequence space $\ell^\pp$, we show that this piece can be further decomposed into two spaces, one of which can be characterized in terms of a generalization of finitely additive measures. As part of our work, we also considered the question of dense subsets in $\Lp$ for unbounded exponents. We constructed two examples, one for general variable Lebesgue spaces $\Lp$ and one in the sequence space $\lp$. This gives an answer to another question from~[fiorenza-cruzuribe2013].

We study the problem of dominating the dyadic strong maximal function by $(1, 1)$-type sparse forms based on rectangles with sides parallel to the axes, and show that such domination is impossible. Our proof relies on an explicit construction of a pair of maximally separated point sets with respect to an appropriately defined notion of distance

We present a data resource which can be useful for research purposes on language grounding tasks in the context of geographical referring expression generation. The resource is composed of two data sets that encompass 25 different geographical descriptors and a set of associated graphical representations, drawn as polygons on a map by two groups of human subjects: teenage students and expert meteorologists., Comment: Resource paper, 5 pages, 6 figures, 1 table. Conference: INLG 2018

The usual one third trick allows to reduce problems involving general cubes to a countable family. Moreover, this covering lemma uses only dyadic cubes, which allows to use nice martingale properties in harmonic analysis problems. We consider alternatives to this technique in spaces equipped with nonhomogeneous measures. This entails additional difficulties which forces us to consider martingale filtrations that are not regular. The dyadic covering that we find can be used to clarify the relationship between martingale BMO spaces and the most natural BMO space in this setting, which is the space RBMO introduced by Tolsa., Comment: Comments welcome!

Consider a totally irregular measure $\mu$ in $\mathbb{R}^{n+1}$, that is, the upper density $\limsup_{r\to0}\frac{\mu(B(x,r))}{(2r)^n}$ is positive $\mu$-a.e.\ in $\mathbb{R}^{n+1}$, and the lower density $\liminf_{r\to0}\frac{\mu(B(x,r))}{(2r)^n}$ vanishes $\mu$-a.e. in $\mathbb{R}^{n+1}$. We show that if $T_\mu f(x)=\int K(x,y)\,d\mu(y)$ is an operator whose kernel $K(\cdot,\cdot)$ is the gradient of the fundamental solution for a uniformly elliptic operator in divergence form associated with a matrix with H\"older continuous coefficients, then $T_\mu$ is not bounded in $L^2(\mu)$. This extends a celebrated result proved previously by Eiderman, Nazarov and Volberg for the $n$-dimensional Riesz transform.

We present a novel heuristic approach that defines fuzzy geographical descriptors using data gathered from a survey with human subjects. The participants were asked to provide graphical interpretations of the descriptors `north' and `south' for the Galician region (Spain). Based on these interpretations, our approach builds fuzzy descriptors that are able to compute membership degrees for geographical locations. We evaluated our approach in terms of efficiency and precision. The fuzzy descriptors are meant to be used as the cornerstones of a geographical referring expression generation algorithm that is able to linguistically characterize geographical locations and regions. This work is also part of a general research effort that intends to establish a methodology which reunites the empirical studies traditionally practiced in data-to-text and the use of fuzzy sets to model imprecision and vagueness in words and expressions for text generation purposes., Comment: Conference paper: Accepted for FUZZIEEE-2017. One column version for arXiv (8 pages)

We prove that bilinear forms associated to the rough homogeneous singular integrals $T_\Omega$ on $\mathbb R^d$, where the angular part $\Omega \in L^q (S^{d-1})$ has vanishing average and $1

Given a measure $\mu$ of polynomial growth, we refine a deep result by David and Mattila to construct an atomic martingale filtration of $\mathrm{supp}(\mu)$ which provides the right framework for a dyadic form of nondoubling harmonic analysis. Despite this filtration being highly irregular, its atoms are comparable to balls in the given metric |which in turn are all doubling| and satisfy a weaker but crucial form of regularity. Our dyadic formulation is effective to address three basic questions: i) A dyadic form of Tolsa's RBMO space which contains it. ii) Lerner's domination and $A_2$-type bounds for nondoubling measures. iii) A noncommutative form of nonhomogeneous Calder\'on-Zygmund theory. Our martingale RBMO space preserves the crucial properties of Tolsa's original definition and reveals its interpolation behavior with the $L_p$ scale in the category of Banach spaces, unknown so far. On the other hand, due to some known obstructions for Haar shifts and related concepts over nondoubling measures, our pointwise domination theorem via sparsity naturally deviates from its doubling analogue. In a different direction, matrix-valued harmonic analysis over noncommutative $L_p$ spaces has recently produced profound applications. Our analogue for nondoubling measures was expected for quite some time. Finally, we also find a dyadic form of the Calder\'on-Zygmund decomposition which unifies those by Tolsa and L\'opez-S\'anchez/Martell/Parcet.

In this paper, we introduce a new comprehensive data set on educational attainment and inequality measures of education for 142 countries over the period 1970 to 2010. Most of the previous attempts to measure educational attainment have treated education as a categorical variable, whose mean is computed as a weighted average of the official duration of each cycle and attainment rates, thus omitting differences in educational achievement within levels of education. This aggregation into different groups may result in a loss of information introducing, therefore, a potential source of measurement error. We explore here a more nuanced alternative to estimate educational attainment, which considers the continuous nature of the educational variable. This `continuous approach' allows us to impose more plausible assumptions about the distribution of years of schooling within each level of education, and to take into account the right censoring of the data in the estimation, thus leading to more accurate estimates of educational attainment and education inequality. These improved series may help to better understand the role of education on different socio-economic aspects, such as quality of life and human capital formation., Comment: 29 pages, 5 figures, 4 tables

Let $T$ be a multilinear integral operator which is bounded on certain products of Lebesgue spaces on $\mathbb R^n$. We assume that its associated kernel satisfies some mild regularity condition which is weaker than the usual H\"older continuity of those in the class of multilinear Calder\'on-Zygmund singular integral operators. In this paper, given a suitable multiple weight $\vec{w}$, we obtain the bound for the weighted norm of multilinear operators $T$ in terms of $\vec{w}$. As applications, we exploit this result to obtain the weighted bounds for certain singular integral operators such as linear and multilinear Fourier multipliers and the Riesz transforms associated to Schr\"odinger operators on $\mathbb{R}^n$. Our results are new even in the linear case.