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We investigate nontrigonometric forms of Riesz transforms in the context of Schur multipliers. This refines Grothendieck-Haagerup's endpoint criterion with a new condition for the Schatten p-boundedness of Schur multipliers and strengthens Potapov/Sukochev's solution of Arazy's conjecture. We recover as well dimension-free estimates for trigonometric Riesz transforms. Our discrete approach is much simpler than previous harmonic analysis and probabilistic approaches. As an application, we find a very simple proof of recent criteria for Schur multipliers of H\"ormander-Mikhlin and Marcinkiewicz type., Comment: Minor improvements

Understanding and predicting the electricity demand responses to prices are critical activities for system operators, retailers, and regulators. While conventional machine learning and time series analyses have been adequate for the routine demand patterns that have adapted only slowly over many years, the emergence of active consumers with flexible assets such as solar-plus-storage systems, and electric vehicles, introduces new challenges. These active consumers exhibit more complex consumption patterns, the drivers of which are often unobservable to the retailers and system operators. In practice, system operators and retailers can only monitor the net demand (metered at grid connection points), which reflects the overall energy consumption or production exchanged with the grid. As a result, all "behind-the-meter" activities-such as the use of flexibility-remain hidden from these entities. Such behind-the-meter behavior may be controlled by third party agents or incentivized by tariffs; in either case, the retailer's revenue and the system loads would be impacted by these activities behind the meter, but their details can only be inferred. We define the main components of net demand, as baseload, flexible, and self-generation, each having nonlinear responses to market price signals. As flexible demand response and self generation are increasing, this raises a pressing question of whether existing methods still perform well and, if not, whether there is an alternative way to understand and project the unobserved components of behavior. In response to this practical challenge, we evaluate the potential of a data-driven inverse optimization (IO) methodology. This approach characterizes decomposed consumption patterns without requiring direct observation of behind-the-meter behavior or device-level metering [...]

We study decision problems under uncertainty, where the decision-maker has access to $K$ data sources that carry {\em biased} information about the underlying risk factors. The biases are measured by the mismatch between the risk factor distribution and the $K$ data-generating distributions with respect to an optimal transport (OT) distance. In this situation the decision-maker can exploit the information contained in the biased samples by solving a distributionally robust optimization (DRO) problem, where the ambiguity set is defined as the intersection of $K$ OT neighborhoods, each of which is centered at the empirical distribution on the samples generated by a biased data source. We show that if the decision-maker has a prior belief about the biases, then the out-of-sample performance of the DRO solution can improve with $K$ -- irrespective of the magnitude of the biases. We also show that, under standard convexity assumptions, the proposed DRO problem is computationally tractable if either $K$ or the dimension of the risk factors is kept constant.

Let $\mathbf{E}_n: \mathcal{M} \to \mathcal{M}_n$ and $\mathbf{E}_m: \mathcal{N} \to \mathcal{N}_m$ be two sequences of conditional expectations on finite von Neumann algebras. The optimal weak Orlicz type of the associated strong maximal operator $\mathcal{E} = (\mathbf{E}_n\otimes \mathbf{E}_m)_{n,m}$ is not yet known. In a recent work of Jose Conde and the two first-named authors, it was show that $\mathcal{E}$ has weak type $(\Phi, \Phi)$ for a family of functions including $\Phi(t) = t \, \log^{2+\varepsilon} t$, for every $\varepsilon > 0$. In this article, we prove that the weak Orlicz type of $\mathcal{E}$ cannot be lowered below $L \log^2 L$, meaning that if $\mathcal{E}$ is of weak type $(\Phi, \Phi)$, then $\Phi(s) \not\in o(s \, \log^2 s)$. Our proof is based on interpolation. Namely, we use recent techniques of Cadilhac/Ricard to formulate a Marcinkiewicz type theorem for maximal weak Orlicz types. Then, we show that a weak Orlicz type lower than $L \log^2 L$ would imply a $p$-operator constant for $\mathcal{E}$ smaller than the known optimum as $p \to 1^{+}$., Comment: 16 pages; Minor changes, acknowledgment added

Let $T_m$ be a noncommutative Fourier multiplier. In recent work, Mei and Ricard introduced a noncommutative analogue of Cotlar's identity in order to prove that certain multipliers are bounded on the non-commutative $L_p$-spaces of a free group. Here, we study Cotlar type identities in full generality, giving a closed characterization for them in terms of $m$: \[ \big( m(g h) - m(g) \big) \, \big( m(g^{-1}) - m(h) \big) = 0, \; \forall g \in \mathrm{G} \setminus \{e\}, h \in \mathrm{G}. \] We manage to prove, using a geometric argument, that if $X$ is a tree -- or more generally an $\mathbb{R}$-tree -- on which $\mathrm{G}$ acts and $m$ lifts to a function $\widetilde{m}: X \to \mathbb{C}$ that is constant on the connected subsets of $X \setminus \{x_0\}$, then $m$ satisfies Cotlar's identity and thus $T_m$ is bounded in $L_p$ for $1 < p < \infty$. This result establishes a new connection between groups actions on $\mathbb{R}$-trees and Fourier multipliers. We show that $m$ is trivial when the action has global fixed points. This machinery allows us to simultaneously generalize the free group transforms of Mei and Ricard and the theory of Hilbert transforms in left-orderable groups, which follows from Arveson's subdiagonal algebras. Using Bass-Serre theory, we construct new examples of Fourier multipliers in groups. Those include new families like Baumslag-Solitar groups. We also show that a natural Hilbert transform in $\mathrm{PSL}_2(\mathbb{C})$ satisfies Cotlar's identity when restricted to the Bianchi group $\mathrm{PSL}_2(\mathbb{Z}[\sqrt{-1}])$., Comment: Minor mistakes corrected. The introduction and Sections 1 and 4 are slightly improved. A Remark on Schur multipliers has been added to Section 1. In Section 4, Theorem 4.2 has been added

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 develop a distributionally robust chance-constrained formulation of the Optimal Power Flow problem (OPF) whereby the system operator can leverage contextual information. For this purpose, we exploit an ambiguity set based on probability trimmings and optimal transport through which the dispatch solution is protected against the incomplete knowledge of the relationship between the OPF uncertainties and the context that is conveyed by a sample of their joint probability distribution. We provide a tractable reformulation of the proposed distributionally robust chance-constrained OPF problem under the popular conditional-value-at-risk approximation. By way of numerical experiments run on a modified IEEE-118 bus network with wind uncertainty, we show how the power system can substantially benefit from taking into account the well-known statistical dependence between the point forecast of wind power outputs and its associated prediction error. Furthermore, the experiments conducted also reveal that the distributional robustness conferred on the OPF solution by our probability-trimmings-based approach is superior to that bestowed by alternative approaches in terms of expected cost and system reliability., Comment: 44 pages, 37 figures

We consider stochastic programs conditional on some covariate information, where the only knowledge of the possible relationship between the uncertain parameters and the covariates is reduced to a finite data sample of their joint distribution. By exploiting the close link between the notion of trimmings of a probability measure and the partial mass transportation problem, we construct a data-driven Distributionally Robust Optimization (DRO) framework to hedge the decision against the intrinsic error in the process of inferring conditional information from limited joint data. We show that our approach is computationally as tractable as the standard (without side information) Wasserstein-metric-based DRO and enjoys performance guarantees. Furthermore, our DRO framework can be conveniently used to address data-driven decision-making problems under contaminated samples and naturally produces distributionally robust versions of some local nonparametric predictive methods, such as Nadaraya-Watson kernel regression and $K$-nearest neighbors, which are often used in the context of conditional stochastic optimization. Finally, the theoretical results are illustrated using a single-item newsvendor problem and a portfolio allocation problem with side information., Comment: 24 figures, 76 pages

Let $\Gamma \curvearrowright \Omega$ be a measure-preserving action and $\mathcal{L} \Gamma \hookrightarrow L^\infty(\Omega) \rtimes \Gamma$ the natural inclusion of the group von Neumann algebra into the crossed product. When $\mu(\Omega) = \infty$, we have that this natural embedding is not trace-preserving and therefore does not extends boundedly to the associated noncommutative $L^p$-spaces. Nevertheless, we show that when $\Omega$ has an invariant mean there is an isometric embedding of $L^p(\mathcal{L} \Gamma)$ into an ultrapower of $L^p(\Omega \rtimes \Gamma)$ that intertwines Fourier multipliers and is $\mathcal{L} \Gamma$-bimodular. As a consequence we obtain the lower transference bound \[ \big\| T_m: L^p(\mathcal{L} \Gamma) \to L^p(\mathcal{L} \Gamma) \big\| \leq \big\| (\mathrm{id} \rtimes T_m): L^p(\Omega \rtimes \Gamma) \to L^p(\Omega \rtimes \Gamma) \big\|, \] and the same follows for complete norms. The condition of having an invariant mean is quite restrictive. Therefore, we explore whether other equivariant embeddings $\Phi: \mathcal{L} \Gamma \to L^\infty(\Omega)$ yield a more general transference result. We show that the transference proof above works verbatim whenever $\Phi$ is completely positive, amenable (in the sense of inducing an amenable correspondence) and intertwines Fourier multipliers at the $L^2$-level. Although no new transference results are obtained, both the classification of equivariant maps and the study their amenability may be of independent interest to some readers.

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 define H\"older classes $\Lambda_\alpha$ associated with a Markovian semigroup and prove that, when the semigroup satisfies the $\Gamma^2 \geq 0$ condition, the Riesz transforms are bounded between the H\"older classes. As a consequence, this bound holds in manifolds with nonnegative Ricci curvature. We also show, without the need for extra assumptions on the semigroup, that a version of the Morrey inequalities is equivalent to the ultracontractivity property. This result extends the semigroup approach to the Sobolev inequalities laid by Varopoulos. After that, we study certain families of operators between the homogeneous H\"older classes. One of these families is given by analytic spectral multipliers and includes the imaginary powers of the generator, the other, by smooth multipliers analogous to those in the Marcienkiewicz theorem. Lastly, we explore the connection between the H\"older norm and Campanato's formula for semigroups.

In this paper we wish to tackle stochastic programs affected by ambiguity about the probability law that governs their uncertain parameters. Using optimal transport theory, we construct an ambiguity set that exploits the knowledge about the distribution of the uncertain parameters, which is provided by: (1) sample data and (2) a-priori information on the order among the probabilities that the true data-generating distribution assigns to some regions of its support set. This type of order is enforced by means of order cone constraints and can encode a wide range of information on the shape of the probability distribution of the uncertain parameters such as information related to monotonicity or multi-modality. We seek decisions that are distributionally robust. In a number of practical cases, the resulting distributionally robust optimization (DRO) problem can be reformulated as a finite convex problem where the a-priori information translates into linear constraints. In addition, our method inherits the finite-sample performance guarantees of the Wasserstein-metric-based DRO approach proposed by Mohajerin Esfahani and Kuhn (Math Program 171(1-2):115-166. https://doi.org/10.1007/s10107-017-1172-1, 2018), while generalizing this and other popular DRO approaches. Finally, we have designed numerical experiments to analyze the performance of our approach with the newsvendor problem and the problem of a strategic firm competing \`a la Cournot in a market.

Hurricanes are cyclones circulating about a defined center whose closed wind speeds exceed 75 mph originating over tropical and subtropical waters. At landfall, hurricanes can result in severe disasters. The accuracy of predicting their trajectory paths is critical to reduce economic loss and save human lives. Given the complexity and nonlinearity of weather data, a recurrent neural network (RNN) could be beneficial in modeling hurricane behavior. We propose the application of a fully connected RNN to predict the trajectory of hurricanes. We employed the RNN over a fine grid to reduce typical truncation errors. We utilized their latitude, longitude, wind speed, and pressure publicly provided by the National Hurricane Center (NHC) to predict the trajectory of a hurricane at 6-hour intervals. Results show that this proposed technique is competitive to methods currently employed by the NHC and can predict up to approximately 120 hours of hurricane path.

In this paper, we establish the core of singular integral theory and pseudodifferential calculus over the archetypal algebras of noncommutative geometry: quantum forms of Euclidean spaces and tori. Our results go beyond Connes' pseudodifferential calculus for rotation algebras, thanks to a new form of Calder\'on-Zygmund theory over these spaces which crucially incorporates nonconvolution kernels. We deduce $L_p$-boundedness and Sobolev $p$-estimates for regular, exotic and forbidden symbols in the expected ranks. In the $L_2$ level both Calder\'on-Vaillancourt and Bourdaud theorems for exotic and forbidden symbols are also generalized to the quantum setting. As a basic application of our methods, we prove $L_p$-regularity of solutions for elliptic PDEs., Comment: 87 pages

The concept of quantum relation $\mathcal{R}$ over a von Neumann algebra $\mathcal{M}$ has been recently introduced by Nik Weaver. When $\mathcal{M}$ is either finite dimensional or discrete and abelian, $\mathcal{R}$ is given by an orthogonal projection in $\mathcal{M} \otimes \mathcal{M}_\mathrm{op}$. Here, we generalize such result to general von Neumann algebras, proving that quantum relations are in bijective correspondence with weak-$\ast$ closed left ideals inside $\mathcal{M} \otimes_{e h} \mathcal{M}$, where $\otimes_{e h}$ is the extended Haagerup tensor product. The correspondence between the two is given by identifying $\mathcal{M} \otimes_{e h} \mathcal{M}$ with $\mathcal{M}'$-bimodular operators and proving a double annihilator relation. Given an action of a group/quantum group on $\mathcal{M}$ we give a definition for invariant quantum relations and prove that, in the case of group von Neumann algebras $\mathcal{L} G$, invariant quantum relations are left ideals in the measure algebra $M G$. At the end we explore possible applications to noncommutative harmonic analysis, in particular noncommutative Gaussian bounds.

We investigate Fourier multipliers with smooth symbols defined over locally compact Hausdorff groups. Our main results in this paper establish new H\"ormander-Mikhlin criteria for spectral and non-spectral multipliers. The key novelties which shape our approach are three. First, we control a broad class of Fourier multipliers by certain maximal operators in noncommutative $L_p$ spaces. This general principle ---exploited in Euclidean harmonic analysis during the last 40 years--- is of independent interest and might admit further applications. Second, we replace the formerly used cocycle dimension by the Sobolev dimension. This is based on a noncommutative form of the Sobolev embedding theory for Markov semigroups initiated by Varopoulos, and yields more flexibility to measure the smoothness of the symbol. Third, we introduce a dual notion of polynomial growth to further exploit our maximal principle for non-spectral Fourier multipliers. The combination of these ingredients yields new $L_p$ estimates for smooth Fourier multipliers in group algebras.