Spectral multipliers in group algebras and noncommutative Calderón-Zygmund theory

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 L2-form of Hörmander's kernel condition suffices for the weak type (1,1) of Calderón-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 L1 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 L1 endpoint inequalities for singular Fourier and Schur multipliers over nonabelian groups. Given a locally compact group G equipped with a conditionally negative length ψ:G→R+, we prove that Herz-Schur multipliers with symbol m∘ψ satisfying a Mikhlin condition in terms of the ψ-cocycle dimension are of weak type (1,1). Our result extends to Fourier multipliers for amenable groups and imposes sharp regularity conditions on the symbol. The proof crucially combines our new CZ methods with novel forms of recent transference techniques. This L1 endpoint gives a very much expected inequality which complements the L∞→BMO estimates proved in 2014 by Junge, Mei and ParcetJ. Parcet and J.M. Conde-Alonso were partially supported by Spanish Grant PID2019-107914GB-I00 and Severo Ochoa Programme for Centres of Excellence in R&D CEX2019-000904-S , funded by: MCIN/ AEI / 10.13039/501100011033. J.M. Conde-Alonso has also been supported by the Madrid Government ( Comunidad de Madrid -Spain) under the Multiannual Agreement with Universidad Autónoma de Madrid in the line of action encouraging youth research doctors, in the context of the V PRICIT, project SI1/PJI/2019-00514 . L. Cadilhac was supported by the French ( Agence Nationale de la Recherche ) grant ANR-19-CE40-0002

Balanced measures, sparse domination and complexity-dependent weight classes

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

Noncommutative strong maximals and almost uniform convergence in several directions

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 page

Experimental Feeding and Growth of Turbot (Scophthalmus maximus L.) from 0.5 To 2.7 Kg in Galicia (NW Spain)

Two groups of turbot, Scophthalmus maximus L., of 0.5 and 1.2 kg initial mean weight were fed a semimoist diet containing fish, fish meal and vitamin-mineral complex. The trial was carried out in tanks of 16 cubic meters, for aperiod of a year. Resul ts on feed conversion index and growth in weight are give

Effects of external ventricular drainage decompression of intracranial hypertension on rebleeding of brain aneurysms: A fluid structure interaction study

Objectives: The treatment of hydrocephalus using external ventricular drainage (EVD) seems to favour rebleeding of an untreated ruptured aneurysm. FSI studies are valuable to study this environment. Patients and methods: From December 2014 to December 2017, 61 patients with SAH required EVD due to hydrocephalus, 6 patients had aneurysm rebleeding after the procedure. Two controls for each case was included. DSA studies were used for fluid–structure interaction simulations using two scenarios high ICP (5332 Pa) and low ICP (133 Pa). Results: Maximum displacement of the wall in HICP was 0.34 mm and 0.26 mm in rebleeding and no rebleeding cases respectively, after EVD (LICP), it was 0.36 mm and 0.27 mm. The difference after implantation of EVD (HICP-LICP) had an average of 0.01567 mm and 0.00683 mm in rebleeding and no rebleeding cases (p = 0.05). This measure in low shear areas of the aneurysm was 0.026 and 0.0065 mm in rebleeding and no rebleeding cases (p = 0.01). Effective stress in the HICP was 4.77 MPa and 3.26 MPa in rebleeding and no rebleeding cases (p = 0.25). In LICP condition, this measure was 2.28 MPa and 1.42 MPa respectively (p = 0.33). TAWSS had no significant differences in the conditions of HICP and LICP. Conclusion: Changes after EVD placement includes an increase in the wall displacement with greater differences over low shear areas, this had a strong association with rebleeding.Xunta de Galicia | Ref. POS-A/2013/161Xunta de Galicia | Ref. ED481B 2016/047-0Xunta de Galicia | Ref. ED481D 2017/01

Study of the synergistic impact of Fe3O4, Na2CO3 and organic C on kaolin-based lightweight aggregates by a DOE (Mixture Experiments) approach

The compositional synergies involved in the thermal formation of lightweight aggregates (LWAs) have been investigated through four pure phases: non-expansive kaolin (K); cork powder (C); sodium carbonate (N) and magnetite-Fe3O4 (M). Mixture Experiments has been applied for formulation, modeling and optimization. LWAs have been manufactured from 36 starting mixtures and the main technological properties have been characterized: bloating index (BI), particle density (ρrd), water absorption (WA24) and crushing strength (S). Maximum BI and WA24 together with minimum density are associated with the addition of a significant amount of iron phase in combination with small proportions of organic carbon (Optimal [BI > 60%]: 56.0% K + 40.2% M + 3.9% C + 0% N), while S increases antagonistically with expansion. Iron reduction by incomplete combustion of C appears to be critical in pore formation and concomitant bloating. N has enhanced the sphericity of the expanded specimens. The contrast between experimental and estimated data has shown that the models have generally performed very well.This research was conducted as a part of the ECO-MET-AL Project, PID2019-109520RB-I00 / AEI / 10.13039/501100011033, “Can industrial and mining metalliferous wastes produce green lightweight aggregates? Applying the Circular Economy” funded by the Spanish Ministry of Science, Innovation and Universities and ERDF funds, framed in the “Ayudas a Proyectos I + D + i en el marco de los Programas Estatales de Generación de Conocimiento y Fortalecimiento Científico y Tecnológico del Sistema de I + D + i y de I + D + i orientada a los Retos de la Sociedad, Convocatoria 2019”

Can statistical methods optimize complex multicomponent mixtures for sintering ceramic granular materials? A case of success with synthetic aggregates

The relationship between the proportions of multicomponent mixtures with the technological properties of ceramic granular materials (synthetic aggregates) has been studied using statistical methods. The four phases involved in the formulations have been: kaolin (K) as aluminosilicate source; cork powder (C) as organic carbon source; sodium carbonate (N) as flux and pyrite (P) as source of iron and sulfur. The Mixture Experiments - Design of Experiments (ME-DOE) has been the statistical methodology applied from the initial configuration of the 36 starting formulations to the final validation of the models and optimums. After granulation, artificial aggregates have been obtained by sintering in a rotary kiln, and their main technological properties have been determined. Bloating index (BI), particle density (ρrd), water absorption (WA24) and crushing strength (S) were selected as the four key characteristics to be modeled and optimized, using response surface and effect plots to assess the effect of K, C, N and P on such properties. 32 out of 36 starting varieties met the density criteria for lightweight aggregates. In the optimum formulations obtained, the minimum percentage of K was 83 wt%, so that the variations in the percentages of P, C and N were the critical variables for determining the final properties of the aggregate. The contrast between experimental and estimated data has shown that the models fit adequately, indicating that this type of approach may have enormous potential for future research on artificial aggregates and other ceramic materialsThis research was conducted as a part of the ECO-MET-AL Project, PID2019-109520RB-I00 / AEI / 10.13039/501100011033, “Can industrial and mining metalliferous wastes produce green lightweight aggregates? Applying the Circular Economy” funded by the Spanish Ministry of Science, Innovation and Universities and ERDF funds, framed in the “Ayudas a “Proyectos I + D + i" en el marco de los Programas Estatales de Generación de Conocimiento y Fortalecimiento Científico y Tecnológico del Sistema de I + D + i y de I + D + i orientada a los Retos de la Sociedad, Convocatoria 2019”. Thanks also to the SCAI of the University of Jaén, the University of Castilla-La Mancha and the University of Málaga for their service