The best constant for centered Hardy-Littlewood maximal inequality

We find the exact value of the best possible constant $C$ for the weak type $(1,1)$ inequality for the one dimensional centered Hardy-Littlewood maximal operator. We prove that $C$ is the largest root of the quadratic equation $12C^{2}-22C+5=0$ thus obtaining $C=1.5675208...$. This is the first time the best constant for one of the fundamental inequalities satisfied by a centered maximal operator is precisely evaluated.Comment: 42 pages, published versio

Local lower norm estimates for dyadic maximal operators and related Bellman functions

We provide lower $L^q$ and weak $L^p$-bounds for the localized dyadic maximal operator on $R^n$, when the local $L^1$ and the local $L^p$ norm of the function are given. We actually do that in the more general context of homo- geneous tree-like families in probability spaces.Comment: 9 page

Estimates for Bellman functions related to dyadic-like maximal operators on weighted spaces

We provide some new estimates for Bellman type functions for the dyadic maximal opeator on $R^n$ and of maximal operators on martingales related to weighted spaces. Using a type of symmetrization principle, introduced for the dyadic maximal operator in earlier works of the authors we introduce certain conditions on the weight that imply estimate for the maximal operator on the corresponding weighted space. Also using a well known estimate for the maximal operator by a double maximal operators on different m easures related to the weight we give new estimates for the above Bellman type functions.Comment: 10 pages. arXiv admin note: text overlap with arXiv:1511.0611

Sharp Lorentz estimates for dyadic-like maximal operators and related Bellman functions

We precisely evaluate Bellman type functions for the dyadic maximal opeator on $R^n$ and of maximal operators on martingales related to local Lorentz type estimates. Using a type of symmetrization principle, introduced for the dyadic maximal operator in earlier works of the authors we precisely evaluate the supremum of the Lorentz quasinorm of the maximal operator on a function $\phi$ when the integral of $\phi$ is fixed and also the same Lorentz quasinorm of $\phi$ is fixed. Also we find the corresponding supremum when the integral of $\phi$ is fixed and several weak type conditions are given.Comment: 11 page

Dyadic weights on RnR^n and reverse Holder inequalities

We prove that for any weight $\phi$ defined on $[0,1]^n$ that satisfies a reverse Holder inequality with exponent p > 1 and constant $c\ge1$ upon all dyadic subcubes of $[0,1]^n$, it's non increasing rearrangement satisfies a reverse Holder inequality with the same exponent and constant not more than $2^nc-2^n + 1$, upon all subintervals of $[0; 1]$ of the form $[0; t]$. This gives as a consequence, according to the results in [8], an interval $[p; p_0(p; c)) = I{p,c}$, such that for any $q \in I{p,c}$, we have that $\phi$ is in $L^q$.Comment: 10 page

On weak type inequalities for dyadic maximal functions

AbstractWe obtain sharp estimates for the localized distribution function of the dyadic maximal function Mϕd, given the local L1 norms of ϕ and of G○ϕ where G is a convex increasing function such that G(x)/x→+∞ as x→+∞. Using this we obtain sharp refined weak type estimates for the dyadic maximal operator

Intracisternal delivery of NFκB-inducible scAAV2/9 reveals locoregional neuroinflammation induced by systemic kainic acid treatment.

We have previously demonstrated disease-dependent gene delivery in the brain using an AAV vector responding to NFκB activation as a probe for inflammatory responses. This vector, injected focally in the parenchyma prior to a systemic kainic acid (KA) injection mediated inducible transgene expression in the hippocampus but not in the cerebellum, regions, respectively, known to be affected or not by the pathology. However, such a focal approach relies on previous knowledge of the model parameters and does not allow to predict the whole brain response to the disease. Global brain gene delivery would allow to predict the regional distribution of the pathology as well as to deliver therapeutic factors in all affected brain regions. We show that self-complementary AAV2/9 (scAAV2/9) delivery in the adult rat cisterna magna allows a widespread but not homogenous transduction of the brain. Indeed, superficial regions, i.e., cortex, hippocampus, and cerebellum were more efficiently transduced than deeper regions, such as striatum, and substantia nigra. These data suggest that viral particles penetration from the cerebrospinal fluid (CSF) into the brain is a limiting factor. Interestingly, AAV2/9-2YF a rationally designed capsid mutant (affecting surface tyrosines) increased gene transfer efficiency approximately fivefold. Neurons, astrocytes, and oligodendrocytes, but not microglia, were transduced in varying proportions depending on the brain region and the type of capsid. Finally, after a single intracisternal injection of scAAV2/9-2YF using the NFκB-inducible promoter, KA treatment induced transgene expression in the hippocampus and cortex but not in the cerebellum, corresponding to the expression of the CD11b marker of microglial activation. These data support the use of disease-inducible vectors administered in the cisterna magna as a tool to characterize the brain pathology in systemic drug-induced or transgenic disease models. However, further improvements are required to enhance viral particles penetration into the brain

The impact of temperature changes on summer time ozone and its precursors in the Eastern Mediterranean

Changes in temperature due to variability in meteorology and climate change are expected to significantly impact atmospheric composition. The Mediterranean is a climate sensitive region and includes megacities like Istanbul and large urban agglomerations such as Athens. The effect of temperature changes on gaseous air pollutant levels and the atmospheric processes that are controlling them in the Eastern Mediterranean are here investigated. The WRF/CMAQ mesoscale modeling system is used, coupled with the MEGAN model for the processing of biogenic volatile organic compound emissions. A set of temperature perturbations (spanning from 1 to 5 K) is applied on a base case simulation corresponding to July 2004. The results indicate that the Eastern Mediterranean basin acts as a reservoir of pollutants and their precursor emissions from large urban agglomerations. During summer, chemistry is a major sink at these urban areas near the surface, and a minor contributor at downwind areas. On average, the atmospheric processes are more effective within the first 1000 m above ground. Temperature increases lead to increases in biogenic emissions by 9±3% K<sup>−1</sup>. Ozone mixing ratios increase almost linearly with the increases in ambient temperatures by 1±0.1 ppb O<sub>3</sub> K<sup>−1</sup> for all studied urban and receptor stations except for Istanbul, where a 0.4±0.1 ppb O<sub>3</sub> K<sup>−1</sup> increase is calculated, which is about half of the domain-averaged increase of 0.9±0.1 ppb O<sub>3</sub> K<sup>−1</sup>. The computed changes in atmospheric processes are also linearly related with temperature changes