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

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

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

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

Non-Volatile Particle Number Emission Measurements with Catalytic Strippers: A Review

Vehicle regulations include limits for non-volatile particle number emissions with sizes larger than 23 nm. The measurements are conducted with systems that remove the volatile particles by means of dilution and heating. Recently, the option of measuring from 10 nm was included in the Global Technical Regulation (GTR 15) as an additional option to the current >23 nm methodology. In order to avoid artefacts, i.e., measuring volatile particles that have nucleated downstream of the evaporation tube, a heated oxidation catalyst (i.e., catalytic stripper) is required. This review summarizes the studies with laboratory aerosols that assessed the volatile removal efficiency of evaporation tube and catalytic stripper-based systems using hydrocarbons, sulfuric acid, mixture of them, and ammonium sulfate. Special emphasis was given to distinguish between artefacts that happened in the 10–23 nm range or below. Furthermore, studies with vehicles' aerosols that reported artefacts were collected to estimate critical concentration levels of volatiles. Maximum expected levels of volatiles for mopeds, motorcycles, light-duty and heavy-duty vehicles were also summarized. Both laboratory and vehicle studies confirmed the superiority of catalytic strippers in avoiding artefacts. Open issues that need attention are the sulfur storage capacity and the standardization of technical requirements for catalytic strippers

Evolution of perturbations in 3D air quality models

The deterministic approach of sensitivity analysis is applied on the solution vector of an Air Quality Model. In particular, the photochemical CAMx code is augmented with derivatives utilising the automatic differentiation software ADIFOR. The enhanced with derivatives version of the model is then adopted in a study of the effect of perturbations at the boundary conditions on the predicted ozone concentrations. The calculated derivative matrix provides valuable information e.g., on the ordering of the infl uential factors or the localisation of highly affected regions. Two fundamentally different domains of the Auto-Oil II programme were used as test cases for the simulations, namely Athens and Milan. The results suggest that ozone concentration be highly affected by its own boundary conditions and subsequently, with an order of magnitude less, by the boundary conditions of NOX and VOC

A methodology to calculate the friction coefficient in the transition regime: Application to straight chains

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