Weighted norm inequalities for the geometric maximal operator

We consider two closely related but distinct operators,This extends the work of X. Shi; H. Wei, S. Xianliang and S. Qiyu; X. Yin and B. Muckenhoupt; and C. Sbordone and I. Wik. F I W e give sufficient conditions for the two operators to be equal and show that these conditions are sharp. We also prove two-weight, weighted norm inequalities for both operators using our earlier results about weighted norm inequalities for the minimal operator: \ text{\mgran{m}} f(x) = \inf_{I \ni x} \frac{1}{ \ align M_0f(x)&= \sup_{I\ni x}\exp\left(\frac{1}{\ ,dy\right) \quad\text{and}\\M_0^*f(x) &= \lim_{r\rightarrow0} \sup_{I\ni x}\left(\frac{1}{ \ ,dy. ^ r\,dy\right)^{1/r}.\endalign } \int_I\log } \int_

Norm inequalities for the minimal and maximal operator, and differentiation of the integral

We study the weighted norm inequalities for the minimal operator, a new operator analogous to the Hardy-Littlewood maximal operator which arose in the study of reverse Hölder inequalities. We characterize the classes of weights which govern the strong and weak-type norm inequalities for the minimal operator in the two weight case, and show that these classes are the same. We also show that a generalization of the minimal operator can be used to obtain information about the differentiability of the integral in cases when the associated maximal operator is large, and we give a new condition for this maximal operator to be weak (1,1)

Norm inequalities for the minimal and maximal operator, and differentiation of the integral

We study the weighted norm inequalities for the minimal operator, a new operator analogous to the Hardy-Littlewood maximal operator which arose in the study of reverse Hölder inequalities. We characterize the classes of weights which govern the strong and weak-type norm inequalities for the minimal operator in the two weight case, and show that these classes are the same. We also show that a generalization of the minimal operator can be used to obtain information about the differentiability of the integral in cases when the associated maximal operator is large, and we give a new condition for this maximal operator to be weak (1,1)

Mercury removal in wastewater by iron oxide nanoparticles

Mercury is one of the persistent pollutants in wastewater; it is becoming a severe environmental and public health problem, this is why nowadays its removal is an obligation. Iron oxide nanoparticles are receiving much attention due to their properties, such as: great biocompatibility, ease of separation, high relation of surface-area to volume, surface modifiability, reusability, excellent magnetic properties and relative low cost. In this experiment, Fe3O4 and γ-Fe2O3 nanoparticles were synthesized using iron salts and NaOH as precipitation agents, and Aloe Vera as stabilizing agent; then these nanoparticles were characterized by three different measurements: first, using a Zetasizer Nano ZS for their size estimation, secondly UV-visible spectroscopy which showed the existence of resonance of plasmon at λmax∼360 nm, and lastly by Scanning Electron Microscopy (SEM) to determine nanoparticles form. The results of this characterization showed that the obtained Iron oxides nanoparticles have a narrow size distribution (∼100nm). Mercury removal of 70% approximately was confirmed by atomic absorption spectroscopy measurements

Boundedness of Pseudodifferential Operators on Banach Function Spaces

We show that if the Hardy-Littlewood maximal operator is bounded on a separable Banach function space $X(\mathbb{R}^n)$ and on its associate space $X'(\mathbb{R}^n)$, then a pseudodifferential operator $\operatorname{Op}(a)$ is bounded on $X(\mathbb{R}^n)$ whenever the symbol $a$ belongs to the H\"ormander class $S_{\rho,\delta}^{n(\rho-1)}$ with $0<\rho\le 1$, $0\le\delta<1$ or to the the Miyachi class $S_{\rho,\delta}^{n(\rho-1)}(\varkappa,n)$ with $0\le\delta\le\rho\le 1$, $0\le\delta0$. This result is applied to the case of variable Lebesgue spaces $L^{p(\cdot)}(\mathbb{R}^n)$.Comment: To appear in a special volume of Operator Theory: Advances and Applications dedicated to Ant\'onio Ferreira dos Santo

Pulsed laser deposition growth of heteroepitaxial YBa2Cu3O7/La0.67Ca0.33MnO3 superlattices on NdGaO3 and Sr0.7La0.3Al0.65Ta0.35O3 substrates

Heteroepitaxial superlattices of [YBa2Cu3O7(n)/ La0.67Ca0.33MnO3(m)]x, where n and m are the number of YBCO and LCMO monolayers and x the number of bilayer repetitions, have been grown with pulsed laser deposition on NdGaO3 (110) and Sr0.7La0.3Al0.65Ta0.35O3 (LSAT) (001). These substrates are well lattice matched with YBCO and LCMO and, unlike the commonly used SrTiO3, they do not give rise to complex and uncontrolled strain effects due to structural transitions at low temperature. The growth dynamics and the structure have been studied in-situ with reflection high energy electron diffraction (RHEED) and ex-situ with scanning transmission electron microscopy (STEM), x-ray diffraction, and neutron reflectometry. The individual layers are found to be flat and continuous over long lateral distances with sharp and coherent interfaces and with a well-defined thickness of the individual layer. The only visible defects are antiphase boundaries in the YBCO layers that originate from perovskite unit cell height steps at the interfaces with the LCMO layers. We also find that the first YBCO monolayer at the interface with LCMO has an unusual growth dynamics and is lacking the CuO chain layer while the subsequent YBCO layers have the regular Y-123 structure. Accordingly, the CuO2 bilayers at both the LCMO/YBCO and the YBCO/LCMO interfaces are lacking one of their neighboring CuO chain layers and thus half of their hole doping reservoir. Nevertheless, from electric transport measurements on asuperlattice with n=2 we obtain evidence that the interfacial CuO2 bilayers remain conducting and even exhibit the onset of a superconducting transition at very low temperature. Finally, we show from dc magnetization and neutron reflectometry measurements that the LCMO layers are strongly ferromagnetic

Electron Beam Irradiation Dose Dependently Damages the Bacillus Spore Coat and Spore Membrane

Effective control of spore-forming bacilli begs suitable physical or chemical methods. While many spore inactivation techniques have been proven effective, electron beam (EB) irradiation has been frequently chosen to eradicate Bacillus spores. Despite its widespread use, there are limited data evaluating the effects of EB irradiation on Bacillus spores. To study this, B. atrophaeus spores were purified, suspended in sterile, distilled water, and irradiated with EB (up to 20 kGy). Irradiated spores were found (1) to contain structural damage as observed by electron microscopy, (2) to have spilled cytoplasmic contents as measured by spectroscopy, (3) to have reduced membrane integrity as determined by fluorescence cytometry, and (4) to have fragmented genomic DNA as measured by gel electrophoresis, all in a dose-dependent manner. Additionally, cytometry data reveal decreased spore size, increased surface alterations, and increased uptake of propidium iodide, with increasing EB dose, suggesting spore coat alterations with membrane damage, prior to loss of spore viability. The present study suggests that EB irradiation of spores in water results in substantial structural damage of the spore coat and inner membrane, and that, along with DNA fragmentation, results in dose-dependent spore inactivation