Hardy-Sobolev inequalities in the unit ball for double phase functionals

Author(s):  
Yoshihiro Mizuta ◽  
Tetsu Shimomura
2021 ◽  
Vol 47 (1) ◽  
pp. 23-37
Author(s):  
Yoshihiro Mizuta ◽  
Tetsu Shimomura

Our aim in this paper is to deal with boundary growth of spherical means of Sobolev functions of monotone type for the double phase functional \(\Phi_{p,q}(x,t) = t^{p} + (b(x) t)^{q}\) in the unit ball B of \(\mathbb{R}^n\), where \(1 < p < q < \infty\) and \(b(\cdot)\) is a non-negative bounded function on B which is Hölder continuous of order \(\theta \in (0,1]\).


2019 ◽  
pp. 1-34 ◽  
Author(s):  
YOSHIHIRO MIZUTA ◽  
TAKAO OHNO ◽  
TETSU SHIMOMURA

Our aim in this paper is to deal with integrability of maximal functions for Herz–Morrey spaces on the unit ball with variable exponent$p_{1}(\cdot )$approaching$1$and for double phase functionals$\unicode[STIX]{x1D6F7}_{d}(x,t)=t^{p_{1}(x)}+a(x)t^{p_{2}}$, where$a(x)^{1/p_{2}}$is nonnegative, bounded and Hölder continuous of order$\unicode[STIX]{x1D703}\in (0,1]$and$1/p_{2}=1-\unicode[STIX]{x1D703}/N>0$. We also establish Sobolev type inequality for Riesz potentials on the unit ball.


Author(s):  
P.A. Crozier ◽  
M. Pan

Heterogeneous catalysts can be of varying complexity ranging from single or double phase systems to complicated mixtures of metals and oxides with additives to help promote chemical reactions, extend the life of the catalysts, prevent poisoning etc. Although catalysis occurs on the surface of most systems, detailed descriptions of the microstructure and chemistry of catalysts can be helpful for developing an understanding of the mechanism by which a catalyst facilitates a reaction. Recent years have seen continued development and improvement of various TEM, STEM and AEM techniques for yielding information on the structure and chemistry of catalysts on the nanometer scale. Here we review some quantitative approaches to catalyst characterization that have resulted from new developments in instrumentation.HREM has been used to examine structural features of catalysts often by employing profile imaging techniques to study atomic details on the surface. Digital recording techniques employing slow-scan CCD cameras have facilitated the use of low-dose imaging in zeolite structure analysis and electron crystallography. Fig. la shows a low-dose image from SSZ-33 zeolite revealing the presence of a stacking fault.


2019 ◽  
Author(s):  
Julio Ignacio Urzúa ◽  
Sandra Campana ◽  
Massimo Lazzari ◽  
Mercedes Torneiro

Tetraphenylmethane has emerged as a recurrent building block for advanced porous materials such as COFs, PAFs and hypercrosslinked polymers. Guided by a similar design principle, we have previously synthesized shape-persistent dendrimers with tetraphenylmethane nodes and ethynylene linkers. Here we report the generality of our approach by describing new dendritic architectures built from tetraphenylmethane. First, we prepared expanded dendrimers where the tetrahedral units are bonded through larger rigid rod spacers. Among the different synthetic strategies tested, the convergent route, with alternating steps of Pd-catalyzed Sonogashira coupling and alkyne activation by removal of TMS masking groups, efficiently afforded the first- and second-generation dendrimers. A second type of compounds having a linear diyne at the core is also described. The dendrimers of generations 1-2 were also synthesized by a convergent approach, with the diyne being assembled in the last step of the synthesis by a Glaser oxidative homocoupling of the corresponding dendrons bearing a terminal alkyne at the focal point. A third-generation dendrimer was also successfully prepared by a double-phase strategy.<br>


2008 ◽  
Vol 2 (1) ◽  
pp. 1-6 ◽  
Author(s):  
Jung Sub Lee ◽  
Seong-Jang Kim ◽  
Kuen Tak Suh ◽  
In-Ju Kim ◽  
Yong-Ki Kim

Author(s):  
Josip Globevnik
Keyword(s):  

It is shown that if V is a closed submanifold of the open unit ball of ℂ2 biholomorphically equivalent to a disc, then the area of V ∩ r can grow arbitrarily rapidly as r ↗ 1. It is also shown that if V is a closed submanifold of ℂ2 biholomorphically equivalent to a disc, then the area of V ∩ r can grow arbitrarily rapidly as r ↗ ∞.


Author(s):  
Zoltán M. Balogh ◽  
Cristian E. Gutiérrez ◽  
Alexandru Kristály

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