optimal constants
Recently Published Documents


TOTAL DOCUMENTS

48
(FIVE YEARS 9)

H-INDEX

7
(FIVE YEARS 1)

Mathematics ◽  
2021 ◽  
Vol 9 (9) ◽  
pp. 946
Author(s):  
Xiao Zhang ◽  
Feng Liu ◽  
Huiyun Zhang

In this paper, we introduce and study the Hardy–Littlewood maximal operator MG→ on a finite directed graph G→. We obtain some optimal constants for the ℓp norm of MG→ by introducing two classes of directed graphs.


2021 ◽  
Vol 61 ◽  
pp. 7-12
Author(s):  
Arvydas Karbonskis ◽  
Eugenijus Manstavičius

The variance of a linear statistics on multisets of necklaces is explored. The upper and lower bounds with optimal constants are obtained


2020 ◽  
Vol 20 (2) ◽  
pp. 277-291
Author(s):  
Jean Dolbeault ◽  
Maria J. Esteban

AbstractFor exponents in the subcritical range, we revisit some optimal interpolation inequalities on the sphere with carré du champ methods and use the remainder terms to produce improved inequalities. The method provides us with lower estimates of the optimal constants in the symmetry breaking range and stability estimates for the optimal functions. Some of these results can be reformulated in the Euclidean space using the stereographic projection.


2020 ◽  
Vol 13 (2) ◽  
pp. 595-625
Author(s):  
Clara Antonucci ◽  
Massimo Gobbino ◽  
Matteo Migliorini ◽  
Nicola Picenni

2020 ◽  
Vol 2 (1) ◽  
pp. 157-202 ◽  
Author(s):  
Jeffrey Galkowski ◽  
Euan A. Spence ◽  
Jared Wunsch

2018 ◽  
Vol 108 (1) ◽  
pp. 98-119
Author(s):  
DEREK W. ROBINSON

We establish existence of weighted Hardy and Rellich inequalities on the spaces $L_{p}(\unicode[STIX]{x1D6FA})$, where $\unicode[STIX]{x1D6FA}=\mathbf{R}^{d}\backslash K$ with $K$ a closed convex subset of $\mathbf{R}^{d}$. Let $\unicode[STIX]{x1D6E4}=\unicode[STIX]{x2202}\unicode[STIX]{x1D6FA}$ denote the boundary of $\unicode[STIX]{x1D6FA}$ and $d_{\unicode[STIX]{x1D6E4}}$ the Euclidean distance to $\unicode[STIX]{x1D6E4}$. We consider weighting functions $c_{\unicode[STIX]{x1D6FA}}=c\circ d_{\unicode[STIX]{x1D6E4}}$ with $c(s)=s^{\unicode[STIX]{x1D6FF}}(1+s)^{\unicode[STIX]{x1D6FF}^{\prime }-\unicode[STIX]{x1D6FF}}$ and $\unicode[STIX]{x1D6FF},\unicode[STIX]{x1D6FF}^{\prime }\geq 0$. Then the Hardy inequalities take the form $$\begin{eqnarray}\int _{\unicode[STIX]{x1D6FA}}c_{\unicode[STIX]{x1D6FA}}|\unicode[STIX]{x1D6FB}\unicode[STIX]{x1D711}|^{p}\geq b_{p}\int _{\unicode[STIX]{x1D6FA}}c_{\unicode[STIX]{x1D6FA}}\,d_{\unicode[STIX]{x1D6E4}}^{-p}|\unicode[STIX]{x1D711}|^{p}\end{eqnarray}$$ and the Rellich inequalities are given by $$\begin{eqnarray}\int _{\unicode[STIX]{x1D6FA}}|H\unicode[STIX]{x1D711}|^{p}\geq d_{p}\int _{\unicode[STIX]{x1D6FA}}|c_{\unicode[STIX]{x1D6FA}}\,d_{\unicode[STIX]{x1D6E4}}^{-2}\unicode[STIX]{x1D711}|^{p}\end{eqnarray}$$ with $H=-\text{div}(c_{\unicode[STIX]{x1D6FA}}\unicode[STIX]{x1D6FB})$. The constants $b_{p},d_{p}$ depend on the weighting parameters $\unicode[STIX]{x1D6FF},\unicode[STIX]{x1D6FF}^{\prime }\geq 0$ and the Hausdorff dimension of the boundary. We compute the optimal constants in a broad range of situations.


Sign in / Sign up

Export Citation Format

Share Document