On the monotonicity of the best constant of Morrey’s inequality in convex domains

2021 ◽  
Author(s):  
Maria Fărcăşeanu ◽  
Mihai Mihăilescu
Keyword(s):  
Convex Domains ◽  
Best Constant

ESTIMATES OF BEST CONSTANTS FOR WEIGHTED POINCARÉ INEQUALITIES ON CONVEX DOMAINS

2006 ◽  
Vol 93 (1) ◽  
pp. 197-226 ◽  
Author(s):  
SENG-KEE CHUA ◽  
RICHARD L. WHEEDEN
Keyword(s):  
Poincaré Inequality ◽  
Concave Function ◽  
Continuous Functions ◽  
Directional Derivatives ◽  
Convex Domains ◽  
Poincare Inequality ◽  
Poincaré Inequalities ◽  
Best Constant ◽  
Lipschitz Continuous ◽  
Poincare Inequalities

Let $1 \le q \le p <\infty$ and let $\mathcal{C}$ be the class of all bounded convex domains $\Omega$ in $\mathbb{R}^n$. Following the approach in `An optimal Poincaré inequality in $L^1$ for convex domains', by G. Acosta and R. G. Durán (Proc. Amer. Math. Soc. 132 (2003) 195–202), we show that the best constant $C$ in the weighted Poincaré inequality$$ \| f - f_{av} \|_{L^q_w (\Omega)} \le C w(\Omega)^{\frac{1}{q} - \frac{1}{p}} \mbox{diam}(\Omega) \| \nabla f \|_{L^p_w(\Omega)} $$for all $\Omega \in \mathcal{C}$, all Lipschitz continuous functions $f$ on $\Omega$, and all weights $w$ which are any positive power of a non-negative concave function on $\Omega$ is the same as the best constant for the corresponding one-dimensional situation, where $\mathcal{C}$ reduces to the class of bounded intervals. Using facts from `Sharp conditions for weighted 1-dimensional Poincaré inequalities', by S.-K. Chua and R. L. Wheeden (Indiana Math. J. 49 (2000) 143–175), we estimate the best constant. In the case $q = 1$ and $1 <\infty$, our estimate is between the best constant and twice the best constant. Furthermore, when $p = q = 1$ or $p = q = 2$, the estimate is sharp. Finally, in the case where the domains in $\mathbb{R}^n$ are further restricted to be parallelepipeds, we obtain a slightly different form of Poincaré's inequality which is better adapted to directional derivatives and the sidelengths of the parallelepipeds. We also show that this estimate is sharp for a fixed rectangle.

scholarly journals Conditions for the validity of a class of optimal Hilbert type multiple integral inequalities with nonhomogeneous kernels

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Bing He ◽  
Yong Hong ◽  
Zhen Li
Keyword(s):  
Integral Inequality ◽  
Function Method ◽  
Sufficient Conditions ◽  
Integral Operators ◽  
Multiple Integral ◽  
Constant Factor ◽  
Best Constant ◽  
Necessary And Sufficient ◽  
Best Constant Factor ◽  
Hilbert Type

AbstractFor the Hilbert type multiple integral inequality $$ \int _{\mathbb{R}_{+}^{n}} \int _{\mathbb{R}_{+}^{m}} K\bigl( \Vert x \Vert _{m,\rho }, \Vert y \Vert _{n, \rho }\bigr) f(x)g(y) \,\mathrm{d} x \,\mathrm{d} y \leq M \Vert f \Vert _{p, \alpha } \Vert g \Vert _{q, \beta } $$ ∫ R + n ∫ R + m K ( ∥ x ∥ m , ρ , ∥ y ∥ n , ρ ) f ( x ) g ( y ) d x d y ≤ M ∥ f ∥ p , α ∥ g ∥ q , β with a nonhomogeneous kernel $K(\|x\|_{m, \rho }, \|y\|_{n, \rho })=G(\|x\|^{\lambda _{1}}_{m, \rho }/ \|y\|^{\lambda _{2}}_{n, \rho })$ K ( ∥ x ∥ m , ρ , ∥ y ∥ n , ρ ) = G ( ∥ x ∥ m , ρ λ 1 / ∥ y ∥ n , ρ λ 2 ) ($\lambda _{1}\lambda _{2}> 0$ λ 1 λ 2 > 0 ), in this paper, by using the weight function method, necessary and sufficient conditions that parameters p, q, $\lambda _{1}$ λ 1 , $\lambda _{2}$ λ 2 , α, β, m, and n should satisfy to make the inequality hold for some constant M are established, and the expression formula of the best constant factor is also obtained. Finally, their applications in operator boundedness and operator norm are also considered, and the norms of several integral operators are discussed.

scholarly journals A Note on Universal Bilinear Portfolios

2021 ◽  
Vol 9 (1) ◽  
pp. 11
Author(s):  
Alex Garivaltis
Keyword(s):  
Growth Factor ◽  
Trading Strategy ◽  
Weighted Averaging ◽  
Asset Market ◽  
Best Constant ◽  
Capital Growth ◽  
Universal Portfolios ◽  
Universal Portfolio ◽  
Technical Sense ◽  
Constant Rebalanced Portfolios

This note provides a neat and enjoyable expansion and application of the magnificent Ordentlich-Cover theory of “universal portfolios”. I generalize Cover’s benchmark of the best constant-rebalanced portfolio (or 1-linear trading strategy) in hindsight by considering the best bilinear trading strategy determined in hindsight for the realized sequence of asset prices. A bilinear trading strategy is a mini two-period active strategy whose final capital growth factor is linear separately in each period’s gross return vector for the asset market. I apply Thomas Cover’s ingenious performance-weighted averaging technique to construct a universal bilinear portfolio that is guaranteed (uniformly for all possible market behavior) to compound its money at the same asymptotic rate as the best bilinear trading strategy in hindsight. Thus, the universal bilinear portfolio asymptotically dominates the original (1-linear) universal portfolio in the same technical sense that Cover’s universal portfolios asymptotically dominate all constant-rebalanced portfolios and all buy-and-hold strategies. In fact, like so many Russian dolls, one can get carried away and use these ideas to construct an endless hierarchy of ever more dominant H-linear universal portfolios.

scholarly journals The Nonlinear Schrödinger Equation for Orthonormal Functions II: Application to Lieb–Thirring Inequalities

2021 ◽  
Author(s):  
Rupert L. Frank ◽  
David Gontier ◽  
Mathieu Lewin
Keyword(s):  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Bound State ◽  
Nonlinear Schrodinger Equation ◽  
Best Constant ◽  
Nonlinear Schrödinger ◽  
The One ◽  
Bound State Case ◽  
Cubic Nonlinear Schrödinger Equation

AbstractIn this paper we disprove part of a conjecture of Lieb and Thirring concerning the best constant in their eponymous inequality. We prove that the best Lieb–Thirring constant when the eigenvalues of a Schrödinger operator $$-\Delta +V(x)$$ - Δ + V ( x ) are raised to the power $$\kappa $$ κ is never given by the one-bound state case when $$\kappa >\max (0,2-d/2)$$ κ > max ( 0 , 2 - d / 2 ) in space dimension $$d\ge 1$$ d ≥ 1 . When in addition $$\kappa \ge 1$$ κ ≥ 1 we prove that this best constant is never attained for a potential having finitely many eigenvalues. The method to obtain the first result is to carefully compute the exponentially small interaction between two Gagliardo–Nirenberg optimisers placed far away. For the second result, we study the dual version of the Lieb–Thirring inequality, in the same spirit as in Part I of this work Gontier et al. (The nonlinear Schrödinger equation for orthonormal functions I. Existence of ground states. Arch. Rat. Mech. Anal, 2021. https://doi.org/10.1007/s00205-021-01634-7). In a different but related direction, we also show that the cubic nonlinear Schrödinger equation admits no orthonormal ground state in 1D, for more than one function.

scholarly journals q ‐pseudoconvex and q ‐holomorphically convex domains

2019 ◽  
Vol 292 (12) ◽  
pp. 2619-2623
Author(s):  
George‐Ionuţ Ioniţă ◽  
Ovidiu Preda
Keyword(s):  
Convex Domains

Ramifications of Landau's inequality

1980 ◽  
Vol 86 (3-4) ◽  
pp. 175-212 ◽  
Author(s):  
Man Kam Kwong ◽  
A. Zettl
Keyword(s):  
Best Constant

SynopsisThe problem of determining the best constant κ in the inequality ‖y′‖≦K ‖y‖ ‖y″‖ is discussed in the context of the classical Lp spaces, 1 ≦ p ≦ ∞.

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