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Register a new userThe best constants in the Multiple Khintchine Inequality
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Diana Serrano-Rodr\\'iguez M
Publication date
2017
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English
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In this work we provide the best constants of the multiple Khintchine inequality. This allows us, among other results, to obtain the best constants of the mixed $left( ell_{frac{p}{p-1}},ell_{2}right) $-Littlewood inequality, thus ending completely a work started by Pellegrino in cite{pell}.
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In this paper we prove that the complex polynomial Bohnenblust-Hille constant for $2$-homogeneous polynomials in ${mathbb C}^2$ is exactly $sqrt[4]{frac{3}{2}}$. We also give the exact value of the real polynomial Bohnenblust-Hille constant for $2$-homogeneous polynomials in ${mathbb R}^2$. Finally, we provide lower estimates for the real polynomial Bohnenblust-Hille constant for polynomials in ${mathbb R}^2$ of higher degrees.
The best constants of two kinds of discrete Sobolev inequalities on the C60 fullerene buckyball are obtained. All the eigenvalues of discrete Laplacian $A$ corresponding to the buckyball are found. They are roots of algebraic equation at most degree $4$ with integer coefficients. Green matrix $G(a)=(A+a I)^{-1} (0<a<infty)$ and the pseudo Green matrix $G_*=A^{dagger}$ are obtained by using computer software Mathematica. Diagonal values of $G_*$ and $G(a)$ are identical and they are equal to the best constants of discrete Sobolev inequalities.
We show that a recent interpolative new proof of the Bohnenblust--Hille inequality, when suitably handled, recovers its best known constants. This seems to be unexpectedly surprising since the known interpolative approaches only provide constants having exponential growth. This preprint is no longer an independent submission, it is now contained in the preprint arXiv 1310.2834.
For the scalar field $mathbb{K}=mathbb{R}$ or $mathbb{C}$, the multilinear Bohnenblust--Hille inequality asserts that there exists a sequence of positive scalars $(C_{mathbb{K},m})_{m=1}^{infty}$ such that %[(sumlimits_{i_{1},...,i_{m}=1}^{N}|U(e_{i_{^{1}}}%,...,e_{i_{m}})|^{frac{2m}{m+1}})^{frac{m+1}{2m}}leq C_{mathbb{K},m}sup_{z_{1},...,z_{m}inmathbb{D}^{N}}|U(z_{1},...,z_{m})|] for all $m$-linear form $U:mathbb{K}^{N}times...timesmathbb{K}% ^{N}rightarrowmathbb{K}$ and every positive integer $N$, where $(e_{i})_{i=1}^{N}$ denotes the canonical basis of $mathbb{K}^{N}$ and $mathbb{D}^{N}$ represents the open unit polydisk in $mathbb{K}^{N}$. Since its proof in 1931, the estimates for $C_{mathbb{K},m}$ have been improved in various papers. In 2012 it was shown that there exist constants $(C_{mathbb{K},m})_{m=1}^{infty}$ with subexponential growth satisfying the Bohnenblust-Hille inequality. However, these constants were obtained via a complicated recursive formula. In this paper, among other results, we obtain a closed (non-recursive) formula for these constants with subexponential growth.
Yuan and Leng (2007) gave a generalization of Ky Fans determinantal inequality, which is a celebrated refinement of the fundamental Brunn-Minkowski inequality $(det (A+B))^{1/n} ge (det A)^{1/n} +(det B)^{1/n}$, where $A$ and $B$ are positive semidefinite matrices. In this note, we first give an extension of Yuan-Lengs result to multiple positive definite matrices, and then we further extend the result to a larger class of matrices whose numerical ranges are contained in a sector. Our result improves a recent result of Liu [Linear Algebra Appl. 508 (2016) 206--213].
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