اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدA closed formula for subexponential constants in the multilinear Bohnenblust--Hille inequality
143
0
0.0
(
0
)
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
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.
قيم البحث
اقرأ أيضاً
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 hav
ing exponential growth. This preprint is no longer an independent submission, it is now contained in the preprint arXiv 1310.2834.
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 $2$-h
omogeneous 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 Bohnenblust-Hille inequality and its variants have found applications in several areas of Mathematics and related fields. The control of the constants for the variant for complex $m$-homogeneous polynomials is of special interest for applications
in Harmonic Analysis and Number Theory. Up to now, the best known estimates for its constants are dominated by $kappaleft(1+varepsilonright) ^{m}$, where $varepsilon>0$ is arbitrary and $kappa>0$ depends on the choice of $varepsilon$. For the special cases in which the number of variables in each monomial is bounded by some fixed number $M$, it has been shown that the optimal constant is dominated by a constant depending solely on $M$. In this note, based on a deep result of Bayart, we prove an inequality for any subset of the indices, showing how summability of arbitrary restrictions on monomials can be related to the combinatorial dimension associated with them.
Using elementary techniques, we prove sharp anisotropic Hardy-Littlewood inequalities for positive multilinear forms. In particular, we recover an inequality proved by F. Bayart in 2018.
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}.
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
