arXiv subjects

Daniel Nunez-Alarcon

Publications and source records attributed to Daniel Nunez-Alarcon.

6 recordsLinked to original sources

arXiv2026

Bombieri--Weyl Contractivity and Rigidity for Homogeneous Polynomials

We determine the dimension-free threshold for the comparison between the Bombieri--Weyl norm and the supremum norm of complex homogeneous polynomials on $\ell_p^n$. For $m$-homogeneous polynomials, the critical scale is $p=2m$: below this threshold no dimension-free comparison is possible, while at $p=2m$ we obtain $\|P\|_{\mathrm{BW}}\le (1+C/m)\|P\|_{2m}$. Moreover, there exist absolute constants $A>0$ and $m_0\in\mathbb N$ such that, for every $m\ge m_0$ and $p\ge 2m+A$, $\|P\|_{\mathrm{BW}}\le\|P\|_p$, with optimal constant one. Equality holds precisely for coordinate pure powers. We also obtain a quantitative stability statement for near-extremizers. The proof is based on a decomposition by multiplicity patterns, contractive orbit projections, Hardy--Littlewood estimates for reduced multilinear forms, and Wiener-type slice estimates. Finally, we show that the corresponding contractive phenomenon fails over the real scalar field.

math.FA↗

arXiv2026

The phase diagram of injective-to-projective tensor distortion

For finite-dimensional Banach spaces $E$ and $F$, set \[ ρ(E,F):=\sup_{0\neq z\in E\otimes F}\frac{π(z)}{\varepsilon(z)}. \] The square growth of $ρ(\ell_p^d,\ell_q^d)$ was determined by Bonet, Defant, Peris and Ramanujan. We study the rectangular problem with the two dimensions varying independently. If $r=\min\{n,m\}$ and $η(t)=\min\{1/t,1/t'\}$, then, whenever $p$ and $q$ lie on the same side of $2$, \[ ρ(\ell_p^n,\ell_q^m)\asymp_{p,q} r^{1/2}\min\{n^{η(p)},m^{η(q)}\}. \] Thus the classical square exponent splits into two dimensional scales. In the mixed range $1\le p\le2\le q\le\infty$, writing $a=1/p$ and $b=1/q$, we obtain the lower estimate \[ ρ_{p,q}(n,m)\gtrsim_{p,q} \max\!\left\{r^{\min\{a+b,2-a-b\}}, m^{b-1/2}r^{1/2}\min\{n^{1-a},m^{1/2}\}\right\}, \] and the upper estimate \[ ρ_{p,q}(n,m)\lesssim_{p,q} \min\!\left\{n^{1-a}r^{1-b},m^b r^a,r\right\}. \] Moreover, if $(n-m)(a+b-1)\le0$, then \[ ρ_{p,q}(n,m)\asymp_{p,q} r^{\min\{a+b,\,2-a-b\}}. \] In the interior mixed range $1<p<2<q<p'$, if \[ α_{p,q}:= \frac{(a+b-1)(\frac12-b)}{a-\frac12}, \] then \[ ρ_{p,q}(n,m)\lesssim_{p,q}m^{1-α_{p,q}}, \qquad \sup_{n\ge1}ρ_{p,q}(n,m)\asymp_{p,q}m^{1-α_{p,q}}. \] The corresponding statements in the reversed mixed range follow by duality and symmetry.

math.FA↗

arXiv2026

The Geometry of Real Anisotropic Bohnenblust--Hille Constants

We determine the growth scale of the optimal constants in the real anisotropic Bohnenblust--Hille inequality. For an exponent vector $\mathbf q^{(m)}$, write $C_{\mathbf q^{(m)}}^{(m)}$ for its optimal constant and $d_m$ for its diameter. These constants are superpolynomial precisely when $m d_m/\log m\to\infty$; throughout this regime, their logarithm has the sharp scale $m d_m$. If also $d_m\to0$, then $\log C_{\mathbf q^{(m)}}^{(m)}/(m d_m)$ lies asymptotically in the interval $\left[\frac{\log 2}{4},\frac{2-\log 2-γ}{4}\right]$, where $γ$ is the Euler--Mascheroni constant; this interval has width less than $10^{-2}$. We also solve the extremal problems at fixed diameter and fixed total deficit. The normalized reciprocal-deficit profiles order the canonically arranged optimal constants by majorization, yield exact formulas on a full-dimensional region, and recover the lower coefficient $(\log 2)/4$ throughout a broad class of anisotropic regimes.

math.FA↗

arXiv2016

Some applications of the Hölder inequality for mixed sums

We use the Hölder inequality for mixed exponents to prove some optimal variants of the generalized Hardy--Littlewood inequality for $m$-linear forms on $\ell _{p}$ spaces with mixed exponents. Our results extend recent results of Araujo et al.

math.FA↗

arXiv2016

The optimal constants for the real Hardy--Littlewood inequality for bilinear forms on $c_{0}\times\ell_{p}$

For $p,q\geq2$, the Hardy and Littlewood inequalities for real bilinear forms, in its unified formulation, assert that there is a constant $C_{p,q}\geq1$ such that \begin{equation} \left(\sum\limits_{j=1}^{\infty}\left(\sum\limits_{k=1}^{\infty}\left\vert A(e_{j},e_{k})\right\vert ^{2}\right) ^{\fracλ{2}}\right) ^{\frac {1}λ}\leq C_{p,q}\left\Vert A\right\Vert, \end{equation} with sharp exponent $λ=\frac{pq}{pq-p-q},$ for all continuous bilinear forms $A:\ell_{p}\times\ell_{q}\rightarrow\mathbb{R}$ (as usual, $c_{0}$ replaces $\ell_{p}$ or $\ell_{q}$ when $p=\infty$ or $q=\infty$)$.$ In this note, among other results, we show that the sharp constants $C_{p,\infty}$ are precisely \[ C_{p,\infty}=2^{\frac{1}{2}-\frac{1}{p}}% \] whenever $p\geq\frac{p_{0}}{p_{0}-1}\approx2.18.$ The number $p_{0}\in(1,2)$ is the unique real number satisfying \[ Γ\left(\frac{p_{0}+1}{2}\right) =\frac{\sqrtπ}{2}. \] In the remaining case, i.e., for $2<p<\frac{p_{0}}{p_{0}-1}\approx 2.18,$ we obtain almost optimal constants, with better precision than $4\cdot10^{-4}$. This last result extends a result from Diniz et al. giving the sharp constant of the famous Littlewood's $4/3$ theorem for real scalars.

math.NT↗

arXiv2012

There exist multilinear Bohnenblust-Hille constants $(C_{n})_{n=1}^{\infty}$ with $\displaystyle \lim_{n\rightarrow \infty}(C_{n+1}-C_{n}) =0.$

The $n$-linear Bohnenblust-Hille inequality asserts that there is a constant $C_{n}\in\lbrack1,\infty)$ such that the $\ell_{\frac{2n}{n+1}}$-norm of $(U(e_{i_{^{1}}},...,e_{i_{n}}))_{i_{1},...i_{n}=1}^{N}$is bounded above by $C_{n}$ times the supremum norm of $U,$ regardless of the $n$-linear form $U:\mathbb{C}^{N}\times...\times\mathbb{C}^{N}% \rightarrow\mathbb{C}$ and the positive integer $N$ (the same holds for real scalars). The power $2n/(n+1)$ is sharp but the values and asymptotic behavior of the optimal constants remain a mystery. The first estimates for these constants had exponential growth. Very recently, a new panorama emerged and the importance, for many applications, of the knowledge of the optimal constants (denoted by $(K_{n})_{n=1}^{\infty}$) was stressed. The title of this paper is part of our Fundamental Lemma, one of the novelties presented here. It brings surprising new (and precise) information on the optimal constants (for both real and complex scalars). For instance, [K_{n+1}-K_{n}<\frac{0.87}{n^{0.473}}] for infinitely many $n$'s. In the case of complex scalars we present a curious formula, where $π,e$ and the famous Euler--Mascheroni constant $γ$ appear together: [K_{n}<1+(\frac{4}{\sqrtπ}(1-e^{γ/2-1/2}) {\sum\limits_{j=1}^{n-1}}j^{^{\log_{2}(e^{-γ/2+1/2}) -1}%})] for all $n\geq2$. Numerically, the above formula shows a surprising low growth, [K_{n}<1.41(n-1)^{0.305}-0.04] for every integer $n \geq2$. We also provide a brief discussion on the interplay between the Kahane-Salem-Zygmund and the Bohnenblust-Hille (polynomial and multilinear) inequalities.

math.FA↗