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Eduardo Teixeira

Publications and source records attributed to Eduardo Teixeira.

6 recordsLinked to original sources
arXiv2026

Hardy--Littlewood Constants Beyond Polarization

The polynomial Hardy--Littlewood inequalities bound coefficient norms of homogeneous polynomials on $\ell_p$ balls by their supremum norms, with constants independent of the dimension. Their $p=\infty$ endpoint is the Bohnenblust--Hille inequality. A longstanding obstruction to connecting their constants is the polarization loss in the usual multilinear approach. We overcome this obstruction by proving $D_{m,p_m}\le(1+o(1))D_m$ whenever $p_m/m^2\to\infty$, where $D_{m,p}$ and $D_m$ are the optimal complex polynomial Hardy--Littlewood and Bohnenblust--Hille constants. At the opposite boundary, we determine the optimal endpoint constant exactly, $D_{m,m}=m$. We then prove the full first-order asymptotic $D_{m,p_m}\sim m$ whenever $p_m\ge m$ and $p_m-m\to0$, and the exact polynomial growth $D_{m,p_m}=m^{1+o(1)}$ on the wider scale $p_m-m=o((m/\log m)^{1/3})$.

math.FA↗
arXiv2026

Sharp Summability on Supports of Prescribed Combinatorial Dimension

We solve four questions raised by Bayart concerning coefficient summability for multilinear forms with prescribed supports. For every $m\ge 2$ and $d\in[1,m]$, we determine the product-summability exponent and the multilinear summability invariant: \[ \mathrm{prod}(m,d) =\min\left\{\frac{m}{d},\,m-\lceil d\rceil+1\right\}, \qquad γ_{\mathrm{mult}}(m,d) =\min\left\{m-\lceil d\rceil+1,\frac{2m}{d+1}\right\}. \] In particular, $γ_{\mathrm{mult}}(4,2)=8/3$, showing that the multilinear invariant need not be an integer. We also prove that, for every $d\in[1,m]$, there is a single infinite support of exact combinatorial dimension $d$ on which the dimensional Hardy--Littlewood bound is attained over both scalar fields for every anisotropic parameter $\mathbf p=(p_1,\ldots,p_m)$ with $\sum_j 1/p_j<1$, simultaneously across the two regimes separated by $\sum_j 1/p_j=1/2$.

math.FA↗
arXiv2020

On a two-phase free boundary problem ruled by the infinity Laplacian

In this paper we consider a two-phase free boundary problem ruled by the infinity Laplacian. Our main result states that bounded viscosity solutions in $B_1$ are universally Lipschitz continuous in $B_{1/2}$, which is the optimal regularity for the problem. We make a new use of the Ishii-Lions' method, which works as a surrogate for the lack of a monotonicity formula and is bound to be applicable in related problems.

math.AP↗
arXiv2016

Towards sharp Bohnenblust--Hille constants

We investigate the optimality problem associated with the best constants in a class of Bohnenblust--Hille type inequalities for $m$--linear forms. While germinal estimates indicated an exponential growth, in this work we provide strong evidences to the conjecture that the sharp constants in the classical Bohnenblust--Hille inequality are universally bounded, irrespectively of the value of $m$; hereafter referred as the \textit{Universality Conjecture}. In our approach, we introduce the {notions of entropy and complexity}, designed to measure, to some extent, the complexity of such optimization problems. We show that the notion of entropy is critically connected to the Universality Conjecture; for instance, that if the entropy grows at most exponentially with respect to $m$, then the optimal constants of the $m$% --linear Bohnenblust--Hille inequality for real scalars are indeed bounded universally in $m$. It is likely that indeed the entropy grows as $4^{m-1}$, and in this scenario, we show that the optimal constants are precisely $2^{1-\frac{1}{m}} $. In the bilinear case, $m=2$, we show that any extremum of the Littlewood's $4/3$-inequality has entropy $4$ and complexity $2$, and thus we are able to classify all extrema of the problem. We also prove that, for any mixed $\left( \ell _{1},\ell _{2}\right) $% --Littlewood inequality, the entropy do grow exponentially and the sharp constants for such a class of inequalities are precisely $(\sqrt{2})^{m-1}$. In addition to the {notions of entropy and complexity}, the approach we develop in this work makes decisive use of a family of strongly non-symmetric $m$--linear forms, which has further consequences to the theory, as we explain herein.

math.FA↗