arXiv subjects
Publications and source records attributed to Daniel Nuñez-Alarcón.
Recently, in paper published in the Annals of Mathematics, it was shown that the Bohnenblust-Hille inequality for (complex) homogeneous polynomials is hypercontractive. However, and to the best of our knowledge, there is no result providing (nontrivial) lower bounds for the optimal constants for n-homogeneous polynomials (n > 2). In this short note we provide lower bounds for these famous constants.
The power $\frac{2m}{m+1}$ in the polynomial (and multilinear) Bohnenblust--Hille inequality is optimal. This result is well-known but its proof highly nontrivial. In this note we present a quite simple proof of this fact.
Let $(K_{n})_{n=1}^{\infty}$ be the optimal constants satisfying the multilinear (real or complex) Bohnenblust--Hille inequality. The exact values of the constants $K_{n}$ are still waiting to be discovered since eighty years ago; recently, it was proved that $(K_{n})_{n=1}^{\infty}$ has a subexponential growth. In this note we go a step further and address the following question: Is it true that \[ \lim_{n\rightarrow\infty}(K_{n}-K_{n-1}) =0? \] Our main result is a Dichotomy Theorem for the constants satisfying the Bohnenblust--Hille inequality; in particular we show that the answer to the above problem is essentially positive in a sense that will be clear along the note. Another consequence of the dichotomy proved in this note is that $(K_{n})_{n=1}^{\infty}$ has a kind of subpolynomial growth: if $p(n)$ is any non-constant polynomial, then $K_{n}$ \textit{is} \textit{not} asymptotically equal to $p(n).$ Moreover, if \[ q>\log_{2}(\frac{e^{1-1/2γ}}{\sqrt{2}}) \approx0.526, \] then \[ K_{n}\nsim n^{q}. \]