Bayart, Pellegrino and Rueda proved that every continuous $m$-linear operator from $(\ell_1)^m$ into $\ell_p$ is absolutely $(r_{m,p};1)$-summing, for explicit exponents $r_{m,p}$. The optimality of these exponents was known for $2\le p\le\infty$. We prove optimality in the remaining range $1\le p<2$. Our argument is finite-dimensional and is based on convolution on $\mathbb F_2^d$ and the Walsh character system. The same construction yields a local obstruction theorem for range spaces containing $\ell_p^n$ uniformly. As applications, we recover the sharp exponent $2/m$ for $L_1[0,1]$-valued mappings and show that cotype alone does not determine the optimal absolute $(r;1)$-coincidence exponent.
Harwit and Sloane conjectured that every nonsingular entrywise-nonnegative matrix $A\in\mathbb R^{n\times n}$ satisfies $\|A^{-1}\|_F\ge 2n(n+1)^{-1}\|A\|_{\max}^{-1}$, with equality precisely for positive multiples of $S$-matrices. Cheng proved the conjecture in odd dimensions, while Frankel and Urschel proved the eve...
In 1940, Levin\footnotemark[1] and Milman proved that a closed linear subspace of $C[0,1]$ whose elements all have bounded Jordan variation must be finite-dimensional. We prove its analogue for variation in the sense of Young and, more generally, for every finite-valued nondecreasing gauge $\varphi:[0,\infty)\to[0,\inf...
N. Albuquerque, D. Bugajewska, E. Demétrio et al.· 0 citations
For every $1\leq p<\infty$ and even integer $m\geq4$, we determine the optimal order of the $p$-moment torus inequality for $L_1$: it is $m^p n^{(1-p/2)_+}+n$, with comparison constants independent of $p,m,n$ after taking $p$-th roots. Consequently, for every $2\leq q<\infty$ and $1\leq p\leq q$, the corresponding metr...
We study real multilinear forms with coefficients in $\{-1,1\}$ on finite-dimensional Hilbert spaces. Every trilinear sign form on $\ell_2^r\times\ell_2^n\times\ell_2^n$ has norm at least $\sqrt n$. Writing $K_{r,n}$ for the least norm divided by $\sqrt n$, we prove that $K_{r,n}=1$ exactly when a Hadamard matrix of or...
We investigate the typical dynamical properties of hypercyclic operators in $\mathcal{L}_M(X)$, the set of all bounded linear operators on $X$ whose norms are at most $M$, when $X=\ell_p$, $1<p<\infty$. We show that, with respect to SOT$^*$, a typical operator $T\in \mathcal{L}_M(X)$ is weakly mixing, is weakly disjoin...
Let $S_k(\alpha;K)$ denote the exponential sum over $k$-free integers in the short interval $(N-K,N]$. For $s>0$, we prove essentially tight bounds on the $s$-th moments of $S_k(\alpha;K)$ whenever $K \gg N^{\theta_{k,s}+\epsilon}$ for some $\theta_{k,s}<1/2$. As an immediate consequence, we obtain a lower bound for th...
B. Doyle· 0 citations
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