arXiv subjects
Publications and source records attributed to Daniel Pellegrino.
For a fixed $0<\varepsilon<1$, F. Bayart asked in 2024 whether there exists an operator $T$ such that, for every $0<δ<1$, $T$ is $δ$-hypercyclic if and only if $δ\in[\varepsilon,1)$. We answer this question affirmatively by constructing a weighted backward shift on $\ell_2(\mathbb N_0,\ell_2(\mathbb N_0))$ with this property.
We prove polynomial growth bounds for the optimal constants in the complex polynomial Hardy--Littlewood inequality whenever $p\geq c m^2/\log m$, for every fixed $c>0$. This extends the recently established polynomial growth of the complex polynomial Bohnenblust--Hille constants at $p=\infty$ to finite values of $p$. Moreover, when $p/m^2\to\infty$, the Hardy--Littlewood constants are bounded by $(1+o(1))$ times the corresponding Bohnenblust--Hille constants. For real scalars, whenever $p_m/m\to\infty$, the optimal constants satisfy $H^{\rm pol}_{m,p_m}(\mathbb R)=2^{m+o(m)}$.
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.
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 $\sum_j 1/p_j=1/2$.
In $1977$, G. Bennett proved, by means of non-deterministic methods, an inequality which plays a fundamental role in a series of optimization problems. More precisely, Bennett's inequality shows that, for $p_{1},p_{2} \in\lbrack1,\infty]$ and all positive integers $n_{1},n_{2}$, there exists a bilinear form $A_{n_{1},n_{2}}\colon\left( \mathbb{R}^{n_{1}},\left\Vert \cdot\right\Vert _{p_{1}}\right) \times\left( \mathbb{R}^{n_{2}},\left\Vert \cdot\right\Vert _{p_{2}}\right) \longrightarrow\mathbb{R}$ with coefficients $\pm1$ satisfying \[ \left\Vert A_{n_{1},n_{2}}\right\Vert \leq C_{p_{1},p_{2}}\max\left\{ n_{1}^{1-\frac{1}{p_{1}}}n_{2}^{\max\left\{ \frac{1}{2}-\frac{1}{p_{2} },0\right\} },n_{2}^{1-\frac{1}{p_{2}}}n_{1}^{\max\left\{ \frac{1}{2} -\frac{1}{p_{1}},0\right\} }\right\} \] for a certain constant $C_{p_{1},p_{2}}$ depending just on $p_{1},p_{2}$; moreover, the exponents of $n_{1},n_{2}$ cannot be improved. In this paper, using a constructive approach, we prove that $C_{p_{1},p_{2}}\leq\sqrt{8/5}$ whenever $p_{1},p_{2}\in\left[ 2,\infty\right] $ or $p_{1}=p_{2}=p\in\left[ 1,\infty\right] $. Our techniques are applied to provide new upper bounds for the constants of a combinatorial game, known as Gale--Berlekamp switching game or unbalancing lights problem. As a consequence, we improve estimates obtained by Brown and Spencer in $1971$ and by Carlson and Stolarski in $2004$.
We prove that any solution of a degenerate elliptic PDE is of class $C^1$, provided the inverse of the equation's degeneracy law satisfies an integrability criterium, viz. $σ^{-1} \in L^1\left (\frac{1}λ {\bf d}λ\right )$. The proof is based upon the construction of a sequence of converging tangent hyperplanes that approximate $u(x)$, near $x_0$, by an error of order $\text{o}(|x-x_0|)$. Explicit control of such hyperplanes is carried over through the construction, yielding universal estimates upon the ${C}^1$--regularity of solutions. Among the main new ingredients required in the proof, we develop an alternative recursive algorithm for the renormalization of approximating solutions. This new method is based on a technique tailored to prevent the sequence of degeneracy laws constructed through the process from being, itself, degenerate.
The investigation of regularity/summability properties of the coefficients of bilinear forms in sequence spaces was initiated by Littlewood in $1930$. Nowadays, this topic has important connections with other fields of Pure and Applied Mathematics as Complex Analysis, Quantum Information Theory, Theoretical Computer Science and Combinatorial Games. In this paper we explore a regularity technique to obtain optimal parameters for several results in this framework, extending/generalizing theorems of Osikiewicz and Tonge ($2001$), Albuquerque \textit{et al.} ($2016$), Aron \textit{et al.} ($2017$), Albuquerque and Rezende ($2018$), Paulino ($2020$), among others.
In [18] we have shown that, for $p_{1},p_{2}\in(2,\infty]$, the constants of Bennett's inequality on unimodular bilinear forms on $\ell_{p_{1}}^{n_{1} }\times\ell_{p_{2}}^{n_{2}}$ are asymptotically bounded by $1$. In the present paper we use a different approximation technique to investigate the remaining cases $p_{1},p_{2}\in\lbrack1,\infty].$ This new approach also provides a stronger asymptotic control, in the sense that the constants are "uniformly" asymptotically bounded by $1$, with no dependence on $p_{1},p_{2}.$
The Kahane--Salem--Zygmund inequality for multilinear forms in $\ell_{\infty}$ spaces claims that, for all positive integers $m,n_{1},...,n_{m}$, there exists an $m$-linear form $A\colon\ell_{\infty}^{n_{1}}\times\cdots\times \ell_{\infty}^{n_{m}}\longrightarrow\mathbb{K}$ ($\mathbb{K}=\mathbb{R}$ or $\mathbb{C}$) of the type \[ A(z^{(1)},...,z^{(m)})=\sum_{j_{1}=1}^{n_{1}}\cdots\sum_{j_{m}=1}^{n_{m}}\pm z_{j_{1}}^{\left( 1\right) }\cdots z_{j_{m}}^{\left( m\right) }\text{,} \] satisfying \[ \Vert A\Vert\leq C_{m}\max\left\{ n_{1}^{1/2},\ldots,n_{m}^{1/2}\right\} {\textstyle\prod\limits_{j=1}^{m}}n_{j}^{1/2}\text{,} \] for \[ C_{m}\leqκ\sqrt{m\log m}\sqrt{m!} \] and a certain $κ>0.$ Our main result shows that given any $ε>0$ and any positive integer $m,$ there exists a positive integer $N$ such that \[ C_{m}<1+ε\text{,} \] when we consider $n_{1},...,n_{m}>N$. In addition, while the original proof of the Kahane--Salem--Zygmund relies in highly non-deterministic arguments, our approach is constructive. We also provide the same asymptotic bound (which is shown to be optimal in some cases) for the constant of a related non-deterministic inequality proved by G. Bennett in 1977. Applications to Berlekamp's switching game are given.
In 1947, M. S. Macphail constructed a series in $\ell_{1}$ that converges unconditionally but does not converge absolutely. According to the literature, this result helped Dvoretzky and Rogers to finally answer a long standing problem of Banach Space Theory, by showing that in all infinite-dimensional Banach spaces, there exists an unconditionally summable sequence that fails to be absolutely summable. More precisely, the Dvoretzky--Rogers Theorem asserts that in every infinite-dimensional Banach space $E$ there exists an unconditionally convergent series ${\textstyle\sum}x^{(j)}$ such that ${\textstyle\sum}\Vert x^{(j)}\Vert^{^{2-\varepsilon}}=\infty$ for all $\varepsilon>0.$ Their proof is non-constructive and Macphail's result for $E=\ell_{1}$ provides a constructive proof just for $\varepsilon\geq1.$ In this note we revisit Machphail's paper and present two alternative constructions that work for all $\varepsilon>0.$
We prove that several results of lineability/spaceability in the framework of sequence spaces are valid in a stricter sense.
We investigate algebraic structures within sets of surjective and injective linear operators between sequence spaces, completing results of Aron et al.
A famous result of S. Kwapień asserts that a linear operator from a Banach space to a Hilbert space is absolutely $1$-summing whenever its adjoint is absolutely $q$-summing for some $1\leq q<\infty$; this result was recently extended to Lipschitz operators by Chen and Zheng. In the present paper we show that Kwapień's and Chen--Zheng theorems hold in a very relaxed nonlinear environment, under weaker hypotheses. Even when restricted to the original linear case, our result generalizes Kwapień's theorem because it holds when the adjoint is just almost summing. In addition, a variant for $\mathcal{L}_{p}$-spaces, with $p\geq2$, instead of Hilbert spaces is provided.
We determine the number of isomorphism classes of elementary gradings by a finite group on an algebra of upper block-triangular matrices. As a consequence we prove that, for a finite abelian group $G$, the sequence of the numbers $E(G,m)$ of isomorphism classes of elementary $G$-gradings on the algebra $M_{m}(\mathbb{F})$ of $m\times m$ matrices with entries in a field $\mathbb{F}$ characterizes $G$. A formula for the number of isomorphism classes of gradings by a finite abelian group on an algebra of upper block-triangular matrices over an algebraically closed field, with mild restrictions on its characteristic, is also provided. Finally, if $G$ is a finite abelian group, $\mathbb{F}$ is an algebraically closed field and $N(G,m)$ is the number of isomorphism classes of $G$-gradings on $M_{m}% (\mathbb{F})$ we prove that $N(G,m)\sim\frac{1}{\left\vert G\right\vert !}m^{\left\vert G\right\vert -1}\sim E(G,m)$.
We revisit the Bohnenblust--Hille multilinear and polynomial inequalities and prove some new properties. Our main result is a multilinear version of a recent result on polynomials whose monomials have a uniformly bounded number of variables.
We propose a continuous version of the classical Gale--Berlekamp switching game. We also study a weighted version of this new continuous game. The main results of this paper concern growth estimates for the corresponding optimization problems. The methods developed in this article are deterministic in nature and in some special cases the estimates obtained are optimal.
The Kahane--Salem--Zygmund inequality is a probabilistic result that guarantees the existence of special matrices with entries $1$ and $-1$ generating unimodular $m$-linear forms $A_{m,n}:\ell_{p_{1}}^{n}\times \cdots\times\ell_{p_{m}}^{n}\longrightarrow\mathbb{R}$ (or $\mathbb{C}$) with relatively small norms. The optimal asymptotic estimates for the smallest possible norms of $A_{m,n}$ when $\left\{ p_{1},...,p_{m}\right\} \subset\lbrack2,\infty]$ and when $\left\{ p_{1},...,p_{m}\right\} \subset\lbrack1,2)$ are well-known and in this paper we obtain the optimal asymptotic estimates for the remaining case: $\left\{ p_{1},...,p_{m}\right\} $ intercepts both $[2,\infty]$ and $[1,2)$. In particular we prove that a conjecture posed by Albuquerque and Rezende is false and, using a special type of matrices that dates back to the works of Toeplitz, we also answer a problem posed by the same authors.
In this paper we obtain the optimal constants of some classical inequalities, such as the multiple Khinchine inequality for Steinhaus variables and the mixed Littlewood inequality for complex scalars.