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Martino Fassina

Publications and source records attributed to Martino Fassina.

14 recordsLinked to original sources
arXiv2021

Two-parameter Identities for Divisor Sums in Algebraic Number Fields

In a one-page fragment published with his lost notebook, Ramanujan stated two double series identities associated, respectively, with the famous Gauss Circle and Dirichlet Divisor problems. The identities contain an "extra" parameter, and it is possible that Ramanujan derived these identities with the intent of attacking these famous problems. Similar famous unsolved problems are connected with $f_K(n)$, the number of integral ideals of norm $n$ in an algebraic number field $K$. In this paper we establish Riesz sum identities containing an "extra" parameter and involving $f_K(n)$, or divisor functions associated with $K$. Upper bounds for the sums as the upper index tends to infinity are also established.

math.NT↗
arXiv2021

Balanced Derivatives, Identities, and Bounds for Trigonometric and Bessel Series

Motivated by two identities published with Ramanujan's lost notebook and connected, respectively, with the Gauss circle problem and the Dirichlet divisor problem, in an earlier paper, three of the present authors derived representations for certain sums of products of trigonometric functions as double series of Bessel functions. These series are generalized in the present paper by introducing the novel notion of balanced derivatives, leading to further theorems. As we will see below, the regions of convergence in the unbalanced case are entirely different than those in the balanced case. From this viewpoint, it is remarkable that Ramanujan had the intuition to formulate entries that are, in our new terminology, "balanced". If $x$ denotes the number of products of the trigonometric functions appearing in our sums, in addition to proving the identities mentioned above, theorems and conjectures for upper and lower bounds for the sums as $x\to\infty$ are established.

math.NT↗
arXiv2020

Algorithms for subelliptic multipliers in $\mathbb{C}^2$

We give examples of pseudoconvex domains of finite type in $\mathbb{C}^2$ where the Kohn algorithm for subelliptic estimates fails to yield an effective lower bound for the order of subellipticity in terms of the type. We show how to modify the algorithm to obtain an effective procedure to prove subellipticity on domains of finite type in $\mathbb{C}^2$ with real analytic boundary satisfying a condition slightly stronger than pseudoconvexity. We close with a generalization to higher dimensions.

math.CV↗
arXiv2019

An extension theorem for regular functions of two quaternionic variables

For functions of two quaternionic variables that are regular in the sense of Fueter, we establish a result similar in spirit to the Hanges and Trèves theorem. Namely, we show that a ball contained in the boundary of a domain is a propagator of regular extendability across the boundary.

math.CV↗
arXiv2019

On the Ekeland-Hofer symplectic capacities of the real bidisc

In $\mathbb{C}^2$ with the standard symplectic structure we consider the bidisc $D^2\times D^2$ constructed as the product of two open real discs of radius $1$. We compute explicit values for the first, second and third Ekeland-Hofer symplectic capacity of $D^2\times D^2$. We discuss some applications to questions of symplectic rigidity.

math.CV↗
arXiv2019

Supnorm estimates for $\bar\partial$ on product domains in $\mathbb{C}^n$

Let $Ω\subset\mathbb{C}^n$ be a product of one-dimensional open bounded domains with $C^{1,α}$ boundary, where $0<α<1$. Using methods from complex analysis in one variable, we construct an integral operator that solves $\bar\partial$ in $Ω$ with supnorm estimates when the datum is in $C^{n-1,α}(Ω)$.

math.CV↗
arXiv2019

Orthogonal testing families and holomorphic extension from the sphere to the ball

Let $\mathbb{B}^2$ denote the open unit ball in $\mathbb{C}^2$, and let $p\in \mathbb{C}^2\setminus\overline{\mathbb{B}^2}$. We prove that if $f$ is an analytic function on the sphere $\partial\mathbb{B}^2$ that extends holomorphically in each variable separately and along each complex line through $p$, then $f$ is the trace of a holomorphic function in the ball.

math.CV↗
arXiv2018

A remark on two notions of order of contact

We recall two measurements of the order of contact of an ideal in the ring of germs of holomorphic functions at a point and we provide a class of examples in which they differ.

math.CV↗
arXiv2014

Regularity of the $\bar\partial$-Neumann problem by means of superlogarithmic multipliers

This thesis starts from a review on current research on the local hypoellipticity of the $\bar\partial$-Neumann problem. It presents the classical method of regularity from estimates of the energy: subelliptic as well as superlogarithmic. More recent material is included in which the regularity of the solution is obtained from the geometry of the singularities of the Levi form. The new contribution to this discussion consists in a general weighted Kohn-Hörmander-Morrey formula twisted by a pseudodifferential operator. As an application, a new class of domains for which the $\bar\partial$-Neumann problem is locally regular is exhibited.

math.CV↗