Related papers: On differentiation of integrals in the infinite-di…

We study the problem of differentiation of integrals for certain bases in the infinite-dimensional torus $\mathbb{T}^\omega$. In particular, for every $p_0 \in [1,\infty)$, we construct a basis $\mathcal{B}$ which differentiates…

In this note we will show a Calder\'on--Zygmund decomposition associated with a function $f\in L^1(\mathbb{T}^{\omega})$. The idea relies on an adaptation of a more general result by J. L. Rubio de Francia in the setting of locally compact…

We present some simple counterexamples, based on quadratic forms in infinitely many variables, showing that the implication $f\in C^{(\infty}(\mathbb{T}^\omega)\Longrightarrow\sum_{\bar{p}\in\mathbb{Z}^\infty}|\widehat{f}(\bar{p})|<\infty$…

We discuss generalizations of Rubio de Francia's inequality for Triebel--Lizorkin and Besov spaces, continuing the research from [5]. Two versions of Rubio de Francia's operator are discussed: it is shown that a rotation factor is needed…

For the importance of differentiation theorems in metric spaces (starting with Pansu Rademacher type theorem in Carnot groups) and relations with rigidity of embeddings see the section 1.2 in Cheeger and Kleiner paper arXiv:math/0611954 and…

We prove that every perfect torsion theory for a ring $R$ is differential (in the sense of [P. E. Bland, Differential torsion theory, Journal of Pure and Applied Algebra 204 (2006) 1 -- 8]). In this case, we construct the extension of a…

This article provides the basic algebraic background on infinitesimal deformations and presents the proof of the well-known fact that the non-trivial infinitesimal deformations of a $K$-algebra $R$ are parameterized by the elements of…

It is well known that in $R^n$ , G{\^a}teaux (hence Fr{\'e}chet) differ-entiability of a convex continuous function at some point is equivalent to the existence of the partial derivatives at this point. We prove that this result extends…

We extend the theorem of Liouville on integration in finite terms to include dilogarithmic integrals. The results provide a necessary and sufficient condition for an element of the base field to have an antiderivative in a field extension…

We study a generalized Abreu Equation in $n$-dimensional polytopes and prove some differential inequalities for homogeneous toric bundles.

We study maximal operators related to bases on the infinite-dimensional torus $\mathbb{T}^\omega$. {For the normalized Haar measure $dx$ on $\mathbb{T}^\omega$ it is known that $M^{\mathcal{R}_0}$, the maximal operator associated with the…

We study the dependence of the embedding calculus Taylor tower on the smooth structures of the source and target. We prove that embedding calculus does not distinguish exotic smooth structures in dimension 4, implying a negative answer to a…

The irrational torus, $\mathrm{T}_\alpha$, originally introduced as a geometric model for quasicrystals, is a foundational object in the theory of diffeology. This paper, after recalling its main algebraic properties, provides a…

We establish new conditions that prevent the existence of (weak) normal integral bases in tame Galois extensions of number fields. This leads to the following result: under appropriate technical hypotheses, the existence of a normal…

Ordinary differential equations of the first order on the torus have been investigated in detail by H. Poincar\'e and A. Denjoy. The long-standing problem of generalising these results for the equations of the order $k>1$ (or for the…

Let $\Omega$ be a collection of disjoint dyadic squares $\omega$, let $\pi_\omega$ denote the non-smooth bilinear projection onto $\omega$ \[ \pi_\omega (f,g)(x):=\int\int \mathbf{1}_{\omega}(\xi,\eta) \widehat{f}(\xi) \widehat{g}(\eta)…

For an arbitrary infinite-dimensional Banach space $\X$, we construct examples of strongly-measurable $\X$-valued Pettis integrable functions whose indefinite Pettis integrals are nowhere weakly differentiable; thus, for these functions the…

We show that the Lagrangian torus in the cotangent bundles of the 2-sphere obtained by applying the geodesic flow to the unit circle in a fibre is not displaceable by computing its Lagrangian Floer homology. The computation is based on a…

We show that Laplace isospectral deformations within a conformal class of generic Liouville metrics on the two-dimensional torus that are linear in the deformation parameter are necessarily trivial. Two of the main ingredients in our proof…

The ring of Fermat reals is an extension of the real field containing nilpotent infinitesimals, and represents an alternative to Synthetic Differential Geometry in classical logic. In the present paper, our first aim is to study this ring…