Related papers: Extensions of divergence-free fields in $\mathrm{L…

I begin from a particular field of generalised Puiseux series and investigate a class of nonlinear differential equations in the field. It is appeared that the main part of differential equation determines solvability and positions of…

We use the well-posedness of transmission problems on classes of two-sided Sobolev extension domains to give variational definitions for (boundary) layer potential operators and Neumann-Poincar{\'e} operators. These classes of domains…

Many elliptic boundary value problems exhibit an interior regularity property, which can be exploited to construct local approximation spaces that converge exponentially within function spaces satisfying this property. These spaces can be…

In this paper we propose a new supersymmetric extension of conformal mechanics. The Grassmannian variables that we introduce are the basis of the forms and of the vector-fields built over the symplectic space of the original system. Our…

We construct, for any given ${\ell}=\frac{1}{2}+{\mathbb{N}}_0$, the second-order, linear PDEs which are invariant under the centrally extended Conformal Galilei Algebra. \par At the given ${\ell}$, two invariant equations in one time and…

We obtain an optimal deviation from the mean upper bound \begin{equation} D(x)\=\sup_{f\in \F}\mu\{f-\E_{\mu} f\geq x\},\qquad\ \text{for}\ x\in\R\label{abstr} \end{equation} where $\F$ is the class of the integrable, Lipschitz functions on…

On bounded domains $\Omega \subset \mathbb{R}^d , d \geq 2$, reaching far beyond the scope of Lipschitz domains, we consider an elliptic system of order $2 m$ in divergence form with complex $\mathrm{L}^{\infty}$-coefficients complemented…

This work offers a new prospective on asymptotic perturbation theory for varying self-adjoint extensions of symmetric operators. Employing symplectic formulation of self-adjointness we obtain a new version of Krein formula for resolvent…

In the context of a finite measure metric space whose measure satisfies a growth condition, we prove "T1" type necessary and sufficient conditions for the boundedness of fractional integrals, singular integrals, and hypersingular integrals…

In this paper, we study the relationship between Sobolev extension domains and homogeneous Sobolev extension domains. Precisely, we obtain the following results. 1- Let $1\leq q\leq p\leq \infty$. Then a bounded $(L^{1, p}, L^{1,…

In this paper, we discuss about monotone vector fields, which is a typical extension to the theory of convex functions, by exploiting the tangent space structure. This new approach to monotonicity in CAT(0) spaces stands in opposed to the…

Smooth vector fields on $\mathbb{R}^n$ can be decomposed into the sum of a gradient vector field and divergence-free (solenoidal) vector field under suitable hypotheses. This is called the Helmholtz-Hodge decomposition (HHD), which has been…

We consider certain non-integer base $\beta$-expansions of Parry's type and we study various properties of the transfer (Perron-Frobenius) operator $\mathcal{P}:L^p([0,1])\mapsto L^p([0,1])$ with $p\geq 1$ and its associated composition…

A one-dimensional discrete Stark Hamiltonian with a continuous electric field is constructed by extension theory methods. In absence of the impurities the model is proved to be exactly solvable, the spectrum is shown to be simple,…

A well-known result is that any Lipschitz domain is an extension domain for $W^{s,p}$. This paper extends this result to Lipschitz subsets of compact Lipschitz submanifolds of $\mathbb{R}^n$. We adapt the construction of an extension…

Within the setting of metric spaces equipped with a doubling measure and supporting a $p$-Poincar\'e inequality, establishing existence of solutions to Dirichlet problem in a bounded domain in such a metric space is accomplished via direct…

We investigate $L^1(\mathbb R^n)\to L^\infty(\mathbb R^n)$ dispersive estimates for the Schr\"odinger operator $H=-\Delta+V$ when there is an eigenvalue at zero energy in even dimensions $n\geq 6$. In particular, we show that if there is an…

We consider approximating analytic functions on the interval $[-1,1]$ from their values at a set of $m+1$ equispaced nodes. A result of Platte, Trefethen \& Kuijlaars states that fast and stable approximation from equispaced samples is…

Let $\Omega\subset\mathbb{R}^n$ be a bounded $(\varepsilon,\infty)$-domain with $\varepsilon\in(0,1]$, $X(\mathbb{R}^n)$ a ball Banach function space satisfying some extra mild assumptions, and $\{\rho_\nu\}_{\nu\in(0,\nu_0)}$ with…

This paper investigates the well-posedness of linear elliptic equations, focusing on the divergence-free transformation introduced in the author's recent work [J. Math. Anal. Appl. 548 (2025), 129425]. By comparing this approach with…