arXiv subjects

Marta de León-Contreras

Publications and source records attributed to Marta de León-Contreras.

5 recordsLinked to original sources

arXiv2026

Extension theorems for logarithmic Schrödinger and discrete Laplacian operators

In this paper we consider logarithmic operators in two different contexts: the adapted to (continuous) Schrödinger operators and the classical discrete setting. The Schrödinger operator $\mathcal L_V$ on $\mathbb R^d$ is defined as $\mathcal L_V=-Δ+V$, where the potential $V$ is nonnegative and satisfies a reverse Hölder inequality and, as usual, $Δ$ denotes the Euclidean Laplacian, while the discrete Laplacian $Δ_d$ on $\mathbb Z$ is given by $(Δ_df)(n)=f(n+1)-2f(n)+f(n-1)$, $n\in \mathbb Z$. Both logarithmic operators $\log \mathcal L_V$ and $\log (-Δ_d)$ are nonlocal operators and we will define them through suitable extension problems. The extension problems for logarithmic operators are inspired by the one introduced by Caffarelli and Silvestre for the fractional Laplacian but, in this case, the logarithmic operators are obtained as the boundary values of the extension in a more involved way.

math.CA↗

arXiv2022

The quasi-static plasmonic problem for polyhedra

We characterize the essential spectrum of the plasmonic problem for polyhedra in $\mathbb{R}^3$. The description is particularly simple for convex polyhedra and permittivities $ε< - 1$. The plasmonic problem is interpreted as a spectral problem through a boundary integral operator, the direct value of the double layer potential, also known as the Neumann--Poincaré operator. We therefore study the spectral structure of the the double layer potential for polyhedral cones and polyhedra.

math.FA↗

arXiv2017

Fractional powers of the parabolic Hermite operator. Regularity properties

Let $\mathcal{L}= \partial_t- Δ_x+|x|^2$. Consider its Poisson semigroup $e^{-y\sqrt{\mathcal{L}}}$. For $α>0$ define the Parabolic Hermite-Zygmund spaces $$ Λ^α_{\mathcal{L}}=\left\{f: \:f\in L^\infty(\mathbb{R}^{n+1})\:\; {\rm and} \:\; \left\|\partial_y^k e^{-y\sqrt{\mathcal{L}}} f \right\|_{L^\infty(\mathbb{R}^{n+1})}\leq C_k y^{-k+α},\;\: {\rm with }\, k=[α]+1, y>0. \right\}, $$ with the obvious norm. It is shown that these spaces have a pointwise description of Hölder type. The fractional powers $\mathcal{L}^{\pm β}$ are well defined in these spaces and the following regularity properties are proved: \begin{eqnarray*} α, β>0, \quad \|\mathcal{L}^{-β} f\|_{ Λ^{α+2β}_{\mathcal{L}}}\le C \|f\|_{ Λ^α_{\mathcal{L}}}. \end{eqnarray*} \begin{eqnarray*} 0< 2β< α, \quad \|\mathcal{L}^βf\|_{Λ_{\mathcal{L}}^{α-2β}}\le C \|f\|_{Λ^α_{\mathcal{L}}}. \end{eqnarray*} Parallel results are obtained for the Hermite operator $- Δ+|x|^2.$ The proofs use in a fundamental way the semigroup definition of the operators $\mathcal{L}^{\pm β}$ and $(-Δ+|x|^2)^{\pm β}$. The non-convolution structure of the operators produce an extra difficulty of the arguments.

math.FA↗

arXiv2017

Parabolic equations involving Bessel operators and singular integrals

In this paper we consider the evolution equation $\partial_t u=Δ_μu+f$ and the corresponding Cauchy problem, where $Δ_μ$ represents the Bessel operator $\partial_x^2+(\frac{1}{4}-μ^2)x^{-2}$, for every $μ>-1$. We establish weighted and mixed weighted Sobolev type inequalities for solutions of Bessel parabolic equations. We use singular integrals techniques in a parabolic setting.

math.AP↗

arXiv2016

Non-local fractional derivatives. Discrete and continuous

We prove maximum and comparison principles for fractional discrete derivatives in the integers. Regularity results when the space is a mesh of length $h$, and approximation theorems to the continuous fractional derivatives are shown. When the functions are good enough, these approximation procedures give a measure of the order of approximation. These results also allows us to prove the coincidence, for good enough functions, of the Marchaud and Grünwald-Letnikov derivatives in every point and the speed of convergence to the Grünwald-Letnikov derivative. The fractional discrete derivative will be also described as a Neumann-Dirichlet operator defined by a semi-discrete extension problem. Some operators related to the Harmonic Analysis associated to the discrete derivative will be also considered, in particular their behavior in the Lebesgue spaces $\ell^p(\mathbb{Z}).$

math.AP↗