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Cristian Tacelli

Publications and source records attributed to Cristian Tacelli.

14 recordsLinked to original sources
arXiv2026

Higher order Schrödinger operators

In this paper we consider higher order Schrödinger operators $$\mathcal L u=Lu+Vu,$$ where $L$ denotes a fourth order operator and $V\geq 0$ a suitable potential. We initiate our analysis by considering the constant coefficients differential operator $L=Δ^2$. Subsequently, we extend our results to more general operators $L$ featuring suitable variable coefficients. We are interested in domain characterization and generation properties of these operators in $L^p(\mathbb{R}^N)$ for $p \in (1, \infty)$. To address this problems we employ a noncommutative version of the Dore-Venni theorem due to Monniaux and Prüss and we prove that the $L^p$-realization of $\mathcal L$ is quasi sectorial and, consequently, generates an analytic semigroup. Furthermore, this approach allows for a sharp characterization of the operator's domain as the intersection of the domains of the bilaplacian and the multiplication operator. The required assumptions allow to treat potentials that grow at infinity like $|x|^r$ for some $r<4$.

math.AP↗
arXiv2025

Fourth-order operators with unbounded coefficients in $L^1$ spaces

We prove that operators of the form $A=-a(x)^2Δ^{2}$, with suitable growth conditions on the coefficient $a(x)$, generate analytic semigroups in $L^1(\mathbb{R}^N)$. In particular, we deduce generation results for the operator $A :=- (1+|x|^2)^α Δ^{2}$, $0\leqα\leq2$. Moreover, we characterise the maximal domain of $A$ in $L^1(\mathbb{R}^N)$.

math.FA↗
arXiv2024

Kernel estimates for a class of fractional Kolmogorov operators

Assuming a weighted Nash type inequality for the generator $-A$ of a Markov semigroup, we prove a weighted Nash type inequality for its fractional power and deduce non-uniform bounds on the transition kernel corresponding to the Markov semigroup generated by $-A^α$.

math.DS↗
arXiv2024

Fourth-order operators with unbounded coefficients

We prove that operators of the form $A=-a(x)^2Δ^{2}$, with $|D a(x)|\leq c a(x)^\frac{1}{2}$, generate analytic semigroups in $L^p(\mathbb{R}^N)$ for $1<p\leq\infty$ and in $C_b(\mathbb{R}^N)$. In particular, we deduce generation results for the operator $A :=- (1+|x|^2)^α Δ^{2}$, $0\leqα\leq2$. Moreover, we characterize the maximal domain of such operators in $L^p(\mathbb{R}^N)$ for $1<p<\infty$.

math.AP↗
arXiv2022

Fourth-order Schrödinger type operator with unbounded coefficients in $L^2(\mathbb{R}^N)$

In this paper we study generation results in $L^2(\mathbb{R}^N)$ for the fourth order Schrödinger type operator with unbounded coefficients of the form $$A=a^{2} Δ^2+V^{2}$$ where $a(x)=1+|x|^α$ and $V=|x|^β$ with $α>0$ and $β>(α-2)^+$. We obtain that $(-A,D(A))$ generates an analytic strongly continuous semigroup in $L^2(\mathbb{R}^N)$ for $N\geq5$. Moreover, the maximal domain $D(A)$ can be characterized for $N>8$ by the weighted Sobolev space \[ D_2(A)=\{u\in H^{4}(\mathbb{R}^N)\,:\,V^{2}u\in L^{2}(\mathbb{R}^N), |x|^{2α-h}D^{4-h}u\in L^{2}(\mathbb{R}^N) \text{ for } h=0,1,2,3,4\}. \]

math.AP↗
arXiv2021

Bi-Kolmogorov type operators and weighted Rellich's inequalities

In this paper we consider the symmetric Kolmogorov operator $L=Δ+\frac{\nabla μ}μ\cdot \nabla$ on $L^2(\mathbb R^N,dμ)$, where $μ$ is the density of a probability measure on $\mathbb R^N$. Under general conditions on $μ$ we prove first weighted Rellich's inequalities with optimal constants and deduce that the operators $L$ and $-L^2$ with domain $H^2(\mathbb R^N,dμ)$ and $H^4(\mathbb R^N,dμ)$ respectively, generate analytic semigroups of contractions on $L^2(\mathbb R^N,dμ)$. We observe that $dμ$ is the unique invariant measure for the semigroup generated by $-L^2$ and as a consequence we describe the asymptotic behaviour of such semigroup and obtain some local positivity properties. As an application we study the bi-Ornstein-Uhlenbeck operator and its semigroup on $L^2(\mathbb R^N,dμ)$.

math.AP↗
arXiv2021

Some results on second-order elliptic operators with polynomially growing coefficients in $L^p$-spaces

In this paper we study minimal realizations in $L^p(\mathbb{R}^N)$ of the second order elliptic operator \begin{equation*} { A_{b,c}} := (1+|x|^α)Δ+ b|x|^{α-2}x\cdot\nabla - c |x|^{α-2} - |x|^β , \quad x \in \mathbb{R}^N, \end{equation*} where $N\geq3$, $α\in[0,2)$, $β>0$, and $b, c$ are real numbers. We use quadratic form methods to prove that $\left(A_{b,c},C_c^\infty\left(\mathbb{R}^N\setminus \{0\}\right)\right)$ admits an extension that generates an analytic $C_0-$semigroup for all $p\in(1,\infty)$. Moreover, we give conditions on the coefficients under which this extension is precisely the closure of $\left(A_{b,c},C_c^\infty\left(\mathbb{R}^N\setminus \{0\}\right)\right)$.

math.AP↗
arXiv2017

Weighted Hardy's inequalities and Kolmogorov-type operators

We give general conditions to state the weighted Hardy inequality \[ c\int_{\mathbb{R}^N}\frac{φ^2} {|x|^2}dμ\leq\int_{\mathbb{R}^N}|\nabla φ|^2 dμ+C\int_{\mathbb{R}^N} φ^2dμ,\quad φ\in C_c^{\infty}(\mathbb{R}^N),\,c\leq c_{0,μ}, \] with respect to a probability measure $dμ$. Moreover, the optimality of the constant $c_{0,μ}$ is given. The inequality is related to the following Kolmogorov equation perturbed by a singular potential \[ Lu+Vu=\left(Δu+\frac{\nabla μ}μ\cdot \nabla u\right)+\frac{c}{|x|^2}u \] for which the existence of positive solutions to the corresponding parabolic problem can be investigated. The hypotheses on $dμ$ allow the drift term to be of type $\frac{\nabla μ}μ= -|x|^{m-2}x$ with $m> 0$.

math.AP↗
arXiv2016

Kernel estimates for Schrödinger type operators with unbounded diffusion and potential terms

We prove that the heat kernel associated to the Schrödinger type operator $A:=(1+|x|^α)Δ-|x|^β$ satisfies the estimate $$k(t,x,y)\leq c_1e^{λ_0t}e^{c_2t^{-b}}\frac{(|x||y|)^{-\frac{N-1}{2}-\frac{β-α}{4}}}{1+|y|^α} e^{-\frac{2}{β-α+2}|x|^{\frac{β-α+2}{2}}} e^{-\frac{2}{β-α+2}|y|^{\frac{β-α+2}{2}}} $$ for $t>0,|x|,|y|\ge 1$, where $c_1,c_2$ are positive constants and $b=\frac{β-α+2}{β+α-2}$ provided that $N>2,\,α\geq 2$ and $β>α-2$. We also obtain an estimate of the eigenfunctions of $A$.

math.AP↗
arXiv2016

Optimal kernel estimates for a Schrödinger type operator

In the paper the principal result obtained is the estimate for the heat kernel associated to the Schrödinger type operator $(1+|x|^α)Δ-|x|^β$ \[ k(t,x,y)\leq Ct^{-\fracθ{2}}\frac {φ(x)φ(y)}{1+|x|^α}, \] where $φ=(1+|x|^α)^{\frac{2-θ}{4}+\frac{1}α\frac{θ-N}{2}}$, $θ\geq N$ and $0 2$, $α> 2$ and $β>α-2$. This estimate improves a similar estimate in \cite {can-rhan-tac2} with respect to the dependence on spatial component.

math.AP↗
arXiv2010

On the interpolation of discontinuous functions

Given a sequence of real numbers, we consider its subsequences converging to possibly different limits and associate to each of them an index of convergence which depends on the density of the associated subsequences. This index turns out to be useful for a complete description of some phenomena in interpolation theory at points of discontinuity of the first kind. In particular we give some applications to Lagrange and Shepard operators.

math.FA↗