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Luigi D'Onofrio

Publications and source records attributed to Luigi D'Onofrio.

9 recordsLinked to original sources
arXiv2026

$W^{2,1}$ approximation of planar Sobolev homeomorphisms by smooth diffeomorphisms

The approximation of Sobolev homeomorphisms by smooth diffeomorphisms is well understood in first-order spaces $W^{1,p}$, but remains largely open in the second-order space $W^{2,1}$ due to a fundamental tension between curvature control and injectivity. In this paper we isolate and resolve the local analytical component of this problem. We construct explicit local regularisations both across flat interfaces and near multi-cell vertices, and prove convergence in $W^{2,1}$ together with quantitative preservation of the Jacobian. The resulting maps are $C^{1}$ on the whole domain and smooth inside each cell of the partition; in particular they are $C^{2}$ away from the interfaces. These local constructions are combined into a global smoothing theorem: any piecewise quadratic $C^{1}$-compatible planar homeomorphism satisfying a quantitative bi-Lipschitz condition can be approximated in $W^{2,1}$ by maps that are $C^{1}$, injective, and have positive Jacobian. As a consequence, we show that the general $W^{2,1}$ approximation problem reduces to a purely geometric question: the construction of piecewise quadratic approximations with quantitative injectivity and nondegeneracy.

math.FA↗
arXiv2026

Regularity for Elliptic Equations with Coefficients of Small Mean Oscillation

We give a detailed proof of interior $W^{2,p}$ regularity for uniformly elliptic equations in nondivergence form \[ a^{ij}(x)D_{ij}u+b^i(x)D_i u=f. \] The leading matrix is assumed to have sufficiently small mean oscillation on the balls under consideration. The proof uses a modern real-variable argument: a constant-coefficient harmonic replacement yields a sharp-function estimate for the Hessian, and the Fefferman--Stein and Hardy--Littlewood theorems permit the coefficient error to be absorbed. A parameter estimate, obtained by Agmon's auxiliary-variable argument, supplies a consistent resolvent on all $L^p$ spaces and makes the subsequent gain of integrability non-circular. The first-order term is retained throughout and is controlled by the scale-invariant quantity $R^{1-n/q}\|b\|_{L^q(B_R)}$, with $q>\max\{n,p\}$. As a consequence, coefficients in $\VMO_{\rm loc}$ give the usual local $W^{2,p}$ regularity for every finite $p$.

math.FA↗
arXiv2026

A scalar $c_0$-approximation criterion and gap-coordinate preduals for Lipschitz-free spaces

We recover a scalar $c_0$-approximation criterion for Banach spaces canonically embedded into $\ell^\infty$ by a countable norming family. It identifies the dual of the associated $c_0$-subspace with the prescribed atomic predual precisely when bounded coordinatewise approximation by $c_0$-elements is available. The formulation is a self-contained scalar version of the atomic predual and two-stars framework of D'Onofrio--Greco--Perfekt--Sbordone--Schiattarella. We then apply the criterion to Lipschitz-free spaces. The weak-star density of Lipschitz functions with bounded support is due to Aliaga--Pernecká--Petitjean--Procházka; in proper spaces this yields a canonical quotient from the bidual of the norm-closed span of compactly supported Lipschitz functions onto $\Lip_0(M)$. The main concrete application concerns countable proper subsets $K\subset [0,\infty)$ with the Euclidean metric. The connected components $(a_j,b_j)$ of the complement of $K$ give gap coordinates \[ Δ_j f=\frac{f(b_j)-f(a_j)}{b_j-a_j}. \] We prove that, in the infinite case, these coordinates identify $\Lip_0(K)$ with $\ell^\infty$, that the corresponding $c_0$-subspace coincides with Dalet's predual $S(K)$, and hence that \[ S(K)^*\cong \F(K),\qquad S(K)^{**}\cong \Lip_0(K) \] canonically and isometrically. This gives a coordinate $c_0$ realization, for a non-discrete class with accumulation points, of the preduality covered abstractly by Dalet's theorem for countable proper metric spaces. No new proof of Dalet's theorem in full generality is claimed.

math.FA↗
arXiv2025

Diffeomorphic approximation of piecewise affine homeomorphisms

Given any $f$ a locally finitely piecewise affine homeomorphism of $Ω\subset \mathbb{R}^d$ onto $Δ\subset \mathbb{R}^d$ (for $d=3, 4$) such that $f\in W^{1,p}(Ω, \mathbb{R}^d)$ and $f^{-1}\in W^{1,q}(Δ, \mathbb{R}^d)$, $1\leq p ,q < \infty$ and any $ε>0$ we construct a diffeomorphism $\tilde{f}$ such that $$\|f-\tilde{f}\|_{W^{1,p}(Ω,\mathbb{R}^d)} + \|f^{-1}-\tilde{f}^{-1}\|_{W^{1,q}(Δ,\mathbb{R}^d)} < ε.$$

math.AP↗
arXiv2020

Preduals and double preduals of some Banach spaces

In this paper we study ways to establish when a Banach space can be identified as the dual or the double dual of another Banach space. To obtain these results, we relate these spaces with other, concrete Banach spaces - tipically $\ell^1$ and $\ell^\infty$ - and show that under suitable assumptions we can transfer properties of these spaces to the space we consider. In particular, we show how these results can be used to obtain in a simple way interesting results about spaces such as $BMO$ and $BV$.

math.FA↗
arXiv2020

On BV homeomorphisms

We obtain the rectifiability of the graph of a bounded variation homeomorphism $f$ in the plane and relations between gradients of $f$ and its inverse. Further, we show an example of a bounded variation homeomorphism $f$ in the plane which satisfies the $(N)$ and $(N^{-1})$ properties and strict positivity of Jacobian of both itself and its inverse, but neither $f$ nor $f^{-1}$ is Sobolev.

math.CA↗
arXiv2019

Duality and distance formulas in Lipschitz-Hölder spaces

For a compact metric space $(K, ρ)$, the predual of $Lip(K, ρ)$ can be identified with the normed space $M(K)$ of finite (signed) Borel measures on $K$ equipped with the Kantorovich-Rubinstein norm, this is due to Kantorovich [20]. Here we deduce atomic decomposition of $M(K)$ by mean of some results from [10]. It is also known, under suitable assumption, that there is a natural isometric isomorphism between $Lip(K, ρ)$ and $(lip(K, ρ))_{**}$ [15]. In this work we also show that the pair $(lip(K, ρ), Lip(K, ρ))$ can be framed in the theory of o-O type structures introduced by K. M. Perfekt.

math.FA↗
arXiv2019

Atomic decompositions, two stars theorems, and distances for the Bourgain-Brezis-Mironescu space and other big spaces

Given a Banach space $E$ with a supremum-type norm induced by a collection of operators, we prove that $E$ is a dual space and provide an atomic decomposition of its predual. We apply this result, and some results obtained previously by one of the authors, to the function space $\mathcal{B}$ introduced recently by Bourgain, Brezis, and Mironescu. This yields an atomic decomposition of the predual $\mathcal{B}_\ast$, the biduality result that $\mathcal{B}_0^\ast = \mathcal{B}_\ast$ and $\mathcal{B}_\ast^\ast = \mathcal{B}$, and a formula for the distance from an element $f \in \mathcal{B}$ to $\mathcal{B}_0$.

math.FA↗