Stefano Meda
Current Organizations / Institutes
The organizations/institutes from the scholar's latest article
Research Subjects
Stefano Meda has a total of 28 co-authors. Here are the top co-authors by number of shared publications:
- Giancarlo Mauceri
Università di Genova
13 shared publications - Maria Vallarino
Politecnico di Torino, Italy
11 shared publications - Maria Vallarino
Dipartimento di Matematica e Applicazioni, Università di Milano–Bicocca, Milan, Italy
7 shared publications - Alessio Martini
Politecnico di Torino, Italy
4 shared publications - Giona Veronelli
Dipartimento di Matematica e Applicazioni, Università di Milano Bicocca, Milan, Italy
4 shared publications - Błażej Wróbel
Instytut Matematyczny, Uniwersytet Wrocławski, Wrocław, Poland
3 shared publications
Latest Publications
- Open AccessSpider's webs and sharp $L^p$ bounds for the Hardy--Littlewood maximal operator on Gromov hyperbolic spacesIn this paper we prove that if $1<a\leq bτ$, and it is of weak type $(τ,τ)$, where $τ:= \log_ab$. A key step in the proof is a new structural theorem for Gromov hyperbolic spaces with $(a,b)$-pinched exponential growth at infinity, consisting in a discretisation of $X$ by means of certain graphs, introduced in this paper and called spider's webs, with ``good connectivity properties". Our result applies to trees with bounded geometry, and Cartan--Hadamard manifolds of pinched negative curvature, providing new boundedness results in these settings. The index $τ$ is optimal in the sense that if $p<τ$, then there exists $X$ satisfying the assumptions above such that $\mathcal M$ is not of weak type $(p,p)$. Furthermore, if $b>a^2$, then there are examples of spaces $X$ satisfying the assumptions above such that $\mathcal M$ bounded on $L^p(X)$ if and only if $p=\infty$.Open AccessSpider's webs and sharp $L^p$ bounds for the Hardy--Littlewood maximal operator on Gromov hyperbolic spacesIn this paper we prove that if $1<a\leq bτ$, and it is of weak type $(τ,τ)$, where $τ:= \log_ab$. A key step in the proof is a new structural theorem for Gromov hyperbolic spaces with $(a,b)$-pinched exponential growth at infinity, consisting in a discretisation of $X$ by means of certain graphs, introduced in this paper and called spider's webs, with ``good connectivity properties". Our result applies to trees with bounded geometry, and Cartan--Hadamard manifolds of pinched negative curvature, providing new boundedness results in these settings. The index $τ$ is optimal in the sense that if $p<τ$, then there exists $X$ satisfying the assumptions above such that $\mathcal M$ is not of weak type $(p,p)$. Furthermore, if $b>a^2$, then there are examples of spaces $X$ satisfying the assumptions above such that $\mathcal M$ bounded on $L^p(X)$ if and only if $p=\infty$.
- Open Access
Hardy--Littlewood maximal operators on certain manifolds with bounded geometry
In this paper we study the $L^p$ boundedness of the centred and the uncentred Hardy--Littlewood maximal operators on certain Riemannian manifolds with bounded geometry. Our results complement those of various authors. We show that, under mild assumptions, $L^p$ estimates for the centred operator are ``stable'' under conformal changes of the metric, and prove sharp~$L^p$ estimates for the centred operator on Riemannian models with pinched negative scalar curvature. Furthermore, we prove that the centred operator is of weak type $(1,1)$ on the connected sum of two space forms with negative curvature, whereas the uncentred operator is, perhaps surprisingly, bounded only on $L^\infty$. We also prove that if two locally doubling geodesic metric measure spaces enjoying the uniform ball size condition are strictly quasi-isometric, then they share the same boundedness properties for both the centred and the uncentred maximal operator. Finally, we discuss some $L^p$ mapping properties for the centred operator on a specific Riemannian surface introduced by Strömberg, providing new interesting results.Open AccessHardy--Littlewood maximal operators on certain manifolds with bounded geometry
In this paper we study the $L^p$ boundedness of the centred and the uncentred Hardy--Littlewood maximal operators on certain Riemannian manifolds with bounded geometry. Our results complement those of various authors. We show that, under mild assumptions, $L^p$ estimates for the centred operator are ``stable'' under conformal changes of the metric, and prove sharp~$L^p$ estimates for the centred operator on Riemannian models with pinched negative scalar curvature. Furthermore, we prove that the centred operator is of weak type $(1,1)$ on the connected sum of two space forms with negative curvature, whereas the uncentred operator is, perhaps surprisingly, bounded only on $L^\infty$. We also prove that if two locally doubling geodesic metric measure spaces enjoying the uniform ball size condition are strictly quasi-isometric, then they share the same boundedness properties for both the centred and the uncentred maximal operator. Finally, we discuss some $L^p$ mapping properties for the centred operator on a specific Riemannian surface introduced by Strömberg, providing new interesting results. - Open Access
- 8 April 2024
- Published by European Mathematical Society - EMS - Publishing House GmbH in Revista Matemática Iberoamericana
- Vol. 40 (3) , 803-826
$L^{p}$ gradient estimates and Calderón–Zygmund inequalities under Ricci lower bounds
In this paper, we investigate the validity of first and second order L^{p} estimates for the solutions of the Poisson equation depending on the geometry of the underlying manifold. We first present L^{p} estimates of the gradient under the assumption that the Ricci tensor is lower bounded in a local integral sense, and construct the first counterexample showing that they are false, in general, without curvature restrictions. Next, we obtain L^{p} estimates for the second order Riesz transform (or, equivalently, the validity of L^{p} Calderón–Zygmund inequalities) on the whole scale 1<p<+\infty by assuming that the injectivity radius is positive and that the Ricci tensor is either pointwise lower bounded, or non-negative in a global integral sense. When 1<p \leq 2 , analogous L^{p} bounds on higher even order Riesz transforms are obtained provided that also the derivatives of Ricci are controlled up to a suitable order.
- 8 April 2024
- Published by European Mathematical Society - EMS - Publishing House GmbH in Revista Matemática Iberoamericana
- Vol. 40 (3) , 803-826
Open Access$L^{p}$ gradient estimates and Calderón–Zygmund inequalities under Ricci lower bounds
In this paper, we investigate the validity of first and second order L^{p} estimates for the solutions of the Poisson equation depending on the geometry of the underlying manifold. We first present L^{p} estimates of the gradient under the assumption that the Ricci tensor is lower bounded in a local integral sense, and construct the first counterexample showing that they are false, in general, without curvature restrictions. Next, we obtain L^{p} estimates for the second order Riesz transform (or, equivalently, the validity of L^{p} Calderón–Zygmund inequalities) on the whole scale 1<p<+\infty by assuming that the injectivity radius is positive and that the Ricci tensor is either pointwise lower bounded, or non-negative in a global integral sense. When 1<p \leq 2 , analogous L^{p} bounds on higher even order Riesz transforms are obtained provided that also the derivatives of Ricci are controlled up to a suitable order.
- Open Access
Triangular maximal operators on locally finite trees
We introduce the centred and the uncentred triangular maximal operators $\mathcal T$ and $\mathcal U$, respectively, on any locally finite tree in which each vertex has at least three neighbours. We prove that both $\mathcal T$ and $\mathcal U$ are bounded on $L^p$ for every $p$ in $(1,\infty]$, that $\mathcal T$ is also bounded on $L^1(\mathfrak T)$, and that $\mathcal U$ is not of weak type $(1,1)$ on homogeneous trees. Our proof of the $L^p$ boundedness of $\mathcal U$ hinges on the geometric approach of A. Córdoba and R. Fefferman. We also establish $L^p$ bounds for some related maximal operators. Our results are in sharp contrast with the fact that the centred and the uncentred Hardy--Littlewood maximal operators (on balls) may be unbounded on $L^p$ for every $p<\infty$ even on some trees where the number of neighbours is uniformly bounded.Open AccessTriangular maximal operators on locally finite trees
We introduce the centred and the uncentred triangular maximal operators $\mathcal T$ and $\mathcal U$, respectively, on any locally finite tree in which each vertex has at least three neighbours. We prove that both $\mathcal T$ and $\mathcal U$ are bounded on $L^p$ for every $p$ in $(1,\infty]$, that $\mathcal T$ is also bounded on $L^1(\mathfrak T)$, and that $\mathcal U$ is not of weak type $(1,1)$ on homogeneous trees. Our proof of the $L^p$ boundedness of $\mathcal U$ hinges on the geometric approach of A. Córdoba and R. Fefferman. We also establish $L^p$ bounds for some related maximal operators. Our results are in sharp contrast with the fact that the centred and the uncentred Hardy--Littlewood maximal operators (on balls) may be unbounded on $L^p$ for every $p<\infty$ even on some trees where the number of neighbours is uniformly bounded. - Open Access
- 9 November 2023
- Published by Elsevier in Journal of Functional Analysis
- Vol. 286 (3)
Inclusions and noninclusions of Hardy type spaces on certain nondoubling manifolds
- 9 November 2023
- Published by Elsevier in Journal of Functional Analysis
- Vol. 286 (3)
Open AccessInclusions and noninclusions of Hardy type spaces on certain nondoubling manifolds
- Open Access
Hardy-Littlewood maximal operators on trees with bounded geometry
In this paper we study the $L^p$ boundedness of the centred and the uncentred Hardy--Littlewood maximal operators on the class $Υ_{a,b}$, $2\leq a\leq b$, of trees with $(a,b)$-bounded geometry. We find the sharp range of $p$, depending on $a$ and $b$, where the centred maximal operator is bounded on $L^p(\mathfrak T)$ for all $\mathfrak T$ in $Υ_{a,b}$. We show that there exists a tree in $Υ_{a,b}$ for which the uncentred maximal function is bounded on $L^p$ if and only if $p=\infty$. We also extend these results to graphs which are strictly roughly isometric, in the sense of Kanai, to trees in the class $Υ_{a,b}$.Open AccessHardy-Littlewood maximal operators on trees with bounded geometry
In this paper we study the $L^p$ boundedness of the centred and the uncentred Hardy--Littlewood maximal operators on the class $Υ_{a,b}$, $2\leq a\leq b$, of trees with $(a,b)$-bounded geometry. We find the sharp range of $p$, depending on $a$ and $b$, where the centred maximal operator is bounded on $L^p(\mathfrak T)$ for all $\mathfrak T$ in $Υ_{a,b}$. We show that there exists a tree in $Υ_{a,b}$ for which the uncentred maximal function is bounded on $L^p$ if and only if $p=\infty$. We also extend these results to graphs which are strictly roughly isometric, in the sense of Kanai, to trees in the class $Υ_{a,b}$. - 8 February 2023
- Published by Elsevier in Journal of Differential Equations
- Vol. 356, 163-187
Schrödinger equation on non-compact symmetric spaces
- 8 February 2023
- Published by Elsevier in Journal of Differential Equations
- Vol. 356, 163-187
Schrödinger equation on non-compact symmetric spaces
- Open Access
Schrödinger equation on noncompact symmetric spaces
We establish sharp-in-time kernel and dispersive estimates for the Schr\"odinger equation on non-compact Riemannian symmetric spaces of any rank. Due to the particular geometry at infinity and the Kunze-Stein phenomenon, these properties are more pronounced in large time and enable us to prove the global-in-time Strichartz inequality for a larger family of admissible couples than in the Euclidean case. Consequently, we obtain the global well-posedness for the corresponding semilinear equation with lower regularity data and some scattering properties for small powers which are known to fail in the Euclidean setting. The crucial kernel estimates are achieved by combining the stationary phase method based on a subtle barycentric decomposition, a subordination formula of the Schr\"odinger group to the wave propagator and an improved Hadamard parametrix.
Open AccessSchrödinger equation on noncompact symmetric spaces
We establish sharp-in-time kernel and dispersive estimates for the Schr\"odinger equation on non-compact Riemannian symmetric spaces of any rank. Due to the particular geometry at infinity and the Kunze-Stein phenomenon, these properties are more pronounced in large time and enable us to prove the global-in-time Strichartz inequality for a larger family of admissible couples than in the Euclidean case. Consequently, we obtain the global well-posedness for the corresponding semilinear equation with lower regularity data and some scattering properties for small powers which are known to fail in the Euclidean setting. The crucial kernel estimates are achieved by combining the stationary phase method based on a subtle barycentric decomposition, a subordination formula of the Schr\"odinger group to the wave propagator and an improved Hadamard parametrix.
- Open Access
- 5 January 2022
- Published by Springer Nature in The Journal of Geometric Analysis
- Vol. 32 (2) , 1-57
Local Riesz Transform and Local Hardy Spaces on Riemannian Manifolds with Bounded Geometry
- 5 January 2022
- Published by Springer Nature in The Journal of Geometric Analysis
- Vol. 32 (2) , 1-57
Open AccessLocal Riesz Transform and Local Hardy Spaces on Riemannian Manifolds with Bounded Geometry
- Open Access
- 30 August 2021
- Published by Springer Nature in Mathematische Zeitschrift
- Vol. 300 (2) , 1705-1739
Maximal characterisation of local Hardy spaces on locally doubling manifolds
We prove a radial maximal function characterisation of the local atomic Hardy space $${{\mathfrak {h}}}^1(M)$$ on a Riemannian manifold M with positive injectivity radius and Ricci curvature bounded from below. As a consequence, we show that an integrable function belongs to $${{\mathfrak {h}}}^1(M)$$ if and only if either its local heat maximal function or its local Poisson maximal function is integrable. A key ingredient is a decomposition of Hölder cut-offs in terms of an appropriate class of approximations of the identity, which we obtain on arbitrary Ahlfors-regular metric measure spaces and generalises a previous result of A. Uchiyama.- 30 August 2021
- Published by Springer Nature in Mathematische Zeitschrift
- Vol. 300 (2) , 1705-1739
Open AccessMaximal characterisation of local Hardy spaces on locally doubling manifolds
We prove a radial maximal function characterisation of the local atomic Hardy space $${{\mathfrak {h}}}^1(M)$$ on a Riemannian manifold M with positive injectivity radius and Ricci curvature bounded from below. As a consequence, we show that an integrable function belongs to $${{\mathfrak {h}}}^1(M)$$ if and only if either its local heat maximal function or its local Poisson maximal function is integrable. A key ingredient is a decomposition of Hölder cut-offs in terms of an appropriate class of approximations of the identity, which we obtain on arbitrary Ahlfors-regular metric measure spaces and generalises a previous result of A. Uchiyama.