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Andrea Carbonaro

Publications and source records attributed to Andrea Carbonaro.

12 recordsLinked to original sources
arXiv2023

Boundedness of Riesz transforms on $\RCD(K, \infty)$ spaces

For $1<p<\infty$, we prove the $L^p$-boundedness of the Riesz transform operators on metric measure spaces with Riemannian Ricci curvature bounded from below, without any restriction on their dimension. This large class of spaces include e.g. that of Hilbert spaces endowed with a log-concave probability measure. As a consequence, we extend the range of the validity of the Lusin-type approximation of Sobolev by Lipschitz functions, previously obtained by L. Ambrosio, E. Bruè and the third author in the quadratic case, i.e. $p=2$. The proofs are analytic and rely on computations on an explicit Bellman function.

math.MG↗
arXiv2022

On semigroup maximal operators associated with divergence-form operators with complex coefficients

Let $L_{A}=-{\rm div}(A\nabla)$ be an elliptic divergence form operator with bounded complex coefficients subject to mixed boundary conditions on an arbitrary open set $Ω\subseteq\mathbb{R}^{d}$. We prove that the maximal operator ${\mathscr M}^{A} f=\sup_{t>0}|\exp(-tL_{A})f|$ is bounded in $L^{p}(Ω)$, whenever $A$ is $p$-elliptic in the sense of [10]. The relevance of this result is that, in general, the semigroup generated by $-L_{A}$ is neither contractive in $L^{\infty}$ nor positive, therefore neither the Hopf--Dunford--Schwartz maximal ergodic theorem [15, Chap.~VIII] nor Akcoglu's maximal ergodic theorem [1] can be used. We also show that if $d\geq 3$ and the domain of the sesquilinear form associated with $L_{A}$ embeds into $L^{2^{*}}(Ω)$ with $2^{*}=2d/(d-2)$, then the range of $L^{p}$-boundedness of ${\mathscr M}^{A}$ improves into the interval $(rd/((r-1)d+2),rd/(d-2))$, where $r\geq 2$ is such that $A$ is $r$-elliptic. With our method we are also able to study the boundedness of the two-parameter maximal operator $\sup_{s,t>0}|T^{A_{1}}_{s}T^{A_{2}}_{t}f|$.

math.FA↗
arXiv2022

Sharp $L^p$ estimates of powers of the complex Riesz transform

Let $R_{1,2}$ be scalar Riesz transforms on $\mathbb{R}^2$. We prove that the $L^p$ norms of $k$-th powers of the operator $R_2+iR_1$ behave exactly as $|k|^{1-2/p}p$, uniformly in $k\in\mathbb{Z}\backslash\{0\}$, $p\geq2$. This gives a complete asymptotic answer to a question suggested by Iwaniec and Martin in 1996. The main novelty are the lower estimates, of which we give three different proofs. We also conjecture the exact value of $\|(R_2+iR_1)^k\|_p$. Furthermore, we establish the sharp behaviour of weak $(1,1)$ constants of $(R_2+iR_1)^k$ and an $L^\infty$ to $BMO$ estimate that is sharp up to a logarithmic factor.

math.CA↗
arXiv2021

Trilinear embedding for divergence-form operators with complex coefficients

We prove a dimension-free $L^p(Ω)\times L^q(Ω)\times L^r(Ω)\rightarrow L^1(Ω\times (0,\infty))$ embedding for triples of elliptic operators in divergence form with complex coefficients and subject to mixed boundary conditions on $Ω$, and for triples of exponents $p,q,r\in(1,\infty)$ mutually related by the identity $1/p+1/q+1/r=1$. Here $Ω$ is allowed to be an arbitrary open subset of $\mathbb{R}^d$. Our assumptions involving the exponents and coefficient matrices are expressed in terms of a condition known as $p$-ellipticity. The proof utilizes the method of Bellman functions and heat flows. As a corollary, we give applications to (i) paraproducts and (ii) square functions associated with the corresponding operator semigroups, moreover, we prove (iii) inequalities of Kato--Ponce type for elliptic operators with complex coefficients. All the above results are the first of their kind for elliptic divergence-form operators with complex coefficients on arbitrary open sets. Furthermore, the approach to (ii),(iii) through trilinear embeddings seems to be new.

math.AP↗
arXiv2019

Bilinear embedding for divergence-form operators with complex coefficients on irregular domains

Let $Ω\subseteq \mathbb{R}^{d}$ be open and $A$ a complex uniformly strictly accretive $d\times d$ matrix-valued function on $Ω$ with $L^{\infty}$ coefficients. Consider the divergence-form operator ${\mathscr L}^{A}=-{\rm div}(A\nabla)$ with mixed boundary conditions on $Ω$. We extend the bilinear inequality that we proved in [16] in the special case when $Ω=\mathbb{R}^{d}$. As a consequence, we obtain that the solution to the parabolic problem $u^{\prime}(t)+{\mathscr L}^{A}u(t)=f(t)$, $u(0)=0$, has maximal regularity in $L^{p}(Ω)$, for all $p>1$ such that $A$ satisfies the $p$-ellipticity condition that we introduced in [16]. This range of exponents is optimal for the class of operators we consider. We do not impose any conditions on $Ω$, in particular, we do not assume any regularity of $\partialΩ$, nor the existence of a Sobolev embedding. The methods of [16] do not apply directly to the present case and a new argument is needed.

math.AP↗
arXiv2019

Convexity of power functions and bilinear embedding for divergence-form operators with complex coefficients

We introduce a condition on accretive matrix functions, called $p$-ellipticity, and discuss its applications to the $L^p$ theory of elliptic PDE with complex coefficients. Our examples are: (i) generalized convexity of power functions (Bellman functions), (ii) dimension-free bilinear embeddings, (iii) $L^p$-contractivity of semigroups and (iv) holomorphic functional calculus. Recent work by Dindoš and Pipher (arXiv:1612.01568v3) established close ties between $p$-ellipticity and (v) regularity theory of elliptic PDE with complex coefficients. The $p$-ellipticity condition arises from studying uniform positivity of a quadratic form associated with the matrix in question on one hand, and the Hessian of a power function on the other. Our results regarding contractivity extend earlier theorems by Cialdea and Maz'ya.

math.CA↗
arXiv2016

Bounded holomorphic functional calculus for nonsymmetric Ornstein-Uhlenbeck operators

We study bounded holomorphic functional calculus for nonsymmetric infinite dimensional Ornstein-Uhlenbeck operators ${\mathscr L}$. We prove that if $-{\mathscr L}$ generates an analytic semigroup on $L^{2}(γ_{\infty})$, then ${\mathscr L}$ has bounded holomorphic functional calculus on $L^{r}(γ_{\infty})$, $1 θ^{*}_{r}$, where $γ_{\infty}$ is the associated invariant measure and $θ^{*}_{r}$ the sectoriality angle of ${\mathscr L}$ on $L^{r}(γ_{\infty})$. The angle $θ^{*}_{r}$ is optimal. In particular our result applies to any nondegenerate finite dimensional Ornstein-Uhlenbeck operator, with dimension-free estimates.

math.FA↗
arXiv2013

Bellman function and linear dimension-free estimates in a theorem of Bakry

By using an explicit Bellman function, we prove a bilinear embedding theorem for the Laplacian associated with a weighted Riemannian manifold $(M,μ_ϕ)$ having the Bakry-Emery curvature bounded from below. The embedding, acting on the cartesian product of $L^p(M,μ_ϕ)$ and $L^q(T^*M,μ_ϕ)$, $1/p+1/q=1$, involves estimates which are independent of the dimension of the manifold and linear in $p$. As a consequence we obtain linear dimension-free estimates of the $L^p$ norms of the corresponding shifted Riesz transform. All our proofs are analytic.

math.FA↗
arXiv2011

Local Hardy Spaces of Differential Forms on Riemannian Manifolds

We define local Hardy spaces of differential forms $h^p_{\mathcal D}(\wedge T^*M)$ for all $p\in[1,\infty]$ that are adapted to a class of first order differential operators $\mathcal D$ on a complete Riemannian manifold $M$ with at most exponential volume growth. In particular, if $D$ is the Hodge--Dirac operator on $M$ and $Δ=D^2$ is the Hodge--Laplacian, then the local geometric Riesz transform ${D(Δ+aI)^{-{1}/{2}}}$ has a bounded extension to $h^p_D$ for all $p\in[1,\infty]$, provided that $a>0$ is large enough compared to the exponential growth of $M$. A characterisation of $h^1_{\mathcal D}$ in terms of local molecules is also obtained. These results can be viewed as the localisation of those for the Hardy spaces of differential forms $H^p_D(\wedge T^*M)$ introduced by Auscher, McIntosh and Russ.

math.DG↗
arXiv2008

H^1 and BMO for certain nondoubling metric measure spaces

Suppose that (M,d,m) is an unbounded metric measure space, which possesses two geometric properties, called "isoperimetric property" and "approximate midpoint property", and that the measure m is locally doubling. The isoperimetric property implies that the volume of balls grows at least exponentially with the radius. Hence the measure m is not globally doubling. In this paper we define an atomic Hardy space H1(m), where atoms are supported only on "small balls", and a corresponding space BMO(m) of functions of bounded mean oscillation, where the control is only on the oscillation over small balls. We prove that BMO(m) is the dual of H1(m) and that an inequality of John-Nirenberg type on small balls holds for functions in BMO(m). Furthermore, we show that the Lp(m) spaces are intermediate spaces between H1(m) and BMO(m), and we develop a theory of singular integral operators acting on function spaces on M. Finally, we show that our theory is strong enough to give H1(m)-L1(m) and L1(m)-BMO(m) estimates for various interesting operators on Riemannian manifolds and symmetric spaces which are unbounded on L1(m) and on L\infty(m).

math.FA↗