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Nicola Garofalo

Publications and source records attributed to Nicola Garofalo.

At least 37 records · Page 2Linked to original sources
arXiv2024

Higher order Boundary Schauder Estimates in Carnot Groups

In his seminal 1981 study D. Jerison showed the remarkable negative phenomenon that there exist, in general, no Schauder estimates near the characteristic boundary in the Heisenberg group $\mathbb H^n$. On the positive side, by adapting tools from Fourier and microlocal analysis, he developed a Schauder theory at a non-characteristic portion of the boundary, based on the non-isotropic Folland-Stein Hölder classes. On the other hand, the 1976 celebrated work of Rothschild and Stein on their lifting theorem established the central position of stratified nilpotent Lie groups (nowadays known as Carnot groups) in the analysis of Hörmander operators but, to present date, there exists no known counterpart of Jerison's results in these sub-Riemannian ambients. In this paper we fill this gap. We prove optimal $Γ^{k,α}$ ($k\geq 2$) Schauder estimates near a $C^{k,α}$ non-characteristic portion of the boundary for $Γ^{k-2, α}$ perturbations of horizontal Laplacians in Carnot groups.

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arXiv2024

A fundamental solution for a subelliptic operator in Finsler geometry

We introduce a class of nonlinear partial differential equations in a product space which are at the interface of Finsler and sub-Riemannian geometry. To such equations we associate a non-isotropic Minkowski gauge $Θ$ for which we introduce a suitable notion of Legendre transform $Θ^0$. We compute the action of the relevant nonlinear PDEs on ``radial" functions, i.e., functions of $Θ^0$, and by exploiting it we are able to compute explicit fundamental solutions of such PDEs.

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arXiv2023

An observation on eigenfunctions of the Laplacian

In his seminal 1943 paper F. Rellich proved that, in the complement of a cavity $Ω= \{x\in \mathbb R^n\mid |x|>R_0\}$, there exist no nontrivial solution $f$ of the Helmholtz equation $Δf = - λf$, when $λ>0$, such that $\int_Ω |f|^2 dx < \infty$. In this note we generalise this result by showing that if $\int_Ω |f|^p dx < \infty$ for some $0 \frac{2n}{n-1}$, eigenfunctions do exist in $Ω$.

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arXiv2023

Overdetermined problems in groups of Heisenberg type: conjectures and partial results

In this paper we formulate some conjectures in sub-Riemannian geometry concerning a characterisation of the Koranyi-Kaplan ball in a group of Heisenberg type through the existence of a solution to suitably overdetermined problems. We prove an integral identity that provides a rigidity constraint for one of the two problems. By exploiting some new invariances of these Lie groups, for domains having partial symmetry we solve these problems by converting them to known results for the classical $p$-Laplacian

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arXiv2023

Some inequalities for the Fourier transform and their limiting behaviour

We identify a one-parameter family of inequalities for the Fourier transform whose limiting case is the restriction conjecture for the sphere. Using Stein's method of complex interpolation we prove the conjectured inequalities when the target space is $L^2$, and show that this recovers in the limit the celebrated Tomas-Stein theorem.

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arXiv2023

A sub-Riemannian maximum modulus theorem

In this note we prove a sub-Riemannian maximum modulus theorem in a Carnot group. Using a nontrivial counterexample, we also show that such result is best possible, in the sense that in its statement one cannot replace the right-invariant horizontal gradient with the left-invariant one.

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arXiv2023

On the forward in time propagation of zeros in fractional heat type problems

In this short note we prove that if $u$ solves $(\partial_t - Δ)^s u = Vu$ in $\mathbb R^n_x \times \mathbb R_t$, and vanishes to infinite order at a point $(x_0, t_0)$, then $u \equiv 0$ in $\mathbb R^n_x \times \mathbb R_t$. This sharpens (and completes) our earlier result that proves $u(\cdot, t) \equiv 0$ for $t \leq t_0$ if it vanishes to infinite order at $(x_0, t_0)$.

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arXiv2022

A universal heat semigroup characterisation of Sobolev and BV spaces in Carnot groups

In sub-Riemannian geometry there exist, in general, no known explicit representations of the heat kernels, and these functions fail to have any symmetry whatsoever. In particular, they are not a function of the control distance, nor they are for instance spherically symmetric in any of the layers of the Lie algebra. Despite these unfavourable aspects, in this paper we establish a new heat semigroup characterisation of the Sobolev and $BV$ spaces in a Carnot group by means of an integral decoupling property of the heat kernel.

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arXiv2022

A note on monotonicity and Bochner formulas in Carnot groups

In this note we prove two monotonicity formulas for solutions of $Δ_H f = c$ and $Δ_H f - \p_t f = c$ in Carnot groups. Such formulas involve the right-invariant \emph{carré du champ} of a function and they are false for the left-invariant one. The main results, Theorems 1.1 and 1.2, display a resemblance with two deep monotonicity formulas respectively due to Alt-Caffarelli-Friedman for the standard Laplacian, and to Caffarelli for the heat equation. In connection with this aspect we ask the question whether an "almost monotoniccity" formula be possible. In the last section we discuss the failure of the nondecreasing monotonicity of an Almgren type functional.

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arXiv2022

On the space-like analyticity in the extension problem for nonlocal parabolic equations

In this note we give an elementary proof of the space-like real analyticity of solutions to a degenerate evolution problem that arises in the study of fractional parabolic operators of the type $(\partial_t - div_x(B(x)\nabla_x))^s$, $0<s<1$. Our primary interest is in the so-called \emph{extension variable}. We show that weak solutions that are even in such variable, are in fact real-analytic in the totality of the space variables. As an application of this result we prove the weak unique continuation property for nonlocal parabolic operators of the type above, where $B(x)$ is a uniformly elliptic matrix-valued function with real-analytic entries.

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arXiv2022

Space-like strong unique continuation for some fractional parabolic equations

In this paper we establish the \emph{space-like} strong unique continuation for nonlocal equations of the type $(\partial_t - Δ)^s u= Vu$, for $0<s <1$. The proof of our main result, Theorem 1.1, is achieved via a conditional elliptic type doubling property for solutions to the appropriate extension problem, followed by a blowup analysis.

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arXiv2021

Heat kernels for a class of hybrid evolution equations

The aim of this paper is to construct (explicit) heat kernels for some hybrid evolution equations which arise in physics, conformal geometry and subelliptic PDEs. Hybrid means that the relevant partial differential operator appears in the form $\mathscr L_1 + \mathscr L_2 - \partial_t$, but the variables cannot be decoupled. As a consequence, the relative heat kernel cannot be obtained as the product of the heat kernels of the operators $\mathscr L_1 - \partial_t$ and $\mathscr L_2 - \partial_t$. Our approach is new and ultimately rests on the generalised Ornstein-Uhlenbeck operators in the opening of Hörmander's 1967 groundbreaking paper on hypoellipticity.

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