arXiv subjects

Gianmarco Brocchi

Publications and source records attributed to Gianmarco Brocchi.

6 recordsLinked to original sources

arXiv2026

Global quadratic estimates for degenerate elliptic operators on cylinders

On $d$-dimensional cylinders $\mathcal{C}= \mathbb{R}^k\times N$, with a closed manifold $N$ as base and large scale dimension $k\in[1,d)$, we prove quadratic estimates in weighted $L^2$ space for Dirac operators perturbed by bounded, measurable and accretive coefficients. This gives in particular homogeneous Kato square root estimates on $\mathcal{C}$ for Riesz transforms associated with second order divergence form elliptic operators, having measurable coefficients with degeneracy governed by a Muckenhoupt $A_2$ weight. By localisation and scaling, it also yields local quadratic estimates for perturbed Dirac operators on general manifolds with locally thin cylindrical geometry, and possibly with zero injectivity radius.

math.AP↗

arXiv2025

The metric for matrix degenerate Kato square root operators

We prove a Kato square root estimate with anisotropically degenerate matrix coefficients. We do so by doing the harmonic analysis using an auxiliary Riemannian metric adapted to the operator. We also derive $L^2$-solvability estimates for boundary value problems for divergence form elliptic equations with matrix degenerate coefficients. Main tools are chain rules and Piola transformations for fields in matrix weighted $L^2$ spaces, under $W^{1,1}$ homeomorphism.

math.AP↗

arXiv2023

Refined Two Weight Estimates for the Bergman Projection

We prove sufficient conditions for the two-weight boundedness of the Bergman projection on the unit ball. The first condition is in terms of Orlicz averages of the weights, while the second condition is in terms of the mixed $B_{\infty}$-$B_2$ characteristics.

math.CV↗

arXiv2020

Quadratic sparse domination and Weighted Estimates for non-integral Square Functions

We prove a quadratic sparse domination result for general non-integral square functions $S$. That is, we prove an estimate of the form \begin{equation*} \int_{M} (S f)^{2} g \, \mathrm{d}μ\le c \sum_{P \in \mathcal{S}} \left(\frac{1}{\lvert 5P \rvert}\int_{5 P} \lvert f\rvert^{p_{0}} \, \mathrm{d}μ\right)^{2/p_{0}} \left(\frac{1}{\lvert 5P \rvert} \int_{5 P} \lvert g\rvert^{q_{0}^*}\,\mathrm{d}μ\right)^{1/q_{0}^*} \lvert P\rvert, \end{equation*} where $q_{0}^{*}$ is the Hölder conjugate of $q_{0}/2$, $M$ is the underlying doubling space and $\mathcal{S}$ is a sparse collection of cubes on $M$. Our result will cover both square functions associated with divergence form elliptic operators and those associated with the Laplace-Beltrami operator. This sparse domination allows us to derive optimal norm estimates in the weighted space $L^{p}(w)$.

math.CA↗

arXiv2020

A sparse quadratic $T1$ theorem

We show that any Littlewood--Paley square function $S$ satisfying a minimal local testing condition is dominated by a sparse form, \begin{equation*} \langle (Sf)^2,g \rangle\le C \sum_{I \in \mathscr{S}} \langle \lvert f\rvert\rangle_I^2 \langle \lvert g\rvert\rangle_I \lvert I\rvert . \end{equation*} This implies strong weighted $L^p$ estimates for all $A_p$ weights with sharp dependence on the $A_p$ characteristic. The proof uses random dyadic grids, decomposition in the Haar basis, and a stopping time argument.

math.CA↗

arXiv2018

Sharp Strichartz inequalities for fractional and higher order Schrödinger equations

We investigate a class of sharp Fourier extension inequalities on the planar curves $s=|y|^p$, $p>1$. We identify the mechanism responsible for the possible loss of compactness of nonnegative extremizing sequences, and prove that extremizers exist if $1 4$. In particular, this resolves the dichotomy of Jiang, Pausader & Shao concerning the existence of extremizers for the Strichartz inequality for the fourth order Schrödinger equation in one spatial dimension. One of our tools is a geometric comparison principle for $n$-fold convolutions of certain singular measures in $\mathbb{R}^d$, developed in a companion paper. We further show that any extremizer exhibits fast $L^2$-decay in physical space, and so its Fourier transform can be extended to an entire function on the whole complex plane. Finally, we investigate the extent to which our methods apply to the case of the planar curves $s=y|y|^{p-1}$, $p>1$.

math.CA↗