arXiv subjects

Francesco D'Emilio

Publications and source records attributed to Francesco D'Emilio.

4 recordsLinked to original sources

arXiv2026

Martingale Transforms and Compensated Bellman Estimates for Dunkl Riesz Transforms

We develop a martingale-transform framework for Dunkl harmonic analysis and apply it to prove $L^p$ estimates for the Dunkl Riesz transforms using the martingale decomposition of the Dunkl process. A fundamental difference from the classical Brownian setting is that the martingale representing the Dunkl Riesz transform of a function is not, in general, differentially subordinated to the Poisson martingale of the function itself. Our main argument bypasses this obstruction by applying Burkholder's Bellman function directly. The possible positive defect of the continuous Itô drift is compensated by the negative contribution produced by the reflection jumps. This yields $L^p$ estimates for single Dunkl Riesz transforms for $1<p<\infty$, and vector-valued estimates for $p\geq2$ with a spectral dependence on the root system. For $G$-invariant functions, differential subordination can be recovered and the resulting estimates are independent of the root system.

math.CA↗

arXiv2026

Predictable subordination, sharp martingale inequalities and applications

We introduce a new method for obtaining sharp $L^p$ estimates for martingales under predictable analogues of differential subordination. By allowing suitable comparisons between continuous and jump variation adapted to the relevant range of $p$, we retain sharp constants in settings where pathwise differential subordination fails. The key idea is to study together the continuous and jump contributions arising from the appropriate Bellman functions. Although these contributions need not be nonpositive separately, we quantify their defects and show that they compensate at the predictable level. This compensation mechanism is inspired by the author's earlier work. As an application, we substantially improve the explicit dimension-free bounds for the Riesz vectors on the Hamming cube and on $\mathbb{Z}^n$.

math.PR↗

arXiv2025

Optimal Sparse Bounds and Commutator Characterizations Without Doubling

We examine dyadic paraproducts and commutators in the non-homogeneous setting, where the underlying Borel measure $μ$ is not assumed to be doubling. We first establish a pointwise sparse domination for dyadic paraproducts and related operators with symbols $b \in \textrm{BMO}(μ)$, improving upon an earlier result of Lacey, where the symbol $b$ was assumed to satisfy a stronger Carleson-type condition, that coincides with $\textrm{BMO}$ only in the doubling setting. As an application of this result, we obtain sharpened weighted inequalities for the commutator of a dyadic Hilbert transform $\mathcal{H}$ previously studied by Borges, Conde Alonso, Pipher, and the third author. We also characterize the symbols for which the commutator $[\mathcal{H},b]$ is bounded on $L^p(μ)$ for $1<p<\infty$ and provide some interesting examples to prove that this class of symbols strictly depends on $p$ and is nested between symbols satisfying the $p$-Carleson packing condition and symbols belonging to martingale BMO (even in the case of absolutely continuous measures).

math.CA↗

arXiv2025

Matrix Weighted $L^p$ Estimates in the Nonhomogeneous Setting

We establish a modified pointwise convex body domination for vector-valued Haar shifts in the nonhomogeneous setting, strengthening and extending the scalar case developed in arXiv:2309.13943. Moreover, we identify a subclass of shifts, called $L^1$-normalized, for which the standard convex body domination holds without requiring any regularity assumption on the measure. Finally, we extend the best-known matrix weighted $L^p$ estimates for sparse forms to the nonhomogeneous setting. The key difficulty here is the lack of a reverse-Hölder inequality for scalar weights, which was used in arXiv:1710.03397 to establish $L^p$ matrix weighted estimates and only works in the doubling setting. Our approach relies instead on a generalization of the weighted Carleson embedding theorem which allows to control not only a fixed weight, but also collections of weights localized on different dyadic cubes that satisfy a certain compatibility condition.

math.CA↗