arXiv subjects

Marcus Pasquariello

Publications and source records attributed to Marcus Pasquariello.

3 recordsLinked to original sources

arXiv2026

From weighted paraboloid restriction to $k$-stars and distance graphs

In this paper, we study pinned $k$-star distance sets associated to compact subsets of $\mathbb{R}^n$, $n\geq 2$. For pins $x_1,\dots,x_k\in E$, the pinned $k$-star distance set is \[ Δ_{x_1,\dots,x_k}^{k\text{-star}}(E) = \{(|x_1-x|,\dots,|x_k-x|):x\in E\}\subset\mathbb{R}^k. \] We obtain improved Hausdorff-dimension thresholds on $E$ guaranteeing that pinned $k$-star distance sets have positive $k$-dimensional Lebesgue measure. The main analytic input is a reformulation of the connection, first observed in \cite{IPPS22}, between $k$-stars in $\mathbb{R}^n$ and pinned dot products on the paraboloid in $\mathbb{R}^{n+1}$. In our framework, $L^2(\mathbb{R}^k)$ estimates for the densities of pinned $k$-star distance measures are reduced to a weighted Fourier extension estimate for the paraboloid whose weight is defined explicitly in terms of Frostman measures on $E$. For $1\leq k α_{+}(n,k):=\frac{n^2+nk+k}{2n+1}=\frac{n+k-1}{2}+\frac14 +\frac{2k+1}{4(2n+1)}.\] Using the graph-building machinery of \cite{BFOPR2026}, our positive-measure results for $k$-stars can be used as building blocks for finite distance graph configurations with prescribed pins. As a consequence, we improve the best-known positive-measure thresholds for pinned $k$-simplices in every dimension $n\geq 3$ and for necklace graphs (cycles) in every dimension $n\geq 3$. We further prove nonempty interior results for $k$-stars. In the special case $k=1$, corresponding to the pinned nonempty interior of the distance set $Δ_{x}(E)=\{|x-y|\colon y\in E\}$, we use a sharper argument to improve the pinned nonempty-interior thresholds of \cite{BFOP2026} in all dimensions $n\geq 4$.

math.CA↗

arXiv2025

Medians, Oscillations, and Distance Functions

Vasin (for $n=1$) and Anderson, Lehrbäck, Mudarra, and Vähäkangas (arXiv:2209.06284) (for $n>1$) provided a geometric characterization of the sets $E \subset \mathbb{R}^n$ so that $w = \text{dist}(\cdot, E)^{-α}$ is a Muckenhoupt $A_1$ weight for some $α> 0$. In this paper, we provide a geometric characterization of the sets $E \subset \mathbb{R}^n$ (which we call median porous sets) so that $w = \text{dist}(\cdot, E)^{-α}$ is a Muckenhoupt $A_p$ weight for some $α> 0$ (given any $1 < p \leq \infty$). Given $1 < p \leq \infty$, we also find the precise range of exponents $α$ so that $w = \text{dist}(\cdot, E)^{-α} \in A_p$, in analogy to the $p=1$ case done in arXiv:2209.06284. With our characterization we prove that $\mathbb{R}^n \setminus E$ supports a Hardy-Sobolev inequality if $E$ is an appropriate median porous set. All previous such results that we are aware of make the strictly stronger assumption that the set $E$ is porous, e.g. arXiv:1705.01360, arXiv:1502.01190. As far as we know, this is the first instance in the literature that the ``porosity barrier" is broken in this context. Examples of such appropriate median porous (but not porous) sets were known. We provide further such examples, additional applications to weighted Poincaré inequalities, and a geometric characterization of the nonnegative Hölder continuous functions $w$ such that $\log (w) \in BMO$. We prove that two of the methods we use ($A_p$ and Riesz potential methods) are sharp, i.e. they cannot be improved beyond the results we obtain. The proofs rely on a new median-value characterization of $BMO$: For a real-valued measurable function on $\mathbb{R}^n$ and constants $0 < s < t < 1$, \[\|f\|_{BMO} \approx_{s, t, n} \sup_{Q}[M_t(f, Q) - M_s(f, Q)]\] where $M_s(f, Q)$ denotes the $s$-median value of $f$ on $Q$.

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↗