arXiv subjects

Sean R. Sovine

Publications and source records attributed to Sean R. Sovine.

3 recordsLinked to original sources

arXiv2025

Sparse bounds for maximal triangle and bilinear spherical averaging operators

We show that the method in recent work of Roncal, Shrivastava, and Shuin can be adapted to show that certain $L^p$-improving bounds in the interior of the boundedness region for the bilinear spherical or triangle averaging operator imply sparse bounds for the corresponding lacunary maximal operator, and that $L^p$-improving bounds in the interior of the boundedness region for the corresponding single-scale maximal operators imply sparse bounds for the correpsonding full maximal operators. More generally we show that the framework applies for bilinear convolutions with compactly supported finite Borel measures that satisfy appropriate $L^p$-improving and continuity estimates. This shows that the method used by Roncal, Shrivastava, and Shuin can be adapted to obtain sparse bounds for a general class of bilinear operators that are not of product type, for a certain range of $L^p$ exponents.

math.CA↗

arXiv2023

The Triangle Operator

We examine the averaging operator $T$ corresponding to the manifold in $\mathbb{R}^{2d}$ of pairs of points $(u,v)$ satisfying $|u| = |v| = |u - v| = 1$, so that $\{0,u,v\}$ is the set of vertices of an equilateral triangle. We establish $L^p \times L^q \rightarrow L^r$ boundedness for $T$ for $(1/p, 1/q, 1/r)$ in the convex hull of the set of points $\lbrace (0, 0, 0) ,\, (1, 0 , 1) ,\, (0, 1, 1) , \, ({1}/{p_d}, {1}/{p_d}, {2}/{p_d}) \rbrace$, where $p_d = \frac{19d-4}{11d - 12}$ and $d\geq 7$.

math.CA↗

arXiv2021

Simplex Averaging Operators: Quasi-Banach and $L^p$-Improving Bounds in Lower Dimensions

We establish some new $L^p$-improving bounds for the $k$-simplex averaging operators $S^k$ that hold in dimensions $d \geq k$. As a consequence of these $L^p$-improving bounds we obtain nontrivial bounds $S^k\colon L^{p_1}\times\cdots\times L^{p_k}\rightarrow L^r$ with $r < 1$. In particular we show that the triangle averaging operator $S^2$ maps $ L^{\frac{d+1}{d}}\times L^{\frac{d+1}{d}} \rightarrow L^{\frac{d+1}{2d}}$ in dimensions $d\geq 2$. This improves quasi-Banach bounds obtained by Palsson and Sovine and extends bounds obtained by Greenleaf, Iosevich, Krauss, and Liu for the case of $k = d = 2$.

math.CA↗