Survey on bilinear spherical averages and associated maximal operators

TL;DR

This survey consolidates progress on $L^{p}$ bounds for bilinear spherical averages and associated maximal operators, including the bilinear spherical maximal function $\mathcal{M}$, its lacunary variant $\mathcal{M}_{lac}$, and the localized version $\tilde{\mathcal{M}}$. It presents sharp (or near-sharp) regions of boundedness, endpoint and restricted weak-type estimates, Sobolev smoothing, and sparsity-driven bounds, with a detailed account of dimension-dependent phenomena and analytic techniques such as slicing, smoothing, and interpolation. The work also covers extensions to general surfaces $S_k$, general dilation sets, and multilinear generalizations, highlighting open questions at boundaries and in low dimensions. Overall, the article maps current knowledge, tools, and open directions for bilinear harmonic analysis on sphere-adjacent averaging, providing a foundation for weighted and multilinear developments at the interface of harmonic analysis and geometric measure theory.

Abstract

In this survey, we collect recent progress in the understanding of $L^{p}$ bounds for bilinear spherical averages and some associated maximal functions like the bilinear spherical maximal function and its lacunary counterpart. We describe necessary conditions satisfied by triples in the $L^{p}$ improving region of a bilinear spherical averaging operator and the localized bilinear spherical maximal function, as well as describe the best-known boundedness regions to date. We state some open questions along the way to motivate future research on this topic, and we exploit some possible generalizations.

Figures (7)

• Figure 1: $L^{p}(\mathbb{R}^d)\times L^{q}(\mathbb{R}^d)\rightarrow L^{\frac{pq}{p+q}}(\mathbb{R}^d)$ bounds for $\mathcal{M}$, $d\geq 2$.
• Figure 2: $L^{p}(\mathbb{R})\times L^{q}(\mathbb{R})\rightarrow L^{\frac{pq}{p+q}}(\mathbb{R})$ bounds for $\mathcal{M}$ in $d=1$.
• Figure 3: Hölder bounds for the lacunary bilinear spherical maximal function in $d\geq 2$ in the left, and for $d=1$ in the right.
• Figure 4: $L^p$ improving boundedness region for $\tilde{\mathcal{S}}$.
• Figure 5: The light blue region illustrates the known sufficient region for $\|\tilde{\mathcal{M}}\|_{L^1\times L^q\rightarrow L^r}<\infty$, while the closure of the polygons corresponds to the known necessary conditions.

Theorems & Definitions (12)

• Theorem 2.2
• Theorem 2.3
• Theorem 3.1: borgeslacunary in $d\geq 2$ and ChristZhou in $d=1$
• Theorem 4.2
• Proposition 4.4
• Proposition 4.5: JL
• Proposition 4.6
• Theorem 5.1
• Proposition 5.3
• Theorem 6.1: BFOPZ