Asymmetric $L^p$-improving and maximal estimates for the triangle averaging operator

Introduction

Let $d\ge2$, and let

be the submanifold of $\mathbb{R}^{2d}$ consisting of all pairs $(y,z)$ for which $\{0,y,z\}$ is the set of vertices of an equilateral triangle of side length one. For $t>0$ we define the bilinear triangle averaging

The first author was supported in part by NSF grant DMS-2506858 and a Simons Fellowship in Mathematics in 2026. operator by

where $\sigma$ denotes the normalized surface measure on $\mathcal{I}$. Thus $T_t$ averages the product $f(x-ty)g(x-tz)$ over all equilateral triangles of side length $t$ with one vertex at $x$, as in Figure 1. Any two pairs of unit vectors with the same inner product differ by an orthogonal transformation, so the orthogonal group $O(d)$ acts transitively on $\mathcal{I}$, and $\sigma$ is the unique $O(d)$-invariant probability measure on $\mathcal{I}$. This description of $\sigma$ will be used several times.

Averaging operators over curved manifolds have been studied extensively in harmonic analysis. The classical example is the spherical averaging operator

where $\sigma_{\mathbb{S}^{d-1}}$ is the normalized surface measure on the unit sphere. Its $L^p$-improving properties were established by Strichartz [30] (see also Littman [Lit73]), and the associated maximal operator was studied by Stein [29] for $d\ge3$ and by Bourgain [8] for $d=2$.

Bilinear spherical averages and the related maximal operators have attracted considerable attention in recent years [11 BGHHO18, 22]. Subsequent developments include sparse domination and weighted estimates [27, 7, 26], lacunary and fractal-dilation maximal estimates [5 BFO25, 9], and endpoint estimates [3 Ver26], and the survey [4] offers a broader account of the area.

Triangle averaging operators were studied in [15, 25, 21]. Maximal estimates for simplices were obtained by Cook, Lyall and Magyar [10], with more recent developments by Borges and Foster [5], by Borges, Foster and Ou [BFO25], and by Palsson and Sovine [26]. These operators are closely connected with Falconer-type configuration problems [15, 14].

Notation

We write $\mathbb{N}_0 = \{0,1,2,\ldots\}$, we use $\mathcal{S}(\mathbb{R}^n)$ for the Schwartz space, $\delta_0$ for the Dirac mass at the origin of the ambient Euclidean space and $\Gamma$ for the Gamma function, and we use the convention $\frac{1}{\infty}=0$. The Fourier transform is normalized by

and for $s\in\mathbb{R}$ the inhomogeneous Sobolev space $H^s(\mathbb{R}^n)$ carries the norm

where $\langle\xi\rangle=(1+|\xi|^2)^{\frac{1}{2}}$. For $s<0$ the elements of $H^s(\mathbb{R}^n)$ are understood as tempered distributions.

We write $\|f\|_p=\|f\|_{L^p(\mathbb{R}^d)}$, and for $0<r<1$ the symbol $\|\cdot\|_r$ denotes the usual $L^r$ quasi-norm. The Hölder conjugate of $1<p<\infty$ is $p'=\frac{p}{p-1}$, and $1'=\infty$, $\infty'=1$. The notation $X\lesssim Y$ means that $X\le CY$ for a constant $C$ independent of the functions and of the frequency indices under consideration, and the dependence of $C$ on geometric parameters and covering constants is indicated in the relevant statements. We write $X\approx Y$ when $X\lesssim Y$ and $Y\lesssim X$.

Fixed-radius $L^p$-improving estimates

We first consider $L^p$-improving properties of the fixed-radius operator $T_1$. Palsson and Sovine [25] obtained Hölder type $L^p\times L^q\to L^r$ bounds for triangle averages in dimensions $d\ge5$, including quasi-Banach target spaces. Sharp $L^p$-improving estimates for the triangle averaging operator in the Banach range were obtained in the plane by Greenleaf, Iosevich, Krause and Liu [16], and Iosevich, Palsson and Sovine [21] subsequently extended these results to higher dimensions, establishing $L^p$-improving estimates for all $d\ge2$ that include

for $\frac{d+1}{2d}\le r\le1$. Unlike the planar results, these estimates are not expected to be sharp in general.

It is convenient to record such estimates through reciprocal exponents, and we say that $T_1$ is bounded at the point $\left(\frac{1}{p},\frac{1}{q},\frac{1}{r}\right)$ if $\|T_1(f,g)\|_r\lesssim\|f\|_p\|g\|_q$. To measure what our results add, we collect the estimates available from earlier work into a comparison region. Since $T_1$ is an average with respect to a probability measure, it is bounded at $(0,0,0)$, $(1,0,1)$ and $(0,1,1)$. The pointwise bound $|T_1(f,g)|\le\|g\|_\infty A_1|f|$, together with the spherical estimate $A_1:L^{\frac{d+1}{d}}\to L^{d+1}$ of Strichartz [30] and Littman [Lit73], shows that $T_1$ is bounded at $(\alpha,0,b)$ and, by symmetry, at $(0,\alpha,b)$, where $\alpha=\frac{d+1}{d}$ and $b=\frac{1}{d+1}$, while the estimates of Iosevich, Palsson and Sovine quoted above correspond to the points $(\alpha,\alpha,1)$ and $(\alpha,\alpha,2\alpha)$. This leads to the polytope

A second source of estimates is the product bound of Cook, Lyall and Magyar [10]. Let $m = \left\lfloor\frac{d}{2} \right\rfloor$ and $\vartheta= \frac{m-1}{m}$. In the form recorded in [21 Theorem 2.1], their estimate reads