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

Abstract

Let $d\ge2$ and $t>0$. We study the bilinear triangle averaging operator $$\mathcal T_t(f,g)(x)=\int_{\mathcal I}f(x-ty)g(x-tz) dσ(y,z),$$ where $σ$ is the normalized surface measure on the manifold $\mathcal I=\{(y,z)\in\mathbb R^d\times\mathbb R^d:|y|=|z|=|y-z|=1\}$ of equilateral triangles of side length one with a vertex at the origin. We prove the strong-type asymmetric $L^p$-improving estimate $$\mathcal T_1:L^{\frac{d+1}{d-1}}(\mathbb R^d)\times L^{\frac{d^2-1}{d^2-3d+4}}(\mathbb R^d)\longrightarrow L^{\frac{d+1}{2}}(\mathbb R^d)$$ in every dimension $d\ge4$. This enlarges the range of exponents $(p,q,r)$ for which the triangle averaging operator is known to map $L^p(\mathbb R^d)\times L^q(\mathbb R^d)$ to $L^r(\mathbb R^d)$, and symmetrization produces new symmetric estimates in every such dimension as well. The proof combines Sobolev smoothing estimates, obtained through a microlocal decomposition of the triangle incidence relation, with complex interpolation in an analytic family of averages over triangles of varying shape that contains the triangle averaging operator. We also establish maximal estimates for triangle averages. Let $E\subset[1,2]$ be a nonempty set of upper Minkowski dimension $β$, and let $\mathcal T_E(f,g)=\sup_{t\in E}|\mathcal T_t(f,g)|$. We prove that $$\|\mathcal T_E(f,g)\|_2\lesssim\|f\|_{H^{-s_1}}\|g\|_{H^{-s_2}}$$ whenever $s_1,s_2\ge0$, $s_1+s_2<\frac{d-3-β}{2}$ and $d>3+β$. This improves the previously known Sobolev smoothing estimate by half a derivative and lowers the dimension requirement. We also obtain asymmetric localized maximal estimates of the form $\mathcal T_E:L^p(\mathbb R^d)\times L^q(\mathbb R^d)\to L^r(\mathbb R^d)$ in an enlarged range of exponents.