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

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 sid