Smoothing inequalities for corner-type bilinear averages: geometric characterization and applications

We study Sobolev smoothing inequalities for bilinear averages associated with corner-type configurations \[ A_{\gamma,\rho}(f_1,f_2)(x_1,x_2) =\int f_1(x_1+\gamma_1(t),x_2)f_2(x_1,x_2+\gamma_2(t))\rho(t)\,dt. \] In the real-analytic setting, we obtain a complete geometric characterization of the curves $\gamma=(\gamma_{1},\gamma_{2})$ for which $A_{\gamma,\rho}(f_1,f_2)$ satisfies the smoothing inequality \[ \|A_{\gamma,\rho}(f_1,f_2)\|_{L^1} \lesssim \left\|f_1 \right\|_{H^{(-\varepsilon,0)}} \cdot \left\|f_2 \right\|_{H^{(0,-\varepsilon)}}\, \] for some $\varepsilon >0$.
For general $C^4 $ embedded curves, we establish analogous quantitative statement involving purely geometric conditions that encode certain uniform complexity bounds and allows degeneracies on the various curvature conditions. For definable families of curves in an arbitrary o-minimal expansion of the real field, the relevant complexity parameters are uniformly finite, leading to smoothing inequalities that hold uniformly across the family.
As applications, we obtain bounds for triangular Hilbert transforms along a large family of curves, their associated maximal operators, and corner-type lacunary spherical maximal operators. We further prove the existence of configurations of the form
$(x,y)$, $(x+\gamma_{1}(t),y)$, $(x,y+\gamma_{2}(t))$ inside sets of positive measure, together with a quantitative lower bound on the gap $t$.