  We prove a quantitative Roth-type theorem for polynomial corners in
$\mathbb{R}^2$. Let $P_1$ and $P_2$ be two linearly independent polynomials
with zero constant term. We show that any measurable subset of $[0,1]^2$ with
positive measure contains three points $(x,y)$, $(x+P_1(t),y)$, $(x,y+P_2(t))$
with a gap estimate on $t$. We also prove boundedness results for a variant of
the triangular Hilbert transform involving two polynomials and its associated
maximal function. These results extend some earlier work of Christ, Durcik and
Roos. The key of the proof is to establish certain smoothing inequalities
involving two polynomials. To accomplish that we give sublevel set estimates
with general polynomials, explicit exponents and simplified proofs.
