Thiele's Problem 13: the symmetric L3 bound for the triangular Hilbert transform

Principal manuscript: for complex $F,G\in L^3(\mathbb R^2)$, $\|\sup_{0<\varepsilon<R<\infty}|B_{\varepsilon,R}(F,G)|\|_{L^{3/2}}\le C\|F\|_3\|G\|_3$, where $B_{\varepsilon,R}(F,G)(x,y)=\int_{\varepsilon<|t|<R}F(x+t,y)G(x,y+t)\,dt/t$, with joint principal values a.e. and in $L^{3/2}$; pairing with a third $L^3$ function gives the scalar bound of Problem 13. Companions (October 5): $r$-variation bounds for every $r>2$, and $\sum_k\sum_{\mathbf I}|L_{\mathbf I}|\le40\prod_v\|F_v\|_3$ for Thiele's dyadic model with no structural assumption. Only the symmetric exponent point is treated; other triples with $1/p+1/q+1/r=1$ and the four-linear simplex form are not.

Principal manuscript: for complex $F,G\in L^3(\mathbb R^2)$, $\|\sup_{0<\varepsilon<R<\infty}|B_{\varepsilon,R}(F,G)|\|_{L^{3/2}}\le C\|F\|_3\|G\|_3$, where $B_{\varepsilon,R}(F,G)(x,y)=\int_{\varepsilon<|t|<R}F(x+t,y)G(x,y+t)\,dt/t$, with joint principal values a.e. and in $L^{3/2}$; pairing with a third $L^3$ function gives the scalar bound of Problem 13. Companions (October 5): $r$-variation bounds for every $r>2$, and $\sum_k\sum_{\mathbf I}|L_{\mathbf I}|\le40\prod_v\|F_v\|_3$ for Thiele's dyadic model with no structural assumption. Only the symmetric exponent point is treated; other triples with $1/p+1/q+1/r=1$ and the four-linear simplex form are not.