Brief description (by Sergei Treil): Singular integrals made easy
In the theory of singular integral operators significant effort is often required to rigorously define such an operator. This is due to the fact that the kernels of such operators are not locally integrable on the diagonal $s=t$, so the integral formally defining the operator $T$ or its bilinear form $\Langle Tf, g \Rangle$ is not well defined (the integrand in not in $L^1$) even for `nice' $f$ and $g$.
However, as it turned out, the situation with interpretation of singular integral operators is much simpler, than it seems; to investigate the boundedness one only needs to study an elementary and well defined restricted bilinear form. While our result is not a replacement for the hard analysis still necessary to investigate the boundedness of a singular integral operator, it simplifies the setup significantly.
The main idea is embarrassingly simple, and we should be ashamed that we did not arrive to it much earlier. In our defense we can only say that this idea was overlooked by many harmonic analysts before us.
This is joint work with Sergei Treil.