  We discuss the L^p-boundedness of maximal singular integrals in the plane
over a finite set V of N directions. Logarithmic bounds are established for a
set V of arbitrary structure in the 2<=p<infinity range. Sharp bounds are
proved for lacunary and Vargas sets of directions. The latter include the case
of uniformly distributed directions and the finite truncations of the Cantor
set.
  We make use of both classical harmonic analysis methods and product-BMO based
time-frequency analysis techniques. As a further application of the latter, we
derive an L^p almost orthogonality principle for Fourier restrictions to cones.

Classical Analysis and ODEs