We develop an Lp theory for outer measures which differs from classical examples in that it replaces the outer measure of the set where a given function exceeds a given threshold by a more general quantity.
The theory gives an alternative description of some well known themes such as Carleson embedding theorems and paraproduct estimates. It is also behind time-frequency-analysis as orginated in Carleson's proof of a.e. convergence of Fourier series and can be used to prove bounds for the bilinear Hilbert transform. This is joint work with Yen Do.