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We prove pointwise variational Lp bounds for a bilinear Fourier integraloperator in a large but not necessarily sharp range of exponents. This resultis a joint strengthening of the corresponding bounds for the classical Carlesonoperator, the bilinear Hilbert transform, the variation norm Carleson operator,and the bi-Carleson operator. Terry Lyon's rough path theory allows forextension of our result to multilinear estimates. We consider our result aproof of concept for a wider array of similar estimates with possibleapplications to ordinary differential equations.

Variational estimates for the bilinear iterated Fourier integral