A multilinear generalisation of the Hilbert transform and fractional integration

We study a multilinear analogue of the Hilbert transform. As can be expected, the finiteness of the form depends on cancellation properties in the kernel and care must be taken in the definition of the form. We show how to define the form in terms of distributions and prove L p bounds for that form. In the second part, we study an analogous form on the level of frac- tional integration. This has been studied in one form by Drury. We note the L p bounds for it and find the optimal constant for this bound in the case with the most symmetries. We also determine all functions which are optimisers for this inequality. Finally, we consider analogues of the fractional integration form in directions similar to those of Beckner's approach for multilinear multi- linear Hardy-Littlewood-Sobolev inequalities.