Compensation phenomena

In principle, the only way to estimate the integrability of a product of two square-integrable functions is to use  Hoelder's inequality, which yields that such a product is integrable. Easy examples show that this estimate is sharp. However, there are particular expressions (among them Jacobian determinants), which at the first sight are only integrable, but, due to their particular structure, have slightly better integrability properties.  The first such observations were made by H.Wente in late 1960, with later improvements by Tartar, Mueller, culminating in the celebrated div-curl lemma of Coifman, Lions, Meyer and Semmes. In the lecture I will prove the div-curl lemma and show some applications.