Colloquium

The talk will begin with an illustration of the concept of fractional differentiation. We are interested in calculating the fractional derivative of the product of two functions. We estimate the \(L^r\) norm of the Bessel potential \(J^s=(1-\Delta)^{s/2}\) (or Riesz potential \(D^s = (-\Delta)^{s/2}\)) of the product of two functions in terms of the product of the \(L^{p}\) norm of one function and the \(L^{q}\) norm of the the Bessel potential \(J^s\) (resp. Riesz potential \(D^s\)) of the other function. Here the indices \(p\), \(q\), and \(r\) are related as in Hölder's inequality \(1/p+1/q=1/r\) and they satisfy \(1\leq p,q \leq \infty\) and \(1/2\leq r<\infty\) and \(s>\max(0,rac {n}{r-n})\). The last condition is sharp in terms of the range of \(s\).

Note: the talk will be accessible to a general math audience.