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.
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Last modified: April 11 2016 - 18:14:43