**Pointwise interpolation inequalities for integer and
fractional derivatives and their applications**

I overview recent results, obtained together with V.Maz'ya,
concerning interpolation inequalities for functional and
fractional derivatives. A typical example is the Landau type
inequality on the real line

|*u*'(*x*)|^{2} ≤
(8/3) **M***u*(*x*) **M***u*''(*x*),
where the constant 8/3 is best possible and **M** is the
Hardy-Littlewood maximal operator.
Similar inequalities are used in an elementary proof of a
theorem by H.Brezis and P.Mironescu on the continuity of the
composition operator in fractional Sobolev spaces.

New limiting properties of fractional Sobolev spaces initiated
recently by Bourgain, Brezis, and Mironescu will be discussed
as well.

*Tatyana Shaposhnikova*

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