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) Mu(x) Mu''(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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