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A sharp Sobolev trace inequality for the fractional-order derivatives. (English) Zbl 1096.46019

The author shows that the particular case \[ \|f\|_{\frac{2n}{n-2\alpha}}\leq C(n,\alpha)\|(-\Delta)^\alpha f\|_2: = C(n,\alpha)|||x^\alpha f(x)|||_2, \quad 0<\alpha<\tfrac{n}{2} \] of the Sobolev inequality (with the known exact constant \(C(n,\alpha)\)) in the case \(0<\alpha<1\) may be given in the form \[ \|f\|_{\frac{2n}{n-2\alpha}}\leq C_1(n,\alpha)\left(\int_{\mathbb{R}^{n+1}_+}|\nabla f(x,t)|^2 t^{1-2\alpha}\right). \] The proof is based on the observation that the right-hand sides of the inequalities coincide, which follows from direct calculations.
Some logarithmic versions of the last inequality are also given.

MSC:

46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
26D10 Inequalities involving derivatives and differential and integral operators
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