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.


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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