## 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
Full Text:

### References:

 [1] Beckner, W., Sharp inequalities on the sphere and the Moser-Trudinger inequality, Ann. of math., 138, 213-242, (1993) · Zbl 0826.58042 [2] Beckner, W.; Pearson, M., On sharp Sobolev embedding and the logarithmic Sobolev inequality, Bull. London math. soc., 30, 80-84, (1998) · Zbl 0921.58072 [3] Escobar, J.F., Sharp constant in a Sobolev trace inequality, Indiana univ. math. J., 37, 687-698, (1988) · Zbl 0666.35014 [4] Escobar, J.F., The Yamabe problem on manifolds with boundary, J. differential geom., 35, 21-48, (1992) · Zbl 0771.53017 [5] Essén, M.; Janson, S.; Peng, L.; Xiao, J., Q spaces of several real variables, Indiana univ. math. J., 49, 575-615, (2000) · Zbl 0984.46020 [6] Fefferman, C.; Stein, E., $$H^p$$ spaces of several variables, Acta math., 129, 137-193, (1972) · Zbl 0257.46078 [7] Gross, L., Logarithmic Sobolev inequalities, Amer. J. math., 97, 1061-1083, (1975) · Zbl 0318.46049 [8] John, F.; Nirenberg, L., On functions of bounded Mean oscillation, Comm. pure appl. math., 18, 415-426, (1965) · Zbl 0102.04302 [9] Lieb, E., Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. of math., 118, 349-374, (1983) · Zbl 0527.42011 [10] Lieb, E.; Loss, M., Analysis, Graduate studies in mathematics, vol. 14, (2001), Amer. Math. Soc. Providence, RI [11] Park, Y.J., Logarithmic Sobolev trace inequality, Proc. amer. math. soc., 132, 2075-2083, (2004) · Zbl 1054.26015 [12] Shatah, J.; Struwe, M., Geometric wave equations, Courant lecture notes in math., vol. 2, (1998), Courant Institute of Mathematical Sciences, New York Univ., Amer. Math. Soc. Providence, RI · Zbl 0993.35001 [13] Sobolev, S.L., On a theorem of functional analysis, Mat. sb. (N.S.), AMS transl. ser. 2, 34, 39-68, (1963) · Zbl 0131.11501 [14] Stein, E.M., Singular integrals and differentiability properties of functions, (1970), Princeton University Press Princeton, NJ · Zbl 0207.13501
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.