On the constants for some fractional Gagliardo–Nirenberg and Sobolev inequalities

Abstract

We consider the inequalities of Gagliardo–Nirenberg and Sobolev in Rd, formulated in terms of the Laplacian Δ and of the fractional powers Dn≔−Δn with real n⩾0; we review known facts and present novel, complementary results in this area. After illustrating the equivalence between these two inequalities and the relations between the corresponding sharp constants and maximizers, we focus the attention on the ℒ2 case where, for all sufficiently regular f:Rd→C, the norm ‖Djf‖ℒr is bounded in terms of ‖f‖ℒ2 and ‖Dnf‖ℒ2, for 1∕r=1∕2−(ϑn−j)∕d, and suitable values of j,n,ϑ (with j,n possibly noninteger). In the special cases ϑ=1 and ϑ=j∕n+d∕2n (i.e., r=+∞), related to previous results of Lieb and Ilyin, the sharp constants and the maximizers can be found explicitly; we point out that the maximizers can be expressed in terms of hypergeometric, Fox and Meijer functions. For the general ℒ2 case, we present two kinds of upper bounds on the sharp constants: the first kind is suggested by the literature, the second one is an alternative proposal of ours, often more precise than the first one. We also derive two kinds of lower bounds. Combining all the available upper and lower bounds, the sharp constants are confined to quite narrow intervals. Several examples are given, including the numerical values of the previously mentioned bounds.