Fractional Gagliardo–Nirenberg and Hardy inequalities under Lorentz norms

Abstract

Keywords:Fractional Laplacian Gagliardo-Nirenberg inequality Sobolev inequality Logarithmic Sobolev inequality Hardy inequality a b s t r a c tIn this paper, we establish the Gagliardo-Nirenberg inequality under Lorentz norms for fractional Laplacian. Based on special cases of this inequality under Lebesgue norms, we prove the L p -logarithmic Gagliardo-Nirenberg and Sobolev inequalities. Motivated by the L 2 -logarithmic Sobolev inequality, we obtain a fractional logarithmic Sobolev trace inequality in terms of the restriction τ k u of u from R n to R n−k . Finally, we prove the fractional Hardy inequality under Lorentz norms.

lnfinitely many solutions for fractional Schrödinger equations with perturbation via variational methods

“…By the Fractional Gagliardo-Nirenberg inequality ( [29],corollary 2.3]) and the definition of the norm in X , we know that…”

Flows and functional inequalities for fractional operators

“…In the above statement D(R d ) denotes the space of smooth functions with compact support, but it is simple to extend it by density to the space w ∈H α 2 (R d ) : [26,Theorem 2.1] for details. When p = q 2 = d d−α , then ϑ = 1 and C GNS = S d,α .…”

Weighted inequalities for the fractional Laplacian and the existence of extremals

“…It should be mentioned that there is increasing literature devoted to the study of improved versions of the Sobolev-Gagliardo-Nirenberg and related inequalities for their own sake. Besides the above mentioned articles [9] and [16], we can mention [1,4,11,12,13,19], among others.…”