Extended Caffarelli–Kohn–Nirenberg inequalities and superweights for Lp-weighted Hardy inequalities

Abstract In this paper, we give an extension of the classical Caffarelli–Kohn–Nirenberg inequalities with respect to the range of parameters. We also establish best constants for large families of parameters. Moreover, we also obtain anisotropic versions of these inequalities which can be conveniently formulated in the language of Folland and Stein's homogeneous groups. We also establish sharp Hardy type inequalities in L p , 1 p ∞ , with superweights.

Theorem 1.1. Let n ∈ N and let p, q, r, a, b, d, δ ∈ R be such that p, q ≥ 1, r > 0, 0 ≤ δ ≤ 1, and holds for all f ∈ C ∞ 0 (R n ), if and only if the following conditions hold: The first goal of the present paper is to extend the CKN inequalities with respect to widening the range of indices in (1).
Moreover, another improvement will be achieved by replacing the full gradient ∇ f in (2) by the radial derivative R f = ∂ f ∂r . It turns out that such generalised versions hold both in the isotropic and anisotropic settings with sharp constants. We also establish the critical case of the CKN inequality with logarithmic terms. These results are presented in Section 2.
Although the obtained results in this paper are new already in the usual setting of R n , our techniques apply well also for anisotropic structures. Consequently, it is convenient to work in the setting of homogeneous groups developed by Folland and Stein [3] with an idea of emphasising general results of harmonic analysis depending only of the group and dilation structures. In particular, in this way we obtain results on the anisotropic R n , on the Heisenberg group, general stratified groups, graded groups, etc. In the special case of stratified groups (or homogeneous Carnot groups), other formulations using horizontal gradient are possible, and we refer to [7] (see also [5]) and especially to [6] for versions of such results and the discussion of the corresponding literature. Let us very briefly discuss some key notions of homogeneous groups following Folland and Stein [3] as well as a recent treatise [2]. A connected simply connected Lie group G is called a homogeneous group if its Lie algebra g is equipped with a family of the following dilations: where A is a diagonalisable positive linear operator on g, and every D λ is a morphism of g, that is, A homogeneous group is a nilpotent (Lie) group and exponential mapping exp G : g → G of this group is a global diffeomorphism. Thus, this implies the dilation structure, and this dilation is denoted by D λ x or just by λx, on homogeneous groups. Moreover, every homogeneous group G has a homogeneous quasi-norm, which will be denoted by | · |. Another key property of a homogeneous group is that it allows polarisation. We also use a derivative operator R on G with respect to the corresponding quasi-norm.
The classical Hardy inequalities and their sharp extensions, such as the sharp CKN inequalities, usually involve the weights of the form 1 |x| m . In this note, we also consider the weights of the form allowing for different choices of α and β, which give sharp constants. If α = 0 or β = 0, this reduces to traditional weights. So, we are interested in Section 3 in the situation when αβ = 0 and, in fact, we obtain two families of sharp inequalities depending on whether αβ > 0 or αβ < 0. Moreover, | · | in these expressions can be an arbitrary homogeneous quasi-norm and the constants for the obtained inequalities are sharp. The freedom in choosing parameters α, β, a, b, m and a quasi-norm led us to calling these weights the 'superweights' in this context.

Extended Caffarelli-Kohn-Nirenberg inequalities
In this section, we introduce new CKN-type inequalities in the Euclidean setting of R n as well as on homogeneous groups. δ). Then, for any homogeneous quasi-norm | · | on a homogeneous group G of homogeneous dimension Q we have the following CKN-type inequalities for all f ∈ C ∞ 0 (G\{0}): where R := d d|x| . The constant in the inequality (6) is sharp for p = q with a − b = 1 or p = q with p(1 − a) + bq = 0. Moreover, the constants in (6) and (7) are sharp for δ = 0 or δ = 1.
Note that the conditions δr (3) of Theorem 1.1, and in our case a − d = 1. In the Abelian case G = (R n , +) and Q = n, (7) implies a new type of the CKN inequality for any quasi-norm on R n : Let 1 < p, q < ∞, 0 < r < ∞ with p + q ≥ r and δ ∈ [0, 1] ∩ r−q r , p r and a, b, c ∈ R. Assume that δr p + (1−δ)r q = 1, n = p(1 − a) and c = δ(a − 1) + b(1 − δ). Then we have the CKN-type inequality for any function f ∈ C ∞ 0 (R n \{0}) and for any homogeneous quasi-norm | · |: By the Schwarz inequality with the standard Euclidean distance given by |x| = x 2 1 + x 2 2 + ... + x 2 n , we obtain the Euclidean form of the CKN-type inequality: where ∇ is the standard gradient in R n . Similarly, we write the inequality (6) in the Abelian case: Let 1 < p, q < ∞, 0 < r < ∞ with p +q ≥ r and δ ∈ [0, 1] ∩ r−q r , p r and a, b, c ∈ R. Assume that δr p + (1−δ)r q = 1, n = p(1 −a) and c = δ(a −1) +b(1 −δ). Then we have a CKN-type inequality for any function f ∈ C ∞ 0 (R n \{0}) and for any homogeneous quasi-norm | · |: Then, using the Schwarz inequality with the standard Euclidean distance, we obtain the Euclidean form of the CKN-type inequality: with sharp constant, where ∇ is the standard gradient in R n . Since we have 1 q + b n = 1 p + 1 n − n p = 0, we see that (1) fails, so that the inequality (12) is not covered by Theorem 1.1.

L p -Hardy inequalities with super weights
We now give versions of Hardy inequalities with more general weights, that we call superweights, due to their rather general form and the freedom in choosing the parameters. Theorem 3.1. Let a, b > 0 and 1 < p < ∞. For any homogeneous quasi-norm | · | on a homogeneous group G of homogeneous dimension Q .
As it is mentioned in the introduction, the weights in the inequalities (13)  in the Abelian (isotropic or anisotropic) case G = (R n , +), we have Q = n, so for any quasi-norm | · | on R n , all a, b > 0 and 1 < p < ∞ these imply new inequalities. Thus, if αβ > 0 and pm ≤ n − p, then for all f ∈ C ∞ 0 (R n \{0}), we have with the constant being sharp for n = pm + p.
If αβ < 0 and pm − αβ ≤ n − p, then for all f ∈ C ∞ 0 (R n \{0}), we have with the sharp constant for n = pm + p − αβ. In the case of the standard Euclidean distance |x| = x 2 1 + . . . + x 2 n , by using the Schwartz inequality from the inequalities (15) and (16), we obtain that if αβ > 0 and pm ≤ n − p, then for all with the constant sharp for n = pm + p.
If αβ < 0 and pm − αβ ≤ n − p, then for all f ∈ C ∞ 0 (R n \{0}), we have with the sharp constant for n = pm + p − αβ. The L 2 -version, that is, for p = 2 the inequalities (17) and (18) were obtained in [4]. We also shall note that these inequalities have interesting applications in the theory of ODEs (see [4,Theorem 2.1]).