Extended Caffarelli-Kohn-Nirenberg inequalities, and remainders, stability, and superweights for $L^{p}$-weighted Hardy inequalities

In this paper we give an extension of the classical Caffarelli-Kohn-Nirenberg inequalities: we show that for $1<p,q<\infty$, $0<r<\infty$ with $p+q\geq r$, $\delta\in[0,1]\cap\left[\frac{r-q}{r},\frac{p}{r}\right]$ with $\frac{\delta r}{p}+\frac{(1-\delta)r}{q}=1$ and $a$, $b$, $c\in\mathbb{R}$ with $c=\delta(a-1)+b(1-\delta)$, and for all functions $f\in C_{0}^{\infty}(\mathbb{R}^{n}\backslash\{0\})$ we have $$ \||x|^{c}f\|_{L^{r}(\mathbb{R}^{n})} \leq \left|\frac{p}{n-p(1-a)}\right|^{\delta} \left\||x|^{a}\nabla f\right\|^{\delta}_{L^{p}(\mathbb{R}^{n})} \left\||x|^{b}f\right\|^{1-\delta}_{L^{q}(\mathbb{R}^{n})} $$ for $n\neq p(1-a)$, where the constant $\left|\frac{p}{n-p(1-a)}\right|^{\delta}$ is sharp for $p=q$ with $a-b=1$ or $p\neq q$ with $p(1-a)+bq\neq0$. In the critical case $n=p(1-a)$ we have $$ \left\||x|^{c}f\right\|_{L^{r}(\mathbb{R}^{n})} \leq p^{\delta} \left\||x|^{a}\log|x|\nabla f\right\|^{\delta}_{L^{p}(\mathbb{R}^{n})} \left\||x|^{b}f\right\|^{1-\delta}_{L^{q}(\mathbb{R}^{n})}. $$ Moreover, we also obtain anisotropic versions of these inequalities which can be conveniently formulated in the language of Folland and Stein's homogeneous groups. Consequently, we obtain remainder estimates for $L^{p}$-weighted Hardy inequalities on homogeneous groups, which are also new in the Euclidean setting of $\mathbb{R}^{n}$. The critical Hardy inequalities of logarithmic type and uncertainty type principles on homogeneous groups are obtained. Moreover, we investigate another improved version of $L^{p}$-weighted Hardy inequalities involving a distance and stability estimates. We also establish sharp Hardy type inequalities in $L^{p}$, $1<p<\infty$, with superweights, i.e. with the weights of the form $\frac{(a+b|x|^{\alpha})^{\frac{\beta}{p}}}{|x|^{m}}$ allowing for different choices of $\alpha$ and $\beta$.

MICHAEL RUZHANSKY, DURVUDKHAN SURAGAN, AND NURGISSA YESSIRKEGENOV Abstract. In this paper we give an extension of the classical Caffarelli-Kohn-Nirenberg inequalities: we show that for 1 < p, q < ∞, 0 < r < ∞ with p + q ≥ r, δ ∈ [0, 1] ∩ r−q r , p r with δr p + (1−δ)r q = 1 and a, b, c ∈ R with c = δ(a − 1) + b(1 − δ), and for all functions f ∈ C ∞ 0 (R n \{0}) we have Moreover, we also obtain anisotropic versions of these inequalities which can be conveniently formulated in the language of Folland and Stein's homogeneous groups. Consequently, we obtain remainder estimates for L p -weighted Hardy inequalities on homogeneous groups, which are also new in the Euclidean setting of R n . The critical Hardy inequalities of logarithmic type and uncertainty type principles on homogeneous groups are obtained. Moreover, we investigate another improved version of L p -weighted Hardy inequalities involving a distance and stability estimates. The relation between the critical and the subcritical Hardy inequalities on homogeneous groups is also investigated. We also establish sharp Hardy type inequalities in L p , 1 < p < ∞, with superweights, i.e. with the weights of the form (a+b|x| α )  The aim of this paper is to give an extension of the classical Caffarelli-Kohn-Nirenberg (CKN) inequalities [CKN84] with respect to ranges of parameters and to investigate the remainders and stability of the weighted L p -Hardy inequalities. Moreover, our methods also provide sharp constants for the CKN inequality for known ranges of parameters as well as give an improvement by replacing the full gradient by the radial derivative. We also obtain the critical case of the CKN inequality with logarithmic terms, and investigate the remainders and other properties in the case when CKN inequalities reduce to the weighted Hardy inequalities. For the latter, we also establish L p weighted Hardy inequalities with more general weights of the form Theorem 1.1. Let n ∈ N and let p, q, r, a, b, d, δ ∈ R such that p, q ≥ 1, r > 0, 0 ≤ δ ≤ 1, and 1 p + a n , where c = δd + (1 − δ)b. Then there exists a positive constant C such that holds for all f ∈ C ∞ 0 (R n ), if and only if the following conditions hold: (1.5) The first aim of this paper is to extend the CKN-inequalities for general functions with respect to widening the range of indices (1.1). Moreover, another improvement will be achieved by replacing the full gradient ∇f in (1.2) by the radial derivative Rf = ∂f ∂r . It turns out that such improved versions can be establsihed with sharp constants, and to hold both in the isotropic and anisotropic settings.
To compare with Theorem 1.1 let us first formulate the isotropic version of our extension in the usual setting of R n . Theorem 1.2. Let n ∈ N, 1 < p, q < ∞, 0 < r < ∞, with p + q ≥ r, δ ∈ [0, 1] ∩ r−q r , p r and a, b, c ∈ R. Assume that δr (1.6) In the critical case n = p(1 − a) for any function f ∈ C ∞ 0 (R n \{0}) we have for any homogeneous quasi-norm | · |. If | · | is the Euclidean norm on R n , inequalities (1.6) and (1.7) imply, respectively, and for n = p(1 − a). The inequality (1.6) holds for any homogeneous quasi-norm | · |, and the constant where ∇ is the standard gradient in R n . Since we have we see that (1.1) fails, so that the inequality (1.10) is not covered by Theorem 1.1. Moreover, in this case, p = q with a − b = 1 hold true, so that the constant p n−p(1−a) δ = 1 in the inequality (1.10) is sharp.
Although these results are new already in the usual setting of R n , our techniques apply well also for the anisotropic structures. Consequently, it is convenient to work in the setting of homogeneous groups developed by Folland and Stein [FS82] 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 [RS17a] and especially to [RS17] for versions of such results and the discussion of the corresponding literature.
The improved versions of the Caffarelli-Kohn-Nirenberg inequality for radially symmetric functions with respect to the range of parameters was investigated in [NDD12]. In [ZHD15] and [HZ11], weighted Hardy type inequalities were obtained for the generalised Baouendi-Grushin vector fields, which is when γ = 0 gives the standard gradient in R n . We also refer to [HNZ11], [Han15] for weighted Hardy inequalities on the Heisenberg group, to [HZD11] and [ZHD14] on the H-type groups, and a recent paper [Yac17] on Lie groups of polynomial growth as well as to references therein.
In Section 2 we very briefly recall the necessary notions and fix the notation in more detail. Assuming the notation there, Theorem 1.2 is the special case of the following theorem that we prove in this paper: Theorem 1.4. Let G be a homogeneous group of homogeneous dimension Q. Let | · | be an arbitrary homogeneous quasi-norm on G. Let 1 < p, q < ∞, 0 < r < ∞ with p + q ≥ r, δ ∈ [0, 1] ∩ r−q r , p r and a, b, c ∈ R. Assume that δr p + (1−δ)r q = 1 and c = δ(a − 1) + b(1 − δ). Then for all f ∈ C ∞ 0 (G\{0}) we have the following Caffarelli-Kohn-Nirenberg type inequalities, with R := d d|x| being the radial derivative: where the constant Moreover, the constants 1.2. L p -weighted Hardy inequalities. Let us recall the following L p -weighted Hardy inequality for every function f ∈ C ∞ 0 (R n ), where −∞ < α < n−p p and 2 ≤ p < n. The inequality (1.11) is a special case of the Caffarelli-Kohn-Nirenberg inequalities [CKN84], recalled also in Theorem 1.1. Since in this paper we are also interested in remainder estimates for the L p -weighted Hardy inequality, let us introduce known results in this direction. Overall, the study of remainders in Hardy and other related inequalities is a classical topic going back to [BL85,BM97,BV97].
Ghoussoub and Moradifam [GM08] proved that there exists no strictly positive function V ∈ C 1 (0, ∞) such that the inequality holds for any f ∈ W 1,2 (R n ). Cianchi and Ferone [CF08] showed that for all 1 < p < n there exists a constant C = C(p, n) such that holds for all real-valued weakly differentiable functions f in R n such that f and |∇f | ∈ L p (R n ) go to zero at infinity. Here with p * = np n−p , and L τ,σ (R n ) is the Lorentz space for 0 < τ ≤ ∞ and 1 ≤ σ ≤ ∞. In the case of a bounded domain Ω, Wang and Willem [WW03] for p = 2 and Abdellaoui, Colorado and Peral [ACP05] for 1 < p < ∞ investigated the improved type of (1.11) (see also [ST15a] and [ST15b] for more details).
For more general Lie group discussions of above inequalities we refer to recent papers [RS17a], [RS16c] and [RS17] as well as references therein.
Sometimes the improved versions of different inequalities, or remainder estimates, are called stability of the inequality if the estimates depend on certain distances: see, e.g. [BJOS16] for stability of trace theorems, [CFW13] for stability of Sobolev inequalities, etc.
We also note that Sano and Takahashi obtained the improved version of (1.11) in [ST15a] for Ω = R n and α = 0 and then in [ST15b] for any −∞ < α < n−p p : Let . Then there exists a constant C > 0 such that the inequality holds for any radially symmetric function f ∈ W 1,p 0,α (R n ), f = 0. For the convenience of the reader we now briefly recapture the main results of this part of the paper, formulating them directly in the anisotropic cases following the notation recalled in Section 2. Thus, we show that for a homogeneous group G of homogeneous dimension Q and any homogeneous quasi-norm | · | we have the following results: • (Remainder estimates for the L p -weighted Hardy inequality) , r = |x|, R := d d|x| is the radial derivative, c p is defined in Lemma 2.1, f α and d R (·, ·) are defined in (4.1) and (4.2), respectively. • (Critical Hardy inequalities of logarithmic type) Let 1 < γ < ∞ and let max{1, γ − 1} < p < ∞. Then for all f ∈ C ∞ 0 (G\{0}) and all R > 0 we have is the radial derivative, and the constant p γ−1 is optimal. In the abelian case, this result was obtained in [MOW15]. In the case γ = p this result on the homogeneous group was proved in [RS16b]. • (Uncertainty inequalities) Let 1 < p < ∞ and q > 1 be such that 1 p + 1 q = 1 2 . Let 1 < γ < ∞ and max{1, γ − 1} < p < ∞. Then for any R > 0 and f ∈ C ∞ 0 (G\{0}) we have the uncertainty inequalities where R := d d|x| is the radial derivative (see (2.4)). Moreover, holds for 1 p + 1 p ′ = 1.

• (Relation between critical and subcritical Hardy inequalities) Let
Q ≥ m + 1, m ≥ 2. Let | · | be a homogeneous quasi-norm. Then for any nonnegative radial function holds true, where R := d d|x| is the radial derivative, |σ| and | σ| are Q − 1 and m − 1 dimensional surface measure of the unit sphere, respectively.
1.3. L p -Hardy inequalities with superweights. The classical Hardy inequalities and their extensions, such as the Caffarelli-Kohn-Nirenberg inequalities, usually involve the weights of the form 1 |x| m . In this paper, we also consider the weights of the form (a+b|x| α ) β p |x| m allowing for different choices of α and β. If α = 0 or β = 0, this reduces to traditional weights. So, we are interested in the case when αβ = 0 and, in fact, we obtain two families of 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.
Again, the obtained estimates will include both the isotropic and anisotropic settings of R n , for which our range of obtained estimates appears also to be new. Namely, already in the Euclidean case of R n with the Euclidean norm, they extend the inequalities that have been known for p = 2 for some range of parameters from [GM11] to the full range of 1 < p < ∞.
Therefore, we can again work on the homogeneous groups. To summarise, on a homogeneous group G with homogeneous dimension Q for any homogeneous quasinorm | · | on G, all a, b > 0 and 1 < p < ∞ we prove that . (1.12) . (1.13) As noted before, the weights in the inequalities (1.12) and (1.13) are called superweights since the constants Q−pm−p p in (1.12) and Q−pm+αβ−p p in (1.13) are sharp for arbitrary homogeneous quasi-norm | · | of G and wide range of choices of the allowed parameters α, β, a, b and m. Directly from the inequalities (1.12) and (1.13), choosing different α, β, a, b, m and Q, one can obtain a number of Hardy type inequalities which have various consequences and applications. For instance, 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 (1.14) with the constant being sharp for n = pm + p.
If αβ < 0 and pm − αβ ≤ n − p, then for all f ∈ C ∞ 0 (G\{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 (1.14) and (1.15) we obtain that if αβ > 0 and pm ≤ Q − p, then for all f ∈ with the constant sharp for n = pm + p.
If αβ < 0 and pm − αβ ≤ n − p, then for all f ∈ C ∞ 0 (G\{0}), we have with the sharp constant for n = pm + p − αβ. The L 2 -version, that is, when p = 2 the inequalities (1.16) and (1.17) were obtained in [GM11]. We also shall note that these inequalities have interesting applications in theory of ODE (see [GM11, Theorem 2.1]). In Section 8 we give the main result of this part and give its short proof. Some higher order versions of the obtained inequalities are discussed briefly in Section 9.
In Section 2 we briefly recall the main concepts of homogeneous groups and fix the notation. In Section 5 we present critical Hardy inequalities of logarithmic type and uncertainty type principles on homogeneous groups. The remainder estimates for L p -weighted Hardy inequalities on homogeneous groups are proved in Section 3. Moreover, in Section 4 we also investigate another improved version of L p -weighted Hardy inequalities involving a distance. In Section 6 the relation between the critical and the subcritical Hardy inequalities on homogeneous groups is investigated. In Section 7 we introduce Caffarelli-Kohn-Nirenberg type inequalities on homogenous groups and prove their extended version.

Preliminaries
In this section we very briefly recall the necessary notation concerning the setting of homogeneous groups following Folland and Stein [FS82] as well as a recent treatise [FR16]. We also recall a few other facts that will be used in the proofs. 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, , holds. We recall that Q := Tr A is called the homogeneous dimension of G.
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.
Then we have Here dx is the Haar measure on homogeneous groups G and |S| is the volume of a measurable set S ⊂ G. The Haar measure on a homogeneous group G is the standard Lebesgue measure for R n (see, for example [FR16, Proposition 1.6.6]). Let | · | be a homogeneous quasi-norm on G. Then the quasi-ball centred at x ∈ G with radius R > 0 is defined by B(x, R) := {y ∈ G : |x −1 y| < R}.
The following notation will be also used in this paper We refer to [FS82] for the proof of the following important polar decomposition on homogeneous Lie groups, which can be also found in [FR16, Section 3.1.7]: there is a (unique) positive Borel measure σ on the unit quasi-sphere so that for every f ∈ L 1 (G) we have Let us now fix a basis {X 1 , . . . , X n } of a Lie algebra g such that for every k, so that the matrix A can be taken to be A = diag(ν 1 , . . . , ν n ). Then every X k is homogeneous of degree ν k and Q = ν 1 + · · · + ν n .
We use the notation for any homogeneous quasi-norm |x| on G. Let us recall the following lemma, which will be used in our proof. ((1 − t) p − t p + pt p−1 ) is sharp in this inequality.

On remainder estimates of anisotropic L p -weighted Hardy inequalities
In this section we obtain a family of remainder estimates in the weighted L p -Hardy inequalities, with a freedom of choosing the parameter b ∈ R. The obtained remainder estimates are new already in the standard setting of R n .
Remark 3.2. Since the inequality (3.1) holds for any b ∈ R, choosing b = Q(p−1) p so that C p = 0, we obtain the L p -weighted Hardy inequalities on homogeneous groups: for all functions f ∈ C ∞ 0 (G\{0}). In the abelian case G = (R n , +) with Q = n, the inequality (3.2) gives the L p -weighted Hardy inequalities for any quasi-norm on R n : For any function f ∈ C ∞ 0 (R n \{0}) we have where −∞ < α < n−p p and 2 ≤ p < n. By Schwarz's inequality with the standard Euclidean distance |x| = x 2 1 + x 2 2 + ... + x 2 n , we obtain the Euclidean form of the L p -weighted Hardy inequalities on R n : for any function f ∈ C ∞ 0 (R n \{0}), where ∇ is the standard gradient in R n . Remark 3.3. We also note that in the abelian case, (3.1) implies a new remainder estimate for any quasi-norm on R n : For any function f ∈ C ∞ 0 (R n \{0}) and for any b ∈ R, we obtain As in Remark 3.2, by Schwarz's inequality with the standard Euclidean distance, we obtain the Euclidean version of the remainder estimate for L p -weighted Hardy inequalities: for every function f ∈ C ∞ 0 (R n \{0}) and for any b ∈ R, where ∇ is the standard gradient in R n .
Thus, we note that the remainder estimate (3.4) is new already in the standard setting of R n . Since f = f (r) ∈ C ∞ 0 (0, ∞) and α < Q−p p , we obtain g(0) = 0 and g(+∞) = 0. We set g(x) = g(|x|) for x ∈ G. Introducing polar coordinates (r, y) = (|x|, x |x| ) ∈ (0, ∞) × S on G and using (2.3), we have where |σ| is the Q − 1 dimensional surface measure of the unit quasi-sphere. Here applying Lemma 2.1 to the integrand of the first term in the last expression above, we get Since g(0) = g(+∞) = 0 and p ≥ 2, we note that This gives a "ground state representation" ( [FS08]) of the Hardy difference J: Putting a = Q−p p in (2.6), we obtain for any b ∈ R, that It gives that Taking into account that g(x) = g(|x|), x ∈ G, and (3.5), one calculates On the other hand, Putting these into (3.8), we obtain Now let us prove it for non-radial functions. We consider the radial function for a non-radial function f : (3.10) Using the Hölder inequality, we calculate Thus, we have It follows that In view of (3.10), we obtain for any θ ∈ R. Then, it is easy to see that (3.11) and (3.12) imply that (3.1) holds also for all non-radial functions.

Stability of anisotropic L p -weighted Hardy inequalities
In this section we establish a remainder estimate in the L p -weighted Hardy inequality involving the distance to the set of extremisers: estimates of such type are known as stability estimates in the literature. Let us denote for −∞ < α < Q−p p , and we set for functions f and g for which the integral in (4.2) is finite.
Proof of Theorem 4.1. Since p ≥ 2, as in (3.6) in the proof of Theorem 3.1, we have By Theorem 3.1 in [RS16b] or Remark 5.3 with γ = p, we obtain for any R > 0. Here using f (x) = f (r), r = |x|, one calculates yielding (4.3).

Critical Hardy inequalities of logarithmic type and uncertainty principle
In this section, we present critical Hardy inequalities of logarithmic type on the homogeneous group G. In the abelian isotropic case, the following result was obtained in [MOW15]. In the case γ = p this result on the homogeneous group was proved in [RS16b].
Proof of Theorem 5.1. First, let us consider the integrals in (5.1) restricted to B(0, R). Introducing polar coordinates (r, y) = (|x|, x |x| ) ∈ (0, ∞) × S on G, where S is the sphere as in (8.3), and using (2.3), we have where p − γ + 1 > 0, so that the boundary term at r = R vanishes due to inequalities Then, by the Hölder inequality, we get The inequalities (5.2) and (5.3) imply (5.1). Now let us prove the optimality of the constant p γ−1 in (5.1). The inequality (5.1) gives that  (5.4) It is enough to prove the optimality of the constant p γ−1 in (5.4). As in the abelian case (see [MOW15, Section 3]), we define the following sequence of functions γ−1 p , when R 2 < r < R. Denoting by |σ| the Q − 1 dimensional surface measure of the unit sphere, by a direct calculation one has where C γ,p := 2 p (log 2) γ−1 |σ| log 2 0 s p−γ e −ps ds.
Since p − γ + 1 > 0, we get C γ,q < +∞. On the other hand, we see The inequality log R r ≥ R−r R for all r ≤ R and the assumption p − γ > −1, imply C R,γ,p < +∞. Then, by (5.5) and (5.6), we have as k → ∞, which implies that the constant p γ−1 in (5.4) is optimal.

Critical and subcritical Hardy inequalities
In this section, we study the relation between the critical and the subcritical Hardy inequalities on homogeneous groups.
Moreover, the constants p n−p(1−a) δ and p δ are sharp for δ = 0 or δ = 1. If (7.7) is not satisfied, then the inequality (7.6) is not covered by Theorem 1.1 because condition (1.1) fails. So we obtain a new range of the Caffarelli-Kohn-Nirenberg inequality [CKN84].
Thus, the inequalities (7.4) and (7.6) are new already in the abelian case and, moreover, (7.1) and (7.2) hold for any choice of homogeneous quasi-norm.
The proof of Theorem 7.1 will be based on the following family of weighted Hardy inequalities that was obtained in [RSY16, Theorem 3.4], where E = |x|R is the Euler operator.
Theorem 7.4 ( [RSY16]). Let G be a homogeneous group of homogeneous dimension Q and let α ∈ R. Then for all complex-valued functions f ∈ C ∞ 0 (G\{0}), 1 < p < ∞, and any homogeneous quasi-norm | · | on G for αp = Q we have where the constant p is sharp.
We briefly recall its proof for the convenience of the reader but also since it will be useful in our argument.
where C ∈ R and C = 0. Then by a direct calculation we obtain Here we note that when p = q and a − b = 1 Hölder's equality condition is held for any function. We also note that in the case p = q the function h(x) = |x| 1 (p−q) (p(1−a)+bq) (7.14) satisfies Hölder's equality condition: , 1 < p < ∞.

L p -Hardy inequalities with super weights
We now discuss versions of Hardy inequalities with more general weights, that we call superweights. The following is the main result of this section.
Theorem 8.1. Let G be a homogeneous group of homogeneous dimension Q and let | · | be a homogeneous quasi-norm on G. Let a, b > 0 and 1 < p < ∞, Q ≥ 1.
(i) If αβ > 0 and pm ≤ Q − p, then for all f ∈ C ∞