Chapter 4 Fractional Hardy Inequalities

In this chapter we present results concerning fractional forms of Hardy inequalities. Such a topic is well investigated in the Abelian Euclidean setting and we will be providing relevant references in the sequel. For a general survey of fractional Laplacians in the Euclidean setting see, e.g., [Gar17]. However, as usual, the general approach based on homogeneous groups allows one to get insights also in the Abelian case, for example, from the point of view of the possibility of choosing an arbitrary quasi-norm. Moreover, another application of the setting of homogeneous groups is that the results can be equally applied to both elliptic and subelliptic problems.

Definition 4.1.2 (Gagliardo seminorm and fractional Sobolev spaces). For a measurable function u : G → R, its Gagliardo seminorm is defined as [u]   Moreover, the Sobolev space W s,p 0 (Ω) is defined as the completion of C ∞ 0 (Ω) with respect to the norm u W s,p (Ω) .

Fractional Hardy inequalities on homogeneous groups
In the present section we establish fractional Hardy inequalities on homogeneous groups.
Remark 4.2.2. In [AB17] the authors studied the weighted fractional p-Laplacian and established the following weighted fractional L p -Hardy inequality: where β < N −ps 2 , u ∈ C ∞ 0 (R N ), C > 0 is a positive constant, and | · | E is the Euclidean distance in R N .
Before we prove Theorem 4.2.1, let us establish the following two lemmas that will be instrumental in the proof.
Proof of Theorem 4.2.1. Let u ∈ C ∞ 0 (G) and γ < Q−ps p−1 . By Lemma 4.2.4 and Lemma 4.2.3 we readily obtain that This completes the proof of Theorem 4.2.1.

Fractional Sobolev inequalities on homogeneous groups
In this section we establish fractional Sobolev inequalities on homogeneous groups.
with | · | E being the Euclidean distance in R N .
To prove the above analogue of the fractional Sobolev inequality, first we present the following two lemmas.
Lemma 4.3.3. Let p > 1, s ∈ (0, 1), and let K ⊂ G be a Haar measurable set. Fix x ∈ G and a quasi-norm | · | on G. Then we have , where ω Q is a surface measure of the unit quasi-ball on G. For the corresponding quasi-ball B(x, δ) = B |·| (x, δ) centred at x with radius δ, we have where | · | (by abuse of notation, only in this proof) is the Haar measure on G. Then, By using (4.18) we obtain Now using the polar decomposition formula in Proposition 1.2.10 we obtain that completing the proof.
Proof of Lemma 4.3.4. We define and By the assumption u ∈ L ∞ (G) is compactly supported, a k and d k are bounded and vanish when k is large enough. Also, we notice that the D k 's are disjoint, therefore, l∈Z, l≤k (4.24) and l∈Z, l≥k (4.25) From (4.25) it follows that Since a k and d k are bounded and vanish when k is large enough, (4.26) and (4.27) are convergent. We define the convergent series (4.30) Notice that for any x, y ∈ G. If we fix i ∈ Z and x ∈ D i , then for any j ∈ Z with j ≤ i − 2, for any y ∈ D j using the above inequality, we obtain that Then, using (4.24), we have (4.31) Now using (4.31) and Lemma 4.3.3, we obtain that (4.33) By (4.32) and (4.28) it follows that (4.34) Then, by using (4.30), (4.33) and (4.34), we obtain that This means that for some constant C > 0. By symmetry and using (4.35), we arrive at Suppose also that u ∈ L ∞ (G). If (4.36) is satisfied for bounded functions, it holds also for the function u n , obtained from u by cutting at levels −n and n, that is, for u n := max{min{u(x), n}, −n}, for any n ∈ R and x ∈ G. Thus, using the fact that lim n→+∞ u n L p (G) = u L p (G) , 1 < p < ∞, and by using (4.36) with the dominated convergence theorem, we obtain that (4.37) Defining a k and A k as in Lemma 4.3.4, we have Recall the following fact from [DNPV12, Lemma 6.2]: let T, p > 1 and s ∈ (0, 1) be such that Q > sp, m ∈ Z, and assume that a k is a bounded, decreasing, non-negative sequence with a k = 0 for any k ≥ m; then we have for some positive constant C = C(Q, s, p, T ). Then, with p/p * = 1 − sp/Q < 1 and T = 2 p , this fact yields (4.39) for a positive constant C = C(Q, p, s, | · |). Finally, using Lemma 4.3.4 we arrive at Theorem 4.3.1 is proved.

Fractional Gagliardo-Nirenberg inequalities
In this section we discuss an analogue of the fractional Gagliardo-Nirenberg inequality on homogeneous groups. As it can be partly expected, in its proof we will use an already established version of a fractional Sobolev inequality on the homogeneous groups.

Fractional Caffarelli-Kohn-Nirenberg inequalities
In this section we discuss the weighted fractional Caffarelli-Kohn-Nirenberg inequalities on homogeneous groups. First, let us define a weighted version of fractional Sobolev spaces from Definition 4.1.2.
To prove the fractional weighted Caffarelli-Kohn-Nirenberg inequality on G we will use Theorem 4.4.1 in the proof of the following lemma.
Let λ > 0 and 0 < r < R and set Then, for all u ∈ C 1 (Ω), we have where C r,R is a positive constant independent of u and λ.
where C r,R > 0. Let us apply the above inequality to u(λx) instead of u(x). This yields This completes the proof.
Proof of Theorem 4.5.2. First let us consider the case (4.46), that is, β − σ ≤ s and 1 τ + γ Q = 1 p + β−s Q . By using Lemma 4.5.4 with λ = 2 k , r = 1, R = 2 and Ω = A k , we get Here and below A k is the quasi-annulus defined by for k ∈ Z. Now by using (4.50) we obtain Then, from (4.51) we get Here by (4.45), we have and by summing over k from m to n, we get where k, m, n ∈ Z and m ≤ n − 2.
To prove (4.47) let us choose n such that where B 2 n is a quasi-ball of G with the radius 2 n . Let us consider the following integral On the other hand, a direct calculation gives (4.57) From (4.57) and Lemma 4.5.4, we obtain By using this fact, taking τ = 1 we have On the other hand (see, e.g., [NS18b, Lemma 2.2]), there exists a positive constant C depending ξ > 1 and η > 1 such that 1 < ζ < ξ, (4.60) Thus, by using with η = τ , ζ = 2 γτ +Q c, where c = 2 1+2 γτ+Q < 1, since γτ + Q > 0, from the previous inequality we have . By summing over k from m to n and by using (4.56) we have (4.61) By using (4.61), we compute This yields From (4.55) and (4.62), we have (4.64) By using this inequality in (4.63) with s = τ a Inequality (4.47) is proved.

Lyapunov inequalities on homogeneous groups
In this section we give an application of the preceding results to derive a Lyapunov type inequality for the fractional p-sub-Laplacian with homogeneous Dirichlet boundary condition on homogeneous groups. First, we summarize the basic results concerning the classical Lyapunov inequality.
1. In [Lya07], Lyapunov obtained the following result for the one-dimensional homogeneous Dirichlet boundary value problem. Consider the second-order ordinary differential equation has a non-trivial solution u for ω ∈ L 1 (a, b), (4.83) Obviously, taking p = 2 in (4.83), we recover (4.81). 3. In [JKS17], the following Lyapunov inequality was obtained for the multidimensional fractional p-Laplacian (−Δ p ) s , 1 < p < ∞, s ∈ (0, 1), with a homogeneous Dirichlet boundary condition, that is, for the equation where Ω ⊂ R N is a measurable set, 1 < p < ∞, and s ∈ (0, 1). More precisely, let ω ∈ L θ (Ω) with N > sp, N sp < θ < ∞, be a non-negative weight. Suppose that the problem (4.84) has a non-trivial weak solution u ∈ W s,p 0 (Ω). Then we have where C > 0 is a universal constant and r Ω is the inner radius of Ω.
The appearance of the inner radius in (4.85) motivates one to define its analogue also in the setting of homogeneous groups.
Definition 4.6.2 (Inner quasi-radius). Let p > 1 and s ∈ (0, 1) be such that Q > sp. Let Ω ⊂ G be a Haar measurable set. We denote by r Ω,q the inner quasi-radius of Ω, that is, r Ω = r Ω,|·| := sup{|x| : x ∈ Ω}. (4.86) Clearly, the exact values depend on the choice of a homogeneous quasi-norm | · |. As before, if the quasi-norm is fixed, we can omit it from the notation.

Lyapunov type inequality for fractional p-sub-Laplacians
We now turn our attention to the Lyapunov inequalities for the fractional psub-Laplacian (−Δ p ) s from Definition 4.1.1. Let us consider the boundary value problem where ω ∈ L ∞ (Ω). A function u ∈ W s,p 0 (Ω) is called a weak solution of the problem (4.87) if we have Theorem 4. 6

.3 (Lyapunov inequality for fractional p-sub-Laplacian). Let G be a homogeneous group of homogeneous dimension Q.
Let Ω ⊂ G be a Haar measurable set. Let ω ∈ L θ (Ω) be a non-negative function with Q sp < θ < ∞, and with Q > ps. Suppose that the boundary value problem (4.87) has a non-trivial weak solution u ∈ W s,p 0 (Ω). Then we have Proof of Theorem 4.6.3. We fix a homogeneous quasi-norm | · | thus also eliminating it from the notation. Let α := θ−θ/sp θ−1 ∈ (0, 1) and let p * be the Sobolev conjugate exponent as in Theorem 4.3.1. Let us define Let β = pθ with 1/θ + 1/θ = 1. Then we have On the other hand, the Hölder inequality with exponents ν = α −1 and its conju- Further, by using Theorem 4.3.1 and Theorem 4.2.1, we get That is, we obtain Thus, from (4.89) we have Finally, we arrive at C Theorem 4.6.3 is proved.
As an application of the Lyapunov inequality, let us consider the spectral problem for the (nonlinear) fractional p-sub-Laplacian (−Δ p ) s , 1 < p < ∞, s ∈ (0, 1), with the Dirichlet boundary condition: (4.90) We define the corresponding Rayleigh quotient by . (4.91) Clearly, its precise value may depend on the choice of the homogeneous quasi-norm | · |. As a consequence of Theorem 4.6.3 we have where C is a positive constant given in Theorem 4.6.3 and |Ω| is the Haar measure of Ω.

Lyapunov type inequality for systems
In the previous section we have presented the Lyapunov type inequality for the fractional p-sub-Laplacian with the homogeneous Dirichlet condition. Now we discuss the Lyapunov type inequality for the fractional p-sub-Laplacian system for the homogeneous Dirichlet problem.
Now we present the following analogue of the Lyapunov type inequality for the fractional p-sub-Laplacian system on G.
Theorem 4.6.6 (Lyapunov inequality for fractional p-sub-Laplacian system). Let G be a homogeneous group of homogeneous dimension Q. Let s i ∈ (0, 1) and p i ∈ (1, ∞) be such that Q > s i p i with i = 1, . . . , n. Let ω i ∈ L θ (Ω) be a nonnegative weight and assume that If (4.94)-(4.95) admits a non-trivial weak solution, then we have (4.97) where C is a positive constant.
Proof of Theorem 4.6.6. Set and where p * i = Q Q−sipi is the Sobolev conjugate exponent as in Theorem 4.3.1. Notice that for all i = 1, . . . , n we have γ i ∈ (0, 1) and ξ i = p i θ , where θ = θ θ−1 . Then for every i we have and by using the Hölder inequality with ν i = 1 γi and 1 νi + 1 (4.99) On the other hand, from Theorem 4.3.1, we obtain for every i = 1, . . . , n. Therefore, by using the Hölder inequality with exponents θ and θ , we obtain Again by using the Hölder inequality and (4.93), we get That is, we have Thus, for every e i > 0 we have This implies where C is a positive constant. Then, we choose e i , i = 1, . . . , n, such that αi n j=1 ej pi − e i = 0. Consequently, from (4.93) we have the solution of this system (4.101) Combining (4.100), (4.98) and (4.101) we arrive at (4.102) Theorem 4.6.6 is proved.
In order to discuss an application of the Lyapunov type inequality for the fractional p-sub-Laplacian system on G, we consider the spectral problem for the system of fractional p-sub-Laplacians: where Ω ⊂ G is a Haar measurable set, ϕ ∈ L 1 (Ω), ϕ ≥ 0 and s i ∈ (0, 1), p i ∈ (1, ∞), i = 1, . . . , n.

Lyapunov type inequality for Riesz potentials
In this section we discuss the Lyapunov type inequality for the Riesz potential operators on homogeneous groups. As an application, we discuss a two-sided estimate for the first eigenvalue of the Riesz potential.
Definition 4.6.9 (Riesz potentials). Let G be a homogeneous group of homogeneous dimension Q with a quasi-norm | · |. The Riesz potential on a Haar measurable set Ω ⊂ G is the operator given by the formula (4.105) The (weighted) Riesz potential is defined by (4.106) A Lyapunov type inequality for the weighted Riesz potentials can be formulated as follows.
Let us now consider the following spectral problem for the Riesz potential: (4.110) We recall the Rayleigh quotient for the Riesz potential: where λ 1 (Ω) is the first eigenvalue of the Riesz potential. A direct consequence of Theorem 4.6.10 is the following estimate for this first eigenvalue.
Euclidean case. Let us now record several applications of the above constructions in the case of the Abelian group (R N , +). With the Euclidean distance | · | E , the Riesz potential is given by (4.114) and the weighted Riesz potential is (4.115) Then, in Theorem 4.6.10, setting G = (R N , +) and taking the standard Euclidean distance instead of the quasi-norm, we obtain Theorem 4.6.12 (Euclidean Lyapunov inequality for the Riesz potential). Let Ω ⊂ R N be a measurable set with |Ω| < ∞, 1 < p < 2 and let N ≥ 2 > 2s > 0. Let .

Hardy inequalities for fractional sub-Laplacians on stratified groups
In this section we discuss the Hardy inequalities involving fractional powers of the sub-Laplacian from a different point of view. First, we observe another way of writing the well-known one-dimensional L p -Hardy inequality Consequently, one can replace f (x) by xf (x) and restate this inequality in terms of the boundedness of the operator or of its dual operator In this section, we discuss this point of view and its extension to the subelliptic setting. For fractional powers of the sub-Laplacian on stratified groups this has been analysed in [CCR15], with subsequent extensions to more general hypoelliptic operators and more general graded groups in [RY18a]. We discuss this matter in the spirit of the former.

Riesz kernels on stratified Lie groups
In this section we briefly recapture some properties of the Riesz kernels of the sub-Laplacians on stratified Lie groups. Historically, these have been consistently developed in [Fol75]. For a detailed unifying description containing also higherorder hypoelliptic operators on general graded groups, their fractional powers, and the corresponding Riesz and Bessel kernels we refer to the exposition in [FR16, Section 4.3] where one can find proofs of most of the properties described in this section.
The analysis of second-order hypoelliptic operators, including the sub-Laplacian, is significantly simpler than that of higher-order operators. This difference is based on the Hunt theorem [Hun56] which asserts that the semigroup e −tL t>0 generated by the sub-Laplacian L (or more precisely, by its unique self-adjoint extension) consists of convolutions with probability measures. Moreover, the hypoellipticity of L implies that these measures are absolutely continuous with respect to the Haar measure and have densities in the Schwartz space [FS82], which are strictly positive by the Bony maximum principle [Bon69]. Therefore, for all t ∈ R + , there exists the heat kernel of L, that is, a function p t ∈ S(G) such that for all x ∈ G and all f ∈ L 2 (G). Since L is homogeneous of order two with respect to the dilations D r of a stratified group G, we also have for all x ∈ G, where P = p 1 , which this is a strictly positive function in the Schwartz class S(G) and G P (x)dx = 1.
Let us define the following fractional integral kernel F α on G\{0} by which converges absolutely and uniformly on compact subsets of G\{0} to a smooth function, homogeneous of degree α − Q. Since P is positive F α is positive when α is real. Thus, the function | · | α , given by the formula is non-negative and homogeneous of degree 1, and vanishes only at the origin. So | · | α is a homogeneous norm, and for further discussions it will be convenient to use the norm | · | 0 , which is a limit of the norms | · | α . The fractional integral F α (x) in (4.117) is holomorphic with respect to α in H Q := {α ∈ C : Re α < Q}, and its derivatives are the absolutely convergent integrals. The functions F α are smooth and homogeneous of degree α − Q and locally integrable on G when 0 < Re α < Q. Thus, the associated distributions are defined by for all φ ∈ C ∞ 0 (G). Now it is clear that I α := Γ(α/2) −1 F α is the Riesz kernel of order α, i.e., the convolution kernel of L −α/2 , that is, with The family of Riesz kernels can be analytically continued as distributions to the half-plane H Q by the identity The analytic continuation to the strip {α ∈ C : −1 < Re α < Q} will be enough for our purpose. An explicit expression is obtained by rewriting (4.119) in the form (4.122) Since Γ (1) = −γ and hence and the equality (4.122) implies that for α near 0, for φ ∈ C ∞ 0 (G) we have where Λ is the distribution defined by (4.125) Thus, (4.124) is the Taylor expansion of I α around 0. By the analytic continuation of (4.122) one obtains the following representation of the distribution Λ in terms of the homogeneous norm | · | 0 defined in (4.118): There exists a > 0 such that (4.126) for all φ ∈ C ∞ 0 . Let 0 < Re α, 0 < Re β and Re(α + β) < Q. Then the convolution of the Riesz potentials of orders α and β is defined pointwise by absolutely convergent integrals as well as I α * I β = I α+β (4.127) is satisfied pointwise and in the sense of distributions. This identity is equivalent to the functional equality L −α/2 L −β/2 = L −(α+β)/2 and can be obtained from (4.117) by using the properties of the heat kernel. This fact can be also extended by using the analytical continuation. That is, holds if Re α > −1, Re β > −1 and Re(α + β) < Q. These distributions will be useful to present a family of Hardy type inequalities in the following sections. We refer to [FR16, Section 4.3] for the detailed discussion of these and other properties of the Riesz kernels and fractional powers of left invariant hypoelliptic operators.

Hardy inequalities for fractional powers of sub-Laplacians
As in this whole section, let L be the sub-Laplacian on the stratified Lie group G of homogeneous dimension Q. Let us define the operator T α on C ∞ 0 (G) by the formula Consequently, the operator for all f, g ∈ C ∞ 0 (G). It turns out that the operator T α is bounded on L p (G) when 1 < p < ∞ and 0 < α < Q/p which, in turn, can be formulated as a version of a Hardy inequality: Theorem 4.7.1 (Hardy inequality for fractional powers of sub-Laplacian). Let G be a stratified Lie group of homogeneous dimension Q. Let 1 < p < ∞ and 0 < α < Q/p. Then the operator T α extends uniquely to a bounded operator on L p (G).
Applying this to L α/2 f rather than f we get that for all f ∈ C ∞ 0 (G), we have f |x| α (4.129) Remark 4.7.2 (Hardy-Sobolev inequalities).
1. Theorem 4.7.1 was established in [CCR15]. It is easy to see that combined with the Sobolev inequality, it implies the following Hardy-Sobolev inequality: Then there exists a positive constant C > 0 such that we have (4.130) Such an inequality can be interpreted as a weighted Sobolev embedding of the homogeneous Sobolev spaceL p a (G) over L p of order a, based on the sub-Laplacian L. The theory of such spaces has been extensively developed by Folland [Fol75]. We refer to [RY18a] for the inequality (4.130). 2. The asymptotic behaviour of T α L p (G)→L p (G) with respect to α will be given in Theorem 4.7.3; in its proof we will follow the original proof in [CCR15], as well as for Theorem 4.7.4. 3. (Critical global Hardy inequality for a = Q/p) Let 1 < p < r < ∞ and p < q < (r − 1)p , where 1/p + 1/p = 1. Then there exists a positive constant C = C(p, q, r, Q) > 0 such that we have (4.132) and such that lim q→∞ sup C(p, Q, β, r, q) < ∞.
Inequalities (4.131) and (4.132) have been obtained in [RY18a], to which we refer for their proofs as well as for the expressions for the asymptotically sharp constants in these inequalities. 5. (Trudinger-Moser inequality on stratified groups) Let Q ≥ 3 and let | · | be a homogeneous quasi-norm on G. Then there exists a constant α Q > 0 such that we have for any β ∈ [0, Q) and α ∈ (0, α Q (1 − β/Q)), where Q = Q/(Q − 1). When α > α Q (1 − β/Q), the integral in (4.133) is still finite for any f ∈ L Q 1 (G), but the supremum is infinite.
. The inequality (4.133) was obtained in [RY18a], also with a rather explicit expression for the constant α Q . We refer to the above paper also for an extensive history of this subject.
Theorem 4.7.3 (Estimate for L p -operator norm). Let G be a stratified Lie group of homogeneous dimension Q. Let 1 < p < ∞ and 0 < α < Q/p. For the particular homogeneous norm | · | 0 , the operator norm T α L p (G)→L p (G) satisfies For the rest of this subsection, we assume that |·| = |·| 0 and that the operator T α is defined using | · | 0 .
Proof of Theorem 4.7.3. The proof of Theorem 4.7.1, and especially (4.135), show that, if (4.140) First let us show that there is a constant C p such that holds for all sufficiently small positive α. Assume that Q/p < γ < Q. Then (4.138) is valid for α in a neighborhood of 0 + . Moreover, with β and β as in (4.136), the constants A α,γ,p and B α,γ,p in (4.140) are bounded by 1 + L β α + O(α 2 ) and 1 + L β α + O(α 2 ), respectively. By (4.121) we obtain which confirms (4.141). Combining this with Theorem 4.7.1 completes the proof of Theorem 4.7.3.
Theorem 4.7.4 (A logarithmic version of uncertainty principle). Let G be a stratified Lie group and let 1 < p < ∞. Then we have Proof of Theorem 4.7.4. Now let us recall the distribution Λ introduced in (4.125).

Landau-Kolmogorov inequalities on stratified groups
In this section we discuss the stratified groups version of the Landau-Kolmogorov inequality, and some of its consequences. We start with a related simpler version of such an inequality.
Now we present the following Landau-Kolmogorov type inequality.
Theorem 4.7.6 (Landau-Kolmogorov type inequality). Let G be a stratified group and let L be a sub-Laplacian on G. If 1 < p, q, r < ∞, and then we have for all f ∈ C ∞ 0 (G). Proof of Theorem 4.7.6. To prove this theorem we will use the well-known complex interpolation methods (see, e.g., [BL76] or [Ste56]). First, we have to see that the operator L iy , where y ∈ R, is bounded on L s (G) with 1 < s < ∞, and that L iy/2 L s →L s ≤ C(s)φ(y) with φ(y) = e γ|y| . = 1.
Further, let us fix f ∈ C ∞ 0 (G) and set h(z) := e z 2 G L z/2 f (x)g z (x)dx.
By the assumptions on f , for each z ∈ S, the function L z/2 f on G is smooth, while g z is a simple function with compact support, and so h(z) is well defined. In addition, if z = η + iy, then |h(z)| ≤ e η 2 −y 2 L z/2 f L s (G) g z L s (G) = e η 2 −y 2 L iy/2 L η/2 f L s (G) ≤ Ce −y 2 φ(y) L η/2 f L s (G) , Moreover, if Re z = 0, then |h(z)| ≤ C f L r (G) , while if Re z = α/θ, then |h(z)| ≤ C L α/2θ f L q (G) . On the other hand, the Phragmen-Lindelölf theorem implies that Since h(α) = G L α/2 f (x)g α (x)dx, and g α is an arbitrary simple function on G with compact support and L p (G)norm equal to 1, this completes the proof.
Let us present the following consequence of the Landau-Kolmogorov inequality.
This completes the proof.
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