A note on the fractional Hardy inequality

We give a direct proof of fractional Hardy inequality by means of Littlewood–Paley decomposition and properties of singular homogeneous kernels of degree -d. A refinement when q>2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q>2$$\end{document} is proved.

The classical Hardy inequality states that when d ≥ 3 (0.1) and it is clearly of fundamental importance in analysis.There are of course many different proofs of (0.1), the simplest one consists in restrict by density to D(R d \ {0}), to observe that ), then to integrate by parts and eventually to apply Cauchy-Schwarz inequality.A natural extension of (0.1) is in the framework of fractional Sobolev spaces Ḣs (R d ).In this setting the following Hardy-type inequality holds (0.2) provided that 0 ≤ s < d 2 .For a compact and nice proof of (0.2) we quote Theorem 2.57 in [1] and the proof given by Tao in the Appendix of [15] while for an improvement involving Besov spaces we quote [2].
If one is interested in proving an L q estimate for |f | |x| s we need to recall the definition of the homogeneous Sobolev norm ||f || Ẇ s,q (R d ) which is defined as |||D| s f || L q (R d ) where ( |D| s f )(ξ) = |2πξ| s u(ξ).In this note we give a direct proof and a refinement when q > 2 for the following class of Hardy-type inequalities that generalize the fractional Hardy inequality (0.2).
Theorem 0.1 (Fractional Hardy inequality).Let 0 < s < d q , 1 < q < ∞ and f ∈ Ẇ s,q (R d ), then The explicit value of the constant C(d, s, q) in (0.3) is due to Herbst [11].The proof of (0.3) goes back to the end of the fifties of the last century thanks to the work of Stein and Weiss [14] who proved an even more general version of (0.3) called Stein-Weiss inequality given by (0.4) where The fact that (0.4) implies (0.3) follows by the fact that and choosing p = q and α = 0, β = s.
In order to state our result we recall the standard definition for Homogeneous Besov norm || • || Ḃs p,q and Tribel-Lizorkin norm || • || Ḟ s p,q (see e.g.[8] for general references).Let f be a tempered distribution such that f ∈ L 1 loc and P N (f ) the Littlewood-Paley projector on the dyadic frequency N, i.e.
Our result is a direct proof of the following Theorem 0.2.Let 0 < s < d q , 1 < q < ∞ then ) can be controlled by homogeneous Besov norms is not a novely, a proof of Theorem 0.2 can be found in [17], see also [18].Here we present a direct proof using the Shur test.We shall remark that our corollary when q > 2 is a refinement of Hardy inequality (0.3).Indeed we have when 2(q − 1) > 2 thanks to square function estimate The case 1 < q < 2 is proved by duality and it requires proving the L q continuity for singular homogeneous kernels of degree -d.This fact is well known and is Lemma 2.1 in [14].We underline however that our strategy in proving Theorem 0.2 permits to skip the more delicate lemmas in the Stein and Weiss paper [14] that are needed to prove (0.3).As a final comment, recalling that |D|f We underline that Corollary 0.2 is a refinement of the classical Hardy inequality involving ∇f In the literature there is a lot of interest in proving improvements for (0.9), typically such improvement (in bounded or unbounded domains) are in the direction to add a negative term in r.h.s of (0.9), see e.g.[3,4,5,6,7,9,10,12].Our refinement, although obtained with different techniques, is more in the spirit of [2] and [16], i.e. to control r.h.s. of (0.9) with terms that are smaller (up to a multiplicative constant) than the Sobolev norms.
1. Proof of Theorem 0.2 A key argument in our proof is given by the following well known version of Shur test provided there exists a sequence of positive numbers p N such that that, thanks to (1.1) and Fubini, implies The strategy of the proof for is an adaptation of proof of Hardy inequality in the case q = 2 given by Tao [15], i.e. to prove the following estimate where P N f are the classical Littlewood-Paley projectors with N a dyadic number.
We devide R d in dyadic shells obtaining such that using the Littlewood-Paley decomposition we get (1.5) By the Bernstein inequality and clearly such that we get The last step is to apply the Schur test given by Proposition 1.1 in order to conclude that Notice that such that (arguing in the same way when summing over R) N min{(NR) −s , (NR) The hypoteses for Shur test given by Proposition 1.1 are hence fulfilled by choosing α N,R = min{(NR) −s , (NR) d q −s } and p N = 1 in Proposition 1.1.This proves (0.3).

Proof of Corollary 0.1
In Theorem 0.2 we proved the following estimate (2.1) where P N f are the classical Littlewood-Paley projectors with N a dyadic number.First we prove that (2.1) implies the Fractional Hardy inequality.We have two cases: q ≥ 2, q < 2. Case q ≥ 2: Thanks to (2.1) we derive from the elementary inequality ( i a p 1 i ) where the last equivalence is nothing but the classical square function estimate, see for instance [13].
To prove (0.7) we notice that by applying twice the Holder's inequality, first in the serie with conjugated exponent (2, 2) and then in the integral with conjugated exponent (q, q q−1 ).By definition .
Case q < 2: For the case q < 2 we use the dual characterization of L q norms, i.e.
|x| s = sup for all g ∈ L q ′ with q ′ > 2 such that we could conclude that Now we prove (2.2).We have (skipping q ′ with q to simplify the notation) By previous estimates using Paley-Littlewood decompostion and the square function equivalence we get when q > 2 Concerning S 2 (g) L q we follow the strategy of Stein and Weiss in [14] proving the L q continuity for singular homogeneous kernels of degree -d.The proof of this fact is Lemma 2.1 in [14]   To conclude the proof it suffices hence to show that ˆRd |Ug| q dx ˆ|g| q dx.
Fixing η ∈ S d−1 and calling |x| = R we define By the substitution r = tR we obtain Let h be the function in L q ′ ((0, +∞); R d−1 dR) of unitary norm such that , where the last integral J converges due to the fact that by our assumptions s < d q ′ (remember that we skipped q ′ with q).Now we estimate L q (R d ) norm of Ug.By Jensen inequality |Ug(R)| q = ˆSd−1 |U η g(R)| dσ η q ≤ {|S d−1 |} q−1 ˆSd−1 |U η g| q dσ η , such that integrating with respect to the measure R d−1 dR we get By the fact that Uf (x) is radial we can conclude that This concludes the proof in the case q < 2.

DECLARATIONS
Conflict of interest.The authors declare that they have no conflict of interest.