CYLINDRICAL HARDY INEQUALITIES ON HALF-SPACES

. We study some versions of the cylindrical Hardy identities and inequalities in the style of Badiale-Tarantello [2]. We show that the best constants of the cylindrical Hardy inequalities can be improved when we consider functions on half-spaces.


Introduction
The main subject of this note is the celebrated Hardy inequality on R N , N ≥ 3: for u ∈ C ∞ 0 (R N ): with optimal constant ( N −2 2 ) 2 .Because of their important roles in many areas of mathematics, the Hardy type inequalities have been well-studied and there is a vast literature.See the monographs [3,25,28,29,40], for instance, that are typical references on the topic.
It is well-known that ( N −2 2 ) 2 in (1.1) is never achieved by nontrivial functions.Therefore, many efforts have been devoted to enhance the Hardy inequalities.One way to do so is to add extra nonnegative terms to the right-hand side of (1.1).The first result in this direction was established in [8] where Brezis and Vázquez proved that for u ∈ W 1,2 0 (Ω).Ω is a bounded domain in R N , N ≥ 3, with 0 ∈ Ω, it holds Here ω N is the volume of the unit ball and z 0 = 2.4048 . . . is the first zero of the Bessel function J 0 (z).The constant z 2 0 ω 2/N N |Ω| −2/N is optimal when Ω is a ball.However, Hence, Brezis and Vázquez also conjectured that dx is just a first term of an infinite series of extra terms that can be added to the right-hand side of (1.2).This question was investigated by many authors.We refer the interested reader to [1,4,9,10,11,12,18,21,22,23,26,37,45,46], to name just a few.
Ghoussoub and Moradifam [24,25] proved the following result to improve, extend and unify several results about the Hardy type inequalities: ) and some c 2 > 0.Here we say that a couple of C 1 -functions (V, W ) is a Bessel pair on (0, R) if the ordinary differential equation has a positive solution on the interval (0, R).See the book [25] for more properties and examples about the Bessel pair.
Another line of research on the improvements of the Hardy type inequalities is to replace the usual ∇ by R := x |x| • ∇.It can be noted that Ru is the radial gradient of u.Indeed, in the polar coordinate, |Ru| = |∂ r u(rσ)| while Actually, the radial derivation plays an important part in the literature.The interested reader is referred to [42] for the roles of the radial derivation R in the functional and geometric inequalities on homogeneous groups.We also mention here that the Hardy type inequalities with radial gradient have been intensively studied recently.See [13,14,15,16,27,30,31,39,41,42,43], for example.
In an effort to unify many results about the Hardy type inequalities with radial derivation, and to compute the exact remainders of the Hardy type inequalities, the authors in [15] have proved the following result.
In [2], for investigating the existence and nonexistence of cylindrical solutions for a nonlinear elliptic equation that has been proposed as a model describing the dynamics of elliptic galaxies, Badiale and Tarantello established the following cylindrical Hardy type inequality, The optimal constant C N,k,p = ( k−p p ) p was also conjectured in [2] and then verified in [44].
Recently, in [17,31], the following result about the cylindrical Hardy type inequalities with Bessel pairs has been set up.
Here ϕ is the positive solution of Because of their geometric meaning, Hardy's inequalities have been also studied extensively on the half-spaces R N + = {x ∈ R N : x 1 > 0}.For instance, Hardy's inequalities with distance to the boundary have been investigated in [6,7,19,32,33], to name just a few.Improved Hardy type inequalities on half-spaces have also been set up in, for instance, [5,34,35,36].
It is interesting to note that when one restricts the domain to R N + , the best constant of the Hardy inequality can be improved.Indeed, we have the Hardy inequality on half-space (see, e.g., [24,38]) Here the constant (N/2) 2 is optimal.However, if we concern the Hardy inequality with radial derivation R on R N + , then it is interesting to note that the best constant is still ((N − 2)/2) 2 .Actually, in [32], the authors showed the following identities to provide a simple interpretation of the aforementioned phenomenon, a direct understanding of the Hardy inequality on half-spaces (1.4) as well as the "virtual" ground state in the sense of Frank and Seiringer [20] More generally, the authors in [32] used the factorizations of suitable differential operators to study a version of Theorem 1.
R the ball centered at 0 with radius R on R k .Then we have the following result in [32].
Here ϕ r N +1 V,r N +1 W ;R is the positive solution of on the interval (0, R).
Motivated by the cylindrical Hardy type inequalities studied in [2,17,31], and the Hardy type inequalities on half-spaces in [32], our principal goal of this paper is to investigate the cylindrical Hardy type inequalities with Bessel pairs and with exact remainder terms on R N + .More precisely, let Our main result reads as follows.
In Section 4, we will present some cylindrical Hardy type inequalities on halfspaces as consequences of our main result.Actually, we can obtain as many cylindrical Hardy type inequalities as we can construct Bessel pairs.For several examples of the Bessel pairs, the interested reader is referred to [25].

Some useful calculations
and We also have (2.3)

Proof of main results
Proof of Theorem 1.5.
. Using (2.1) and polar coordinates we have By integrations by parts, we obtain On the other hand, using polar coordinate agains, we have We note that by Theorem 1.3, Hence, when we restrict the domain to half-spaces, the optimal constant of the cylindrical Hardy inequality has been improved from x 2 1 dx.
We note again that by Theorem 1.3, Hence, in this case, the sharp constant of the cylindrical Hardy type inequality has been improved from ( k−2−α 2 ) 2 to ( k−α 2 ) 2 + α when we consider the functions on half-spaces.Now, since (r k+1 1 r k , r k+1  Email address: huy.bac.nguyen.st@vsb.cz

Corollary 4 . 3 .
For u ∈ C ∞ 0 ({0 < |y| < R} ∩ R N + ):These versions of the cylindrical critical Hardy inequalities on half-spaces seem new in the literature.Huy Bac Nguyen (Corresponding Author) Faculty of Electrical Engineering & Computer Science, Technical University of Ostrava, Czech Republic