A curl-free improvement of the Rellich–Hardy inequality with weight

1.1 Preceding results and motivation

It is well known that the Hardy–Leray inequality

holds for all vector fields u ∈ D γ ( R N ) with the best constant on the left-hand side; when the integral on the right-hand side diverges (namely ∫ R N | ∇ u | 2 | x | 2 γ d x = ∞ ), we understand that inequality (1) holds in the sense that the left-hand this is less than ∞. Historically, this inequality was first proved by Leray [1] for (N, γ) = (3, 0), as a higher-dimensional extension of Hardy’s inequality in one dimension [2], see also the book by Ladyzhenskaya [3]. Costin and Maz’ya [4] improved the best value of the constant by assuming u to be divergence-free (under the additional assumption of axisymmetry for N ≥ 3): for the case N = 2, they showed that the inequality

holds with the best constant C γ = 3 + ( γ − 1 ) 2 1 + ( γ − 1 ) 2 γ 2   | γ + 1 | ≤ 3 γ 2 + 1   | γ + 1 | > 3 for divergence-free vector fields u ∈ D γ ( R 2 ) . Since there is an isometry on R 2 between divergence-free fields and curl-free fields, the same conclusion also applies to the two-dimensional curl-free fields. As a generalization of this result, we have derived in recent papers [5], [6] the Hardy–Leray inequality

for curl-free fields u ∈ D γ ( R N ) with the best constant

Since C _γ = H_2,γ, this result recovers Costin-Maz’ya’s one for N = 2.

On the other hand, the Rellich–Leray inequality is given by

for unconstrained fields u ∈ D γ − 1 ( R N ) , where the constant B_N,γ is sharp when

This was found by Rellich [7] for γ = 0 and Caldiroli-Musina [8] for γ ≠ 0. In recent papers [5], [6], we additionally considered the curl-free improvement of the Rellich–Leray inequality: if u ∈ D γ − 1 ( R N ) is assumed to be curl-free, then the same inequality (4) holds with the best constant

here and hereafter we use the notation

for any s ∈ R .

In this paper, we are interested in another version of Rellich–Leray inequality:

holds with the best constant A N , γ = min ν ∈ N ∪ { 0 } A N , γ , ν , where

We call (8) the Rellich–Hardy inequality. This inequality was first found for N ≥ 5 by Tertikas-Zographopoulos [9], Theorem 1.7]. Subsequently, Beckner [10] and Ghoussoub-Moradifam [11] established the same inequality when N ∈ {3, 4} and γ = 0, with the best constants A 3,0 = 25 36 and A_4,0 = 3. See also Cazacu [12] for the unified proof of the inequality when γ = 0.

Here let us consider the special case where γ satisfies

Then we see that a successive application of Rellich–Hardy and Hardy–Leray inequalities reproduces Rellich–Leray inequality (4): we have

with B_N,γ given by (5). Hence, Rellich–Hardy inequality can be considered as a stronger version of Rellich–Leray inequality and plays a role as an intermediate between Rellich–Leray and Hardy–Leray inequalities.

We have mentioned inequalities (1), (5) and (8) in the context of unconstrained vector fields; this is essentially the same as in the context of scalar fields in the sense that the optimal constants remain unchanged in both contexts. For further details of important contributions and developments on the optimality of these types of inequalities in the scalar case, we refer the reader to [13], [14], [15], [16], [17]. In the context of constrained vector fields such as curl-free fields, however, there seems less developments on the optimality of Rellich–Hardy type inequalities. Then we ask how the best contant in (8) will change when u is assumed to be curl-free; this is the main interest of our present problem.

As a side note, it is also worth mentioning that in a recent paper [18] the best constant of Rellich–Hardy inequality was computed for divergence-free (instead of curl-free) fields.

1.2 Results

Motivated by the observation above, we aim to derive the best constant in Rellich–Hardy inequality for curl-free fields. Now, our main result reads as follows:

In addition, we obtain a stronger inequality by adding a remainder term to the right-hand side of (10).

As a direct consequence of Theorem 3, we can conclude that the best constant C_N,γ of the inequality (10) is never attained in D γ − 1 ( R N ) \ { 0 } :

As another direct consequence, by expressing Theorem 3 in terms of scalar potentials of curl-free fields, we have the following fact:

1.3 Overview of the remaining content of the present paper

The rest of this paper is organized as follows: Section 2 provides basic notations and definitions, and reviews a representation of curl-free fields. Section 3 gives the proof of Theorem 1: we recall from [6] the scalar-potential expression of L² integrals of curl-free fields; after that, we derive Lemma 9 as a key tool for evaluating the ratio of the two integrals in (10), which also plays a computational part in the proof of Theorem 3. The proof of Lemma 9 is separated into two cases. Since both the cases use similar techniques and consist of long calculations, we prove only one case in the same section, and postpone the other case in Section 6. Section 4 proves Theorem 3 by using an operator-polynomial representation of Rellich–Hardy integral quotient and by making full use of Lemma 9. Section 5 observes curl-free improvement phenomena of best constants in some cases.

2.1 Notations and definitions in vector calculus on R ̇ N

Here we summarize the minimum required notations and definitions for the proof of our main theorems. We basically use the notation

For every vector x ∈ R ̇ N , the notation

denotes the radius of x and its unit-vector part, which defines the smooth transformation

together with its inverse

Every vector field u = ( u 1 , u 2 , … , u N ) : R ̇ N → R N has its radial scalar component u _R = u _R ( x ) and spherical vector part u _S = u _S ( x ) given by the formulae

for all x ∈ R ̇ N ; the two fields are explicitly given by

In a similar way, the gradient operator ∇ = ∂ ∂ x 1 , … , ∂ ∂ x N can be decomposed into the radial derivative ∂ _r and the spherical gradient ∇ _σ as

in order that ∂ _r f = (∇f) _R = σ ⋅ ∇f and 1 r ∇ σ f = ( ∇ f ) S hold for all f ∈ C ∞ ( R ̇ N ) . The radial derivative operator which we denote by

allows us to express the gradient operator by the formula

The Laplace operator △ = ∑ k = 1 N ∂ 2 / ∂ x k 2 is known to be expressed in terms of (r, σ ) as

where △ _σ denotes the Laplace–Beltrami operator on S N − 1 . We understand that the action of the operator ∂ _r or ∂ on a vector field u is associated with the function r ↦ u (r σ ) for σ ∈ S N − 1 fixed, whereas ∇ _σ or △ _σ is associated with the function σ ↦ u (r σ ) for r = | x | fixed. As a simple example, the operation of ∇ and △ on the scalar field r = | x | or its powers gives ∇r = σ and △r ^s = α _s r^s−2 for all s ∈ R , where α _s is the same as in (7).

2.2 Radial-spherical-scalar representation of curl-free fields

Every vector field u ∈ C ∞ ( R N ) N is said to be curl-free if

or equivalently if there exists a scalar field ϕ ∈ C ∞ ( R N ) satisfying

In view of this equation, we say that u has a scalar potential ϕ. As another representation of curl-free fields, let us recall the following fact:

Later, we will use Proposition 8 by choosing λ = 2 − N 2 − γ . (See (21)).

3.1 Reduction to the case of compact support on R ̇ N

Let ϕ be a smooth scalar potential of a curl-free field u in D γ − 1 R N satisfying ϕ(0) = 0. By considering Taylor expansion of ϕ( x ) at x = 0, one can check from the integrability condition ∫ R N | u | 2 | x | 2 γ − 4 d x < ∞ that there exists an integer m > 2 − N 2 − γ satisfying

as x → 0. Here and hereafter, for any scalar- or vector-valued functions f (x) and positive-valued functions g(x) of (vector or scalar) variable x, the notation “ f (x) = O(g(x)) as x → a” means that

Then it additionally follows that the integrals

are all finite.

For the purpose of deriving the best constant C_N,γ in inequality (10), it is enough to consider the case where the curl-free field u = ∇ϕ is compactly supported on R ̇ N . Here let us verify this fact. First of all let us define { u n } ⊂ C c ∞ ( R ̇ N ) N as a sequence of curl-free fields by

where ζ ∈ C ∞ ( R ) such that ζ ( t ) = 0  for  t ≤ − 1 1  for  0 ≤ t . We use the abbreviations such as ζ n = ζ 1 n log | x | , ζ n ′ = ζ ′ 1 n log | x | , and ζ n ′ ′ = ζ ′ ′ 1 n log | x | . Notice that the asymptotic formulae

hold as n → ∞. Hereafter, whenever we write such as

for functions f , g and h of ( x , n) and positive-valued functions ξ and η of n, it means that there exist positive constant numbers C and n₀ independent of ( x , n) and functions F and G of ( x , n) satisfying f = g F + h G and

for all n ≥ n₀. Then we have the following calculations:

hold as n → ∞. Therefore, taking the L²(| x |^2γdx) integration yields

with the aid of the integrability conditions (16). This fact shows that the two integrals in (10) can be approximated by curl-free fields with compact support on R ̇ N , as desired.

3.2 Radial- and spherical-scalar expression of the integrals

In the rest of the present section, we use the notation

which serves as an alternative radial coordinate obeying the differential rules

and hence generates the same derivative operator ∂ given in (14). For any real number λ, let f and φ be the scalar fields determined by the curl-free field u , as given in Proposition 8, and we set

as a new vector field in C c ∞ ( R ̇ N ) N . Then the equation

holds on R ̇ N . Here we keep in mind that, by taking the derivative ∂ on both sides of (18) and of (19), one easily get

which are (in this order) the same equations (18) and (19) with v , u , f and φ replaced by ∂ v , ∂ u , ∂f and ∂φ respectively; in other words, the equations (18) and (19) are invariant under the following replacement of the quadruple:

Now, we choose

and let us recall from [6], Section 3.2] that the integral on the right-hand side of the Hardy–Leray inequality (2) can be expressed in terms of ( v , f, φ) as follows:

Here the last equality follows from integration by parts together with the support compactness of v or f, φ. Applying (24) to (20), we also obtain