Sharp Hardy-Leray inequality for solenoidal fields

We compute the best constant in functional integral inequality called the Hardy-Leray inequalities for solenoidal vector fields on $\mathbb{R}^N$. This gives a solenoidal improvement of the inequalities whose best constants are known for unconstrained fields, and develops of the former work by Costin-Maz'ya who found the best constant in the Hardy-Leray inequality for axisymmetric solenoidal fields. We derive the same best constant without any symmetry assumption, whose expression can be simplified in relation to the weight exponent. Moreover, it turns out that the best value cannot be attained in the space of functions satisfying the finiteness of the integrals in the inequality.


Introduction
We study functional inequality called the Hardy-Leray inequality, from the viewpoint that how the best value of the constant changes when we assume that the test functions are vector-valued and solenoidal (namely divergence-free) on Ndimensional Euclidean space R N .
1.1. Basic notation and definitions. Throughout this paper, N denotes an integer and we assume that N ≥ 3 unless otherwise specified. We always use bold letters to denote vectors, e.g., x = (x 1 , x 2 , · · · , x N ) ∈ R N which is an N -dimensional vector. For every open subset Ω of R N , the notation u = (u 1 , · · · , u N ) ∈ C ∞ (Ω) N means that {u 1 , · · · , u N } ⊂ C ∞ (Ω), or equivalently that u : Ω → R N , x → u(x) = (u 1 (x), · · · , u N (x)) is a vector field (namely R N -valued function) with the components belonging to the set C ∞ (Ω) of smooth scalar fields (namely R-valued functions) on Ω. The same also applies to other sets of functions: e.g., the notation u ∈ C ∞ c (Ω) N means that u is a smooth vector field with compact support on Ω.
The symbol ∇ denotes the gradient operator which maps every scalar field f on (a domain of) R N to the vector field ∇f = ∂f ∂x1 , ∂f ∂x2 , · · · , ∂f ∂xN . The same also applies to every vector field u in the componentwise sense: we set ∇u = (∇u 1 , ∇u 2 , · · · , ∇u N ) = ∂u j ∂x k (j,k)∈{1,2,··· ,N } 2 : Ω → R N ×N which is a matrix field. The notation u · v = N k=1 u k v k denotes the standard scalar product of two vector fields, and |u| = √ u · u denotes the modulus of u. The same also applies to matrix fields, e.g., ∇u · ∇v = N j=1 N k=1 ∂u j ∂x k ∂v j ∂x k and |∇u| = √ ∇u · ∇u.

Preceding results and motivation.
Hereafter γ denotes a real number. The classical Hardy-Leray inequality with power weight (or shortly H-L inequality) on R N is given by where the constant number γ + N −2 2 2 is known to be the sharp for unconstrained vector fields u ∈ C ∞ (R N ) N under suitable regularity condition. This inequality was shown firstly for scalar fields by J. Leray [15] for N = 3 and γ = 0 along his study on the Navier-Stokes equations, as an extension of the 1-dimensional inequality by G. H. Hardy [14]. The scalar field version of H-L inequality could be trivially extended to vector fields by applying it to each of the components.
One of the questions of interest is whether the best value of the constant in the H-L inequality can exceed the original one by imposing on u to be solenoidal, that is, Such a problem arises in a natural way in the context of hydrodynamics, as asked by O. Costin and V. G. Maz'ya [3], who derived the H-L inequality for axisymmetric solenoidal fields u with the new constant number which we simply call C-M constant. Here the axisymmetry of a vector field means, in brief, that every component of it along the cylindrical coordinates depends only on the axial distance and the height. Notice that the expression (1.1) can be further simplified as where Indeed, one can check by an elementary numerical calculation that the two numbers (1.1) and (1.3) coincide, see [8] for the details. An advantage of the condition of axisymmetry is that it helps us to easily compute the new best constant, normally without affecting the final result. However, it was also pointed out in [6] that the sharpness of the C-M constant under the axisymmetry condition is not always valid when N ≥ 4; more precisely, it was shown that the best constant for axisymmetric solenoidal fields should be Combining this fact together with the same numerical calculation in [8] or Lemma 4.1, one can precisely check that the case C N,γ = C axis N,γ occurs when γ ∈ I N with N ≥ 4. Nevertheless, we may expect that this gap could be reduced by weakening or excluding the condition of axisymmetry, since such a condition seems to play a technical rather than essential role, in order to purely obtain a solenoidal improvement of H-L inequality.
In view of the observation above, there was an advance in the three-dimensional case: Hamamoto [7] succeeded in deriving the C-M constant of H-L inequality for solenoidal fields u on R 3 without any symmetry assumption at all, after his joint work with Takahashi [10] investigating the same inequality under a weakened axisymmetry condition. Consequently the axisymmetry assumption for N = 3 turned out to be completely removable from the solenoidal improvement of H-L inequality. As a matter of course, it is then expected that the same also applies to the higher dimensional case N ≥ 4 and that the validity of the sharpness of C-M constant could be recovered. This is the main theme of our present study.
As a side note, it is also worthy to consider how the best constants change in H-L and R-H inequalities for curl-free vector fields (instead of solenoidal ones). Some topics related to this problem can be found in the papers [12,13,11].
1.3. Main results. In order to formally state our main theorem, let D γ (R N ) denote the space of all smooth vector fields on R N satisfying the finiteness of the norm: As a matter of fact, one can check that the set of smooth solenoidal fields with compact support on R N \ {0} forms a dense subspace of solenoidal fields in D γ (R N ) with respect to the norm · γ ; for the detailed proof, see e.g. the discussion in [9, §3.1]. Now, our first main result reads as follows: holds with the best constant C N,γ given in (1.3), where the equality holds if and only if u ≡ 0.
Hence the sharpness of the expression of C-M constant turns out to be valid even for N ≥ 4 under no assumption of axisymmetry. Moreover, one may conclude from the theorem that the strict inequality C N,γ > γ + N −2 2 2 holds for all γ = − N −2 2 , which says that the solenoidal improvement (almost) always takes effect.
which reproduces the same result as in [3] with N = 2. Hence, it consequently turns out that Theorem 1.1 is also valid even in the two-dimensional case.

1.4.
An overview of the rest of this paper. A key tool that will be used in the proof of Theorem 1.1 is the so-called poloidal-toroidal decomposition (or shortly PT) theorem, which was used in the previous work [7] for only three-dimensional sharp H-L inequality; our recent work [9] also used PT theorem for sharp Rellich-Leray inequality, by quoting the preprint [5] which corresponds to the present paper.
The PT theorem in our study, which originates from G. Backus [1] on R 3 and serves as a specialized version of N. Weck's [17] on R N , enables us to separate the calculation of the best constant into two computable parts. Some techniques on R 3 , employed in the previous work, are not allowed in the higher-dimensional case: we cannot use the concept of "cross product" of vectors in general R N , and there is no way to represent every toroidal field in terms of a single-scalar potential. To avoid such a difficulty, we derive zero-spherical-mean property of toroidal fields, from which one can readily deduce an appropriate L 2 (S N −1 ) estimate.
In the remaining content of this paper, we will prove the theorem in the organization as follows: Section 2 reviews vector calculus on R N \ {0} in terms of radial-spherical variables. Section 3 gives a systematic introduction to the concept of smooth PT fields and establishes the PT decomposition theorem of solenoidal fields on R N . Section 4 gives the proof of Theorem 1.1, where we compute C N,γ to derive the expression (1.1) by making full use of the PT theorem and FET and show the non-attainability of C N,γ by exploiting the expression (1.3).

Standard Vector Calculus onṘ
We introduce some notations frequently used in this paper and briefly review some basic formulae related to gradient or Laplace operators acting on smooth functions onṘ N , from the viewpoint of radial-spherical variables.
2.1. Radial-spherical decomposition of operators. Hereafter we use the ab-breviationṘ In the sense of differential geometry, this is a smooth manifold diffeomorphic to the product of the half line R + = {r ∈ R ; r > 0} and the unit (N − 1)-sphere S N −1 = x ∈ R N ; |x| = 1 . In our setting, we make the identificatioṅ by the relation x = rσ or equivalently In view of the two equations, we may consider (r, σ) as a pair of positive scalar field and unit vector field onṘ N which are related by the single equation ∇r = σ. Every vector field u = (u 1 , u 2 , · · · , u N ) :Ṙ N → R N has as its components the scalar field u R and vector field u S satisfying which we call the radial-spherical decomposition (or shortly RS decomp.) of u. This pair is unique and specified by the equations The operator denotes the radial derivative resp. its skew L 2 adjoint, in the sense that . When these operators act on vector fields u, the operation is componentwise; it is then easy to check from (2.1) that ∂ r r = 1 and ∂ r σ = 0, and that the radial derivative commutes with the RS decomp. of vector fields.
The operator ∇ σ is defined by which we call the spherical gradient onṘ N . It is easy to check that ∇ σ commutes with ∂ r as well as r. Applying (2.2) to u = ∇f yields and hence we may simply write as which can be understood as a RS decomp. of ∇.
Here △ σ denotes the Laplace-Beltrami operator on S N −1 , which we also call the spherical Laplacian. As is well known, the operators ∇ σ and △ σ can be expressed explicitly in terms of angular coordinates of the spherical polar coordinate system inṘ N ; for details, see e.g. [4, Chapter 2] and also [12,Section 2]. Let us recall that divu = ∇ · u = N k=1 ∂u k ∂x k is the trace of the matrix field ∇u. By extracting its spherical part, we define Here the second equality follows by applying the second equation of (2.3); the operation of ∇ σ on vector fields is again componentwise (in the sense of §1.1). Then a direct calculation yields as a RS decomp. of the divergence operator. By applying the identity to a gradient field ∇f = σ∂ r f + 1 as an analogue to the identity ∇ · ∇f = △f . For later use, we show the following two lemmas: Lemma 2.1 (Spherical integration by parts formula).
where the integrals are taken for any fixed radius. In particular, Proof. Let ζ ∈ C ∞ c (Ṙ N ) be a radially symmetric scalar field. Then integration by parts of (u where the last equality follows from (2.6). Since the choice of ζ is arbitrary in the radial coordinate, we get the desired formula, with the aid of the fundamental lemma of the calculus of variations.

Poloidal-toroidal fields
A precise definition of poloidal-toroidal fields in general dimension was introduced by Weck [17] in the framework of differential forms onṘ N , including advanced tools such as Hodge dual, interior product, codifferential, etc. However, such a framework requires too wide range of concepts to focus on our current issue. In order to solve this gap, we give a more specialized formalization to introduce the definition of PT fields, in the framework of minimum required tools in the standard vector calculus.
3.1. Pre-poloidal fields and toroidal fields onṘ N . We say that a vector field u = u(x) onṘ N is pre-poloidal if there exist two scalar fields f and g satisfying This definition is equivalent to the existence of f, g satisfying Then the set of all pre-poloidal fields, which we denote by P(Ṙ N ), is a linear space with the invariance property where ζ ∈ C ∞ (Ṙ N ) is any radially symmetric scalar field. Indeed, for every u ∈ P(Ṙ N ) the relation {ζu, ∂ r u} ⊂ P(Ṙ N ) easily follows from the second definition formula (3.2), and taking the operation of △ on the first formula (3.1) readily yields △u ∈ P(Ṙ N ) from the Leibniz rule; an easy application of this result also yields △ σ u = r 2 △u − r 2 ∂ ′ r ∂ r u ∈ P(Ṙ N ) from (2.5), whence we arrive at (3.3). A vector field u ∈ C ∞ (Ṙ N ) N is said to be toroidal if it is spherical and solenoidal: We denote by T (Ṙ N ) the set of all toroidal fields. Then the same invariance property (3.3) also applies to T (Ṙ N ). To give an example, let i < j be two integers between 1 and N , and let v (i,j) be a vector field with the k-th component given by for every k = 1, 2, · · · , N ; then it is easy to check that v (i,j) ∈ T (Ṙ N ).
Including the above argument, we summarize some principal properties of the spaces of pre-poloidal fields and toroidal fields: where the integrals are taken for any radius. Moreover, the two kinds of fields always satisfy where ζ ∈ C ∞ (Ṙ N ) is any radially symmetric scalar field; namely, the two spaces P(Ṙ N ) and T (Ṙ N ) are invariant under the the multiplication of ζ and the operations of ∂ r and △ σ .
Proof. It suffices to check the orthogonality. The pre-poloidal property of v says that v = σg + ∇ σ f for some f and g, and hence v · w = w · ∇ σ f follows from the spherical property w R = 0 of the toroidal field w. Then integration by parts of both sides using rdivw = ∇ σ · w = 0 yields This proves the first orthogonality formula. To prove the second, by using (2.4) and Lemma 2.1, integration by parts yields where the last equality follows by applying the first orthogonality formula to the fields {∂ r v, v} ⊂ P(Ṙ N ) and {∂ r w, △ σ w} ⊂ T (Ṙ N ).
As an application of Proposition 3.1, we deduce the following simple but important fact: Proof. For every k ∈ {1, 2, · · · , N }, let e k = ∇x k denote the constant unit vector field parallel to the x k -axis. Then it is clear that e k ∈ P(Ṙ N ). Therefore, the integration of the k-th component u k = e k · u of u = u(rσ) yields e k · u dσ = 0 by applying Proposition 3.1 to the case (e k , u) ∈ P(Ṙ N ) × T (Ṙ N ). Since this equation holds for all k ∈ {1, 2, · · · , N }, we arrive at the desired result.
3.2. PT decomposition of solenoidal fields on R N . While all toroidal fields are solenoidal, pre-poloidal fields are not necessarily so; we say that a pre-poloidal field is poloidal whenever it is solenoidal. Now let u be a solenoidal field smoothly defined on the whole space R N . Notice from the Gauss' divergence theorem that the surface integral of u over S N −1 gives S N −1 σ ·u dσ = 0 for any radius. This equation says that the scalar field u R = σ ·u has zero-spherical mean. Then the Poisson-Beltrami equation (equipped with zerospherical-mean condition) is known to have an unique solution f, which we can express as where {u R,ν } ∞ ν=1 are the spherical harmonics components of u R defined by the equations see also [1,Eq. (13)] or [17, Lemma 3]. The solution f is called the poloidal potential of u. To understand this naming, let us introduce the second-order derivative operator D := σ△ σ − r∂ ′ r ∇ σ , which we call the poloidal-field generator. It maps every scalar field f to a poloidal field; indeed, it is clear that Df ∈ P(Ṙ N ), and that follows from (2.6). Moreover, it is easy to check that the vector field is toroidal whenever u is solenoidal. Hence we have obtained the following fact: Then there exists an unique pair of poloidal-toroidal fields (u P , u T ) ∈ P(Ṙ N ) × T (Ṙ N ) satisfying Here the poloidal part has the explicit expression u P = Df in terms of the poloidalfield generator D acting on the poloidal potential f = △ −1 σ u R .

Proof of main theorem
In order to compute the best H-L constant (namely the best constant in the inequality (1.4)) for solenoidal fields u ∈ D γ (R N ), we can assume that u ∈ C ∞ c (Ṙ N ) N by the density argument, and we will proceed from §4.1 to §4.4 under this assumption.

4.1.
Reduction to the case of PT fields. Let u ∈ C ∞ c (Ṙ N ) N be a solenoidal field, and let u = u P + u T be its PT decomposition in the sense of Proposition 3.2. If u P ≡ 0 and u T ≡ 0, then the ratio of the two integrals in inequality (1.4), which we simply call the H-L quotient, has the following estimate: where the first equality follows by applying the L 2 (S N −1 )-orthogonality in Proposition 3.1, and where C pol N,γ resp. C tor N,γ are the best H-L constant for poloidal resp. toroidal fields: Here the abbreviation u ∈ P resp. u ∈ T under the infimum sign means that u runs over all poloidal resp. toroidal fields in C ∞ c (Ṙ N ) N \ {0}. By taking the infimum on both sides of (4.1) over the test solenoidal fields u, the best H-L constant C N,γ for solenoidal fields is found to satisfy Since the reverse inequality C N,γ ≤ min{C pol N,γ , C tor N,γ } is clear, it then turns out that Therefore, the computation of C N,γ can be decomposed into C pol N,γ and C tor N,γ .

4.2.
Evaluation of C pol N,γ . Throughout this subsection, u ∈ C ∞ c (Ṙ N ) N \ {0} is assumed to be poloidal. Notice from Proposition 3.2 that u can be written as for the poloidal potential g = △ −1 σ u R , where and hereafter we use the abbreviation ∂ = r∂ r .
In the traditional way from [16], let us express the spherical harmonics decomposition of u together with g as Here α (·) denotes the quadratic function given by which is just the ν-th eigenvalue of −△ σ whenever ν is a nonnegative integer. Our goal here is to evaluate the quotient for each ν ∈ N with u ν ≡ 0, which enables us to find a lower estimate of the same quotient for the whole field u since (4.3) For that purpose, we fix a positive integer ν for the time being, and we start with taking the transformation of g ν into f by the formula which stems from Brezis-Vázquez [2]. Then u ν can be expressed in terms of f by the following calculation: Taking the operation of ∂ and △ σ on this equation yields and respectively, where the second last equality follows from Lemma 2.2. By applying the integration by parts formula taking the L 2 integral of (4.6) with respect to the measure 1 r drdσ yields where the last equality follows by using the integration by parts formula: which holds true since f is assumed to have compact support on R N \{0}. Similarly, taking the L 2 (R N )-scalar product of (4.5) and (4.7) with respect to the measure 1 r drdσ yields (4.9) We are now ready to express in terms of f the integrals in (1.4), that is, we have (4.11) In order to numericalize the derivative operators, we now traditionally introduce the 1-D Fourier-Emden transform (henceforth, FET for short) ϕ → ϕ in the radial direction as follows: Notice that this transform changes ∂ into the algebraic multiplication by iτ : Also notice from the L 2 (R) isometry of FET that By applying these formulas to ϕ ∈ {f, ∂f }, we get from (4.10) that where Q 0 (·, ·) denotes the polynomial given by Similarly, by applying FET to ϕ ∈ {f, ∂f, ∂ 2 f } we get from (4.11) that where We are now in the position to begin to evaluate C pol N,γ : taking the ratio of (4.14) to (4.12) yields Combine this result with (4.3), then we get In order to evaluate the right-hand side, we see from (4.13) and (4.15) that and that it is monotone increasing in a ≥ α 1 for each τ ≥ 0; indeed, a straightforward calculation yields ∂ ∂a Therefore, returning to (4.16) we get In other words, it turns out from (4.17) that the inequality holds for all poloidal fields u ∈ C ∞ c (Ṙ N ) N with the constant number 4.3. Optimality of C pol N,γ and a correction term. Here we consider only the case when the minimum in (4.19) is attained by τ = 0: which is equivalent to Under this assumption, let us show the sharpness of the constant C pol N,γ in the inequality (4.18). To do so, define {h n } n∈N ⊂ C ∞ c (Ṙ N ) as a sequence of scalar fields by where ζ : R → R is a smooth function ≡ 0 with compact support on R; notice that the eigenequation −△ σ h n = α 1 h n onṘ N (∀n ∈ N) holds from −△ σ x 1 = α 1 x 1 . In this setting, apply (4.4), (4.12) and (4.14) to the case This proves the desired sharpness of C pol N,γ . Next, we derive a correction term to make the inequality (4.18) stronger. To this end, returning to (4.17), we see that is a monotone function in τ > 0, whence By abbreviating as c N,γ = min {1, G(0)} , (4.22) we then see from (4.12) and (4.14) that Therefore, by taking the summation over ν ∈ N, we eventually get the inequality for all poloidal fields u ∈ C ∞ c (Ṙ N ) N , where R N,γ [ · ] denotes the nonnegative functional given by In particular, as long as c N,γ > 0, the inequality (4.23) is stronger than (4.18). As a side note, the optimality of C pol N,γ in (4.19) holds even when (4.20) is not satisfied, though the proof becomes complicated (see [5, §4.3]) and we do not know whether the difference between both sides of (4.18) can be bounded below by a norm of u. For our present purpose, however, this case is fortunately avoidable .

Evaluation and optimality of C tor
N,γ . Let u be a toroidal field in C ∞ c (Ṙ N ) N and let v be its Brezis-Vázquez type transform given by where the second last equality follows by integration by parts together with the compactness of the support of v and where R N,γ [ · ] is the same functional as in (4.24). On the other hand, recall from Corollary 3.1 that every toroidal field has zero-spherical mean; then, by considering the spherical harmonics expansion of u, we easily get the L 2 (S N −1 ) estimate Therefore, by way of the identity |∇u| 2 = |∂u| 2 + |∇ σ u| 2 |x| −2 , taking both sides of (4.25) plus (4.26) yields with the constant number Notice that the equality in (4.27) holds whenever −△ σ u = α 1 u onṘ N . In particular, we have obtained the H-L inequality which is the same as in (3.4) and satisfies the eigenequation In this setting, for every n ∈ N define v and u by where ζ : R → R is a smooth function ≡ 0 with compact support on R. Then, since the equality condition of (4.27) is satisfied, we get Dividing both sides by R N |u| 2 |x| 2γ−2 dx = R N |v| 2 |x| −N dx, we obtain which proves the desired sharpness of C tor N,γ .

4.5.
Non-attainability of C N,γ . Substituting (4.19) and (4.28) into (4.2), we see that the inequality (1.4) holds for all solenoidal fields u in C ∞ c (Ṙ N ) N and hence in D γ (Ṙ N ) with the best constant number C N,γ = min C pol N,γ , C tor Here the last equality is merely a repeat of (1.3); more precisely, for our present purpose, we shall exploit the following two facts:  If γ satisfies B γ − N 2 ≤ 0, then the inequality C tor N,γ + 2 ≤ C pol N,γ holds true. Along these lemmas, let us show the non-attainability of C N,γ separately in the two cases: The case γ ∈ I N . For every solenoidal field u ∈ C ∞ c (Ṙ N ) N , the calculation of (4.18) u=uP plus (4.27) u=uT yields By the density argument, the same inequality also applies to solenoidal fields u. If a solenoidal field u attains C N,γ in (1.4), then this fact together with the condition C pol N,γ > C tor N,γ (from Lemma 4.1) implies that R N |u P | 2 |x| 2γ−2 dx = 0 and R N,γ [u T ] = 0, and hence that u P ≡ 0 and x · ∇ |x| γ+ N −2 2 u T = 0.
Here the second equation further implies that for some vector field c on S N −1 ; this fact together with the integrability condition u T γ < ∞ forces that c ≡ 0. Therefore, it turns out that u ≡ 0.
The case γ ∈ I N . The calculation of (4.23) u=uP plus (4.27) u=uT yields that holds for all solenoidal fields u in C ∞ c (Ṙ N ) N , and hence in D γ (R N ) by the density argument. On the other hand, in view of (4.22), we find c N,γ > 0 from G(0) > 0; indeed, a direct computation yields from (4.21) that where the last inequality follows by applying the contraposition of Lemma 4.2 to the inequality C tor N,γ ≥ C pol N,γ (which comes from Lemma 4.1 and clearly contradicts to C tor N,γ + 2 ≤ C pol N,γ ). Therefore, any solenoidal field u in D γ (R N ) which attains C N,γ in (1.4) must satisfy R N,γ [u] = 0, and hence we get u ≡ 0 in the same way as the previous case.
In summary, the proof of the non-attainability of C N,γ is now complete, and we finish the proof of Theorem 1.1.