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SHARP UNCERTAINTY PRINCIPLE INEQUALITY

FOR SOLENOIDAL FIELDS

NAOKI HAMAMOTO

Abstract. This paper solves the L2 version of Maz’ya’s open problem [22,Section 3.9] on the sharp uncertainty principle inequality

∫

RN|∇u|2dx

∫

RN|u|2|x|2dx ≥ CN

(∫

RN|u|2dx

)

2

for solenoidal (namely divergence-free) vector fields u = u(x) on RN . The best

value of the constant turns out to be CN = 1

4

(

√

(N − 2)2 + 8 + 2)

2

which

exceeds the original value N2/4 for unconstrained fields. Moreover, we show

the attainability of CN and specify the profiles of the extremal solenoidalfields: for N ≥ 4, the extremals are proportional to a poloidal field thatis axisymmetric and unique up to the invariant transformation the axis ofsymmetry and the scaling transformation; for N = 3, there additionally existextremal toroidal fields.

1. Introduction

We study the functional inequality called the Heisenberg’s Uncertainty Principleinequality for real (not complex) vector fields on RN , with focus on how the bestconstant is changed by imposing differential constraints on the test vector fields.

§1.1. Basic notations. Throughout this paper, we use bold letters to denote vec-tors in the N -dimensional Euclidean space, e.g. x = (x1, x2, · · · , xN ) ∈ RN . Bywriting u = (u1, u2, · · · , uN) ∈ C∞(Ω)N , we mean that

u : Ω → RN , x 7→ u(x) = (u1(x), · · · , uN (x))

is a vector field on Ω with the components u1, · · · , uN ⊂ C∞(Ω); the same alsoapplies to other spaces: e.g., u ∈ C∞

c (Ω)N means that u1, · · · , uN ⊂ C∞c (Ω), i.e.,

u is a smooth vector field with compact support on Ω. We use the gradient operator∇ =

(∂

∂x1, · · · , ∂

∂xN

)not only on scalar fields but also vector fields u: we view∇u =

( ∂uj

∂xk

)j,k∈1,··· ,N

: Ω → RN×N as a matrix field. The notation x · y =

∑Nk=1 xkyk

denotes the standard scalar product of two vectors, which induces |x| = √x · x the

absolute value of x; the same also applies to matrix fields, e.g.,

∇u · ∇v =∑

j,k

∂uj∂xk

∂vj∂xk

and |∇u|2 =∑

j,k

( ∂uj∂xk

)2.

§1.2. Motivation and related known results. Heisenberg’s Uncertainty Prin-ciple [19] states in quantum mechanics that the observed values of the positionand momentum of a particle cannot be determined at the same time. This wasmathematically formulated as an inequality called the Heisenberg-Pauli-Weyl Un-certainty Principle (or shortly HUP, see [25]), which asserts that a wave function

2010 Mathematics Subject Classification. 26D10 (Primary); 35A23, 26D15, 81S07 (Secondary).Key words and phrases. Uncertainty Principle inequality, Solenoidal, Poloidal, Toroidal, Spher-

ical harmonics, Best constant, Laguerre polynomials, Confluent hypergeometric function of thefirst kind.

1

2 N. HAMAMOTO

and its Fourier transform cannot be simultaneously sharply localized at a singlepoint. In the present paper, we employ the expression of HUP inequality in termsof real vector fields on RN , which is precisely described as follows:

∫

RN

|∇u|2dx∫

RN

|u|2|x|2dx ≥ CN

(∫

RN

|u|2dx)2

(1.1)

for u = u(x) ∈ C∞(RN )N with a suitable integrability condition. Here the constant

CN = N2/4 (1.2)

is known to be sharp for unconstrained vector fields u and attained when the compo-nents u1, · · · , uN have the Gaussian profile, that is, each of them is proportional

to e−c|x|2 for some constant c > 0 (see e.g. [6]). Note that the inequality (1.1) isequivalent to its scalar-field version, since the former easily holds by applying thelatter to each of the components of u with the aid of Cauchy-Schwarz inequality;for the details, see the discussion in [3].

Now, a non-trivial problem occurs when u is subject to some differential con-straint, asking how much is the new best value of CN larger than (1.2). When u isassumed to be curl-free, namely when u = ∇φ holds with some scalar potential φ,the answer to the problem was recently obtained by Cazacu-Flynn-Lam [3]: it wasshown for all N ≥ 1 that the inequality (1.1) holds with the sharp constant

CN =1

4(N + 2)2 (1.3)

for curl-free fields u, and that the sharp value is attained when φ has the Gaussianprofile. This result gives a curl-free improvement of the HUP inequality, in the sensethat the best value of CN gets larger by restricting the test vector fields to curl-freefields. In the case N = 2, as is well known, the curl-free fields are isometricallyisomorphic to solenoidal (namely divergence-free) vector fields. Hence the resultof Cazacu-Flynn-Lam also solves the problem of finding the best value of C2 forsolenoidal fields, as a special case of the question asked by Maz’ya in the L2 settingwhich reads as follows:

Open Problem (Maz’ya [22, Section 3.9]). Find the new best value of the constantCN in the inequality (1.1) when u is assumed to be solenoidal.

This problem has attracted a number of mathematicians’ interest and has anexotic feel in the following sense: the uncertainty principle is a concept of quantummechanics, while solenoidal vector fields are objects of fluid dynamics or electro-magnetism; the two are quite different from each other, and the solenoidal conditionis usually not considered as a physical property of wave functions of particles inquantum mechanics; nevertheless, in mathematics, the above open problem doesnot hesitate to connect the two things, which could also be expected to bring a newunknown physical meaning.

From the viewpoint of mathematical analysis, the open problem is consideredas a solenoidal improvement of the HUP inequality. Historically, the solenoidalimprovement of functional inequalities probably first appears in the article [4] byCostin-Maz’ya, who derived the sharp Hardy inequality

∫

RN

|∇u|2dx ≥(N − 2

2

)2(N + 2)2

(N + 2)2 − 8

∫

RN

|u|2|x|2 dx

for axisymmetric solenoidal fields u; here the constant(N−22

)2 (N+2)2

(N+2)2−8 is sharp,

which improves the original best value(N−22

)2in the same inequality (for uncon-

strained fields) found by Leray [20] forN ≥ 3 and Hardy [18] for N = 1. Concerning

SHARP UNCERTAINTY PRINCIPLE INEQUALITY FOR SOLENOIDAL FIELDS 3

such an improvement, Costin-Maz’ya’s result was further refined in a series of recentpapers [14, 9, 7, 10]; in particular, the removal of the axisymmetry condition on thetest solenoidal fields was achieved without changing the best constant, which alsoincludes a solution to Maz’ya’s another open problem [22, Section 9.4]. In addition,other inequalities including Rellich inequality were found in [16, 17, 8, 11, 13] withthe new best constant for curl-free or solenoidal fields. (Incidentally, see also [15]for the two-dimensional logarithmic weighted Hardy inequality.) In any case, thebest constants in such inequalities have turned out to be computable for solenoidalfields on RN . On the other hand, as mentioned in e.g. [2], Hardy’s inequality canalso be considered as another kind of HUP; to see a rough picture of it, suppose

that u(x) is sharply localized at x = 0, then the integral∫RN

|u|2

|x|2 dx and hence∫RN |∇u|2dx gets larger. Then we may expect that the same computability alsoapplies to the best constant in HUP inequality (henceforth, “best HUP constant”for short).

§1.3. Main result. Motivated by the observation above, we try to solve the afore-mentioned open problem for N ≥ 3. We always assume that the test solenoidalfields u are regular, in the sense that the three integrals in the HUP inequality arefinite, which we express as the finiteness of the norm

‖u‖D :=

(∫

RN

|u|2dx+

∫

RN

|u|2|x|2dx +

∫

RN

|∇u|2dx)1/2

. (1.4)

We denote by D(RN ) the Hilbert space completion of C∞c (RN )N with respect to

the norm ‖ · ‖D. Now, our main result reads as follows:

Theorem 1.1. Let u ∈ D(RN ) be a solenoidal field. Then the inequality (1.1)holds with the best constant

CN =1

4

(√(N − 2)2 + 8 + 2

)2.

Moreover, CN is attained in D(RN ) \ 0.Remark. In Section 6, the solenoidal fields u 6≡ 0 satisfying the equality sign in

(1.1), which we say extremal, will be specified; as a result, they are classified intotwo profiles when N = 3 and only one when N ≥ 4. For both the cases, thereexist extremal poloidal fields (in the sense of the so-called poloidal-toroidal fieldsin Section 2) that are proportional to an axisymmetric solenoidal field uniquelydetermined up to the axis of rotation and scaling transform (see (6.2)). For N = 3,there also exist extremal toroidal fields (see (6.4)).

In a comparison between the three HUP best constants shown above for uncon-strained, solenoidal or curl-free fields, it is easy to check that

1

4N2 <

1

4

(√(N − 2)2 + 8 + 2

)2≤ 1

4(N + 2)2,

or that the value of CN in Theorem 1.1 stays between (1.2) and (1.3). This resultagrees with that the solenoidal condition is weaker than the curl-free one in viewof HUP, in the sense that the solenoidal condition consists of only one differen-tial equation while the curl-free condition requires many (or exactly 1

2N(N − 1))equations.

§1.4. Overview of this paper. In the rest of the present paper, we focus on theproof of Theorem 1.1 that consists of the following five parts:

• Section 2. Poloidal-toroidal (or shortly PT) decomposition u = uP + uT

of solenoidal fields u.• Section 3. Computation of the best HUP constant for toroidal fields uT .

4 N. HAMAMOTO

• Section 4. Integral calculation for spherical harmonics components of uP .• Section 5. One-dimensional minimization problem for the functional of theform

R[g] =

∫∞

0 (g′′)2xµ+1dx∫∞

0

(x2(g′)2 − εg2

)xµ−1dx∫∞

0(g′)2xµdx

on the set of test functions g = g(x) : [0,∞) → R with a suitable regularitycondition, where µ and ε are positive parameters in some ranges.

• Section 6. Conclusion.

The PT decomposition theorem in Section 2 historically originates from G. Backus[1] for N = 3 and was generalized by N. Weck [24] to differential forms on RN . Weuse their results in the framework of the standard vector calculus (instead of differ-ential forms) to separate the calculation of CN into two computable parts, namelythe best HUP constants CP,N and CT,N for poloidal and toroidal fields, respec-tively; the same technique was also used in the works [9, 7, 11] on sharp Hardy orRellich inequality for solenoidal fields.

The computation of CT,N in Section 3 will be carried out by reducing it to theproblem of finding the best HUP constant for scalar fields with zero spherical mean.

The calculation of CP,N in Section 4 will be carried out by estimating the in-tegrals in (1.1) for u = uP along the idea of Cazacu-Flynn-Lam [3] that orig-inates from Tertikas-Zographopoulos [23], who used the transformation of typeφ(x) 7→ |x|λφ(x) with the exponent λ varying corresponding to the spherical har-monics components of test fields. By way of this procedure, the estimation for CP,N

can be reduced to one-dimensional minimization problem.The main difficulty lies in Section 5 where we try to solve the minimization prob-

lem for the aforementioned functional R[g]. Problems of the same or more generaltype also appear in recent papers (e.g. [3, 5]), which seem to remain unsolved. Toovercome the difficulty in our case, we make use of a simple calculation method forthe L2 integral (5.7) of Kummer’s equation. However, it should also be noted thatthe discovery of such a method is based on a long and complicated computation ofR[g], carried out in the previous version of the preprint [12] by expressing g in the(general) Laguerre polynomial expansion, where we succeeded in finding that theextremal function of R[g] must be of Kummer type.

2. Preliminary: poloidal-toroidal decomposition

After preparing basic notations (in addition to §1.1), we briefly review poloidal-

toroidal fields in the framework of the standard vector calculus on RN , as a special-ization of Weck’s work [24]. We omit the proofs of the elementary facts related tothe PT fields, since they have already been fully discussed in the previous papers;see [7, 11, 17] for the details. By using the PT decomposition theorem, the bestHUP constant can be separated into two computable parts, as a consequence of theL2(SN−1) orthogonality of PT fields.

§2.1. Basic notations. We make the identification RN ∼= R+ × SN−1, in the

sense that RN = RN \ 0 is a smooth manifold diffeomorphic to the productof the (N − 1)-sphere SN−1 =

x ∈ RN ; |x| = 1

and the open half line R+ =

r ∈ R ; r > 0, and that every point x in RN is uniquely expressed as x = rσin terms of the radius (namely radial coordinate) r = |x| > 0 and the unit vectorσ = x/|x| ∈ SN−1.

For every vector field u = u(x) : RN → RN , there exists an unique pair of ascalar field uR and a vector field uS satisfying

u = σuR + uS and σ · uS = 0 on RN , (2.1)

SHARP UNCERTAINTY PRINCIPLE INEQUALITY FOR SOLENOIDAL FIELDS 5

and these fields have the explicit expressions uR = σ · u and uS = u− σuR whichwe call the radial component and the spherical part of u, respectively.

The gradient operator ∇ =(

∂∂x1

, · · · , ∂∂xN

)and the Laplacian =

∑Nk=1(

∂∂xk

)2

can be decomposed into radial-spherical parts as

∇ = σ∂r +1

r∇σ and = ∂′r∂r +

1

r2σ, (2.2)

where ∇σ and σ denote the spherical gradient and the spherical Laplacian (orLaplace-Beltrami operator on SN−1), respectively, and where

∂r := σ · ∇ =

N∑

k=1

xk|x|

∂

∂xkand ∂′r = ∂r +

N − 1

r

are (in this order) the radial derivative and its skew L2 adjoint, in the sense that∫

RN

f∂rgdx = −∫

RN

(∂′rf)gdx ∀f, g ∈ C∞(RN ).

Applying the gradient formula in (2.2) to a vector field u in (2.1) and taking thetrace part of the matrix field ∇u, one can check that

divu =

N∑

K=1

∂uk∂xk

= ∂′ruR +∇σ · uS ,

where ∇σ · uS denotes the trace part of the matrix field ∇σuS . By using thisidentity, we further obtain ∇σ · ∇σf = σf and the spherical integration by partsformula

∫

SN−1

u · ∇σf dσ = −∫

SN−1

(∇σ · uS)f dσ ∀f ∈ C∞(SN−1)

for any fixed radius (see e.g. [7, Lemma 2.1]). The commutation relations betweenσ and σ or ∇σ are given by the following identities:

σ(σf)− σσf = −(N − 1)σf + 2∇σf,

σ∇σf −∇σσf = −2σσf + (N − 3)∇σf.(2.3)

(For the details of the proof, see e.g., [17, Lemma 3] or [16, Lemma 7].)

§2.2. Poloidal-toroidal fields. We say that a vector field u ∈ C∞(RN )N is pre-poloidal when

uS = ∇σf on RN for some f ∈ C∞(RN ),

and toroidal when uR = divu = 0 on RN . We denote by P(RN ) resp. T (RN ) theset of all pre-poloidal resp. toroidal fields. The principal properties of them aresummarized as follows:

Proposition 2.1. Let v ∈ P(RN) and w ∈ T (RN ). We abbreviate as v = v(rσ)and w(rσ) for any r > 0 and σ ∈ SN−1. Then it holds that

∫

SN−1

v ·w dσ =

∫

SN−1

∇v · ∇w dσ = 0 (∀r > 0),

∫

SN−1

w dσ = 0 (∀r > 0) ,

ζv, ∂rv,σv ⊂ P(RN) and ζw, ∂rw,σw ⊂ T (RN ),

where ζ ∈ C∞(RN ) is any radially symmetric scalar field.

6 N. HAMAMOTO

This proposition says that the spacesP(RN ) and T (RN ) are L2(SN−1)-orthogonaland invariant under radial (derivative) operators and the Laplacians, and that everytoroidal field is of zero spherical mean.

Let us further decompose the space of pre-poloidal fields. For every ν ∈ N, weset

αν := ν(ν +N − 2)

and

Eν(RN ) :=f ∈ C∞(RN ) ; −σf = ανf on R

N,

Pν(RN ) :=

u ∈ P(RN) : u = σf +∇σg for some f, g ∈ Eν(RN )

,

which denote the set of ν-th eigenfunctions of −σ and that of pre-poloidal fieldsgenerated by them. Note that Pν(R

N )ν∈N are invariant under the operation ofζ, ∂r,σ in the sense of Proposition 2.1, and that they are L2(SN−1)-orthogonal:∫

SN−1

u · vdσ =

∫

SN−1

∇u · ∇vdσ = 0 ∀(u,v) ∈ Pν(RN )× Pρ(R

N ) (2.4)

(for any radius r) holds whenever ν 6= ρ.We say that a pre-poloidal field is poloidal whenever it is solenoidal. The poloidal

generator is a second-order differential operator on C∞(RN ) given by

D = σσ − r∂′r∇σ,

which maps every scalar field f to the poloidal field Df ∈ P(RN ). The followingfact is fundamental:

Proposition 2.2. Let u : RN → RN be a smooth solenoidal field on RN . Thenthere exists an unique pair of poloidal-toroidal fields (uP ,uT ) ∈ P(RN ) × T (RN )satisfying

u = uP + uT on RN .

In particular, the poloidal part has the expression

uP = Df.

Here the scalar field f = −1σ uR is an unique solution to the Poisson-Beltrami

equation σf = uR (on RN ) under the condition∫SN−1 f(rσ)dσ = 0, ∀r > 0.

In this proposition, we call f = −1σ uR the poloidal potential of u. When it is

multiplied by a radially symmetric scalar field ζ, the deformation of Df and ∇Dfhas the following L2(SN−1) estimates [11, Lemma 2.3]:

C

∫

SN−1

|D(ζf)− ζDf |2dσ ≤ (∂ζ)2∫

SN−1

|Df |2dσ,

C

∫

SN−1

|∇D(ζf)− ζ∇Df |2dσ ≤((∂ζ)2 + (∂2ζ)2

) ∫

SN−1

|Df |2r2

dσ

(2.5)

for any r > 0, where we abbreviate as

∂ = r∂r (2.6)

and where C is a constant number depending only on N . In particular, this factyields that every solenoidal field u on RN can be L2 approximated by those withcompact support on RN , which we will use in the following form:

Lemma 2.3. Let u ∈ D(RN ) be a solenoidal field, and let ζ0 ∈ C∞c (R) with

ζ0(0) = 1. Define unn∈N ⊂ C∞c (RN )N by

un = D(ζnf) + ζnuT

SHARP UNCERTAINTY PRINCIPLE INEQUALITY FOR SOLENOIDAL FIELDS 7

for the poloidal potential f of uP , where we set ζn(x) = ζ0(1n log |x|

)(∀n ∈ N).

Then it holds that

‖un − u‖D → 0 as n→ ∞.

Proof. It suffices to separately check the cases u = uT and u = uP .Let us consider the case u = uT . Since un = ζnu, it is clear from the domi-

nated convergence theorem that∫RN |un −u|2(1+ |x|2)dx→ 0. Moreover, a direct

calculation yields∫

RN

|∇un − ζn∇u|2dx =

∫

RN

(∂rζn)2 |u|2 dx

=

∫

RN

(∂ζn)2 |u|2|x|2 dx = O(n−2)

∫

RN

|u|2|x|2 dx.

Notice here that the last integral is finite, since u ∈ D(RN ) and N ≥ 3. Therefore,we get

∫RN |∇un− ζn∇u|2dx→ 0. Since

∫RN |ζn∇u−∇u|2dx→ 0 also holds from

the dominated convergence theorem, by way of the L2 triangle inequality we get∫RN |∇un −∇u|2dx→ 0 and hence ‖un − u‖D → 0, as desired.For the case u = uP , it can be written as u = Df . By using (2.5), we have∫

RN

|un − ζnu|2 (1 + |x|2)dx

=

∫

RN

|D(ζnf)− ζnDf |2 (1 + |x|2)dx

≤ 1

C

∫

RN

(∂ζn)2|Df |2(1 + |x|2)dx = O(n−2)

∫

RN

|u|2(1 + |x|2)dx,∫

RN

|∇un − ζn∇u|2 dx =

∫

RN

|∇D(ζnf)− ζn∇Df |2 dx

≤ 1

C

∫

RN

((∂ζn)

2 + (∂2ζn)2) |Df |2

|x|2 dx = O(n−2)

∫

RN

|u|2|x|2 dx.

Notice that both the last integrals are finite, for the same reason as the previouscase. Therefore, since ζnu → u and ζn∇u → ∇u are dominated convergence, weget ‖un − u‖D → 0 by way of the triangle inequality.

§2.3. PT decomposition of the best HUP constant. Let u = uP +uT be thePT decomposition of a solenoidal field u ∈ D(RN ). Then we see from the L2(SN−1)orthogonality of PT fields (in the sense of Proposition 2.1) that∫

RN

|∇u|2dx∫

RN

|u|2|x|2dx

=

(∫

RN

|∇uP |2dx+

∫

RN

|∇uT |2dx)(∫

RN

|uP |2|x|2dx+

∫

RN

|uT |2|x|2dx)

≥(√∫

RN |∇uP |2dx√∫

RN |uP |2|x|2dx+√∫

RN |∇uT |2|x|2dx√∫

RN |uT |2|x|2dx)2

≥(√

CP,N

∫

RN

|uP |2dx+√CT,N

∫

RN

|uT |2dx)2

≥(min

√CP,N ,

√CT,N

(∫

RN

|uP |2dx+

∫

RN

|uT |2dx))2

= min CP,N , CT,N(∫

RN

|u|2dx)2

,

8 N. HAMAMOTO

where the inequality in the third line follows by applying the Cauchy-Schwarz in-equality of the form (a2 + b2)(c2 + d2) ≥ (ac+ bd)2, and where the notation

CP,N := infu∈P

∫RN |∇u|2dx

∫RN |u|2|x|2dx∫

RN |u|2dx

resp. CT,N := infu∈T

∫RN |∇u|2dx

∫RN |u|2|x|2dx∫

RN |u|2dx

denotes the best HUP constant for poloidal resp. toroidal fields in D(RN ) \ 0which we abbreviate as u ∈ P resp. u ∈ T under the infimum sign. Then we findthat the best HUP constant for solenoidal fields satisfies

CN = infdivu=0

∫RN |∇u|2dx

∫RN |u|2|x|2dx∫

RN |u|2dx ≥ min CP,N , CT,N .

Combining this inequality with the reverse CN ≤ min CP,N , CT,N, we get

CN = min CP,N , CT,N . (2.7)

In this way, the problem of finding the value of CN is separated into that of CP,N

and CT,N .

3. The case u = uT : evaluation of CT,N,γ

In this section, we always assume that u = (u1, · · · , uN) ∈ D(RN ) is toroidal.As mentioned in Proposition 2.1, this field is of zero spherical mean:

∫

SN−1

uk(rσ)dσ = 0 ∀r > 0, ∀k ∈ 1, · · · , N.

Then an application of Cauchy-Schwarz inequality yields∫

RN

|∇u|2dx∫

RN

|u|2|x|2dx

=

(N∑

k=1

∫

RN

|∇uk|2dx)(

N∑

k=1

∫

RN

(uk)2|x|2dx

)

≥(

N∑

k=1

√∫

RN

|∇uk|2dx√∫

RN

(uk)2|x|2dx)2

≥(

N∑

k=1

N + 2

2

∫

RN

(uk)2dx

)2

=(N + 2

2

)2(∫

RN

|u|2dx)2

, (3.1)

where the second inequality holds by using the following fact:

Lemma 3.1. Let f ∈ C∞(RN ) be a scalar field with ‖f‖D <∞ and∫

SN−1

f(rσ)dσ = 0 for all r > 0.

We assume that∫SN−1 (f(Rσ))

2dσ = o(R−N ) as R → ∞. Then it holds that

∫

RN

|∇f |2dx∫

RN

f2|x|2dx ≥(N + 2

2

)2(∫

RN

f2dx

)2

.

Here the equality holds when f is of the form f(x) =∑N

k=1Bkxke−B0|x|

2

, withB0 > 0 and B1 · · · , BN ⊂ R being constant numbers.

SHARP UNCERTAINTY PRINCIPLE INEQUALITY FOR SOLENOIDAL FIELDS 9

Proof. First, let ν ∈ N and let us abbreviate as f = f(rσ) for r > 0 and σ ∈SN−1. Then a direct calculation for every R > 0 yields∫ R

0

(∂r(r

−νf))2r2ν+N−1dr =

∫ R

0

(∂rf − νr−1f

)2rN−1dr

=

∫ R

0

( ((∂rf)

2 + ν2r−2f2)rN−1 − 2νrN−2f∂rf

)dr

=

∫ R

0

( ((∂rf)

2 + ν2r−2f2)rN−1 + ν(N − 2)rN−3f2

)dr − ν

[rN−2f2

]r=R

r=0

=

∫ R

0

((∂rf)

2 + ανr−2f2

)rN−1dr − νRN−2(f(Rσ))2,

where the third equality follows by integration by parts. Hence, by letting R → ∞after integration of both sides over SN−1, we get

∫

RN

(∂r(r−νf

))2r2νdx =

∫

RN

((∂rf)

2 + ανr−2f2

)dx.

Second, let

f =∑

ν∈N

fν , fν ∈ Eν(RN ) (∀ν ∈ N)

be the spherical harmonics decomposition of f . Then we have∫

RN

|∇f |2dx =∑

ν∈N

∫

RN

|∇fν |2dx

=∑

ν∈N

∫

RN

((∂rfν)

2 + ανr−2(fν)

2)dx

=∑

ν∈N

∫

RN

(∂r(r−νfν

))2r2νdx,

where the last equality follows from the formula derived in the first step. Hence,an application of Cauchy-Schwarz inequality yields

∫

RN

|∇f |2dx∫

RN

|f |2|x|2dx

=

(∑

ν∈N

∫

RN

(∂r(r−νfν

))2 |x|2νdx)∑

ν∈N

∫

RN

(r−νfν)2|x|2ν+2dx

≥(∑

ν∈N

√∫

RN

(∂r (r−νfν))2 |x|2νdx

√∫

RN

(r−νfν)2|x|2ν+2dx

)2

(3.2)

≥(∑

ν∈N

∫

RN

(∂r(r

−νfν))(r−νfν)|x|2ν+1dx

)2

=

(∑

ν∈N

2ν +N

2

∫

RN

(r−νfν

)2r2νdx− 1

2lim

R→∞RN

∫

SN−1

(f(Rσ))2dσ

)2

=

(∑

ν∈N

2ν +N

2

∫

RN

(r−νfν

)2r2νdx

)2

≥(N + 2

2

)2(∑

ν∈N

∫

RN

(fν)2dx

)2

=(N + 2

2

)2(∫

RN

f2dx

)2

,

10 N. HAMAMOTO

where the equality in the third last line follows by integration by parts with respectto the measure dr, and where the equality in the second last line follows by usingthe assumption of the lemma.

Finally, we specify the condition for the equality sign. In order that all theequalities in the above inequalities are simultaneously realized by f 6≡ 0, it musthold that fν ≡ 0 (∀ν ≥ 2) and that the two integrands in (3.2) are proportional,which leads to

f = f1 and(∂r(r

−1f1))2

= B2(r−1f1)2r2

for some constant B 6= 0, whence

f = f1 and ∂r(r−1f1) = −Br

(r−1f1

).

Since r−1f1(r) is convergent as r → ∞, the solution to this equation must be ofthe form

f(rσ) = Y (σ)re−Br2 with B > 0,

where Y ∈ C∞(SN−1) satisfies the eigenequation−σY = α1Y . Since the eigenspaceof −σ associated with α1 is spanned by σ1, · · · , σN, we consequently have

f(rσ) =

N∑

k=1

Bkσkre−Br2 , that is, f(x) =

N∑

k=1

Bkxke−B|x|2 ,

as desired. Moreover, a direct computation of the integrals for this case yields∫

RN

|∇f |2dx =N + 2

4

( π

2B

)N/2 N∑

k=1

B2k,

∫

RN

f2|x|2dx =N + 2

4(2B)2

( π

2B

)N/2 N∑

k=1

B2k,

∫

RN

f2dx =1

4B

( π

2B

)N/2 N∑

k=1

B2k,

(3.3)

which clearly achieves the equality in the inequality of the lemma.

In order that both the equalities of the two inequalities in (3.1) are simultaneouslyrealized by u 6≡ 0, it must hold that there exists C > 0 such that∫

RN

|∇uk|2dx = C

∫

RN

(uk)2|x|2dx

and

∫

RN

|∇uk|2dx∫

RN

(uk)2|x|2dx =

(N + 2

2

)2(∫

RN

(uk)2dx

)2

for all k ∈ 1, · · · , N. Notice here that the second equation requires u to be

uk(x) =

N∑

j=1

Bk,jxje−Bk,0|x|

2

with constant numbers Bk,0 > 0k∈1,··· ,N and Bk,jk,j∈1,··· ,N ⊂ R; in viewof this together with the first equation, we see from (3.3) that

∫RN |∇uk|2dx∫RN (uk)2dx

=

N+24

(π

2Bk,0

)N/2∑Nj=1 B

2k,j

N+2(2Bk,0)2

(π

2Bk,0

)N/2∑Nj=1 B

2k,j

= B2k,0

must be independent of k, which is equivalent to B1,0 = B2,0 = · · · = BN,0.Therefore, u must be of the form

u(x) = e−c|x|2u0(x), u0(x) = (c1 · x, c2 · x, · · · , cN · x) ,

SHARP UNCERTAINTY PRINCIPLE INEQUALITY FOR SOLENOIDAL FIELDS 11

where c > 0 and where c1, c2 · · · , cN ⊂ RN are N constant vectors. Since u istoroidal if and only if so is u0, it must hold that the constant matrix (c1, · · · , cN)is trace-free and satisfy

x · u0(x) = x1(c1 · x) + · · ·+ xN (cN · x) =N∑

j,k=1

cjkxjxk

=N∑

k=1

c2kkx2k +

∑

j<k

(cjk + ckj)xjxk = 0 ∀x ∈ RN ,

whence cjk = −ckj (∀j, k), that is, (c1, · · · , cN) is antisymmetric. In summary, wehave obtained the following result:

Theorem 3.2. Let u ∈ D(RN ) be a toroidal field. Then the inequality

∫

RN

|∇u|2dx∫

RN

|u|2|x|2dx ≥ CT,N

(∫

RN

|u|2dx)2

holds with the best constant CT,N = (N+2)2

4 . Moreover, CT,N is attained by u = u0

with the profile

u0(x) = (c1 · x, · · · , cN · x) e−c|x|2,

where c is any positive constant number and c1, · · · , cN ⊂ RN are any N constantvectors with the matrix (c1, · · · , cN ) being antisymmetric. Such u0 is unique up tothat profile, under the additional condition limR→∞RN

∫SN−1 |u(Rσ)|2dσ = 0.

In view of this result and (1.3), we find that the toroidal condition gives thesame best HUP constant as the curl-free condition.

4. The case u = uP : evaluation of CP,N

Throughout this section, u is a smooth poloidal field with compact support onRN expressed as

u = Df,

where f ∈ C∞c (RN ) \ 0 is of zero spherical mean.

§4.1. Spherical harmonics decomposition of poloidal fields. In the same wayas in the proof of Lemma 3.1, we again express the spherical harmonics expansionof f as

f =∑

ν∈N

fν , fν ∈ Eν(RN ) (∀ν ∈ N),

and we write as

uν = Dfν (∀ν ∈ N)

which clearly belongs to Pν(RN ). Then an application of Cauchy Schwarz inequality

by way of the L2(SN−1)-orthogonality formulae (2.4) yields that

∫

RN

|∇u|2dx∫

RN

|u|2|x|2dx =

(∑

ν∈N

∫

RN

|∇uν |2dx)(

∑

ν∈N

∫

RN

|uν |2|x|2dx)

≥∑

ν∈N

∫

RN

|∇uν |2dx∫

RN

|uν |2|x|2dx

≥∑

ν∈N

CP,N,ν

∫

RN

|uν |2dx ≥ infν∈N

CP,N,ν

∫

RN

|u|2dx.

12 N. HAMAMOTO

Here, for every ν ∈ N, the notation

CP,N,ν = infu∈DEν

∫RN |∇u|2dx

∫RN |u|2|x|2dx∫

RN |u|2|x|2dx (4.1)

denotes the best HUP constant for the poloidal fields (in C∞c (RN )N ) generated by

Eν(RN ), where the abbreviation u ∈ DEν under the infimum sign means that

u ∈Df ; f ∈ Eν(RN ) ∩ C∞

c (RN ) \ 0.

Then we find the inequality CP,N ≥ infν∈N CP,N,ν from the definition of CP,N .Since the reverse CP,N ≤ infν∈N CP,N,ν also holds true, it turns out that

CP,N = infν∈N

CP,N,ν . (4.2)

Therefore, the problem of computing CP,N is reduced to that of CP,N,ν.

§4.2. Evaluation of CP,N,ν. Here we fix ν ∈ N and assume that u is of the form

u = Df, f ∈ Eν(RN ) ∩ C∞c (RN ).

For simplicity, we use the abbreviations

α = αν , ∂ = r∂r and ∂′ = r∂′r = ∂ +N − 1.

In order to evaluate CP,N,ν , we have to compute the integrals in the HUP inequality.To this end, a direct calculation yields

u = (σσ − r∂′r∇σ) f = −σαf −∇σ∂′f,

∂u = −σα∂f − ∂′∂∇σf,

−σu = ασ(σf) + ∂′σ∇σf

= α (σ (σf − (N − 1)f) + 2∇σf)

+ ∂′ (∇σσf − 2σσf + (N − 3)∇σf)

= α (−σ (α+N − 1) f + 2∇σf)

+ ∂′ (−α∇σf + 2σαf + (N − 3)∇σf)

= ασ (2∂′f − (α+N − 1)f) +∇σ (2αf + (−α+N − 3) ∂′f)

= ασ (2∂f − (α−N + 1)f) +∇σ (2αf + (−α+N − 3)∂′f) ,

where the equality in the fourth line follows by using (2.3). Taking the absolutesquare or scalar product of them yields

|u|2 = α2f2 + |∇σ∂′f |2,

|∂u|2 = α2(∂f)2 + |∇σ∂′∂f |2,

−u · σu = −α2f(2∂f − (α−N + 1)f

)

−∇σ(∂′f) · ∇σ

(2αf + (−α+N − 3)∂′f

).

SHARP UNCERTAINTY PRINCIPLE INEQUALITY FOR SOLENOIDAL FIELDS 13

Integration by parts of them over SN−1 (for any r) then yields the following calcu-lations:

1

α

∫

SN−1

|u|2dσ =

∫

SN−1

(αf2 + (∂′f)

2)dσ,

=

∫

SN−1

(αf2 + (∂f + (N − 1)f)2

)dσ

=

∫

SN−1

( (α+ (N − 1)2

)f2 + (∂f)2 + 2(N − 1)f∂f

)dσ, (4.3)

1

α

∫

SN−1

|∂u|2dσ

=

∫

SN−1

( (α+ (N − 1)2

)(∂f)2 + (∂2f)2 + 2(N − 1)(∂f)∂2f

)dσ (4.4)

and

1

α

∫

SN−1

|∇σu|2dσ =1

α

∫

SN−1

(− α2f

(2∂f − (α−N + 1)f

)

+ (σ∂′f) (2αf + (−α+N − 3)∂′f)

)dσ

=

∫

SN−1

− 2αf∂f + α(α−N + 1)f2

+ (∂′f)(− 2αf + (α−N + 3)∂′f

)

dσ

=

∫

SN−1

(α−N + 3)(∂f)2

+

(− 2α+

(− 2α+ (α−N + 3)(N − 1)

)

+ (N − 1) (α−N + 3)

)f∂f

+(α(α−N + 1) + (N − 1)

(− 2α+ (α−N + 3)(N − 1)

))f2

dσ

=

∫

SN−1

((α−N + 3)(∂f)2 + 2(N − 3)(α−N + 1)f∂f

+ (α−N + 1)(α+ (N − 1)(N − 3)

)f2

)dσ. (4.5)

Now, along the idea of [3, 23], let us introduce the transformation

f = rλg

with λ ∈ R determined later, and notice for any β ∈ R and σ ∈ SN−1 the integration

by parts formulae∫

R+

(f∂f)rβ−1dr = −β2

∫

R+

f2rβ−1dr = −β2

∫

R+

g2r2λ+β−1dr, (4.6)

∫

R+

(∂f)2rβ−1dr =

∫

R+

(λg + ∂g)2rβ+2λ−1dr

=

∫

R+

(λ2g2 + (∂g)2 + 2λg∂g

)rβ+2λdt

=

∫

R+

((∂g)2 − λ (λ+ β) g2

)rβ+2λ−1dr, (4.7)

∫

R+

(∂f)(∂2f)rβ−1dr = −β2

∫

R+

(∂f)2rβ−1dr

= −β2

∫

R+

((∂g)2 − λ(λ + β)g2

)rβ+2λ−1dr, (4.8)

∫

R+

(g∂2g

)rβ−1dr =

∫

R

(∂ (g∂g)− (∂g)2

)rβ−1dr

14 N. HAMAMOTO

=

∫

R

(−βg∂g − (∂g)2

)rβ−1dr =

∫

R+

(β2

2g2 − (∂g)2

)rβ−1dr (4.9)

∫

R+

(∂2f)2rβ−1dr =

∫

R+

(λ2g + 2λ∂g + ∂2g

)2r2λ+β−1dr

=

∫

R+

(λ4g2 + 4λ2(∂g)2 + (∂2g)2 + 4λ3g∂g

+ 2λ2g∂2g + 4λ(∂g)∂2g

)r2λ+β−1dr

=

∫

R+

λ4g2 + 4λ2(∂g)2 + (∂2g)2 − 2λ3(2λ+ β)g2

+ 2λ2(

(2λ+β)2

2 g2 − (∂g)2)− 2λ(2λ+ β)(∂g)2

r2λ+β−1dr

=

∫

R+

(λ2(λ+ β)2g2 − 2λ (λ+ β) (∂g)2 + (∂2g)2

)r2λ+β−1dr, (4.10)

where the second last equality follows by applying (4.9) to 2λ + β instead of β.Then, by way of applying (4.7) and (4.6) to β +N instead of β, the integration of(4.3) with respect to the measure rβ+N−1dr yields

1

α

∫

RN

|u|2|x|βdx =1

α

∫∫

R+×SN−1

|u|2rβ+N−1drdσ

=

∫∫

R+×SN−1

( (α+ (N − 1)2

)f2 + (∂f)2 + 2(N − 1)f∂f

)rβ+N−1dσ

=

∫∫

R+×SN−1

((α+ (N − 1)2

)g2 + (∂g)2

− λ (λ+ β +N) g2 − (N − 1)(β +N)g2

)r2λ+β+N−1drdσ,

=

∫∫

R+×SN−1

((∂g)2 +

(α− αλ+1 − (λ+N − 1)β

)g2)r2λ+β+N−1drdσ.

By applying (4.7), (4.10), (4.8) and (4.6) to the case β = N − 2, the integration of(4.4) + (4.5) with respect to the measure rN−1dr also yields

1

α

∫

RN

|∇u|2dx =1

α

∫∫

R+×SN−1

(|∂u|2 + |∇σu|2

)rN−3drdσ

=

∫∫

R+×SN−1

(2α+N2 − 3N + 4

)(∂f)2 + (∂2f)2

+ 2(N − 1)(∂f)∂2f + 2(N − 3)(α−N + 1)f∂f

+ (α −N + 1)(α+ (N − 1)(N − 3)

)f2

rN−3drdσ

=

∫∫

R+×SN−1

(2α+N2 − 3N + 4

) ((∂g)2 − αλg

2)

+ α2λg

2 − 2αλ(∂g)2 + (∂2g)2

− (N − 1)(N − 2)((∂g)2 − αλg

2)

− (N − 3)(α−N + 1)(N − 2)g2

+ (α −N + 1) (α+ (N − 1)(N − 3)) g2

r2λ+N−3drdσ

=

∫∫

R+×SN−1

((∂2g)2 + 2(α− αλ + 1)(∂g)2

+ (α− αλ−1) (α− αλ+1) g2

)r2λ+N−3drdσ.

Now, we choose

λ = ν − 1

SHARP UNCERTAINTY PRINCIPLE INEQUALITY FOR SOLENOIDAL FIELDS 15

in order that αλ+1 = αν (= α). Then the above results turn into

1

α

∫

RN

|u|2|x|βdx =

∫∫

R+×SN−1

((∂g)2 − (ν +N − 2)βg2

)r2ν+β+N−3drdσ,

1

α

∫

RN

|∇u|2dx =

∫∫

R+×SN−1

((∂2g)2 + 2(2ν +N − 2)(∂g)2

)r2ν+N−5drdσ.

Furthermore, let us change the radial coordinate from r into

s = r2

which obeys the chain rules: ds = 2rdr and ∂ = 2s∂s. Abbreviating as g = g(σ√s)

and ∂sg = ∂s (g(σ√s)), we have the following calculation:

1

αν

∫

RN

|u|2|x|βdx = 2

∫∫

R+×SN−1

((s∂sg)

2 − ν+N−24 βg2

)sν+

β+N−4

2 dsdσ,

1

αν

∫

RN

|∇u|2dx

=

∫∫

R+×SN−1

( ((2s∂s)

2g)2

+ 2(2ν +N − 2)(2s∂sg)2)(√s)2ν+N−6 ds

2dσ

= 4

∫∫

R+×SN−1

(2(s∂s(s∂sg)

)2+ (2ν +N − 2)(s∂sg)

2)sν+

N2−3dsdσ

= 4

∫∫

R+×SN−1

(2(s∂sg + s2∂2sg

)2+ (2ν +N − 2)(s∂sg)

2)sν+

N2−3dsdσ

= 4

∫∫

R+×SN−1

(2(s2∂2sg

)2+ 4s3(∂2sg)∂sg + (2ν +N)s2(∂sg)

2)sν+

N2−3dsdσ

= 8

∫∫

R+×SN−1

(s2∂2sg)2sν+

N2−3dsdσ.

Therefore, we have obtained∫

RN

|u|2|x|2βdx = 2αν

∫

SN−1

Pβ [g] dσ,

∫

RN

|∇u|2dx = 8αν

∫

SN−1

Q[g] dσ

for any β ∈ R, where we put

Pβ [g] :=

∫

R+

((s∂sg)

2 − ν+N−22 βg2

)sν+

N2+β−2ds,

Q[g] :=

∫

R+

(∂2sg)2sν+

N2+1ds,

which we view as functionals on the set of the one-dimensional functionss 7→ g(σ

√s)σ∈SN−1 ⊂ C∞

c (R+)

parameterized by σ. By an application of Cauchy-Schwarz inequality, we then get∫RN |∇u|2dx

∫RN |u|2|x|2dx

4(∫

RN |u|2dx)2 =

∫SN−1 Q[g]dσ

∫SN−1 P1[g]dσ

(∫SN−1 P0[g]dσ

)2

≥(∫

SN−1

√Q[g]P1[g] dσ∫

SN−1 P0[g]dσ

)2

≥ infg∈C∞

c (R+)

(√Q[g]P1[g]

P0[g]

)2

,

16 N. HAMAMOTO

where and hereafter we abbreviate as infg∈C∞

c (R+)instead of inf

g∈C∞

c (R+)\0. Taking

the infimum of both sides over u ∈ DEν then yields from (4.1) the inequality

CP,N,ν ≥ 4 infg∈C∞

c (R+)R[g], where we put R[g] =

Q[g]P1[g]

(P0[g])2 .

On the other hand, for any g ∈ C∞c (R+) \ 0 we set

u(x) = D(|x|ν−1g(|x|2)Y (x/|x|)

)(4.11)

in order that u ∈ DEν , where Y ∈ C∞(SN−1) is any one of the spherical harmonicfunctions of degree ν. Then, after applying to (4.11) the same calculations as above,we see that∫

RN

|u|2|x|2βdx = 2ανPβ [g]

∫

SN−1

Y dσ and

∫

RN

|∇u|2dx = 8ανQ[g]

∫

SN−1

Y dσ,

whence we see that

c

∫

RN

|u|2|x|2βdx = Pβ [g] and c

∫

RN

|∇u|2dx = 4Q[g] (4.12)

hold with some constant c > 0 depending only on N and ν; in particular, we get

CP,N,ν ≤∫RN |∇u|2dx

∫RN |u|2|x|2dx

(∫RN |u|2dx

)2 = 4R[g].

Taking the infimum of both sides over g ∈ C∞c (R+) \ 0 yields

CP,N,ν ≤ 4 infg∈C∞

c (R+)R[g]

as the reverse to the aforementioned inequality, whence we consequently obtain

CP,N,ν = 4 infg∈C∞

c (R+)R[g]. (4.13)

In summary, the evaluation of CP,N,ν is reduced to the one-dimensional mini-mization problem for the functional

R[g] =Q[g]P1[g]

(P0[g])2for g = g(x) ∈ C∞

c (R+) \ 0, (4.14)

where the factors of R[g] are given by

Pβ [g] =

∫ ∞

0

(x2(g′)2 − βεg2

)xµ+β−2dx (β = 0, 1),

Q[g] =

∫ ∞

0

(g′′)2xµ+1dx,

(4.15)

with the notations g′ =dg

dx, g′′ =

d2g

dx2,

µ =N

2+ ν and ε =

ν +N − 2

2. (4.16)

.

SHARP UNCERTAINTY PRINCIPLE INEQUALITY FOR SOLENOIDAL FIELDS 17

5. Solution to the minimization problem for R[g]

The goal of this section is to solve the one-dimensional minimization problemfor the functional (4.14); however, we treat µ and ν as continuous real positive pa-rameters including (4.16) as special cases. Moreover, we consider the test functionsg in a slightly wider space than C∞

c (R+) which in particular admits those withnon-zero values at the origin.

Let µ > 0 and let C∞µ (R+) be the set of R-valued smooth functions g on the half

line R+ = x > 0 satisfying

lim supx→0

(|g(x)|+ |g′(x)|) <∞,

limx→∞

(xµ+1(g′(x))2 + xµ (g(x))2

)= 0,

(5.1)

where and hereafter we use the notations g′ =dg

dxand g′′ =

d2g

dx2. We work on the

space Dµ(R+) defined as the Hilbert space completion of the setg ∈ C∞

µ (R+) ; ‖g‖Dµ<∞

with respect to the norm ‖ · ‖Dµgiven by

‖g‖Dµ:=

(4

∫ ∞

0

(g′′)2xµ+1dx+

∫ ∞

0

(g′)2(x+ 1)xµdx+

∫ ∞

0

g2xµ−1dx

) 12

, (5.2)

in order that all the integrals in (4.15) are well defined for all g ∈ Dµ(R+) in thestandard sense. In the above setting, the answer to the one-dimensional minimiza-tion problem is given as follows:

Theorem 5.1. Let ε > 0 with ε < µ2/4. Then the inequality∫ ∞

0

(g′′)2xµ+1dx

∫ ∞

0

(x2(g′)2 − εg2

)xµ−1dx

≥ 1

4

(√µ2 − 4ε+ 1

)2 ∫ ∞

0

(g′)2xµdx

holds for all g ∈ Dµ(R+) with the best constant factor on the right-hand side.Moreover, all the functions achieving the best constant is given by the set

Ce−λx

1F1(b, µ, λx) ; C ∈ R, λ > 0,

where 1F1(b, µ, ·) is Kummer’s confluent hypergeometric function (see e.g. [21])given by

1F1(b, µ, x) =

∞∑

k=0

(b)k(µ)k

xk

k!with b =

µ−√µ2 − 4ε

2, (5.3)

where (q)k =

1 (k = 0)q(q + 1)(q + 2) · · · (q + k − 1) (k ∈ N)

denotes the rising facto-

rial.

Proof. Step 1. Let us apply to g ∈ Dµ(R+) \ 0 the same notations in (4.14)and (4.15), and we additionally set

R[g] =Q[g] + P1[g]

P0[g].

Here we show that the minimization problem for R[g] = Q[g]P1[g]/(P0[g])2 can be

reduced to that of R[g]; the author of the present paper knew this idea for the first

18 N. HAMAMOTO

time in the recent paper [5] by Duong-Nguyen. To this end, first of all let us checkthat

14

(R[g]

)2≥ R[g] ≥ 1

4 infg∈Dµ

(R[g]

)2, (5.4)

where and hereafter the abbreviation g ∈ Dµ means g ∈ Dµ(R+) \ 0. The firstinequality in (5.4) is easy to verify; to check the second, notice that R[g] is invariantunder the scaling transformation

g(x) 7→ gλ(x) = g(λx)

for any λ > 0. By the change of variables y = λx, we have ∂∂xk

gλ(x) = λ ∂∂yk

g(y),

which leads to

R[gλ] =λ2Q[g] + λ−2P1[g]

P0[g].

Choosing λ =

(P1[g]

Q[g]

)1/4

, we get R[gλ] = 2√R[g] or equivalently

R[g] = 14

(R[gλ]

)2. (5.5)

Combining this equation with(R[gλ]

)2≥ inf

g∈Dµ

(R[g]

)2, we get the second inequal-

ity of (5.4) as desired. Taking the infimum on both sides of (5.4) then yields

infg∈Dµ

R[g] = 14 infg∈Dµ

(R[g]

)2. (5.6)

Next, we derive the relation between the minimizers of R[g] and R[g]. Assume thatg0 ∈ Dµ is a minimizer of R[g]:

R[g0] =14 infg∈Dµ

R[g].

Then, noticing from (5.5) that R[g0] =14

(R[(g0)λ]

)2for some λ > 0, we find from

(5.6) that

14

(R[(g0)λ]

)2= 1

4 infg∈Dµ

(R[g]

)2,

whence

R[(g0)λ] = infg∈Dµ

R[g].

In other words, we conclude that, if g0 ∈ Dµ is a minimizer of R[g] then (g0)λ is

that of R[g] for some λ > 0. Conversely, if g0 ∈ Dµ is a minimizer of R[g] then(g0)λ is that of R[g] for all λ > 0.

Step 2. We start with the nonnegative functional

I[g] :=

∫ ∞

0

(xg′′ + (x+ µ)g′ + (µ− b)g

)2xµ−1dx (5.7)

for g ∈ C∞µ (R+). By expanding the integrand, let I[g] split into

I[g] = I0[g] + I1[g],

where

I0[g] :=

∫ ∞

0

(x2 (g′′)

2+ (x+ µ)2 (g′)

2+ (µ− b)2g2

)xµ−1dx,

I1[g] := 2

∫ ∞

0

(x(x + µ)g′′g′ + (µ− b)

(xg′′g + (x+ µ)g′g

))xµ−1dx.

SHARP UNCERTAINTY PRINCIPLE INEQUALITY FOR SOLENOIDAL FIELDS 19

To compute I1[g] by integration by parts, we have

I1[g] =

∫ ∞

0

(xµ+1 + µxµ

) ((g′)2

)′

+ (µ− b)(2xµg′′g +

(xµ + µxµ−1

) (g2)′)

dx

=

∫ ∞

0

d

dx

((xµ+1 + µxµ

)(g′)2 + (µ− b)

(2xµg′g − µxµ−1g2

+(xµ + µxµ−1)g2

))dx

+

∫ ∞

0

−((µ+ 1)xµ + µ2xµ−1

)(g′)2

+ (µ− b)

(µ(µ− 1)xµ−2g2 − 2(g′)2xµ

−(µxµ−1 + µ(µ− 1)xµ−2

)g2

) dx

=

∫ ∞

0

( ((2b− 3µ− 1)x− µ2

)(g′)2 − (µ− b)µg2

)xµ−1dx,

where the second equality follows with the aid of the identity

xµg′′g =d

dx

(xµg′g − µ

2xµ−1g2

)+

(µ(µ− 1)

2x2g2 − (g′)2

)xµ

and the third follows from the condition (5.1) together with the cancellation of theterms µxµ−1g2. Then we get

I0[g] + I1[g] =

∫ ∞

0

(x2(g′′)2 +

((x + µ)2 + (2b− 3µ− 1)x− µ2

)(g′)2

+((µ− b)2 − (µ− b)µ

)g2

)xµ−1dx

=

∫ ∞

0

(x2(g′′)2 +

(x2 − (µ+ 1− 2b)x

)(g′)2 − b(µ− b)g2

)xµ−1dx.

=

∫ ∞

0

(g′′)2xµ+1dx+

∫ ∞

0

(x2(g′)2 − εg2

)xµ−1dx

−(√

µ2 − 4ε+ 1)∫ ∞

0

(g′)2xµdx

from the definition of b. Therefore, we have obtained the identity

I[g] = Q[g] + P1[g]− cP0[g] with c :=√µ2 − 4ε+ 1

for all g ∈ C∞µ (R+); the same also applies to g ∈ Dµ(R+) by the density argument.

In particular, it is clear from the non-negativity of (5.7) that

Q[g] + P1[g] ≥ cP0[g] or R[g] ≥ c (5.8)

for all g ∈ Dµ(R+), where the equality holds if and only if I[g] = 0 or equivalently

xg′′ + (x+ µ)g′ + (µ− b)g = 0 on R+. (5.9)

In order to specify the equality condition for (5.8) or to solve the equation (5.9),let us take the transformation

g(x) = e−xw(x)

which changes (5.9) into

x(e−xw)′′ + (x+ µ)(e−xw)′ + (µ− b)e−xw

= x(e−xw − 2e−xw′ + e−xw′′

)+ (x+ µ)

(−e−xw + e−xw′

)+ (µ− b)e−xw

= e−x (xw′′ + (µ− x)w′ − bw) = 0.

Then we get the Kummer equation

xw′′ + (µ− x)w′ − bw = 0

20 N. HAMAMOTO

which is known to have the two independent solutions (see e.g. [21]):

ϕ(x) = 1F1(b, µ, x) and ψ(x) = x1−µ1F1(b+ 1− µ, 2− µ, x).

Notice that the function e−xϕ(x) belongs to Dµ(R+), while e−xψ(x) does not since∫∞

0 ((e−xψ(x))′′)2xµ+1dx = ∞ from ψ′′(x) = O(x−1−µ) as x → 0. Therefore,

e−xϕ(x) is the unique solution to (5.9) (up to constant multiplication) when re-stricted to the space Dµ(R+). By recalling Step 1, we have obtained that

14 ming∈Dµ

(R[g]

)2= min

g∈Dµ

R[g] = c2/4,

and that all the minimizers of R[g] are given by the setCe−x

1F1(b, µ, x) ; C ∈ R

which is also a subset of the minimizers of R[g].Finally, let us specify the minimizers g0 of R[g]. According to the conclusion of

Step 2, there exists some λ > 0 such that (g0)λ(x) = g0(λx) is a minimizer of R[g].Then it follows from the above discussion that

(g0)λ(x) = g0(λx) ∝ e−x1F1(b, µ, x) for some λ > 0

or equivalently that

g0(x) ∝ e−λx1F1(b, µ, λx) for some λ > 0.

Therefore, it turns out that the set of all minimizers of R[g] is given byCeλx1F1(b, µ, λx) ; C ∈ R, λ > 0

.

The proof of the theorem is now complete.

For later use, the following fact provides a sufficient condition ensuring that boththe two cases g ∈ C∞

c (R+) and g ∈ Dµ(R+) give the same infimum value of R[g]:

Lemma 5.2. If µ > 2, then C∞c (R+) is dense in Dµ(R+).

Proof. Let g = g(x) ∈ Dµ(R+) ∩ C∞µ (R+) and let ζ0 = ζ0(x) ∈ C∞

c (R) withζ0(0) = 1. Define gnn∈N ⊂ C∞

c (R+) by the formula

gn(x) = ζn(x)g(x), where ζn(x) = ζ0(1n log x

)(∀n ∈ N, ∀x > 0).

In order to see that gn → g in Dµ(R+), a direct calculation yields

g′n − ζng′ = (∂ζn)

g

x,

g′′n − ζng′′ = (∂2ζn − ∂ζn)

g

x2+ 2(∂ζn)

g′

x,

where ∂ = x ddx as in (2.6). Taking the L2 integral of them yields

∫ ∞

0

(g′n − ζng′)2(x+ 1)xµdx = O(n−2)

∫ ∞

0

g2(1 +

1

x

)xµ−1dx,

∫ ∞

0

(g′′n − ζng′′)

2xµ+1dx = O(n−2)

∫ ∞

0

(g2

x2+ (g′)2

)xµ−1dx.

Notice from (5.1) that g(x) is bounded near x = 0; then it follows from ‖g‖Dµ<∞

that both the integrals on the right-hand sides are finite as far as µ > 2. Since the

SHARP UNCERTAINTY PRINCIPLE INEQUALITY FOR SOLENOIDAL FIELDS 21

convergences ζng′ → g′ and ζng

′′ → g′′ are dominated, by way of the L2 triangleinequality, we consequently get

∫ ∞

0

(g′n − g′)2(x+ 1)xµdx→ 0

and

∫ ∞

0

(g′′n − g′′)2xµ+1dx→ 0

as well as∫∞

0 (gn − g)2xµ−1dx → 0 (n → ∞). This fact means ‖gn − g‖Dµ

→ 0,as desired.

6. Conclusion

For every ν ∈ N, we set

µν =N

2+ ν and εν =

ν +N − 2

2.

This setting is the same as in (4.16) with (µ, ε) = (µν , εν), which clearly satisfiesboth the conditions µ > 2 in Lemma 5.2 and 0 < ε < µ2/4 in Theorem 5.1. Byusing their results, it holds from (4.13) that

CP,N,ν = 4 infg∈C∞

c (R+)R[g] = 4 inf

g∈Dµ

R[g] =(√

µ2ν − 4εν + 1

)2

=1

4

(√(2ν +N − 2)2 − 4(N − 3) + 2

)2,

from which we directly find that CP,N,ν+1 > CP,N,ν for all ν ∈ N. Then we getfrom (4.2) that

CP,N = CP,N,1 =

(√µ21 − 4ε1 + 1

)2

=1

4

(√N2 − 4(N − 3) + 2

)2.

Comparing this constant with that in Theorem 3.2, we see from (2.7) that

CN = min

14

(√N2 − 4(N − 3) + 2

)2, 1

4 (N + 2)2

=1

4

(√N2 − 4(N − 3) + 2

)2=

CP,3 = CT,3 (N = 3)

CP,N < CT,N (N ≥ 4), (6.1)

which gives the desired best HUP constant in Theorem 1.1.For the completeness of the proof of Theorem 1.1, it remains to specify the

profile(s) of extremal solenoidal fields which attain the best HUP constant. To this

end, let u0 ∈ C∞(RN )N be the poloidal field of the form

u0(x) = D

(c · x|x| g0(|x|

2)

)(6.2)

with c ∈ RN \ 0 being any constant vector (which serves as the axis of symmetryof u0), where g0 ∈ Dµ1

(R+) is given by

g0(s) = e−λs1F1(b1, µ1, λs) with b1 =

µ1 −√µ21 − 4ε1

2and λ > 0

as in (5.3) with (b, µ, ε) = (b1, µ1, ε1). We have to check that

u0 ∈ D(RN ) and

∫RN |∇u0|2dx

∫RN |u0|2|x|2dx

(∫RN |u0|2dx

)2 = CN .

To do so, define the linear operator D1 : C∞(R+) → C∞(RN )N by

(D1g) (x) = D

(c · x|x| g(|x|

2)

)∀g ∈ C∞(R+),

22 N. HAMAMOTO

in order that D1g0 = u0. Applying (4.11) and (4.12) to the case ν = 1 andY (σ) = c · σ, we then see that the equations

c

∫

RN

|D1g|2|x|2β = Pβ [g], c

∫

RN

|∇D1g|2dx = 4Q[g]

and

∫RN |∇D1g|2dx

∫RN |D1g|2|x|2dx

(∫RN |D1g|2dx

)2 = 4R[g]

∀g ∈ C∞c (R+) (6.3)

hold with c > 0 depending only on N , where Pβ [g], Q[g] and R[g] are the same asin (4.15) and (4.14) with (µ, ε) = (µ1, ε1). In particular, we get

c‖D1g‖2D = P0[g] + P1[g] + 4Q[g] ≤ ‖g‖2Dµ1∀g ∈ C∞

c (R+)

in terms of the notations (1.4) and (5.2). Therefore, by way of the density argumentsin Lemmas 2.3 and 5.2, it holds that

c‖D1g‖D ≤ ‖g‖Dµ1∀g ∈ Dµ1

(R+) ∩ C∞(R+),

which says that D1 is continuous on C∞(R+) from the topology of Dµ1(R+) to

D(RN ). Hence the same formulae in (6.3) are still applicable to

∀g ∈ Dµ1(R+) ∩ C∞(R+);

in particular, by letting g = g0, we obtain ‖u0‖D = ‖D1g0‖D <∞ and∫RN |∇u0|2dx

∫RN |u0|2|x|2dx

(∫RN |u0|2dx

)2 = 4R[g0] =

(√µ21 − 4ε1 + 1

)2

= CN ,

where the second equality holds from Theorem 5.1 with (µ, ε) = (µ1, ε1). Thisproves the desired result.

In view of (6.1), for the case N = 3, there is another profile of the extremalsolenoidal fields: according to Theorem 3.2, the best HUP constant C3 is alsoachieved by the toroidal field

u(x) = (c1 · x, c2 · x, c3 · x) e−c|x|2 (6.4)

with c > 0 and with (c1, c2, c3) ∈ R3×3 being any antisymmetric constant 3 × 3matrix.

The proof of Theorem 1.1 is now complete.

Acknowledgments

This work is supported by JSPS KAKENHI Grant number JP21J00172. The authorthanks to Prof. Y. Kabeya (Osaka Prefecture University) for his great support andencouragement. Additionally, this research was partly supported by JSPS KAK-ENHI Grant-in-Aid for Scientific Research (B) 19H01800 (F. Takahashi) and OsakaCity University Advanced Mathematical Institute (MEXT Joint Usage/ResearchCenter on Mathematics and Theoretical Physics JPMXP0619217849).

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Department of Mathematical Sciences, Osaka Prefecture University,

Sakai, Osaka 599-8531, JapanEmail address: yhjyoe@yahoo.co.jp (N. Hamamoto)