Hardy type inequalities with spherical derivatives · 2020-02-26 · ORIGINAL PAPER Hardy type...

ORIGINAL PAPER

Hardy type inequalities with spherical derivatives

Neal Bez1 • Shuji Machihara1 • Tohru Ozawa2

Received: 3 September 2019 / Accepted: 7 November 2019 / Published online: 23 January 2020� Springer Nature Switzerland AG 2020

AbstractA Hardy type inequality is presented with spherical derivatives in Rn with n� 2 in the

framework of equalities. This clarifies the difference between contribution by radial and

spherical derivatives in the improved Hardy inequality as well as nonexistence of non-

trivial extremizers without compactness arguments.

Mathematics Subject Classification Primary 26D10 � Secondary 35A23 � 46E35

Introduction

In this paper, we study the classical Hardy inequality of the form

n� 2

2

� �2f

jxj

��2

2

�krfk22 ð1Þ

for all f 2 H1ðRnÞ with n� 3, where k � k2 is the standard norm on L2ðRnÞ;rf ¼ðo1f ; . . .; onf Þ is the gradient of fwith ojf ¼ of=oxj; krfk22 is the Dirichlet integral defined by

krf k22 ¼Xnj¼1

kojfk22;

and H1ðRnÞ is the standard Sobolev space of order one built over L2ðRnÞ. There is a hugeliterature on the Hardy inequality and it is impossible to make a list of references which

covers all important papers; for instance, we refer the reader to

[2, 3, 8, 11, 15–17, 19–21, 23, 25] and references therein.

This article is part of the section ‘‘Theory of PDEs’’ edited by Eduardo Teixeira.

& Shuji [email protected]

Neal [email protected]

Tohru [email protected]

1 Department of Mathematics, Faculty of Science, Saitama University, Saitama 338-8570, Japan

2 Department of Applied Physics, Waseda University, Tokyo 169-8555, Japan

SN Partial Differential Equations and Applications

SN Partial Differ. Equ. Appl. (2020) 1:5https://doi.org/10.1007/s42985-019-0001-1(0123456789().,-volV)(0123456789().,-volV)

http://orcid.org/0000-0002-0860-6184

http://crossmark.crossref.org/dialog/?doi=10.1007/s42985-019-0001-1&domain=pdf

https://doi.org/10.1007/s42985-019-0001-1

The viewpoint that we take in this paper is to look at (1) in the framework of equalities.

In particular, we look for an equality with a remainder term which implies (1) once it is

disregarded. For instance, the following equality presented in [8, 11]

n� 2

2

� �2f

jxj

��2

2

¼ krfk22 � rf þ n� 2

2

x

jxj2f

��2

2

ð2Þ

is a typical example in this direction. It is obvious that (2) implies (1) by dropping the

second term on the right hand side of (2). The explicit form of the remainder clarifies the

nonexistence of nontrivial cases of equality in (1) through the first order equations

0 ¼ rf þ n� 2

2

x

jxj2f ¼ jxj1�

n2r jxj

n2�1

f� �

with f 2 H1ðRnÞ. An additional argument establishes that the constant in (1) is best pos-

sible and thus there are no nontrivial extremizers. There are a number of papers on the best

constant in Hardy type inequalities and, for example, the reader may consult [10, 18, 27]

for historical comments and further results in this direction.

A detailed observation has been made in [21] with the equality

n� 2

2

� �2f

jxj

��2

2

¼ korfk22 � orf þn� 2

2jxj f

��2

2

; ð3Þ

where or is the radial derivative defined by

or ¼x

jxj � r ¼Xnj¼1

xj

jxj oj:

Indeed, (3) implies another Hardy inequality

n� 2

2

� �2f

jxj

��2

2

�korfk22; ð4Þ

which in turn implies (1) since the right hand side of (4) is bounded by that of (1). These

investigations (3) and (4) are L2 based. Although there are Lp based studies [15, 16, 22],

we restrict ourselves to L2 based inequalities in this paper.

The associated difference between the right hand sides of (1) and (4) may be taken in a

good shape through the decomposition of the Dirichlet integral

krfk22 ¼ korfk22 þ kLfk22; ð5Þ

where L ¼ r� xjxj or is the spherical derivative and the spherical component of the

Dirichlet integral is defined as

kLfk22 ¼Xnj¼1

kLjfk22

with

Lj ¼ oj �xj

jxj or ¼ oj �Xnk¼1

xjxk

jxj2ok:


5 of 15 SN Partial Differ. Equ. Appl. (2020) 1:5

Meanwhile, a particular account has been taken into the inequality

n2

4

f

jxj

��2

2

�krfk22 ð6Þ

for all f 2 C10 ðRnÞ with n� 2 satisfying

ZSn�1

f ðrxÞdrðxÞ ¼ 0 ð7Þ

for all r� 0, where r is the Lebesgue measure on the unit sphere

Sn�1 ¼ fx 2 Rn; jxj ¼ 1g, [7, 12]. In [12], the inequality (6) is referred to as an improved

Hardy inequality on the basis of the improvement in the coefficient n2

4on the left hand side

of (6), which is larger than the corresponding coefficient n�24

� �2on the left hand side of (1),

as well as of the applicable range of dimensions, in particular, n ¼ 2 is now admissible.

The purpose of this paper is to present a new equality which clarifies why the

improvement in (6) over (1) is realized under (7) on the basis of the separation of con-

tributions by radial and spherical derivatives of functions in H1ðRnÞ. To state our main

results precisely, we introduce the following notation. Some associated basic properties of

those notions are summarized in the next section. We denote by L2radðRnÞ and H1radðRnÞ the

closed subspaces L2ðRnÞ and H1ðRnÞ, respectively, of radial functions:

L2radðRnÞ :¼ ff 2 L2ðRnÞ; There exists u 2 L2ð0;1Þ such that

f ðxÞ ¼ uðjxjÞjxj1�n2 for almost all x 2 Rnnf0gg;

H1radðRnÞ :¼ ff 2 ðH1 \ L2radÞðRnÞ; orf 2 L2radðRnÞg:

For any f 2 L2ðRnÞ, we denote by Pf its radial average over the unit sphere

ðPf ÞðxÞ :¼ 1

rn�1

ZSn�1

f ðjxjxÞdrðxÞ; x 2 Rn:

Then P : f 7!Pf induces the orthogonal projection from L2ðRnÞ onto L2radðRnÞ as well as

from H1ðRnÞ onto H1radðRnÞ (see Proposition 7 below). The operator P? :¼ I � P is the

orthogonal projection onto the orthogonal complement of these spaces in the following

orthogonal decompositions:

L2ðRnÞ ¼ L2radðRnÞ � ðL2radðRnÞÞ?;H1ðRnÞ ¼ H1

radðRnÞ � ðH1radðRnÞÞ?:

We also use the complete orthogonal decomposition

L2ðRnÞ ¼ ak� 0

HkðRnÞ; ð8Þ

where HkðRnÞ is a closed subspace spanned by spherical harmonics of order k multiplied

by radial functions. We denote by Pk the associated orthogonal projection. We refer the

reader to [4–6, 24, 26] for details on the decomposition (8). Here we notice that L2radðRnÞ ¼H0ðRnÞ with P ¼ P0. We now state the main results in this paper.


SN Partial Differ. Equ. Appl. (2020) 1:5 of 15 5

Theorem 1 Let n� 2. Then, the following equality

ðn� 1Þ P?f

jxj

��2

2

¼ kLP?fk22 �X1k¼2

ðk � 1Þðk þ n� 1Þ Pkf

jxj

��2

2

ð9Þ

holds for all f 2 H1ðRnÞ.

Corollary 2 Let n� 2. Then the following inequality

ðn� 1Þ P?f

jxj

��2

2

�kLP?fk22 ð10Þ

holds for all f 2 H1ðRnÞ. Equality holds in (10) if and only if there exist a 2 Cn and

g; h 2 H1radðRnÞ such that

f ðxÞ ¼ ða � xÞgðxÞ þ hðxÞ ð11Þ

for almost all x 2 Rnnf0g, where a � y ¼Pn

j¼1 ajyj for y 2 Rn. In this case, both sides

of (10) are given by

ðn� 1Þ P?f

jxj

��2

2

¼ kLP?f k22 ¼n� 1

nrn�1jaj2

Z 1

0

juðrÞj2dr; ð12Þ

where u 2 L2ð0;1Þ satisfies gðxÞ ¼ uðjxjÞjxj�n�12 for almost all x 2 Rnnf0g and

jaj2 ¼Pn

j¼1 aj�aj.

Theorem 3 Let n� 2. Then, the following equalities

n� 2

2

� �2f

jxj

��2

2

þðn� 1Þ P?f

jxj

��2

2

¼ n� 2

2

� �2Pf

jxj

��2

2

þ n2

4

P?f

jxj

��2

2

¼ krf k22 � or þn� 2

2jxj

� �f

��2

2

�X1k¼2


jxj

��2

2

ð13Þ

hold for all f 2 H1ðRnÞ.

Corollary 4 Let n� 2. Then, the following inequality

n� 2

2

� �2f

jxj

��2

2

þðn� 1Þ P?f

jxj

��2

2

�krf k22 ð14Þ

holds for all f 2 H1ðRnÞ. Equality holds in (14) if and only if f ¼ 0.



Theorem 5 Let n� 2. Then, the following equality

n2

4

P?f

jxj

��2

2

¼ krP?fk22 � or þn� 2

2jxj

� �P?f

��2

2

�X1k¼2


jxj

��2

2

ð15Þ

holds for all f 2 H1ðRnÞ.

Corollary 6 (Improved Hardy inequality [7, 12]) Let n� 2. Then, the following inequality

n2

4

P?f

jxj

��2

2

�krP?f k22 ð16Þ

holds for all f 2 H1ðRnÞ. Equality holds in (16) if and only if f 2 H1radðRnÞ. In this case,

both sides of (16) vanish.

The inequality (14) improved the Hardy inequality (1) in the sense that (14) reveals a

novel term ðn� 1Þ P?fjxj

�� 22on the left hand side and that the improvement in (6) in two

space dimensions arises as a result of the existence of the novel term P?fjxj

�� 22when n ¼ 2.

Moreover, (14) clarifies the contribution by the orthogonal component to H1radðRnÞ with

coefficient n� 1, which together with the standard coefficient n�22

� �2yields the improved

coefficient n2

4in (6) on the basis of the simple identity n�2

2

� �2þðn� 1Þ ¼ n2

4.

In Sect. 2, we prove a density lemma, which enables us to prove the main theorems for

functions in C10 ðRnnf0gÞ. In Sect. 3, we prove the main results stated above. Furthermore,

we also include a justification of the observation that the constant n2

4in the improved Hardy

inequality (16) is best possible and thus we establish the nonexistence of nontrivial

extremizers for the improved Hardy inequality.

Preliminaries

In this section, we collect basic propositions for the proofs of the main theorems. From

now on, we assume that the space dimension n is greater than or equal to 2 unless specified

otherwise. We denote by ð�j�Þ the standard scalar product in L2ðRnÞ.

Proposition 7 The following relations hold:

1. P2 ¼ P; ðP?Þ2 ¼ P?;PP? ¼ P?P ¼ 0.

2. ðPujP?vÞ ¼ 0; kuk22 ¼ kPuk22 þ kP?uk22; u; v 2 L2ðRnÞ.3. orPu ¼ Poru; LPu ¼ 0; u 2 H1ðRnÞ.4. ðrPuÞðxÞ ¼ x

jxj ðPoruÞðxÞ; u 2 H1ðRnÞ; x 2 Rnnf0g.5. rP?u ¼ LP?u; u 2 H1ðRnÞ.6. ðrPujrP?vÞ ¼ 0; kruk22 ¼ krPuk22 þ krP?uk22; u; v 2 H1ðRnÞ.



Each of the claims in the above proposition can be verified by straightforward calculations.

The following proposition states some basic properties of spherical derivative operator

L and the Laplacian on the unit sphere:

DSn�1 :¼X

1� j\k� n

ðxjok � xkojÞ2:

Proposition 8 The following relations hold:

1. ðLjujvÞ ¼ �ðujLjvÞ þ ðn� 1Þ uj xj

jxj2 v� �

; u; v 2 H1ðRnÞ.2. D ¼ o2r þ n�1

jxj or þ 1

jxj2 DSn�1 .

3. DSn�1 ¼ jxj2Xnj¼1

L2j ¼Xnj¼1

ðjxjLjÞ2.

4. DSn�1Pk ¼ �kðk þ n� 2ÞPk; k� 0.

Parts (1)–(3) are easily verified by straightforward calculations. For Part (4), we refer the

reader to, for example, [5, 24].

The following proposition is essential in the proof of the main theorems:

Proposition 9 P?ðC10 ðRnnf0gÞÞ is dense in P?ðH1ðRnÞÞ ¼ ðH1

radðRnÞÞ? if n� 2.

Proof Let n; g 2 C1ðRÞ satisfy 0� n; g� 1; n ¼ 0 on ð�1; 1=2�; n ¼ 1 on ½1;1Þ; g ¼ 1

on ð�1; 1�; g ¼ 0 on ½2;1Þ. For any positive integer j, we define fj 2 C1ðRnÞ by

fjðxÞ ¼ nðjjxjÞgðjxjjÞ; x 2 Rn. Then, supp fj � fx 2 Rn; 1

2j� jxj � 2jg and fjðxÞ ¼ 1 if

1j� jxj � j. Moreover, we have

rfjðxÞ ¼x

jxj jn0ðjjxjÞg jxjj

� �þ 1

jnðjjxjÞg0 jxj

j

� ��

¼

x

jxj jn0ðjjxjÞg jxj

j

� �if jxj 2 1

2j;1

j

;

0 if jxj 2 0;1

2j

[ 1

j; j

[ ½2j;1�;

x

jxj1

jnðjjxjÞg0 jxj

j

� �if jxj 2 j; 2j½ �:

8>>>>>>><>>>>>>>:

This implies

jrfjðxÞj �

1

jxj kn0k1 if jxj 2 1

2j;1

j

;

0 if jxj 2 0;1

2j

[ 1

j; j

[ ½2j;1�;

1

jkg0k1 if jxj 2 j; 2j½ �:

8>>>>>>><>>>>>>>:

ð17Þ



For any f 2 H1ðRnÞ we define ðfjÞj� 1 � C10 ðRnnf0gÞ by fj ¼ qj ðfjP?f Þ, where is the

standard convolution for functions on Rn and qjðxÞ ¼ jnqðjxÞ with q 2 C10 ðRnÞ satisfying

0� q� 1, suppq � fx 2 Rn; jxj � 1=4g, and kqk1 ¼ 1. It suffices to prove that P?fj !P?f in H1ðRnÞ as j ! 1.

For that purpose we write their difference as

P?fj � P?f

¼ P?ðqj ðfjP?f ÞÞ � qj ðP?ðfjP?f ÞÞ

þ qj ðP?ðfjP?f ÞÞ � P?f

¼ ðI � PÞðqj ðfjP?f ÞÞ � qj ððI � PÞfjP?f Þ

þ qj ðfjP?f Þ � P?f

¼ �Pðqj ðfjP?f ÞÞ

þ qj ðfjP?f Þ � P?f

¼ �Pðqj P?f Þ þ Pðqj ðð1� fjÞP?f ÞÞ

� qj ðð1� fjÞP?f Þ þ qj P?f � P?f

¼ �Pðqj P?f Þ � P?ðqj ðð1� fjÞP?f ÞÞ þ qj P?f � P?f ;

ð18Þ

where we have used

P?fjP? ¼ fjP

?P? ¼ fjP?;

PfjP? ¼ fjPP

? ¼ 0:

The first term on the right hand side of the last equality of (18) is rewritten as

Pðqj P?f ÞðxÞ

¼ 1

rn�1

ZSn�1

ðqj P?f ÞðjxjxÞdrðxÞ

¼ 1

rn�1

Zjyj � 1=4

qðyÞZSn�1

ðP?f Þ jxjx� 1

jy

� �drðxÞdy

¼ 1

rn�1

Zjyj � 1=4

qðyÞZSn�1

ðP?f Þ jxjx� 1

jy

� �� ðP?f ÞðjxjxÞ

drðxÞdy;

ð19Þ

which, using the Cauchy–Schwarz and Minkowski inequalities, is estimated in L2 by

kPðqj P?f Þk2 �Zjyj � 1=4

qðyÞks1jyP

?f � P?fk2dy

� supjyj � 1=ð4jÞ

ksyP?f � P?fk2 ! 0ð20Þ



as j ! 1, where ðsyf ÞðxÞ ¼ f ðx� yÞ. We differentiate (19) to have

rPðqj P?f ÞðxÞ

¼ 1

rn�1

Zjyj � 1=4

qðyÞZSn�1

x

jxj x � rP?f� �

jxjx� 1

jy

� ��

� rP?f� �

jxjxð Þ��drðxÞdy;

which is estimated in L2 in the same way as in (20) by

krPðqj P?f Þk2 � supjyj � 1=ð4jÞ

ksyrP?f �rP?f k2 ! 0 ð21Þ

as j ! 1. The other terms on the right hand side of the last equality of (18) are estimated

in L2 as

kP?ðqj ðð1� fjÞP?f ÞÞk2 þ kqj P?f � P?fk2�kð1� fjÞP?f k2 þ kqj P?f � P?f k2 ! 0

ð22Þ

as j ! 1. We differentiate the same terms on the right hand side of the last equality

of (18) to have

�rðP?ðqj ðð1� fjÞP?f ÞÞÞ þ rðqj P?f � P?f Þ¼ �rðqj ðð1� fjÞP?f ÞÞ þ rPðqj ðð1� fjÞP?f Þ þ qj rP?f �rP?f

¼ qj ððrfjÞP?f Þ � x

jxjPðqj ððorfjÞP?f ÞÞ � qj ðð1� fjÞrP?f Þ

þ x

jxjPðqj ðð1� fjÞorP?f ÞÞ þ qj rP?f �rP?f ;

which is estimated by

2kðrfjÞP?fk2 þ 2kð1� fjÞrP?f k2 þ kqj rP?f �rP?fk2: ð23Þ

The second and third terms in (23) tend to zero as j ! 1. By (17), the first norm in (23) is

estimated as

kðrfjÞP?fk22 �kn0k21Zjxj � 1=j

jxj�2jðP?f ÞðxÞj2dxþ 1

j2kg0k21kfk22: ð24Þ

Therefore it remains to prove that

limj!1

Zjxj � 1=j

jxj�2jðP?f ÞðxÞj2dx ¼ 0: ð25Þ

We write the integral in (25) in polar coordinates as

Zjxj � 1=j

jxj�2jðP?f ÞðxÞj2dx

¼ 1

r2n�1

Z 1=j

0

ZSn�1

ZSn�1

ðf ðrxÞ � f ðrx0ÞÞdrðx0Þ��

��2

drðxÞrn�3dr:

ð26Þ



By the Cauchy–Schwarz inequalities,

Zjxj � 1=j

jxj�2��ðP?f ÞðxÞ

��2dx

� 1

rn�1

Z 1=j

0

ZSn�1

ZSn�1

jf ðrxÞ � f ðrx0Þj2drðx0ÞdrðxÞrn�3dr

¼ 1

rn�1

Z 1=j

0

ZSn�1

ZSn�1

Z 1

0

rf ðrðtxþ ð1� tÞx0ÞÞ � ðx� x0Þdt��

��2

drðx0ÞdrðxÞrn�1dr

� 1

rn�1

Z 1=j

0

ZSn�1

ZSn�1

Z 1

0

��rf ðrðtxþ ð1� tÞx0ÞÞ��2

dtjx� x0j2drðx0ÞdrðxÞrn�1dr

� 4

rn�1

Z 1=j

0

ZSn�1

ZSn�1

Z 1

1=2

þZ 1=2

0

!��rf ðrðtxþ ð1� tÞx0ÞÞ��2

dtdrðx0ÞdrðxÞrn�1dr

¼ 4

rn�1

Zjxj � 1=j

Z 1

1=2

ZSn�1

��rf ðtxþ ð1� tÞjxjx0Þ��2drðx0Þdtdx

þ 4

rn�1

Zjxj � 1=j

Z 1=2

0

ZSn�1

��rf ðtjxjxþ ð1� tÞxÞ��2drðxÞdtdx

¼ 8

rn�1

Zjxj � 1=j

Z 1

1=2

ZSn�1

��rf ðtxþ ð1� tÞjxjxÞ��2drðxÞdtdx:

ð27Þ

We consider the mapping in Bð0; 1=jÞ ¼ fx 2 Rn; jxj\1=jg defined by

u : Bð0; 1=jÞ 3 x 7!txþ ð1� tÞjxjx 2 Rn:

The range uðBð0; 1=jÞÞ is in B(0; 1 / j) and therefore u is regarded as a mapping from

B(0; 1 / j) into itself. Moreover, ujBð0; 1=jÞnf0g, the restriction on Bð0; 1=jÞnf0g is

smooth with Jacobian given by

detðu0ðxÞÞ ¼ tn�1 t þ ð1� tÞ x

jxj � x� �

; x 2 Bð0; 1=jÞnf0g;

which is positive for all ðt;xÞ 2 ð1=2; 1� Sn�1. This implies

Zjxj � 1=j

jrf ðuðxÞÞj2dx ¼ZuðBð0;1=jÞÞ

jrf ðyÞj2 1

detðu0ðu�1ðyÞÞÞ dy

�Zjyj � 1=j

jrf ðyÞj2 1

tn�1 t þ ð1� tÞ u�1ðyÞju�1ðyÞj � x

� � dy: ð28Þ

We now estimate the last integral in (27). We consider separately two cases:

(a) 3=4� t� 1,



(b) 1=2� t� 3=4.

If 3=4� t� 1, then the contribution by the Jacobian in (28) is estimated as

tn�1 t þ ð1� tÞ u�1ðyÞju�1ðyÞj � x

� �� tn�1ðt � ð1� tÞÞ ¼ tn�1ð2t � 1Þ

and

Zjxj � 1=j

Z 1

3=4

ZSn�1

jrf ðtxþ ð1� tÞjxjxÞj2drðxÞdtdx

� rn�1

Z 1

3=4

1

tn�1ð2t � 1Þ dt !Z

jxj � 1=j

jrf ðxÞj2dx ! 0

ð29Þ

as j ! 1. In the case 1=2� t� 3=4, we prove that there exists a constant cn [ 0 such that

supx6¼0

Z 3=4

1=2

ZSn�1

1

t þ ð1� tÞ xjxj � xdrðxÞdt� cn: ð30Þ

The integral over the unit sphere is rewritten as

ZSn�1

1

t þ ð1� tÞ xjxj � xdrðxÞ ¼ 2p

n�12

C n�12

� �Z 1

�1

1

t þ ð1� tÞs ð1� s2Þn�32 ds; ð31Þ

so that the required inequality (30) is reduced to the convergence of the following double

integral:

Z 3=4

1=2

Z 1

�1

1

t þ ð1� tÞs ð1� s2Þn�32 dsdt: ð32Þ

We divide (32) into three parts

Z 3=4

1=2

Z 1

�1

1

t þ ð1� tÞs ð1� s2Þn�32 dsdt

¼Z 3=4

1=2

Z 1

1=2

1

t þ ð1� tÞs ð1� s2Þn�32 ds

!dt

þZ 3=4

1=2

Z 1=2

�1=2

1

t þ ð1� tÞs ð1� s2Þn�32 ds

!dt

þZ �1=2

�1

Z 3=4

1=2

1

t þ ð1� tÞs dt !

ð1� s2Þn�32 ds

ð33Þ

and we denote by I, II, III the first, second, third term on the right hand side of (33),

respectively. We estimate I as

I�Z 3=4

1=2

Z 1

1=2

1

t þ ð1� tÞ 12

ð1� s2Þn�32 ds

!dt ¼ 2

Z 3=4

1=2

1

t þ 1dt

Z 1

�1

ð1� s2Þn�32 ds;



where the last integral converges for n� 2. For II, we note that the integrand is continuous

for ðs; tÞ 2 ½�1=2; 1=2� ½1=2; 3=4�. We evaluate III as

III ¼Z �1=2

�1

1

1� slog

sþ 3

2ðsþ 1Þ

� �ð1� s2Þ

n�32 ds;

which is convergent since the singularity at s ¼ �1 is of order ð1þ sÞn�32 logð1þ sÞ, which

is integrable for n� 2. This proves (30), which implies

Zjxj � 1=j

Z 3=4

1=2

ZSn�1

jrf ðtxþ ð1� tÞjxjxÞj2drðxÞdtdx

� cn

Zjxj � 1=j

jrf ðxÞj2dx ! 0

ð34Þ

as j ! 1. By (27), (29), and (34), we have proved (25), as required. This completes the

proof.

Proofs of the main theorems

In this section, we prove Theorems 1, 3, 5 and their corollaries. By a density argument

based on Proposition 9, it suffices to prove the theorems for functions in C10 ðRnnf0gÞ. In

the proofs below, all functions are supposedly elements of C10 ðRnnf0gÞ.

Proof of Theorem 1 Let f 2 C10 ðRnnf0gÞ. By Propositions 7 and 8, the first term on the

right hand side of (9) is represented as

��LP?f��22¼Xnj¼1

��LjP?f��22

¼ �Xnj¼1

L2j P?f jP?f

� �

¼ �Xnj¼1

X1k¼0

L2j P?Pkf jP?f

� �

¼ �Xnj¼1

X1k¼1

L2j P?Pkf jP?f

� �

¼ �X1k¼1

jxj�2DSn�1PkP?f jP?f

� �



and hence

��LP?f��22¼ �

X1k¼1

Z 1

0

ðDSn�1PkP?f Þðrð�ÞjP?f ðrð�ÞÞÞL2ðSn�1Þr

n�3dr

¼X1k¼1

kðk þ n� 2ÞZ 1

0

ðPkP?f Þðrð�ÞjP?f ðrð�ÞÞÞL2ðSn�1Þr

n�3dr

¼X1k¼1

kðk þ n� 2Þ 1

jxjPkP?f

��2

2

¼ ðn� 1ÞX1k¼1

1

jxjPkP?f

��2

2

þX1k¼1

ðkðk þ n� 2Þ

� ðn� 1ÞÞ 1

jxjPkP?f

��2

2

¼ ðn� 1ÞX1k¼0

Pk

P?f

jxj

� ��2

2

þX1k¼2

ðk � 1Þðk þ n� 1Þ 1

jxjPkP?f

��2

2

¼ ðn� 1Þ P?f

jxj

��2

2

þX1k¼2


jxj

��2

2

;

where we have used relations

P0P? ¼ P?P0 ¼ 0;

PkP? ¼ PkðI � P0Þ ¼ Pk for k� 1;

Pk

g

jxj

� �¼ 1

jxjPkg for k� 0:

This completes the proof. h

Proof of Corollary 2 The inequality (10) is a direct consequence of (9). The equality

in (10) holds if and only if Pkfjxj ¼ 0 for all nonnegative integers k with k� 2, namely,

fjxj 2 H0 �H1. This proves (11). Then we take g; h 2 H1

radðRnÞ as in (11). In this case, we

have

ðP?f ÞðxÞ ¼ ða � xÞgðxÞ;

ðLP?f ÞðxÞ ¼ a� x

jxj a � x

jxj

� �� gðxÞ

for almost all x 2 Rnnf0g. Since g is radial and P?fjxj 2 H1, a new function u 2 L2ð0;1Þ is

defined to satisfy gðxÞ ¼ uðjxjÞjxj�n�12 for almost all x 2 Rnnf0g. We evaluate two integrals

in (10) as



P?f

jxj

��2

2

¼Z 1

0

ZSn�1

ja � xj2jgðrxÞj2drðxÞrn�1dr

¼ZSn�1

ja � xj2drðxÞZ 1

0

juðrÞj2dr

¼ rn�1

njaj2

Z 1

0

juðrÞj2dr;

kLP?fk22 ¼Z 1

0

ZSn�1

ja� xða � xÞj2drðxÞjuðrÞj2dr

¼ZSn�1

ðjaj2 � ja � xj2ÞdrðxÞZ 1

0

juðrÞj2dr

¼ 1� 1

n

� �rn�1jaj2

Z 1

0

juðrÞj2dr;

where we have used

ZSn�1

ja � xj2drðxÞ ¼Xnj¼1

Xnk¼1

aj�ak

ZSn�1

xjxkdrðxÞ

¼Xnj¼1

jajj2ZSn�1

x2j drðxÞ

¼Xnj¼1

jajj21

nrn�1 ¼

1

nrn�1jaj2:

This proves (12).

Proof of Theorem 3 The equality (13) follows from (3), (5), (9), and kLP?fk2 ¼ kLfk2.h

Proof of Corollary 4 The equality in (14) holds if and only if (11) and

jxj1�n2orðjxj

n2�1f Þ ¼ or þ

n� 2

2jxj

� �f ¼ 0:

Then f is written as f ðxÞ ¼ jxj1�n2w x

jxj

� �for some function w : Sn�1 ! C, which together

with (11) implies that f ðxÞ ¼ jxj1�n2 a � x

jxj

� �for some a 2 Cn. In this case, f

jxj 2 L2ðRnÞ if

and only if a ¼ 0, which means f ¼ 0. h

Proof of Theorem 5 The equality (15) follows by substituting f by P?f in (13).

Proof of Corollary 6 The equality (16) follows if and only if P?f ¼ 0, which means

f 2 H1radðRnÞ.

We conclude the paper with a justification of the claim that the constant n2

4in (16) is best

possible by making use of an argument in [27]. We first observe that P commutes with the



Fourier transform, hence so does P?, and thus by Plancherel’s theorem, it suffices to show

that the constant in

n2

4ð2pÞn��P

?bfjxj

��2

2

�ZRn

jxj2��P?f ðxÞ

��2dx ð35Þ

is best possible. Here, we are using the Fourier transform

bf ðnÞ ¼ZRn

f ðxÞe�ix�ndx

for appropriate functions f : Rn ! C. To establish optimality of the constant in (35), we

consider fkðxÞ ¼ Y1ð xjxjÞgkðjxjÞ, where Y1 is chosen to be a unit vector inH

1ðRnÞ and gk is tobe chosen momentarily. It is straightforward to see that fk is invariant under the action of

P?, and therefore

ZRn

jxj2��P?fkðxÞ

��2dx ¼Z 1

0

jgkðrÞj2rnþ1dr: ð36Þ

For the norm on the left hand side, we use the well-known expression for dY1dr in terms of

the Bessel function Jn=2 (see, for example, Corollary 5.1 in [26]) to write

bfk ðxÞ ¼Z 1

0

dY1drðrxÞgkðrÞrn�1dr

¼ �ið2pÞn=2Y1ð xjxjÞjxj1�n

2

Z 1

0

Jn=2ðrjxjÞgkðrÞrn=2dr:

Thus, bfk is also invariant under the action of P? and

1

ð2pÞn��P

?bfkjxj

��2

2

¼Z 1

0

jU1gkðsÞj2sn�3ds ð37Þ

where U1 is the operator given by

U1gðsÞ ¼1

sn2�1

Z 1

0

Jn=2ðrsÞgðrÞrn=2dr:

By Lemma 3.8 of [27], we obtain the existence of ðgkÞk� 1 � C10 ðRþÞ such that the

quantity in (36) is equal to 1 for all k and the quantity in (37) converges to 4n2as k ! 1.

This shows that the constant in (35) is best possible and hence so is the constant in (16).

Acknowledgements The first author was supported by JSPS KAKENHI Grant number 16H05995 and16K1377, the second author was supported by JSPS KAKENHI Grant number JP16K05191, the third authorwas supported by JSPS KAKENHI Grant number 19H00644 and 18KK0073.

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