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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)
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:
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5 Page 2 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 Differential Equations and Applications
SN Partial Differ. Equ. Appl. (2020) 1:5 Page 3 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
ðk � 1Þðk þ n� 1Þ Pkf
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.
SN Partial Differential Equations and Applications
5 Page 4 of 15 SN Partial Differ. Equ. Appl. (2020) 1:5
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
ðk � 1Þðk þ n� 1Þ Pkf
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.
SN Partial Differential Equations and Applications
SN Partial Differ. Equ. Appl. (2020) 1:5 Page 5 of 15 5
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
SN Partial Differential Equations and Applications
5 Page 6 of 15 SN Partial Differ. Equ. Appl. (2020) 1:5
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
SN Partial Differential Equations and Applications
SN Partial Differ. Equ. Appl. (2020) 1:5 Page 7 of 15 5
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
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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,
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SN Partial Differ. Equ. Appl. (2020) 1:5 Page 9 of 15 5
(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;
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5 Page 10 of 15 SN Partial Differ. Equ. Appl. (2020) 1:5
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
� �
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SN Partial Differ. Equ. Appl. (2020) 1:5 Page 11 of 15 5
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
ðk � 1Þðk þ n� 1Þ Pkf
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
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5 Page 12 of 15 SN Partial Differ. Equ. Appl. (2020) 1:5
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
SN Partial Differential Equations and Applications
SN Partial Differ. Equ. Appl. (2020) 1:5 Page 13 of 15 5
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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