Remarks on the Caffarelli-Kohn-Nirenberg inequalities of the logarithmic type (The structure of function spaces and its environment)

(1)

Remarks

on

the

Caffarelli‐Kohn‐Nirenberg

inequalities

of

the

_logarithmic

_type

Shuji

Machihara

_l,

Tohru

Ozawa2

and Hidemitsu

Wadade3

1

Department of Mathematics, Saitama University, 255_{Shimookubo, Sakuraku,}Saitama 338‐8570

Japan

(machihar@rimath.

saitama‐u.

_ac.jp)

2Department

of Applied Physics, Waseda _University, _{Shinjuku, Tokyo 169‐8555, Japan}

(txozawa@waseda.jp)

3Faculty

ofMechanical_Engineering, Institute_ofScience and_Engineering, Kanazawa University,

Kakuma, Kanazawa,Ishikawa_920‐1192, _Japan

(wadade@se.

kanazawa‐u.

_ac.jp)

1 Introduction

Let $\Omega$ be adomain in \mathbb{R}^{n} withn _{\geq 3 and} assume 0 \in $\Omega$. The classical

Hardy

inequality

statesthat the

_inequality

(\displaystyle \frac{n-2}{2})^{2}\int_{ $\Omega$}\frac{|f|^{2}}{|x|^{2}}dx\leq\int_{ $\Omega$}|\nabla f|^{2}dx

(1.1)

holds for all

_f

_{\in H_{0}^{1}( $\Omega$)}

,where theconstant

(\displaystyle \frac{n-2}{2})^{2}

is

best‐possible.

It isalso well‐known

that the

_inequality

_(1.1)

admitsnonontrivial

_extremizers,

and thisfact

implies

a

possibility

for

_(1.1)

tobe

_{improved by adding}

someremainderterms. In

_fact,

the authors in

_[9]

proved

thatthe

_{following improved Hardy inequality}

(\displaystyle \frac{n-2}{2})^{2}\int_{ $\Omega$}\frac{|f|^{2}}{|x|^{2}}dx+ $\Lambda$\int_{ $\Omega$}|f|^{2}dx\leq\int_{ $\Omega$}|\nabla f|^{2}dx

(1.2)

holdsfor all

_{f\in H_{0}^{1}( $\Omega$)}

provided

that $\Omega$is

_bounded,

wheretheconstant $\Lambda$ in

_(1.2)

is

_given

by

$\Lambda$= $\Lambda$(n, $\Omega$)

=z_{0}^{2}$\omega$^{\frac{2}{n^{n}}}| $\Omega$|^{-\frac{2}{n}}

, and $\omega$_{n} and

| $\Omega$|

denote the

Lebesgue

measuresof the unit

ball and $\Omega$ on\mathbb{R}^{n}_,

respectively,

and the absoluteconstant z_{0} denotes the first zeroof the

Besselfunction

_{J_{0}(z)}

. Theconstant $\Lambda$ is

optimal

if $\Omega$is a

ball,

but still the

inequality

(1.2)

admitsnonontrivial extremizers. More

generally,

theauthorsin

[9]

obtained the

inequality

(\displaystyle \frac{n-2}{2})^{2}\int_{ $\Omega$}\frac{|f|^{2}}{|x|^{2}}dx+\tilde{ $\Lambda$}(\int_{ $\Omega$}|f|^{p}dx)^{\frac{2}{p}} \leq\int_{ $\Omega$}|\nabla f|^{2}dx

for

_{f\in H_{0}^{1}( $\Omega$)}

,where 1 <p<

\displaystyle \frac{2n}{n-2}

and

\tilde{ $\Lambda$}

isa

positive

constant

independent

ofu. Similar

improvements

have been done for the

Hardy

inequality

not

_only

in the L^{2}

_‐setting

but in

L^{p}

_‐setting

withsomeremainder_terms, seefor instance

[3,

_{7, 8, 21,}

35].

Hardy

type

inequalities

are known asuseful mathematical tools in various fields such

asreal

analysis,

functional

analysis, probability

and

partial

differential

equations.

In

fact,

Hardy

type

inequalities

and their

improvements

are

applied

in_{many contexts.} For

instance,

Hardy

type

inequalities

wereutilizedin

_{investigating}

the

_stability

of solutions of semi‐linear

elliptic

and

_{parabolic equations}

in

_{[9, 11].}

As for the existence and

_asymptotic

behavior of

(2)

referto

_[

_{1, 5, 16,}

1\mathrm{S},

27,

31

]

for the concrete

applications

of

Hardy

type

inequalities.

We

also referto

_[13,

_33]

fora

_{comprehensive}

_{understanding}

of

_Hardy

_type

_{inequalities.}

Based on the historical remarks on the

Hardy

_type

inequalities,

our _{purpose in} this

paper istoestablish the classical

Hardy inequalities

inthe frame work of

equalities

which

immediately

imply

the

_{Hardy inequalities by dropping}

theremainder terms. At thesame

time,

those

_equalities

characterize the form of the

_vanishing

remainderterms. Our method

on the basis of

equalities

presumably provides

a

simple

and direct

understanding

of the

Hardy

type

inequalities

aswellasthe nonexistence of nontrivial extremizers.

Inwhat

_follows,

we

_always

assume $\Omega$=\mathbb{R}^{n} and the standard

L^{2}(\mathbb{R}^{n})

norm is denoted

by

\Vert. \Vert_{2}

. Then the

Hardy

type

inequalities

in L^{2}

‐setting

thatwediscuss inthis paperare

the

_following:

\displaystyle \Vert\frac{f}{|x|}\Vert_{2}\leq\frac{2}{n-2}\Vert\frac{x}{|x|}\cdot\nabla f\Vert_{2} n\geq 3

,

(1.3)

\displaystyle \sup_{R>0}\Vert\frac{f-f_{R}}{|x|^{\frac{n}{2}}\log_{ $\Pi$ x}^{R}}\Vert_{2}\leq 2\Vert\frac{1}{|x|^{\frac{n}{2}-1}}\frac{x}{|x|}\cdot\nabla f\Vert_{2} n\geq 2

,

(1.4)

\displaystyle \int_{0}^{\infty}x^{-p-1}|\int_{0}^{x}f(y)dy|^{2}dx\leq (\frac{2}{p})^{2}\int_{0}^{\infty}x^{-p+1}|f(x)|^{2}dx

,

(1.5)

\displaystyle \int_{0}^{\infty}x^{p-1}|\int_{x}^{\infty}f(y)dy|^{2}dx\leq (\frac{2}{p})^{2}\int_{0}^{\infty}x^{p+1}|f(x)|^{2}dx

,

(1.6)

where

f_{R}(x)=f(R\displaystyle \frac{x}{|x|})

and

_p>0

. The

inequalities

(1.3),

(1.5),

and

(1.6)

arestandard

(see

[19]

for

_instance),

while

_(1.4)

israthernew

(see [28,

30 In

addition,

as wenoticed in

[30],

the

_logarithmic

_{Hardy inequality}

_(1.4)

hasa

scaling

_property.

Westateour main theorems. We denote

by

\partial_{r}

the radial derivative defined

by \partial_{r}

=

\displaystyle \frac{x}{|x|}

\nabla =

\displaystyle \sum_{j=1}^{n}\frac{x_{j}}{|x|}\partial_{j}

. Thespace

D^{1,2}(\mathbb{R}^{n})

denotes the

completion

of

C_{0}^{\infty}(\mathbb{R}^{n})

under the

Dirichletnorm

\Vert\nabla\cdot\Vert_{2}

. Also the notationS^{n-1} denotes the unit

sphere

in\mathbb{R}^{n}endowed with

the

_Lebesgue

measure $\sigma$.

2 _Hardy

_type

_inequalities

in the framework of

_equalities

Inthis

_section,

we shall prove the

Hardy

type

inequalities

in theframeworkof

_equalities.

Our first result statesasfollows:

Theorem 2.1. Let_{n\geq 3}. Then the

equalities

(\displaystyle \frac{n-2}{2})^{2}\Vert\frac{f}{|x|}\Vert_{2}^{2}=\Vert\partial_{r}f\Vert_{2}^{2}-\Vert\partial_{r}f+\frac{n-2}{2|x|}f\Vert_{2}^{2}

(2.1)

=\Vert\partial_{r}f\Vert_{2}^{2}-\Vert|X|^{-\frac{n-2}{2}\partial_{r}(|x|^{\frac{n-2}{2}f)}}\Vert_{2}^{2}

(2.2)

hold

_for

all

_{f\in D^{1,2}(\mathbb{R}^{n})}

.

Moreover,

the secondterm inthe

right

hand side

of

(2.1)

or

(2.2)

vanishes

_if

and

_{only if f}

takes the

_form

f(x)=|x|^{-\frac{n-2}{2}} $\varphi$(\displaystyle \frac{x}{|x|})

(2.3)

for

some

function

_$\varphi$ : S^{n-1} \rightarrow \mathbb{C}, which makes the

left

hand side

of

(2.1)

infinite

unless

\displaystyle \int_{S^{n-1}}| $\varphi$( $\omega$)|^{2}d $\sigma$( $\omega$)=0

:

(3)

We remark thatasin

_(2.4),

functionsof the form

_(2.3)

imply

the nonexistence ofnon‐

trivialextremizersfor

_(1.3).

The

_{corresponding integral diverges}

atboth

_origin

and

_infinity.

A similar result to Theorem 2.1 can be foundin

[6, 15].

_However,

the essential ideas for

the

_proofs

aredifferent.

Indeed,

the

proof

in

[6]

isdone

by

direct calculations withrespect

tothe

_quotient

with the

_optimizer

ofa

Hardy

_type

inequality.

On the other

hand,

weshall

proveTheorem2.1

by

applying

an

orthogonality

_argumentin

general

Hilbert_space

settings.

More

_precisely,

an

_equality

(\displaystyle \frac{n-2}{2})^{2}\Vert\frac{f}{|x|}\Vert_{2}^{2}=\Vert\nabla f\Vert_{2}^{2}-\Vert\nabla f+\frac{n-2}{2}\frac{x}{|x|^{2}}f\Vert_{2}^{2}

(2.5)

has been observed in

_[6,

_15].

We should remark that

_(2.1)

and

_(2.5)

are the same for

radially symmetric

functions and are not the same for nonradial functions. In

fact,

the

Dirichlet

_integral

is

_decomposed

into radial and

spherical

components as

\displaystyle \Vert\nabla f\Vert_{2}^{2}=\Vert\partial_{r}f\Vert_{2}^{2}+\sum_{j=1}^{n}\Vert(\partial_{j}-\frac{x_{j}}{|x|}\partial_{r})f\Vert_{2}^{2}

Next,

we statethe

_logarithmic

_Hardy

_type

_equalities

in the critical

weighted

Sobolev

spaces.

Theorem 2.2. Letn\geq 2. Then the

equalities

\displaystyle \frac{1}{4}\Vert\frac{ff_{R}}{|x|^{\frac{n}{2}}\mathrm{o}\mathrm{g}_{ $\Pi$ x}^{R}}1\Vert_{2}^{2}=\Vert\frac{1}{|x|^{\frac{n}{2}-1}}\partial_{r}f\Vert_{2}^{2}-\Vert\frac{1}{|x|^{\frac{n}{2}-1}} (\displaystyle \partial_{r}f+\frac{f-f_{R}}{2|x|\log\frac{R}{|x|}})\Vert_{2}^{2}

(2.6)

=\displaystyle \Vert\frac{1}{|x|^{\frac{n}{2}-1}}\partial_{r}f\Vert_{2}^{2}-\Vert\frac{|\log\frac{R}{|x|}|^{\frac{1}{2}}}{|x|^{\frac{n}{2}-1}}\partial_{r}

(\displaystyle \frac{f-f_{R}}{|\log\frac{R}{|x|}|^{\frac{1}{2}}}\mathrm{I}\Vert_{2}^{2}

(2.7)

hold

for

all R > 0 and all

f

\in

L_{loc}^{1}(\mathbb{R}^{n})

with

_{\displaystyle \frac{1}{|x|^{\frac{n}{2}-1}}\nabla f}

\in

L^{2}(\mathbb{R}^{n})

, where

f_{R}

is

defined

by

f_{R}(x)

=

f(R\displaystyle \frac{x}{|x|})

.

Moreover,

the secondterm in the

right

hand side

of

(2.6)

or

(2.7)

vanishes

_if

and

_only

_if

_{f-f_{R}}

takes the

_form

f(x)-f_{R}(x)=|\displaystyle \log\frac{R}{|x|}|^{\frac{1}{2}} $\varphi$(\frac{x}{|x|})

(2.8)

for

some

function

_$\varphi$ : S^{n-1} \rightarrow \mathbb{C}, which makes the

left

hand side

of

(2.6)

infinite

unless

\displaystyle \int_{S^{n-1}}| $\varphi$( $\omega$)|^{2}d $\sigma$( $\omega$)=0

:

\displaystyle \frac{|f-f_{R}|^{2}}{|x|^{n}|\log\frac{R}{|x|}|^{2}}=\frac{| $\varphi$(\frac{x}{|x|})|^{2}}{|x|^{n}|\log_{ $\Pi$}^{R}|}\not\in L^{1}(\mathbb{R}^{n})

.

(2.9)

As in

_(2.9),

functions of the form

_(2.8)

_imply

the nonexistence of nontrivial extremizers

for

_(1.4).

The

_{corresponding}

_{integral diverges}

at both

_origin

and

_{infinity and,}

in

_addition,

onthe

_sphere

of radius R>0.

3 _{Caffarelli‐Kohn‐Nirenberg}

_type

_inequalities

in

the

frame‐

work of

_equalities

In this

_section,

we shall_provethe

Caffarelli‐Kohn‐Nirenberg

_type

_inequalities

inthe frame‐ work of

_equalities.

We first recall the

_{Caffarelli‐Kohn‐NirenUerg inequality:}

Let n \in

_\mathbb{N},

1\leq p<\infty, r>0,

_{\displaystyle \frac{1}{r}+\frac{ $\sigma$}{n}}

>0. Then

(4)

taki =r\mathrm{w}\mathrm{e}\mathrm{o}btain t\mathrm{h}\mathrm{e}\mathrm{f}\circ

_llowing

:\mathrm{L}\mathrm{e}\mathrm{t}n\in \mathbb{N}

holds f

\mathrm{o}\mathrm{r}\mathrm{a}\mathrm{l}\mathrm{l}u\in C_{0}^{\infty}(\mathbb{R}^{n})\mathrm{i}\mathrm{f}\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{o}\mathrm{n}\mathrm{l}\mathrm{y}\mathrm{i}\mathrm{f}0\leq $\alpha$- $\sigma$ 1\leq\leq 1p<\mathrm{a}\mathrm{n}\mathrm{d}

_{\displaystyle \infty,+\frac{ $\alpha$-1\frac{1}{p}}{n}.}

\displaystyle \mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e}\frac{1}{r}+\frac{ $\sigma$}{p1\frac {}{}n}=+\frac{ $\alpha$-1}{>^{n}0}.

Then t

Especially,

holds

\Vert|x|^{ $\alpha$-1}u\Vert_{p}\leq C\Vert|x|^{ $\alpha$}|\nabla u|\Vert_{p}

(3.1)

for all

_{u\in C_{0}^{\infty}(\mathbb{R}^{n})}

. We

already proved

the

equality

version of

(3.1)

with $\alpha$=0 and

p=2

asfollows:

Letn\geq 3. Then there holds

(\displaystyle \frac{n-2}{2})^{2}\Vert|x|^{-1}u\Vert_{2}^{2}=\Vert\partial_{r}u\Vert_{2}^{2}-\Vert\partial_{r}u+\frac{n-2}{2|x|}u\Vert_{2}^{2}

(3.2)

for all u\in

_{C_{0}^{\infty}(\mathbb{R}^{n})}

, where

\partial_{r}u(x)

:=

\displaystyle \frac{x}{|x|}

.

\nabla u(x)

for_x

\in \mathbb{R}^{n}\backslash \{0\}

.

Firstly,

weextend the

result of

_(3.2)

for

_general

$\alpha$,which

corresponds

tothe

equality

version of

(3.1)

with

p=2

as stated in Theorem 3.1

(i).

Furthermore,

weshall establish the

_equality

versionof the

Caffarelli‐Kohn‐Nirenberg

type

inequality

of the

_logarithmic

form

_{by applying}

Theorem3.1

(i)

asstated in Theorem 3.1

(ii).

Theorem 3.1.

_(i)

Let n\in \mathrm{N} and $\alpha$>

\displaystyle \frac{2-n}{2}

. Thenthere holds

(\displaystyle \frac{n-2+2 $\alpha$}{2})^{2}\Vert|x|^{ $\alpha$-1}u\Vert_{2}^{2}=\Vert|x|^{ $\alpha$}\partial_{r}u\Vert_{2}^{2}-\Vert|x|^{ $\alpha$}\partial_{r}u+\frac{n-2+2 $\alpha$}{2}|x|^{ $\alpha$-1}u\Vert_{2}^{2}

(3.3)

for

all

_{u\in C_{0}^{\infty}(\mathbb{R}^{n})}

.

(ii)

Let

_{n\in \mathbb{N}, 1< $\gamma$<3}

and R>0. Then there holds

(\displaystyle \frac{ $\gamma$-1}{2})^{2}\Vert|\log(\frac{R}{|x|})|^{-\frac{ $\gamma$}{2}}|x|^{-\frac{n}{2}}(u-u_{R})\Vert_{2}^{2}=\Vert|\log(\frac{R}{|x|})|^{\underline{2}-\neq}|x|^{\frac{2-n}{2}\partial_{r}u\Vert_{2}^{2}}

-\displaystyle \Vert|\log(\frac{R}{|x|})|\rightarrow^{2}-2|x|^{\frac{2-n}{2}} (\partial_{r}u+\frac{ $\gamma$-1}{2}|x|^{-1}|\log(\frac{R}{|x|})|^{-1}(u-u_{R}))\Vert_{2}^{2}

(3.4)

for

all

_{u\in c_{0}\infty(\mathbb{R}^{n})}

, where

u_{R}(x)

:=u(R_{ $\Pi$ x}^{x})

for

x\in \mathbb{R}^{n}\backslash \{0\}.

Proof.

_(i)

It is

_enough

toshow

\displaystyle \Vert|x|^{ $\alpha$-1}u\Vert_{2}^{2}=-\frac{2}{n-2+2 $\alpha$}{\rm Re}\int_{\mathrm{R}^{n}}|x|^{2 $\alpha$-1}u\overline{\partial_{r}u}dx.

Indeed, by integration by

parts,wesee

\displaystyle \Vert|x|^{ $\alpha$-1}u\Vert_{2}^{2}=\int_{\mathbb{R}^{n}}|x|^{2 $\alpha$-2}|u|^{2}dx=\int_{0}^{\infty}r^{2 $\alpha$+n-3}\int_{S^{n-1}}|u(r $\omega$)|^{2}d $\omega$ dr

=-\displaystyle \frac{2}{2 $\alpha$+n-2}{\rm Re}\int_{0}^{\infty}r^{2 $\alpha$+n-2}\int_{S^{n-1}}u(r $\omega$)\overline{ $\omega$\cdot\nabla u(r $\omega$)}d $\omega$ dr

=-\displaystyle \frac{2}{2 $\alpha$+n-2}{\rm Re}\int_{\mathbb{R}^{n}}|x|^{2 $\alpha$-1}u\overline{\partial_{r}u}dx,

wherewe used_{2 $\alpha$+n-2>0.}

(ii)

First,

weestablish

(5)

-\displaystyle \int_{B_{R}(0)}|\partial_{r}u+\frac{ $\gamma$-1}{2}|x|^{-1}

[\displaystyle \log(\frac{R}{|x|})]^{-1}(u-u_{R})|^{2}[\log(\frac{R}{|x|})]^{2- $\gamma$}\frac{dx}{|x|^{n-2}}

(3.5)

for all

_{u\in C_{0}^{\infty}(\mathbb{R}^{n})}

.

By using polar coordinates,

wehave

\displaystyle \int_{B_{R}(0)}|u-u_{R}|^{2}

[\displaystyle \log(\frac{R}{|x|})]^{- $\gamma$}\frac{dx}{|x|^{n}}=\int_{S^{n-1}}\int_{0}^{R}|u(t $\omega$)-u(R $\omega$)|^{2}

[\displaystyle \log(\frac{R}{t})]^{- $\gamma$}\frac{dt}{t}d $\omega$.

Changing

variables

_t=t(r)

:=R\exp(-r^{-\frac{1}{ $\gamma$}})

for

_{r\in(0, \infty)}

,we see

\displaystyle \int_{S^{n-1}}\int_{0}^{R}|u(t $\omega$)-u(R $\omega$)|^{2} [\displaystyle \log(\frac{R}{t})]^{- $\gamma$}\frac{dt}{t}d $\omega$=\frac{1}{ $\gamma$}\int_{S^{n-1}}\int_{0}^{\infty}|u(t(r) $\omega$)-u(R $\omega$)|^{2}r^{-\frac{1}{ $\gamma$}}drd $\omega$

=\displaystyle \frac{1}{2 $\pi \gamma$}\int_{S^{n-1}}\int_{\mathbb{R}^{2}}|u(t(|x|_{2}) $\omega$)-u(R $\omega$)|^{2}|x|_{2}^{-\frac{1}{ $\gamma$}-1}dxd $\omega$=\frac{1}{2 $\pi \gamma$}\int_{S^{n-1}}\Vert|x|^{\frac{ $\gamma$-1}{2^{2 $\gamma$}}-1}f_{ $\omega$}\Vert_{L^{2}(\mathbb{R}^{2})}^{2}d $\omega$,

where

_{|x|_{2}}

denotes the two‐dimensional Euclideannormand

_{f_{ $\omega$}(x)}

_{:=u(t(|x|_{2}) $\omega$)-u(R $\omega$)}

for $\omega$ \in S^{n-1} and x \in\mathbb{R}^{2}. Then

applying

(3.3)

for the function

f_{ $\omega$}

with the dimension 2

and

_{$\alpha$=\displaystyle \frac{ $\gamma$-1}{2 $\gamma$}}

>0,weobtain

\displaystyle \frac{ $\pi$( $\gamma$-1)^{2}}{2 $\gamma$}\int_{B_{R}(0)}|u-u_{R}|^{2}

[\displaystyle \log(\frac{R}{|x|})]^{- $\gamma$}\frac{dx}{|x|^{n}}

=\displaystyle \int_{S^{n-1}}\Vert|x|^{\frac{ $\gamma$-1}{2^{2 $\gamma$}}}\partial_{2,r}f_{ $\omega$}\Vert_{L^{2}(\mathbb{R}^{2})}^{2}d $\omega$-\int_{S^{n-1}}\Vert|x|_{2^{2 $\gamma$}}^{\leftarrow-1}\partial_{2,r}f_{ $\omega$}+\frac{ $\gamma$-1}{2 $\gamma$}|x|^{\frac{ $\gamma$-1}{2^{2 $\gamma$}}-1}f_{ $\omega$}\Vert_{L^{2}(\mathbb{R}^{2})}^{2}d $\omega$,

_(3.6)

where

_{\partial_{2,r}f_{ $\omega$}}

denotes the two‐dimensional radial

_{derivative, i.e.,}

_{\partial_{2,r}f_{ $\omega$}(x)}

_{:=\displaystyle \frac{x}{|x|_{2}}\cdot\nabla_{2}f_{ $\omega$}(x)}

for

_{x\in \mathbb{R}^{2}\backslash \{0\}}

and\nabla_{2}

_{:=(\partial_{1}, \partial_{2})}

.

By

adirect

computation,

we seefor

x\in \mathbb{R}^{2}\backslash \{0\},

\displaystyle \partial_{2,r}f_{ $\omega$}(x)=\frac{R}{ $\gamma$}\partial_{r}u(t(|x|_{2}) $\omega$)\exp(-|x|_{2}^{-\frac{1}{ $\gamma$}})|x|_{2}^{-\frac{1}{ $\gamma$}-1}

and then

_by

_changing

variables

_r=r(t)

:=

[\displaystyle \log(\frac{R}{t})]

- $\gamma$ for

t\in(0, R)

,

\displaystyle \int_{S^{n-1}}

\displaystyle \Vert|x|^{\frac{ $\gamma$-1}{2^{2 $\gamma$}}}\partial_{2,r}f_{ $\omega$}\Vert_{L^{2}(\mathbb{R}^{2})}^{2}d $\omega$=\frac{R^{2}}{$\gamma$^{2}}\int_{S^{n-1}}\int_{\mathrm{R}^{2}}|\partial_{r}u(t(|x|_{2}) $\omega$)|^{2}\exp(-2|x|_{2}^{-\frac{1}{ $\gamma$}})|x|_{2}^{-\frac{3}{ $\gamma$}-1}dxd $\omega$

=\displaystyle \frac{2 $\pi$ R^{2}}{$\gamma$^{2}}\int_{S^{n-1}}\int_{0}^{\infty}|\partial_{r}u(t(r) $\omega$)|^{2}\exp(-2r^{-\frac{1}{ $\gamma$}})r^{-\frac{3}{ $\gamma$}}drd $\omega$

=\displaystyle \frac{2 $\pi$}{ $\gamma$}\int_{S^{n-1}}\int_{0}^{R}|\partial_{r}u(t $\omega$)|^{2}

_{[\displaystyle \log(\frac{R}{t})]^{2- $\gamma$}tdtd $\omega$=\frac{2 $\pi$}{ $\gamma$}\int_{B_{R}(0)}|\partial_{r}u|^{2}}

[\displaystyle \log(\frac{R}{|x|})]^{2- $\gamma$}|x|^{2-n}dx.

(3.7)

Similarly,

we see

\displaystyle \int_{S^{n-1}}\Vert|x|_{2}^{\#^{-1}}\partial_{2,r}f_{ $\omega$}+\frac{ $\gamma$-1}{2 $\gamma$}|x|_{2^{2 $\gamma$}}^{\mapsto^{-1}-1}f_{ $\omega$}\Vert_{L^{2}(\mathrm{R}^{2})}^{2}d $\omega$

=\displaystyle \int_{S^{n-1}}\int_{\mathbb{R}^{2}}|\frac{R}{ $\gamma$}\partial_{r}u(t(|x|_{2}) $\omega$)\exp(-|x|_{2}^{-\frac{1}{ $\gamma$}})|x|_{2}^{-\frac{3}{2 $\gamma$}-\frac{1}{2}}+\frac{ $\gamma$-1}{2 $\gamma$}|x|_{2^{2 $\gamma$}}^{\leftarrow^{-1}-1}(u(t(|x|_{2}) $\omega$)-u(R $\omega$))|^{2}dxd $\omega$

=2 $\pi$\displaystyle \int_{S^{n-1}}\int_{0}^{\infty}|\frac{R}{ $\gamma$}\partial_{r}u(t(r) $\omega$)\exp(-r^{-\frac{1}{ $\gamma$}})r^{-\frac{3}{2 $\gamma$}-\frac{1}{2}}+\frac{ $\gamma$-1}{2 $\gamma$}r^{\frac{ $\gamma$-1}{2 $\gamma$}-1}(u(t(r) $\omega$)-u(R $\omega$))|^{2}rdrd $\omega$

(6)

=\displaystyle \frac{2 $\pi$}{ $\gamma$}\int_{S^{n-1}}\int_{0}^{R}|\partial_{r}u(t $\omega$)+\frac{ $\gamma$-1}{2}t^{-1}

[\displaystyle \log(\frac{R}{t})]^{-1}(u(t $\omega$)-u(R $\omega$))|^{2}[\log(\frac{R}{t})]^{2- $\gamma$}tdtd $\omega$

=\displaystyle \frac{2 $\pi$}{ $\gamma$}\int_{B_{R}(0)}|\partial_{r}u+\frac{ $\gamma$-1}{2}|x|^{-1}

[\displaystyle \log(\frac{R}{|x|})]^{-1}(u-u_{R})|^{2}[\log(\frac{R}{|x|})]^{2- $\gamma$}\frac{dx}{|x|^{n-2}}

.

(3.8)

Plugging

(3.7)

and

_(3.8)

into

_(3.6),

weobtain

(3.5).

In thesame_wayasabove

by replacing

the

change

of variables

by

r=r(t)

:=

[\log (

\displaystyle \frac{t}{R}\mathrm{I}]^{- $\gamma$}

for

_{t\in(R, \infty)}

and

_t=t(r)

:=R\exp(r^{-\frac{1}{ $\gamma$}})

for

_{r\in(0, \infty)}

,weobtain

(\displaystyle \frac{ $\gamma$-1}{2})^{2}\int_{B_{R}(0)^{c}}|u-u_{R}|^{2}

[\displaystyle \log(\frac{|x|}{R})]^{- $\gamma$}\frac{dx}{|x|^{n}}=\int_{B_{R}(0)^{c}}|\partial_{r}u|^{2}

[\displaystyle \log(\frac{|x|}{R})]^{2- $\gamma$}|x|^{2-n}dx

-\displaystyle \int_{B_{R}(0)^{c}}|\partial_{r}u+\frac{ $\gamma$-1}{2}|x|^{-1}

[\displaystyle \log(\frac{|x|}{R})]^{-1}(u-u_{R})|^{2}

[\displaystyle \log(\frac{|x|}{R})]^{2- $\gamma$}\frac{dx}{|x|^{n-2}}

(3.9)

for all u\in

_{C_{0}^{\infty}(\mathbb{R}^{n})}

.

Finally,

adding

the both sides of the

equalities

(3.5)

and

(3.9)

yields

(3.4).

\square

Remark 3.2.

_(i)

We

_easily

see that the

integrals

in

(3.5)

diverge

for

some

_function

in

C_{0}^{\infty}(\mathbb{R}^{n})

when

_$\gamma$\leq

1 or

_{$\gamma$\geq 3}

. On the other

hand,

the

integrals

converge

for

any

function

in

_{C_{0}^{\infty}(\mathbb{R}^{n})}

when 1

_{< $\gamma$<3}

.

Indeed, by using

the estimates

\displaystyle \log(\frac{R}{t})

\geq

\displaystyle \frac{R-t}{R}

for

0<t<

R,

and

_{|u(R $\omega$)-u(t $\omega$)|}

\leq

\Vert\nabla u\Vert_{\infty}(R-t) for

$\omega$\in S^{n-1}

and 0<t<R, wesee

\displaystyle \int_{B_{R}(0)}|u-u_{R}|^{2}

[\displaystyle \log(\frac{R}{|x|})]^{- $\gamma$}\frac{dx}{|x|^{n}}=\int_{S^{n-1}}\int_{0}^{R}|u(t $\omega$)-u(R $\omega$)|^{2}

[\displaystyle \log(\frac{R}{t})]^{- $\gamma$}\frac{dt}{t}d $\omega$

=\displaystyle \int_{S^{n-1}}\int_{0}^{\frac{R}{2}}|u(t $\omega$)-u(R $\omega$)|^{2}

_{[\displaystyle \log(\frac{R}{t})]^{- $\gamma$}\frac{dt}{t}d $\omega$+\int_{S^{n-1}}J_{\frac{R}{2}}^{R}|u(t $\omega$)-u(R $\omega$)|^{2}}

[\displaystyle \log(\frac{R}{t})]^{- $\gamma$}\frac{dt}{t}d $\omega$

\displaystyle \leq 4$\omega$_{n-1}\Vert u||_{\infty}^{2}\int_{0}^{\frac{R}{2}}

_{[\displaystyle \log(\frac{R}{t})]^{- $\gamma$}\frac{dt}{t}+2R^{ $\gamma$-1}$\omega$_{n-1}\Vert\nabla u\Vert_{\infty}^{2}\int_{\frac{R}{2}}^{R}(R-t)^{2- $\gamma$}dt<+\infty}

since

_{1< $\gamma$<3}

. Alsowe have

\displaystyle \int_{B_{R}(0)}|\partial_{r}u|^{2}

[\displaystyle \log(\frac{R}{|x|})]^{2- $\gamma$}|x|^{2-n}dx\leq$\omega$_{n-1}\Vert\nabla u\Vert_{\infty}^{2}\int_{0}^{R}[\log(\frac{R}{t})]^{2- $\gamma$}

tdt<+\infty since

_$\gamma$<3.

(ii)

By restricting

the

_functions

into

_{C_{0}^{\infty}(B_{R}(0))}

, we can remove the condition $\gamma$ < 3.

Precisely,

the

_integrals

in

_(3.5)

converge

for

any

function

in

_{C_{0}^{\infty}(B_{R}(0))}

and the

_inequality

(3.5)

holdsin

_{C_{0^{\infty}}(B_{R}(0))}

_for

all

_$\gamma$>1.

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