Quasiharmonic <span class="mathjax-formula">$L^p$</span>-functions on riemannian manifolds

A NNALI DELLA

S ^CUOLA N ^ORMALE S UPERIORE DI P ^ISA Classe di Scienze

L UNG O CK C HUNG

L EO S ARIO

C ^ECILIA W ^ANG

Quasiharmonic L

^p

-functions on riemannian manifolds

Annali della Scuola Normale Superiore di Pisa, Classe di Scienze 4

^e

série, tome 2, n

^o

3 (1975), p. 469-478

<http://www.numdam.org/item?id=ASNSP_1975_4_2_3_469_0>

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http://www.numdam.org/

Quasiharmonic Lp-Functions

^on

Riemannian Manifolds (*).

LUNG OCK CHUNG - LEO SARIO - CECILIA WANG

Let

Q

^{be the class of}

quasiharmonic functions q,

^defined

by dq = 1,

with

+

^{bd the}

Laplace-Beltrami operator.

^Denote

by P, B, D,

^C

the classes of functions which ^are

positive, bounded,

^Dirichlet

finite,

^or

bounded Dirichlet

finite, respectively.

^For

X = P, B, D, C,

^let

0’ be

^the

classes of Riemannian

N-manifolds, N~2,

^{for which}

QX = Q

^{r1 X = 0. These}

classes are known to be related

by

the strict inclusion relations

for each

N,

^{whereas there is no} inclusion between

OQ

^and

OQ ^{[3, 5].}

In the

present

paper ^we introduce the class

QLp

^of

quasiharmonic

^func-

tions in

Lp,

^with

1 c p

oo; the

value p

^{= oo} will be excluded since is

nothing

^but

O£.

^If

ON signifies

^the

complement

^of

^ON

^with

respect

^to

the

totality

^of Riemannian

N-manifolds,

^{then we shall show that}

for

p~l,

X==P,B;

^and

for p &#x3E;1.

^In

striking

contrast with these

noninclusions,

^{we shall} establish the strict inclusions

and for

1. - We first _{prove the} existence of N-ma~nifolds which _carry neither

QLp

nor

QX

functions :

(*) ^{The work was}

sponsored by

^{the U.S.}

Army

^Research Office, Grant DA- -ARO-31-124-73-G39,

University

^of California, ^Los

Angeles.

MOS Classification : 31 B 30.

Pervenuto alia Redazione il 10

maggio

^{1974 e in forma} definitiva il 9

luglio

^1974.

THEOREM 1. ¹

PROOF. An

example

^is

simply

^{the Euclidean}

N-space ^EN.

^{We first}

show that 0. ^can be written as q ⁼ qo

+ h,

^where

qo ^{= -}

r2/2N

^E

^Q

^{and h}

^belongs

^to the class ~ of harmonic functions. Set h ⁼

h(0) --f- k,

^with

h(0)

^the value of h at the

origin.

^Since

kdV = ^ek(0)

⁼

^0,

we have ^~

in

fact, q, + h(O) _ er2

^{and with A a}

trigono-

metric function of 0 =

(0~,

^...,

To see that

0

^for

p &#x3E; 1,

^{take a function with}

on

{r &#x3E; 1},

^{a a real} constant to be

specified

later. If

p’

is determined

by +I lp’= 1,

^then

-

that for

Suppose

there exists a function

qEQL’P.

Then

( (q, (

^{00. For some}

h E .H,

if

ot &#x3E; - N - 2.

Thus any

a E [- N - 2, - N~p’ ) ^gives

^a

contradiction,

^and

we have

QLP = 0

for every

p ~ 1.

We

proceed

^{to show that for} all X. It suffices to establish

QP

⁼ 0. Write

again

^an

arbitrary q c Q

as q ⁼ qo

+

^{h. Take an}

increasing

sequence

{r.}- ^{with r n}

^-cxJ. For every n there exists ^an 0’ =

(6~

^...,

6N_1)

such that

On)

⁼

h(O).

^Therefore

q(rn, 0")

⁼

⁺ h(O)

^{- -} oo, and

QP.

2. - The relation is trivial in view of the Euclidean N-ball.

We

proceed

^to

give

^a Riemannian N-manifold which carries

Q.Lp

^functions

but no

QX

^functions.

THEOREM 2.

PROOF. Consider the manifold

471

with the

opposite

faces yi ^{= 1}

and y,

^{= - 1} identified for each i

by

^a pa- rallel translation

perpendicular

^{to the} x-axis. Endow T with the metric

For a

quasiharmonic

^function

qo(x)

^{we have}

which is satisfied

by

To estimate

we set a : dt and obtain

Therefore T E for all p.

A harmonic function

ho(x)

^{satisfies which}

gives ho(x)

^{= ax}

+

b. The harmonic measure W of the

boundary component

at a? == o0 on

{~~0}

^{is with harmonic on}

~0 c x c n~.

Thus 0) ^=-

0,

^{the same is true of the harmonic measure of the}

boundary component

at x = - oo, and therefore T

belongs

^{to the class}

O ]

^of

^parabolic

N-manifolds. In view of

(loc. cit.),

^{,ve obtain} for all X.

3. - Our next

problem

^{is to find an} N-manifold which admits

Qx

^func-

tions for X ⁼

P, B,

^{but no}

^QLp

^functions.

THEOREM 3.

PROOF. Let M be the

N-space equipped

with the metric

where

and

the ~i

^are

trigonometric

functions of 6 such that the metric is Euclidean

on

~r c ~ ~.

The function

satisfies the

quasiharmonic equation

as Therefore

qo E QB,

^and

We next prove that

Suppose

^{there exists a Then}

I(q,

^We may

again

write q = qo

+c -E- 7~, k(O)

⁼

0,

^and

Set

then for

and

473

Here

so that for some R &#x3E;

0,

The last

integral

converges, the first

diverges,

^{and therefore}

This contradiction shows that that

is,

To see that for

p &#x3E; 1,

^let

p’

be determined

by

^I

For a

function 99

^E

C’(M)

^with

we have

hence If there exists ^a

q E QLp, then (q,

^{00. But}

(q, q;)

⁼

-

(qo + c,

^{and if} co, the

integrand

ⁱⁿ

(qo +

c,

99)

^is

asymptotically

so

that (qa -~- c, ~p) ~ =

^{oo, a} contradiction.

In the case c ⁼ co ^we observe that for r &#x3E; 1

with

so that as r -~ oo. It follows that the

integrand

ⁱⁿ

is

asymptotically

and therefore This contradiction shows that and the

proof

of Theorem 3 is

complete.

In the

proofs

of Theorems 1-3 we have

actually

shown somewhat more.-

for and

4. - Next we are to find an N-manifold which carries

QD

^{functions but}

no functions. First we consider the case p &#x3E; 1.

THEOREM 4.

PROOF. Take the manifold

with the

opposite

^faces

the metric

identified as in

No. 2,

^{and choose}

7T

where

er, fl

^{are real} constants to be

specified later; they

^will

depend

^on p,

so that ^we shall not have a

generalization

^{of the} theorem in the same manner- as at the end of No. 3.

The function

satisfies the

quasiharmonic equation

provided

475

The Dirichlet

integral

^is

if

For the LP norm we have

if

An

inspection

of the last two

inequalities

^shows

that p

^{=1 is ruled out. For}

all four

inequalities

^are satisfied. In

particular,

The

exponent fl

^{- a in} the volume element is

positive,

^and the constant function 1

belongs

^{to Lp’ for our}

p’ &#x3E; 1. Suppose

there exists a

Then !(~1)!

^00.

Every

^{can be written}

where

/nGnEH, ?

^I

ducts of the form

the _ni

integers &#x3E; 0,

^the

^~~

^pro-

and the

prime

ⁱⁿ

Y’

indicates that in each term at least one ni does not vanish. The harmonic

equation

⁰ is satisfied

by

Suppose

^{first Since}

It follows that the

integrand

ⁱⁿ

( q, 1 ) _ ( qo -~- ho , 1 )

^is

asymptotically

= X-2a+l. A

fortiori, (q, 1 ) ~ =

^{00, a} contradiction.

Now let a ⁼ Since

On the other

hand,

if

Since ,- , the choice

gives

^the

contradiction q~) ~ =

^{oo while}

preserving

^{the earlier}

inequalities.

We conclude that n

OQ ~ ~

^{for all} p &#x3E; ^1.

5. -

For p = 1,

^no

longer

^{true. In}

fact,

^{we even have}

a strict inclusion:

THEOREM 5.

PROOF. To prove ^the inclusion relation suppose For any

regular subregion ,5,

^{the Riesz}

decomposition yields (cf.

e.g. Nakai-

Sario

[3])

-

where is the harmonic function on ,~ with

hp= u

^on

aQ,

and

g!J(x, y)

^{is the Green’s function with}

pole

y.

By

^Stokes’

formula,

477

On

letting

^{Q -R we obtain}

where g

^{is the Green function on R. Since we have}

B

and therefore

By

^Theorem

2,

1(,

hence

6. - It remains to consider the class

QC.

^{Here we have the most}

elegant

case, ^as there is strict inclusion for all _{p :} THEOREM 6.

PROOF. In view of Theorem

2,

^{it suffices to} show that

Suppose

^{and take a}

u E QC.

^{The Riesz}

decomposition

^{of u on 4ib}

implies

On

letting

^{Q - R we obtain From the}

^proof

of Theorem 5

we conclude that

y) dy

^E

C.

Let

Then

and therefore

BIBLIOGRAPHY

[1] ^{L. CHUNG - L.} SARIO, ^Harmonic

Lp-functions

^and

quasiharmonic degeneracy,

J. Indian Math. Soc.

(to

appear).

[2] ^{Y. K. KwoN -} L. SARIO - B. WALSH, ^Behavior

of

biharmonic

functions

^{on Wiener’s}

and

Royden’s compactifications,

^{Ann. Inst. Fourier}

(Grenoble),

²¹

(1971),

pp. ^217-226.

[3] M. NAKAI - L. SARIO,

Quasiharmonic classification of

Riemannian

manifolds,

Proc. Amer. Math. Soc., ³¹

(1972),

pp. ^165-169.

[4] ^L. SARIO, Biharmonic and

quasiharmonic functions

^on Riemannian

manifolds, Duplicated

^{lecture notes} 1968-70,

University

^of California, ^Los

Angeles.

[5] ^L. SARIO,

Quasiharmonic degeneracy of

Riemannian

N-manifolds,

Ködai Math.

Sem.

Rep.,

⁶

(1973),

pp. 1-14.

[6] ^L. SARlO - C. WANG,

Quasiharmonic functions

^{on the Poincaré N-ball, Rend.}

Mat., ⁶

(1973),

pp. ^1-14.

[7]

^L. SARIO - C. WANG,

Negative quasiharmonic functions,

Tôhoku Math. J., ²⁶ (1974), pp. ^85-93.

[8] L. SARlO - C. WANG, ^Radial

quasiharmonic functions,

Pacific J. Math., ⁴⁶ (1973), pp. ^515-522.

[9] ^L. SARIO - C. WANG, Harmonic

Lp-functions

^on Riemannian

manifolds,

^Ködai

Math. Sem.

Rep.,

²⁶ (1975), pp. 204-209.