On the Orlicz space and Diriclilet problem for m-dimensional sphere

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ANNALES SOCIETATIS MATHEMATICAE POLONAE Series I: COMMENTATIONES MATHEMATICAE X X (1978) ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO

Séria I: PRACE M ATEM AT Y CZNE X X (1978)

Z

dzislaw

F

rydrych

(Krakow)

On the Orlicz space and Diriclilet problem for m-dimensional sphere

ч

1. In monograph [2] th e theorem concerning th e convergence of the Poisson integral for boundary continnons fonction is given.

In th e present paper we shall give th e generalization of this theorem.

Assuming th a t the bonndarv function belongs to th e Orlicz space LM* and satisfies d 2 condition we shall prove th a t the solution of th e Dirichlet problem for the ball converges to the boundary function in Orlicz norm.

2. Kow we shall give the definitions and lemma which we shall use in the sequel.

Let M(u) denote th e Orlicz ^ -fu n c tio n and l e t / ( s , t) be th e function defined in th e rectangular P — {(s,t), 0 < s < 2 к, 0 ^ t ^ n j.

De f i n i t i o n 1.

The function M(u) satisfies condition

d 2

if there exists a positive constant k such th a t

M(2u) < k -M(u) f o r w > 0 .

De f i n i t i o n 2.

The sequence f n{s,t) is convergent in mean

t o

the function f {s, t ) in the set P , if

n 2n

sn =

^{f f}

M[ \ f n{s, t ) - f { s , t)\]dsdt->0 as w->oo.

о 0

The norm of th e function и in th e Orlicz space LM* is defined by formula

n 2 7Г

\\u\\M = sup Г Г u{s, t)-v{s, t)dsdt,

e(v,N)*ci 0J о

where N(v) is th e com plem entary function to the function M(u) and

n 2k

q

(

v

, N ) = f f N[v{$, t)]dsdt.

о 0

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312 Z. F ry d ry c h

De f in it io n

3. The sequence of the functions /„(, t) is convergent* in norm to the function f ( s , t ) if

\\fn-fl\M->0 as n-+oo.

We shall prove

Lem m a 1.

I f the functions K r(x,

y ,

s, t) are mesurable for

0 < 1

and for [{x, y), (s, #)] e P x P and if there exists a constant A such that J j \Kr(x, y\ s, t)\dsdt < A , j j \Kr{x, y\ s , t)\dxdy < A

p p

for almost all {x, y) e P and convenably almost all (s, t) e P , moreover, for every function f ( s , t) e H, H being an everywhere dense set in L*

m

, the con

dition

rc 2

^tc

(1) lim f f M[ \ I n{f, s, t } - I m{f, s, t}\]dsdt = 0

nr-H».Q 0 ПНОО

is satisfied, then condition ( 1 ) holds for every function f { s , t ) belonging to the space LM.*

3. I t is known [1] th a t the set of the functions continuous in P is everywhere dense in the space LM for the Ж-Orlicz function satisfying*

Az condition and th a t such space is a complete space. Moreover, in such space the convergence in norm is equivalent with the convergence in mean. By Lemma 1 we get

Th e o r e m

1 . I f the condition

( 2 ) lim J J M[ \ I r{f, s, t } ~ f ( s , t)\]dsdt = 0

T—> \ p

holds for every function f continuous in P, then condition ( 2 ) holds for every function belonging to the class 1 ?МШ -

Let X , Y be arbitrary points of three dimensional space. Let {r,(p, y>) be spherical coordinates, let X = (r, q>, y>), Y — ( l , s , t ) . Let

-

X

f f f(s,

t)

sm

t M t ,

у being the angle between OX and OY. Since I r is the harmonic function in the unit ball, equal 1 for / = 1 , thus

Я ---— sintdsdt — lu ¹ ^{— r2} for 0 < r < 1 [1 + r 2 — 2 r cos у ]3/2

and

/ / [ i + ^ S c ô s ^F s i n y # # = 47t-

Now we shall prove

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Orlicz space and Dirichlet problem 313

Theorem 2.

I f the function M satisfies condition

d 2

and the function f defined i n P belongs to the class LM(Af), then the function I r is the solution* of the Dirichlet problem in the unit ball and I r->f in Orlicz norm as r-> 1.

P ro o f. Let £ be an arbitrary positive number. By continuity of the function / follows th a t there exists a number r0 such th a t 0 < r 0 < 1 , and

!M /> 9b 4>}-f{<Pi VOI < e

uniformly in P for r0 < r < 1. The function being monotone, we get

/ /

<p

,

rp}—f(<p, y>)\]d<pdy

< J j

M{e)d(pdxp

=

M

(fi)2u2,

p ■ p

and be Lemma 1 we get the thesis of Theorem 2.

4. Let X and Y be the points of the ^-dimensional space R m. In spherical coordinates let be X = (r, rlf ..., rw_1), Y = (1, ...,

Let P denote the m —1 dimensional cube 0 < < n, i = 1, 2 , ... , m — 2, 0 < &m- 1 <

Let

i A f J l}

1 - r 2

[1 + r 2 — 2 r cos y ]m/2 • J ’d&x ... d&

_{m —} 1 ?

y being the angle between OX and OY, con the area of the m-dimensional unit sphere and J the convenient jacobian.

Similarly as Theorem 2 we can prove

Theorem 3.

I f the function M satisfies condition A 2 and the function f defined in P belongs to the class LM(A2), then the function I r is the solution* of the Dirichlet problem in unit ball and I r-+f in Orlicz norm as r->l.

References

[1] M. K r a s n o s ie ls k i i J. R u ty c k i, Funkcje wypukle iprzestrzenie Orlicza, Moskwa 1958.

[2] M. K r z y z a n s k i, Bôwnania rôzniczlcowe czqstkowe drugiego rzçdu, T. I, Warszawa 1957.

[3] W. O rlicz, Ein Satz iiber Erweiterung von Linearen Operationen, Studia Math.

5 (1934), p. 127-140.

On the Orlicz space and Diriclilet problem for m-dimensional sphere

Z

F

(Krakow)

On the Orlicz space and Diriclilet problem for m-dimensional sphere

1. In monograph [2] th e theorem concerning th e convergence of the Poisson integral for boundary continnons fonction is given.

In th e present paper we shall give th e generalization of this theorem.

Assuming th a t the bonndarv function belongs to th e Orlicz space L*M and satisfies d 2 condition we shall prove th a t the solution of th e Dirichlet problem for the ball converges to the boundary function in Orlicz norm.

2. Kow we shall give the definitions and lemma which we shall use in the sequel.

Let M(u) denote th e Orlicz ^ -fu n c tio n and l e t / ( s , t) be th e function defined in th e rectangular P — {(s,t), 0 < s < 2 к, 0 ^ t ^ n j.

The function M(u) satisfies condition

if there exists a positive constant k such th a t

M(2u) < k -M(u) f o r w > 0 .

The sequence f n{s,t) is convergent in mean

the function f {s, t ) in the set P , if

n 2n

sn =

M[ \ f n{s, t ) - f { s , t)\]dsdt->0 as w->oo.

The norm of th e function и in th e Orlicz space L*M is defined by formula

\\u\\M = sup Г Г u{s, t)-v{s, t)dsdt,

where N(v) is th e com plem entary function to the function M(u) and

(

, N ) = f f N[v{$, t)]dsdt.

3. The sequence of the functions /„(*, t) is convergent in norm to the function f ( s , t ) if

\\fn-fl\M->0 as n-+oo.

We shall prove

I f the functions K r(x,

s, t) are mesurable for

and for [{x, y), (s, #)] e P x P and if there exists a constant A such that J j \Kr(x, y\ s, t)\dsdt < A , j j \Kr{x, y\ s , t)\dxdy < A

for almost all {x, y) e P and convenably almost all (s, t) e P , moreover, for every function f ( s , t) e H, H being an everywhere dense set in L*

, the con­

dition

rc 2

(1) lim f f M[ \ I n{f, s, t } - I m{f, s, t}\]dsdt = 0

is satisfied, then condition ( 1 ) holds for every function f { s , t ) belonging to the space L*M.

3. I t is known [1] th a t the set of the functions continuous in P is everywhere dense in the space L*M for the Ж-Orlicz function satisfying

Az condition and th a t such space is a complete space. Moreover, in such space the convergence in norm is equivalent with the convergence in mean. By Lemma 1 we get

1 . I f the condition

( 2 ) lim J J M[ \ I r{f, s, t } ~ f ( s , t)\]dsdt = 0

holds for every function f continuous in P, then condition ( 2 ) holds for every function belonging to the class 1 ?МШ -

Let X , Y be arbitrary points of three dimensional space. Let {r,(p, y>) be spherical coordinates, let X = (r, q>, y>), Y — ( l , s , t ) . Let

X

t)

t M t ,

у being the angle between OX and OY. Since I r is the harmonic function in the unit ball, equal 1 for / = 1 , thus

Я ---— sintdsdt — lu 1 — r2 for 0 < r < 1 [1 + r 2 — 2 r cos у ]3/2

and

/ / [ i + ^ S c ô s ^F s i n y # # = 47t-

Now we shall prove

I f the function M satisfies condition

and the function f defined i n P belongs to the class L*M(Af), then the function I r is the solution of the Dirichlet problem in the unit ball and I r->f in Orlicz norm as r-> 1.

P ro o f. Let £ be an arbitrary positive number. By continuity of the function / follows th a t there exists a number r0 such th a t 0 < r 0 < 1 , and

!M /> 9b 4>}-f{<Pi VOI < e

uniformly in P for r0 < r < 1. The function being monotone, we get

/ /

,

< J j

=

(fi)2u2,

and be Lemma 1 we get the thesis of Theorem 2.

4. Let X and Y be the points of the ^-dimensional space R m. In spherical coordinates let be X = (r, rlf ..., rw_1), Y = (1, ...,

Let P denote the m —1 dimensional cube 0 < < n, i = 1, 2 , ... , m — 2, 0 < &m- 1 <

Let

i A f J l}

1 - r 2

[1 + r 2 — 2 r cos y ]m/2 • J ’d&x ... d&

y being the angle between OX and OY, con the area of the m-dimensional unit sphere and J the convenient jacobian.

Similarly as Theorem 2 we can prove

I f the function M satisfies condition A 2 and the function f defined in P belongs to the class L*M(A2), then the function I r is the solution of the Dirichlet problem in unit ball and I r-+f in Orlicz norm as r->l.

Assuming th a t the bonndarv function belongs to th e Orlicz space LM* and satisfies d 2 condition we shall prove th a t the solution of th e Dirichlet problem for the ball converges to the boundary function in Orlicz norm.

The norm of th e function и in th e Orlicz space LM* is defined by formula

3. The sequence of the functions /„(, t) is convergent* in norm to the function f ( s , t ) if

, the con

is satisfied, then condition ( 1 ) holds for every function f { s , t ) belonging to the space LM.*

3. I t is known [1] th a t the set of the functions continuous in P is everywhere dense in the space LM for the Ж-Orlicz function satisfying*

Я ---— sintdsdt — lu ¹ ^{— r2} for 0 < r < 1 [1 + r 2 — 2 r cos у ]3/2

and the function f defined i n P belongs to the class LM(Af), then the function I r is the solution* of the Dirichlet problem in the unit ball and I r->f in Orlicz norm as r-> 1.

I f the function M satisfies condition A 2 and the function f defined in P belongs to the class LM(A2), then the function I r is the solution* of the Dirichlet problem in unit ball and I r-+f in Orlicz norm as r->l.