MULTIPLIERS

IN SPACES WITH "FRACTIONAL" NORMS, AND INNER FUNCTIONS

I. E. Verbitskii

UDC 517.9

INTRODUCTION Recently, criteria have been found for the boundedness of operators of multiplication by a function (multipliers) in Sobolev, Besov, and other spaces (Maz'ya [I, 2], Maz'ya and Shaposhnikova [3, 4], and Stegenga [5]). The main achievements are connected with the attraction to this circle of problems of the concept of the capacity appropriate to the function space being considered. However, the conditions that have been obtained are difficult to verify directly. At the same time, the classes of bounded multipliers introduced in earlier works [6-10] are very narrow (especially in respect of functions with discontinuities of a complicated type). The purpose of the present article is to describe all inner functions (Blaschke products) that are multipliers in the Besov spaces B~ and in the spaces ~ of fractional potentials on the unit circle T. We note that singular inner functions generate unbounded multiplication operators (Lemma 1.4). The simplest space of the type being considered consists unit disk U for which II!IISa = ~

of functions analytic

in the

17(k) 12(k+I) 2 ~ < o o ( a ~ n ) . h~o

(0.I)

-

Here f(k) = f(k)(O)/k! (k ~ O) are the Taylor coefficients of functions satisfying (0.1), and let M D = = { a E D = : f ~ D = ~ a f ~ D = } responding class of multipliers. It is easy to see that a ~ M D = of multiplication by a ( z ) ( z ~ U ) is bounded in D~.

of f. Let D e denote the space , that is, MD e is the corif and only if the operator

It is known [9] that MD~ = D e for e > I/2, and MD e = H ~ for e ~ 0, where H ~ is the Hardy space [6] of functions bounded and analytic in U. Stegenga [5] described MDu for ~ ( 0 , I/2] in the following way. THEOREM A. A function (Ik} of disjoint arcs of T

(0<~
a~MD=

if and only if a ~ H

~ , and for any finite

set

I Ir (z)12(~ -I~ I)~-2~dm ~c.cap2~ (U lh). us(~h) Here S ( I ) = { z = r ~ : ~ l , l-III
Y~l(~) =.[ g(~) T ir

Id~ I ( ~ T ) .

(0.2)

I

The capacity can be defined by cap2~

We recall ~T. form

Let

(e) = inf [ flg~L2(T) 2 : g ~ L2 (T),g~>O;

[6] that a ~ H

~

is called an {nne~ function

{ah}hmo be a sequence of points

in U with

f~g~l

e}..

on

if lim]a(r~)l=l

Z(l--[a~l)< ~.

(0.3) for almost all

An inner function of the

is called a Blaschke product. Our main result

in the case of the spaces D e is the following theorem.

Kishinev. Translated from Sibirskii Matematicheskii Zhurnal, Vol. 26, No. 2, pp. 51-72, March-April, 1985. Original article submitted July 18, 1982.

198

0037-4466/85/2602-0198

$09.50

9 1985 Plenum Publishing

Corporation

THEOREM I. An inner function a~,~/D~ ( 0 < ~ < { / 2 ) if and only if ~ is a B!aschke product and the following condition holds for any finite set of disjoint arcs lj m T:

~E

(~ -~[a~t?-~'<~.ca~o,(U~q).

(o.4)

Let us note that Theorem 1 is not derived directly from Theorem A. In its proof we use (in the same way as in the proof of Theorem A) the Maz'ya--Adams strong capacity inequality (see [I, 11, and 5]):

S cape~ {X :l J J (x)! "~ t} tdt <~ cI!]~[~(T).

(0,5)

8

For ~ /> I/2 the class D~, and consequently M D ~ contains only trivial inner functions (finite Blaschke products) [12]. The inner functions belonging to D~ for ~.(0, I/2) have been described by Carleson (in terms of the zeros {ak(Z)} of the function (
a~D~

then the set of those I for which

,~o

(0.6)

(0.6) is not fulfilled has zero capacity

[13]. The Blaschke products belonging to M/]~ satisfy the well-known condition (CN) of Car!eson--Newman (Lemma 1.5). It can be shown that for such inner functions~ (0.6) simplifies to: a~D~ if and only if Z({ [when

[JI~=,T

this condition

Ia I),-2~< oo

(0,7)

is the same as (0.4)],

In Sec. 2 we obtain a generalization of Theorem I/p) consisting of functions analytic in 8 for which

I to the spaces Bp (I ~< p < oo 0 < u, <

:tI (z)i~(t -[zl)-~-~-~8~n
(o.s)

U

(it is easy to see that B.~ 2 = D~)~ By the general theory of Maz'ya is simplified (it does not contain the capacity).

[14] ~ for p = ~, our result

We observe that the above-mentioned result of Carleson [13] about inner functions from D~ has only recently been generalized (partly) to the spaces B p by Ahern and Clark ([15~ 16]), We use their technique in Secs. I and 2o Section 3 contains another generalization of Theorem I, namely to the spaces ~ [ of fractional potentials, For I < p < 2 this generalization is obtained by usin Z the results of Seco 2 and well-known relations between ~ [ and BP~ For p > 2 we use deeper properties of ( ~ -capacities relating to nonlinear potential theory ([17, 18]). It turns out that for ? > I the classes of inner functions that are multipliers for ~7"~ and B[ coincide. Examples of the verification of the validity of the "capacity" condition (0.4) and also applications to certain questions in multiplier theory will be presented separately, Part of the resu~~t s in the present article was announced in [19]. The author thanks S. A. Vinogradov and V. Go Maz~ya for a useful discussion. ]o

Proof of Theorem

I

Let f be analytic in ~] and ],(~)-----'f(r~),0 ~ r < ~ [6] of functions f for which

#I {p = In what follows

it is convenient

LEi~iA 1.1.

{z~(8, I/2) and

s~p

It L k~m

%~T.

,

<

We denote by H p the Hardy space

~"

to use the following renormalization

of the space D~

[16]. Let

f~,z:

The expression

Ji~,~- I/(0)I~ + { (~III~-lj ]~IF~){~-o~)-~'*~rd~.

(~. ~)

0

199

defines in D a a norm equivalent to the original norm (0.1). LEMMA 1.2.

An inner function

a~MD=

(0<~<~/2)

if and only if

(I .2)

.I(~ - I~ (~) 12)I1 (z)I~ (I - Iz I)-~-== d = ~ c II1 $=. u

Proof.

a ~ MD=.

Assume that

J'u (l

Then by Lemma

-] a (+) I+ ) I/('11

I .I

= (i - l z I)-'-+~' dm

1

S ~ {I ~ (;) I~1! (;)P- I~(rO 121I(r;)] 2} (t

--

OT

The sufficiency

r)-Z-==rdr I a; I ~< IIaf~g <~IllMZ=.

is proved similarly.

COROLLARY 1.1. Let a i (I ~< i ~< n) be inner functions, only if ai~MDa (t-<
COROLLARY 1 . 2 . we h a v e

If

a~MD=

(0<~
, then for

and a = ~ai.

Then

a~MD=

if and

l={~=e+~:lO-.Oo]<~5/2} (6>0)

any arc

/,

.} (t -- l a (z)12) (1 -- I z I) -~-~';' dm ~ c(~ 1-2~. s(D To prove Corollary

1.2 it is enough to put

I (,.) = in

(1.2)

and o b s e r v e

Let

-- ( 1 - - +) e ' % )

that:

~fg= LEMPLA 1.3.

(1.3)

?>0, ~T

=

, and

E (t

h>~o

-

6)~'(k + if= ~< ~.~-~-~.

Sv$(z)=exp(?~_-~).

Then $,$ ~ 7 ~ D ~

for ~ ( 0 ,

I/2).

An assertion for /P-multipliers similar Eo Lemma 1.3 is proved in [20]. The proof given there carries over with obvious changes to the case D~ [it can also be verified directly that ST~ does not satisfy (1.3)]. LEMMA 1.4.

If an inner function

a~MD=

(0<~
, then a is a Blaschke product.

Proof. It is known [6] that any inner function is representable as B-S, where B is a Blaschke product and S is a singular inner function generated by a singular measure o on T:

9S ( z ) = e x p [ ~ z _ ~

(~) .

By Corollary 1.1 it is enough to prove that S~MD~. for otherwise S(z) would contain a multiplier of the form which is impossible by Lemma 1.3 and Corollary 1.1.

(1.4) We may assume that ~ is nonatomie ~T),

ST~=exp{?(zq'~)/(z--~)}(7>0,

We denote by w(~) the modulus of continuity of the nonatomic singular measure ~:

~(6) = { s u p o ( I ) : l - Then ~(6) is a continuous function

arc T, II1 = 8 } .

[21, p. 196], and I i m ~ ( ~ ) / 5 ---- o o .

( t . 5)

8-~0

Assume t h a t

s(z)

Let I0

S~.MD=.

By C o r o l l a r y

.++o_-

s(z)

1.2,

{._o+

for

any a r c

"]i

-

I~T

([![=6)

+

K,. 12

(II01 = 6) be an arc of T such that ~(~) = .q(I0), 0 < r < I/2.

have

--

200

~

88 ~

--.

For

z~S(lo) we

(1.5)

By taking

into account we obtain

~a'-:: ~> ~" (l - 1 s O)l~)(i - I z I) -'-~ s(~o)

dm >7

1

,

f,_oxp(

O(~)('--r))}(t--r) -I-~i dr ~.~c,81-ma(o(~l/8):~

T (t--e-r) rFi-~adr.

483

1-~ k

o

Consequently,

(o.(8)/5) ~r By taking

j" (t_e-r)r-,-~=dr<.~c"

the limit as 6 + 0 we obtain a contradiction

We recall that a finite for any arc I ~ T

to (I .5), and the lemma

is proved.

~ in the disk II is called a Cc~p;~son meas~#

Borel measure

~(S(I) ) <~till. Let

{a~}~0

be a sequence

of points

in U.

~to(e) =

We define a measure

It is known [22] that ~0 is a Carleson measure son--Newman condition (CN)

LEF~A !.5. If a Blaschke the condition (CN). Proof.

By Corollary

product

(1.6) ~0 by the equality

(I .7)

(i--la~I 2) ( e ~ U ) .

~

BEMD~,

if

if and only if {cry} satisfies

then the sequence

the Carle-

{~k} of its zeros

satisfies

1.2~ for any arc I

[ (I - IB(z)l~#(I -- Iz I)-I-~ d e < c IZ 11=~% s(D We use the obvious

Let

estimate

II[ = $ (0 < 5 < I/2).

c0= r <,

In the same way as in the proof

s{, ex( '

,z>

8(1)

of Lemma

1.4 we obtain

)}

ah~8(l) 1~o(8(I))/41 II

6 o

k

48.

o

Consequently,

Oo (s (h)/I 1 !)~-~

~o(S(f))/,tlZl [ 0 -,-9

~-~-~d~ ~<,,~

0

and it follows from here that the measure the condition (CN). The lemma is proved. COROLLARY

1.3.

If a Blaschke

~0 satisfies

product B ~ M D ~

(1.6)~ and the sequence

(0<~
{~k} satisfies

i then

- I B (~) t~/> C~ 0 - ]7~~-f ~19 @ (~ l - J' l~} {~ mU).

(~.~o)

202

We observe that the reverse inequality

(1.11)

(z ~ U)

holds for all Blaschke products

[15].

Proof of Theorem !. Suppose that an inner function tion is equivalent to the embedding theorem

aEMD=.

By Lemma

1.2 this condi-

(1.12) U.

where the measure p is defined

in U by dl~= (i - la(z)lD(1 -

By Stegenga's

theorem [5] the embedding

Izl)-'-~dra.

(1.12) is equivalent

to the estimate

B,(Us(ij)) ~< c- capz~ (Uh), where {lj} is a finite set of disjoint arcs of T. Blaschke product satisfying (I.10). Therefore,

Since for

1.5 and Corollary

By Lemma

!.3,

is a

a~S(l) [ d-

sii)

~ ]-~ am

I a,~ Io:) (~ - j = i~)-20~ I ~ -

> ~ (.~ - t a,~ !)~-%

we have

(1.13)

E (~- --! c~k I)'-+~ ~ c.cap2,~ (U/)). '%=~us(zO

We turn to the proof of the reverse assertion, that is, (1.!I) that it is enough to establish the inequality

1.13)=~(io!2).

It fo!lowsfrom

~.- [% ?

(t.i4)

U

We d e f i n e a measure v in U by

where .u0 i s d e f i n e d by (I . 7 ) .

Then e s t i m a t e s

(I .13)

v(U S(I~)) <~ c .

[ll(z)Pd-I By changing putting

z

and ( 1 . 1 4 )

(1.15)

-~ e ( t - I ~ l) :~ _,. i)-2~ dm (~l-----_~,~(~)~
the order of integration

in the left-hand

tr

into

cap2~'(Ulj),

side of the latter inequality and

o2//(~) ={(|_lL,)~;(t_[zl)-2~lf(~)t~ we rewrite

go, r e s p e c t i v e l y ,

1 t -- -fz 12

d m (z)

}~/2 ,

it as

.( (~ (~))~ d~ (~) -~ r ]tf ~

(1.16)

U

The rest of the proof is devoted to the implication (1.15) ~ (1.16). Here we model, in the same way as in [5], the proof of Carleson's embedding theorem given in [23] (p. 279), but use more complicated maximal functions.

202

Let Pf(z)

denote the Poisson integral of f:

i ! l--lz} ~

Iz-~l<~(l-lzl)}

Let ~ > I and ~ ( ~ ) ~ { z ~ U : maximal function corresponding to

~i(~)

be a Stoltz domain.

We define the nontangential

by

~_mri~(D[v t,t_ ~

)

~dm(z)

t

(1.17)

.

As in the case of an ordinary maximal function connected with a Poisson integral is majorized by the radial maximal function

og{~

[23],

"f In other words, we have the estimate

T),

(1.18)

which follows from the inequality

Finally, we define another maximal function by

(! (t--r)V-l,P l[7__~r~.~]~(r~),2 d~,ff]~/~ where 1 < T ~< 2. Maximal functions of the latter type have been considered by Stein [23] (p. 280). We need a number of auxiliary assertions about the maximal functions introduced above. LEMMA 1.6.

Let 0 < ~ < I/2 and I < y < 2 -- 2~.

Then

(0 < where the constant c does not depend on Proof.

~T

and f.

We use the inequality I t - p r ~ P m cl t - r ~ l ~ ( t - p +

i - r)z-<

We obtain 1

if (l--P~2aLP_f(_r~'~)L2 d~irdr~cf(t--p)2a(l--r){-Y-2a \~] [ t .pr~[2 ( f _ p + t _ _ r ) 2_ ~

o T

o

dr

8o

f!Pl(r~)l~(t--r) "~-z T

=

] t--r~l?

o

The lemma is proved.

LEMM.A 1.7 [23]. LEMMA 1.8. (~) = . ~ ' / ( ~ ) g

I f y > 1, then ll~

Let k ~ ( ~ ) = l l - ~ [

)

L~(T)~C~f[IL2(T)~

~-' ( 0 < ~ < 1 / 2 ,

~T).

Then

-~l,~a(~)~k~ ,0~}~) (~), where

o~A~)

f,g

(~)[d~[ is the convolution of ], g ~ L d T ) .

T

Lemma 1.8 is proved by using the integral form of Minkowski's Now we complete the proof of the implication

(1.15)

=~

inequality.

(1.16),

Let A~i={z:~

Then

J (~1 (z)) 2 dr(z) U

It can be verified follows the estimate

= j v (At) tdt. 0

in the same way as in the proof of Theorem A [5]

that

from

(1.15)

w (At) ~ c.cap2~ {~ : ~ 7 (~) > t}.

203

•

We choose a 7~(], 2--2~) .

Setting

strong

(0.5)

capacity

inequality

f=k=*g,

g~E(T)

, and applying Lemmas

1.6-I .8 and the

we o b t a i n co

.f (~J (~1)~ a,, (~) ~< ~ ,f ~v~ {~: ~aF ) (~) > t} ~at <~ U

o

oo' 2 [~: ks , Gg/(gv)(~)> t} t d t < c @_/{w a L~(T)
IELP(T),

for which

,,~"/Y:,I d~ I < oo.

(2.1)

The class BP(T) is the periodic analogue of the Beslov space BP(R) (see [ 2 3 ] ) . ~ B ~ ( T ) } be the corresponding class of multipliers.

Let

MB~(T)={a~B~(T):f~B~(T)=~a] If ] ~ H

~ ~] B~(T),

then we can define a norm equivalent to (2.1) by the expression

I1/I1~,~ = I l (0) I" + j" I .f' (~) 1~ (~ - I ~ IF -'-=' a~.

[24]

(2.2)

U

This equality is also meaningful realization [25] will be used:

for -oo < ~ < 1.

When ~ < 0 another equivalent renor-

I1]11~,= = ~ II (~) I" (t - I z I) -'-=" d~.

(2. s)

17

We denote by Bp (-~ < ~ < I) the space of functions analytic in U and satisfying lI]ll~.~
Let 0 < ~ < I and I ~< p < ~.

1) a~MB~(T)

if and only if

a~L~(T)

T

2) a ~ f ] l B ~

if

and o n l y

if

and for any

, I ~--i

a~H ~

,f~B~(T)

It+~#

and

I t / (z)I p la' (z)IV(i-- I z I) ,-1-~, dm < c II f [1~,~(1 ~ BE). U

3) If a ~ H ~ , then the conditions a ~ M B P =

and a ~ M B ~ ( T )

are equivalent.

Assertions similar to Lemma 2.1 are well known ([I-5]). Let us explain that 3) follows from !) and the boundedness of the analytic projection onto PP(T) (see [25]). We list several simple properties of the classes MBP:

1~

MBp = Bp for ~ > 1 / p .

2o

MB~P = H oo for ~ < 0.

3 ~ MB:tt

I7 lllBV=~c

MB~

a = (i -- t)o:t + t~z, t / p = (l -- t)lpi + t/p,, t ~ (0, l).

Properties I~ and 2 ~ are proved in the same way as in the case p = 2 [9]; 3 ~ follows from an interpolation theorem for the spaces BP(T) [25]. The main aim of this section is to generalize Theorem I to the spaces B p. Since the nontrivial inner functions belong to Bp only for ~ < I/p (see [26]), in what follows it will be assumed that ~ ( 0 , i/p). Note that the interesting case of the spaces B p requires We denote the BP-capacity of a closed set e =

204

separate consideration.

T by capp,~(e);

it is defined by

THEOREM B [3].

a~MB~(T)

(0<~
if and only if a ~ L ~ ( T )

l
and

, t e; 1r.f! ~I ~(::) - l- I~a+ <;: ~ I , i d~ 1< c.eap~,~ (e) for every closed

set

e ~ T.

The description of MBP(T)

in the case p = I simplifies.

THEOREM C [14]. a ~ M B ~ ( T )

f o r any a r c

I ~

(0<=
if and only i f a ~ L |

and

T.

The following result, which will be required later on, is a direct generalization of a theorem of Stegenga [5, Theorem 2.3] to the case p ~ 2. The proof is hardly different from the p = 2 case, and will not be given here. THEOREM D. Let I < p < o~, U < ~ < I/p, and p be a finite Borel measure lowing assertions are equivalent:

in U.

The fol-

U

2) ~(U S(I~)) <~ c . capv, = (U lj). The proofs of Theorems B and D rely on the periodic analogue of the Maz~ya--Adams strong capacity inequality ([I, 11]) for the spaces BP(R):

o

(the passage from R to T is realized in the same way as in the p = 2 case [5]). equality is used in the proof of the main result of the present section.

!~nis in-

THEOREM 2. For an inner f u n c t i o n a ~ M B ~ (0<=<~/p, 1
~.~

%~ Us(lj)

(t --[ah[)l-=P~c.eap#,=(UIj)

for any finite set of disjoint closed arcs

(2.4)

{Ij},Ij~T.

Proof of Theorem 2. We first prove the necessity of the conditions. Let a~MB~, 1/q = 1 - I/p, and ~ ( 0 , ~) , then a~MBqe by 2 ~ By applying the interpolation property 3 ~ we B (~_e)iz. 2 find that Since B~ = D~, by Lemma 1.4 and Corollary 1.3 cz is a Blaschke product and

i-la(~) I~>~ ~0-l~

~)(~-~ "~) I ~ - :~ I~

(2.5)

(~ ~ ~)"

For the proof of (2.4) we need the following lemma. LEMMA 2.2. Let: a be an inner function, ~ > 0, I ( p < ~, and 0 < 6 < I~ lowing inequality holds: 1

Then the fol-

I

.f (I -- [ a (rE) l)" (l -- ~-:'~Prdr • c f J a' (,;) [' (1 - - r)~-l-a~rdr, 6

8

where the constant c depends only on p and ~. Proof.

We note that for almost all ~ T 1

t -- l a (r;) I ~ S I a' (PDI d~o. .lr

205

Let

(p -- ] -- ~p)/p

(l--

< y < (p -- 1)/p.

la(rO l) p (1 --

r)-'-='rdr

1

By applying

6

o

The lemma

is proved.

By taking

into account

inequality

we obtain

I~' (p~)[p (i " P) dp (t --r)-'-a'rdr

<,

p

~<~.~l,'(p~)l~(l_pF*j(~

H~Ider's

1

-- , .), - ~ - ~ + ~ ) ~ d ~dp < ~ .I I , ' (P~) i~ (l -- p)~-~-~" pdb. 6

Lemmas

2.1 and 2.2

and Theorem

D we obtain

j" (t - I a (z) I )P (t - I ~ I)-~-UPdm ~ c.capp,(z ( U Ij). Us(i~) It follows

from

(2.5)

z))

u We use

that

the inequality

T7----~7~

(1--lzl)=~-~P'dm~c'caPP'~(UIj)"

( ~ bh)v ~ ~ b~(b~~ O, p ~- I) :

(1 --[a/~ I~)p (t --l z ]2)p-l-~p %~Usffj)

S 84 usu~ )

[t

-

-

~hz 12p

drn ~.~ c. capu,~

( U Ij).

that for c~k~S(l)

Let us note

S (1 ]t I z])~ZI2P p-l-~v d r n ~ c ( l - - ] a k l ) 1 - ' - ~ ' .

s(z) Consequently,

(2.4)

holds.

This p r o v e s t h e " n e c e s s i t y "

P r o o f of t h e S u f f i c i e n c y . We use

Let a(z)

part

of Theorem 2.

be a B l a s c h k e p r o d u c t whose z e r o s {ak} s a t i s f y

(2.4).

the estimates

-I%P

i~'(z)l<~o(l-I~l)-L I~'(~)l<~, r i = ~ i , (z~ v). The first of these lows from 2.6) that

~s valid

for any

(2.6)

a ~ H ~ [6]; the second is proved directly.

It fol-

[1-7h~12" By Lemma

2.1

it is enough

to prove

the

inequality

(2.7) We denote by

o'tll(L)(/~B~,

The proof of assertion.

(2.7)

LEMMA

2.3.

%~U)

reduces,

the function

in the same way as in the proof

Let ~ be a finite

Borel

measure

in U,

U

206

I, to the following

I < p < ~, and 0 < e < I/p.

v(US(Ij) ) ~ c 9capp, ~(UI~), then

of T h e o r e m

If

A number

of a u x i l i a r y

assertions

are used

in proving

Lemma

~?) (~) = ~.p ff !P, (,~)i (i - rF -~ where Pf denotes LEMHA 2./4.

the If

Poisson

integral

[ ~H~, l
2.3.

We write

t "~

of f. l
, and 0 < a < I/p,

then

sup ~, (~) <~ c~?) (r (r ~ T), where F ~ ( ~ ) = { z ~ U : ] z - ~ I < ~ ( i - l z i ) }

is a Stoltz

domain.

The proof of Lemma 2.4 hardly differs from a s s e r t i o n for p = 2 is the same as Lemma 1.7. LEMMA

2.5.

For

the p = 2 case

(see See.

I).

The following

! < p < ~ and Y > I we have

(2.8) Estimates

of the type

(2.8)

are known

[23, p. 280].

LEYnMA 2.6. Let T > 1, I < p < ~, 0 < a < I, and /~B[(T) ~ Then ~J}v)~B[(T) and I -~ : ,:,~< c il/ ib,~. / o ~ : ( v ) + o2~/gV), and so Proof. It follows from M i n k o w s k i ~ s inequality that ~/(~) -~/+g~.~:

[ :~ :(V)

Lgt g({)=/(~D(~, ~ T )

~

z o~(v) ({) 9 then q~/(v)(~g):

I].~(v
Consequentiy~

[

141

by Lem_ma 2.5

C T

--

{I/II~+ .fll---u-~ , i : _ ~ : + ~ }{~<~

~

This proves

:~:(v)

,

Lemmm

p

idol

' ll.:jJ~+

T

frl

~

Ivji

l-g!'~

,~41

B:+~P

-

c 1:/ljp.e. ,, ' "

2.6.

Proof of Lemma 2.3. Let 7 ~ ( ~ 2--c~p) , and fr(z) = f(rz) 2.I and 2.6 and the strong c a p a c i t y inequality we obtain

(0 < r < I).

By using

Le:mms

oo

,[ ( % (~)F d~ = S': {~,: % (~) > ~} : - : e t <

iJ

O

co ('

(V).

(~) P

i~

0

with

By taking the limit as r + I we establish it T h e o r e m 2, are proved~ We end

whose

this

section

by c o n s i d e r i n g

E

the constant The proof

c does not depend

is b a s e d

estimate~

Thus Len~na 2~

and

the case p = ~o

T H E O R E M 3. An inner function a~MB~ ( 0 < ~ < i ) zeros {a k} satisfy the condition

%~s('D where

the required

if and only

if ~ is a Blaschke

(~- la~bF~<~1•

on an arc

I c

on the use of Theorem. C

and

product

<2.9)

T. Lemmas

2~I and 2.2; we shall not dwell

it. The f o l l o w i n g a s s e r t i o n c o m p l e m e n t s T h e o r e m of m u l t i p l i e r s for p ~ ( 1 , co) that do not involve

2 and gives conditions the capacity~

for the boundedness

C O R O L L A R Y 2.1. Let ] < p < ~, 0 < ~ < ~/p, B(z) be a Blaschke product and I be an a r b i t r a r y are of T. Then B ~ ~ ]~fB[ if and only if

with

zeros

{~k},

207

on

(l-lahl)~-~ <~cllI ~-~

al~S( I)

(2.10)

for any ~ ( 0 , CZ). The sufficiency of (2.10) follows from Theorem 3 and Property 3~ from Theorem 2 and the estimate cap~,=(f)~c[fl ~-=~ [11]. 3.

Multipliers

the necessity follows

in Spaces of LP-Potentials

We denote by ~ ( T ) (0<~
the class of functions ] ~L~(T)

representable

(3.1) T

where g ~ L r

We set

[I/JJ~m

=ligliLp

andM~(T)= {a~6~(T):[~(T)=~a/~(T)}..

assemble in the following theorem different equivalent definitions of the spaces ~ ( T ) also information about their relation with the spaces BP(T) ([10, 23-25]). Let

'S ! (~) ;~ld~t(k ~

7 (k) = ~

We

, and

Z)

T

be the Fourier coefficients of f. THEOREM E.

1) ~ ( T )

consists of those ]~LP(T)

for which

](k)----(IkE+1)-=g(k)(k~Z), where

g ~ LP(T). 2) For p >/ 2 an equivalent norm is defined in

~(T)

llxllz:m=II

by the expression

IdOl] lagl.

(3,2)

3) The f o l l o w i n g embeddings of ganach spaces h o l d :

We denote by ~P= ( ] < p < o o , - - o o < ~ < I ) fying the condition

llzllg ~-- I/(0)1" + [ THEOREM F [ 2 2 - 2 4 ] . holds. 2) For ~ < 0,

]~*~

(JII'(r~)l~(l--r)~-2%drF

Id;i
1) For 1 < p < ~ and b < ~ < 1, f ~ q ~ ( T ) n Hp

(;

I](l"~)le(t--r)'-2~rdr)P/~,ld~[
(3.4)

for p ~ 2 .

Let l l f c ~ ? P = { a ~ ? P : / ~ . ~ v = e ~ a ] ~ p} be the class of multipliers properties of 7Ff~, analogous to those of MB p in See. 2. o. TFf~__ ~

4 ~.

on ~f~.

We give some

for ~ > I/p. ~

for ~ <~ 0.

P2"~ M .o~, P cz=(i-t)a,+taz, f~M.~%

If a~-H ~ , t h e n a ~ M . 5 ~ ( T ) ~ a ~ M ~ f

i / p = ( l - t ) / p ~ + t / p z , t~(O, t). p, v:~.(O,l)~

The main result of this section is a description of the inner functions pliers in the spaces ~P=. Since the nontrivial inner functions belong to ~

208

(3.3)

of Banach spaces hold:

~q~P~B~ for p ~ 2 ; B ~ . ~ - ~

3~

i f and o n l y i f

and ( 3 . 4 ) a r e e q u i v a l e n t .

3) The following embeddings

2~

(3.3)

i f and o n l y i f I][ll~,~ = ~

t h e norms ( 3 . 3 )

the space of functions f analytic in U and satis-

that are multionly when

I/p).

~, < I/p (see [26]), we shall assume in what follows that ~ ( 0 , capacity of a set e m T:

We define the

~[-

c a p z ~ ( e ) = inf {11Silty: i ~ m~ (T), i 7> 0; & ! (r ~> t o~ e}. (see

(3.5)

Remark 1. L e t ~ : T § R b e t h e n a t u r a l m a p p i n g of T o n t o (--v, ~ ] . It [ 5 , 18]) t h a t eaps i s e q u i v a l e n t t o t h e c a p a c i t y of t h e s e t ~ ( e )

i s e a s y t o show g e n e r a t e d by t h e

(z

Riesz potential

I~1 (x) =

(3.6)

I * --f(g) v i:t-~ d,g

in LP (R). From now on, in the symbol for the capacity it cannot lead to confusion.

(3.5) we shall omit the subscript

~[

when

THEOREM G [4, 5]. Let ~ be a finite Borel measure on T, ~ < p < ~, and 0 < ~ < {/p~ The following conditions are equivalent: ~)

.[ l .f (;) l" dr< <~ ~ll il!~.<=(.,+~

:~(T)

N. C(T));

T

2) ~ ( e ) < c . c a p e f o r any c l o s e d The p r o o f

set

e

~ T.

i s b a s e d on t h e s t r o n g

capacity

inequality

(see

[27])

t cap {~ :1 a,~s<(;) I > *} P-~dt < ~ II S IlL,m,: "

9 (3.7)

0

which will be used below.

Note that Theorem O (Sec. 2) remains valid with B p replaced b y ~

THEOREM 4. For an inner function a~M.9?[ (i < p < o % 0<0~
%=-US(Sj)

(1 - - I ak I)~-~P ~< c'cap(U SD

(3.8)

for any finite set of disjoint arcs (Ij}. Proof. I. Let us note that if a ~ M . ~ ~ then ~ is a B!aschke product whose zeros satisfy the condition (CN). For since a ~ M.~[ ~ ]]g~qo(~qo-~11~,I/p -F I/q = ~) , by Property 3 ~ we have a~M~I===J/ID~/~ = . By Lemma 1.5, the required result follows from here. !I. We now show that the proof of the necessity of (3.8) for p >~ 2 and of its sufficiency for p < 2 reduces to problems which have been considered already in Secso I and 2. Let p /> 2 and a ~ M ~ .

By Theorem F

Consequently,

.[ 1 <~' (z) I"1 i (~)t" (:~ - I z i) '-:'-~<" dm ~< c (11a i7' ~ u In other words, the equality

c

the embedding theorem

~=P~DP(U,~)

dr= la'(z)P(l-

s.s

"

I1i il~g + 11t it" ,,1~<<< It' t-<'~, II ~p. "~'~'.J

'

"~a,

holds, where the measure ~, is defined by

lzlP-'-="dm.

(3.9)

The remainder of the reasoning is a word for word repetition of the proof of the necessity part of Theorem 2 with ~ instead of B p. Now suppose that p < 2 and (3~

li ,v Ilzg


holds.

By Theorem F and Property 2 ~

{I ~ (o)s (o)! + It (,:,t)' Iizg._.~} <.~ {IS (o) 1 + li <,i' Ilzg., - + I1,~'/II ~=,_,} -<.' ~,. {! I@I + + <,

tl s'

+ ffo'sIlo:_,) <

+

209

.

Since

.1 l[ (z)lVla

=

' (z)! ~' (t -- l z !)v-:-~P

dm~

U

the proof that a ~ J f . : . ~ reduces to establishing the embedding theorem ~ , ~ - L ~ ( U , w ) ~ where the measure v is defined by (3.9). The embedding theorem is derived from (3.8) in exactly the same way as for p = 2 in Sec. I. Here we use Lemmas 2.4 and 2.5 from Sec. 2. III. We prove the necessity of (3.8) for p < 2. Suppose that a ~ M ~ . Then by Property 4 ~ a~/~QV~(T) We use Hirschman's interpolation inequality ([28], see also [29])

Let ] ~ ~'~ (T) ~ L ~ , then

Consequently,

'f

I a (if) / (;:-) _ a (;) ~ (;~ I ,

It follows from this inequality and (3.1u) . T

~ ~,+..

~ ~_

I d~ 1 <.

that

If/II~:~.~ II/II~-+ II/ll~z~./.(.~} <.. II/[Iz;:.~ II/1f~- 9

T

Let e be an arbitrary closed subset of T. f0 ~ ~ ( T ) , satisfying

By

~

-capacity theory ([17, 18]) there is an

1o ($) ~ 1, ; ~ e; Itlo if_~(T> = cap (e); 11fo llz~ ~. c = c (p, o~). Setting f = f0 we obtain .

T

]~--~-~ls

d.:~.c.cape.[ol

(~

(3.11)

To complete Part III of the proof we need the following lemma.

LENNA 3.1o

For

aEH 2 the following

inequality

holds:

1

Yla'

(r~)12(i

--

r):-a~dr
O

Proof.

:t

follows

' d~~ I ( ~ T ) "

T

from Cauchy's

theorem that

j:.l~(:'-")t.."

I o' (.:> I< We set g(t) = ]a(~e-"l--a(~)]

.

d,o

Then

1

1

~'a'(r~)[2(i--r)l-=t~dr<.c~ ( 0

0

+eyo1 ( ~Itl>: - r )

g(t)

~

g($)dtte(J--r)-S-=Pdr +

Itl
]

le't--~l-'dt )~(i_r):_~ v ,

d r = A l +

_A2.

By applying the Cauchy--Bunyakowski inequality, we obtain

Ax
Similarly,

23.0

--g

). 0

The lemma is proved.

0

IriSh(I-r)

r) :-~p

dr ~ c j" g2 (t) I t 1-1-~ dr,

We continue with the proof of Part III. Let e=Ul~, where {Ij} is a finite set of disjoint arcs of T. It follows from (3.11) and Lemma 3.1 that

f' [ a' (~) I~'(l -- l z [)~-=v dm < us(~j It can be verified

in exactly

3~

the same way as in the proof of Theorem 2 that

la'(z)l~(i--lzi)~-~dm>~ c By combining

c. cap ( U I J .

E

( l - l a h l ) ~-~'.

(3.13)

(3.12) and (3.13) we obtain (3.8).

IV. We turn to the proof of the sufficiency of condition (3.8) for p >~ 2. We first establish several auxiliary assertions. The proof of the next assertion is based on an idea of Hedberg [29]. LEIiMA 3.2.

~>0, qml, fro0, f~L~(R+).

Let

S ] (t)

t -~-~ dt <

Then

] (t) t-~-f~qdt

(1 q- l/I~)

[lI t1~7~~/q.

0

Proof.

It can be assumed that

O
put

0

~5=

] (t) t=:-~dt

II 111~

Then ~

~

o

o

COROLLARY 3.1.

x~

Suppose that a > O, p ) 2, and

Sla'(r~)12(t--r)l-e~dr~c

a ~ H ".

Then

r)-e'dr] .

Ja'(r~)l(t

0

Proof.

Since

a e I I ~, we have la'(z)l<~c(t-lzl)-t

](r) =

By Lem~na 3.2 y la' (r$)I ~ (i -- r) ~-2~

l a' ((1 --

dr <~~ ] (r) r"X-~dr

0

~ c

We put ~ = 2a, q = p / 2 ,

and

r) ~) I r-~, 0 < r < 1.

1 (r) r-X-~qclr

= c

I a' (r~)[(t -- r) - ~

dr|

0

as we required to prove. Furthermore,

let us note that

Ila]llv~=la(O)/(O)! v + !

(!

I (a])'

(r~)l~(i--r)~-~ardr)'/~ld~l<<

~ ] l ] ~ + ~oS
.

We use Corollary 3.1 and the inequality sup I](r~)l~c~f](~) , where Mf is the Hardy--LittleO
wood maximal function

[k3].

Then the proof that ~ M ~

reduces to establishing the in-

equality 1

IM] (~)lVld~l Sla' (r~)t(1 T

--r)-~Vdr~c~/It~<~

(]~

~(T)).

(3.16)

0

211

Let f = Jag, where g ~ L ~ ( T ) , and IIMgHLp ~< cHg[ILp, t h e e s t i m a t e

then llftlp, a = IIglILP and Mf = MJag ~< JaMg.

(3.14)

S inc e

Mg ~ L ~

f o l l o w s from t h e i n e q u a l i t y

1

[ 1] (0i'1 d;I 0J"In' (r~)] (t T By using estimates

for the derivative

-

r)-~v dr ~.~ c I]/.ll~m.

-

(3.15)

of the Blaschke product

(2.6) we obtain

I

I a' (#'~) I (i - r)

dr <. ~ ~

0

Consequently,

(3.15) follows

~- I ~ I I i - G ; P+<<~ "~

in turn from the inequality

'-I]tL flsmi'Xi,_z:

(3.16)

T

We define a measure v(A) ( A ~ U )

by

v (A) =

~.~ (I -- I ah l) 1-a'. ak~A.

Condition

(3.8)

is rewritten as

v(US(4)) ~ c. cap (UL), and the relation

(3.16)

is rewritten

in the form of the embedding

v ,~r .(' (t

We need to prove the implication By Theorem G the imbedding

(3.17)

I ,~ I )~v

-

(3.17)

(3.18)

=~

theorem

v

(3.!8)

(3.18).

is equivalent

to the relation

~(t-I~1) ~p a~ l~Ijll ~; i§ e.

U

where e is an ordinary closed subset of T. and let F Then we rewrite

(3.19)

--

Let Xe denote the characteristic

function of e,

(z)

(3.19) as

; F (~,) dv (L) ~ c.eap (e).

(3.20)

u

We d e f i n e t h e r a d i a l

and n o n t a n g e n t i a l F(~) =

where ~ T , 8>I; F*, that is,

P~(~)={z~U:

maximal f u n c t i o n s

f o r F(X) b y , r e s p e c t i v e l y ,

sup F(r~), F * ( ~ ) = s u p F(L), o
]z-~I<~(l-lzl)}.

It is easy to verify

[23] that F majorizes

F*(~)~cff(~) ( ~ T ) . In the same way as in the proof of Theorem

(3.21)

I (see also

v{~ ~ U :F(~) ~>.t} ~ c . c a p { ~ T : F * ( ~ )

[5]) we deduce from (3.17)

that

>~ t}.

Consequently, 0o

oo

oo

: F(~,) >.t t}dt ~ c ; cap {~ : F* (~)~ t} d t ~ CY cap {~ : F (~)/> t}.dt. u

0

,

0

It remains to show that for any closed subset e =

T

; cap {g: ~" (g)/I. t} dt ~<5.cap (e), 0

212

(3.22)

For what follows it is technically more convenient to go from T to R using the natural mapping ~:T § (--~, ~]. As we have already observed in Remark I, cap (e) is equivalent to the capacity of the set ~(e) generated by the Riesz potential I~ [18]. Until the end of the proof this latter capacity will be denoted by capp,a(e). Let

@ (x) By well-kno~cn estimates

= sup ~>o

[ ~

Xe (v) T~P

( I 9 -- V I + r)~+~v dy ( d = R, x ~ R).

[23, p. 77]

@(x) < cMx~(x)(x~R), where M denotes the Hardy--Littlewood maximal inequality (3.Z2) follows from the relation

operator.

It is easy to see that the required

Ca,p~.~ {x : M~e (x) >t t} dt <<. c.cap~,= (e),

(3.23)

0

where e is an arbitrary compact

set in R.

We need certain results from the theory of nonlinear potentials [17, 18]. (positive) Borel measure on R. ItN nonlinear potential a~, is defined by =

and energy

~

J

R'

IV - -

t

Y

IY

Let ~ be a

z I~-~dl~ (z) ~/(~-~)

by

THEOREM H [17, 18]. For any compact set e ~ tribution of e) with the properties

R there is a measure ~ (the capacity dis-

1) supp I~ ~ e, 2)

ix(e)

3) ~ 4)

=

eapp, ~(e), >~ l , x ~ e,

~/~(x) ~< c = c(p, u), x ~ R,

5) ~ = eap~ ~ (e), We define the maximal

function My (y ~> I) by M~] = (Mp)'/L of a compact set e.

Let ~ be the capacity distribution

(X" "i/T ~MT"~l~(x)
Consequently,

to prove

(3.23)

By Theorem H

o~) ( z ~ R ) .

it is enough to establish

the inequality

c

eap~,~ {x : M ~ / ~ (x) ~

t '/'~} dt <

A.cap~ ~ (e),

(3.24)

O

where the constant A does not depend on e and ~, but may depend on c~ We need three more lenm~as. LEMMA 3.3 [31, Proposition 4.4]. H~H = ~(R), and p ~ 2. Then

Let ~ be a Borel measure on R with compact

support,

eapp, ~ {x : ~/"(x) ~ t} ~ ct~-~fipil. LEMMA 3.4.

If p ~ 2

and 0 < ~ < l/p,

then

Proof. By applying the integral form of Minkowski's inequality with an exponent I) ~< I, and the formula for the composition of Riesz kernels [23], we obtain

I/(p --

213

This

proves

LEMMA 3 . 5 .

the

lemma.

The

following

inequality

holds

for

Mrg/"(x) ~-c~(x) 1.

Proof. We set

Since My " i n c r e a s e s

monotonically"

p >

2 and

1 ~

Y <

(P -- 1)/(1

-- ap):

(x ~ R ) .

with

increasing

T, we m a y a s s u m e

that y ~ p --

then

x+r

x_

r>O

We first note

)v}lf?

that

4=

S

[ z - y 1 ~ - ~ (y) @ ~ c ~1 y - z f f - ~ ) (y) dy = c~ ~ (x).

lY-xl>~lx-z I

Consequently, i x+r

.~l/v

~7"rj_ '~ dz ) We e s t i m a t e

~ c~ (x) 9

(3.25)

the integral

& =

f

IV-xl<2lX=Zl

Iz - y I~-~ (y)@.

We set ~ = (p -- 1)/(p -- I) -- s, w h e r e the s u f f i c i e n t l y small n u m b e r e > 0 will be c h o s e n below. By a p p l y i n g H ~ i d e r ' s i n e q u a l i t y with an exponent p -- 1 and the c o m p o s i t i o n formula for Riesz k e r n e l s we obtain

12 ~ C[ X __ Z !(~_2)/(,~_1)_~{ ~ Since

[~

d~(u)

}I/(V-I)

z 1(I-~-~)(~-I)-~"

r ~> Ix -- z l we have

i

2-7

I~dz < c

We apply the integral

Ix - - z II + ? f i - Y ( p - 2 ) / ( p - 1 )

f o r m of M i n k o w s k i ' s

(s) i

~+,

I~dz .(~-1>1~~ c

~ X--r

I u -- z

inequality

5(5 dl~ (u)

that the c o m p o s i t i o n

51~-

9

with an e x p o n e n t

[ x - - z 1811 ~

where 5z = Y(P -- 2)/(p - - 1) - - yB - - 1 a n d B2 = ~ / ( P - - 1) tion f o r m u l a again and then L e m m a 3.4, we find that

Note

i(1_~_~)(p_1)_ ~

-- y(1

"

~1

d~)

-

-- ~ -- 5).

T/(p -- I) >i i:

,

Using

the

composi-

formula

~ I=-~ Iy - ~ l~-~a~ = ~i * - yl=+~-i (~, y ~ m

is a p p l i c a b l e in the sense we need u n d e r the c o n d i t i o n s ~ > 0, ~ > O, ~ + 5 < I; this is fulf i l l e d e v e r y w h e r e above p r o v i d e d that O < g < ~ and 0 < e < I/T -- (I -- ~p)/(p -- I), that is, for s u f f i c i e n t l y small e > O. By c o m b i n i n g

214

(3.25)

and

(3.26)

we o b t a i n

the r e q u i r e d

inequality.

The lemma

is proved.

Proof of (3.24).

@

We choose a n u m b e r ~ ~(p--i, (p--i)/(i--~p)). By Lemmas 3.3 and 3.5

0

0

where the constant A2 does not depend on the measure ~. Since ~ is the capacity distribution of the set e~ by Theorem H, M(e) = capped(e). The inequality (3.24) and with it the sufficiency of the condition in Theorem 4 for p ~ 2 are proved. The theorem is proved. COROLLARY 3,2. For inner functions, the conditions of belonging to p < ~, 0 < ~ < I/p) are equivalent.

7Yf~

and MB~ (I <

Corollary 3.2 follows from Theorems 2 and 4 and the connections between ~ and B~capacities (these connections were established in [30] for p ~ 2, and for p < 2 they follow from recent results of P. Nil'sson, which were kindly communicated to the author by V. G. Maz'ya). LITERATURE CITED i. 2.

3. 4.

5. 6. 7. 8. 9. !0. 11. 12. 13. 14.

15. 16. 17. 18. 19. 20. 21. 22.

V. Go Maz'ya, "Strong capacity estimates for 'fractional' norms, '~ J. Soy. Math., 23~ No. ! (1983). V, G. Maz'ya, "Multipliers in Sobolev spaces, ~' in: Applications of the Methods of the Theory of Functions and Functional Analysis to Problems of Mathematical Physics [in Russian], Novosibirsk (1978), pp. 181-189. V. G. Maztya and T. O. Shaposhnikova, "Multipliers in function spaces with fractional derivatives," Dokl. Akad. Nauk SSSR, 244, No. 5, 1065-1067 (1979). V. G. Maz'ya and T. O. Shaposhnikova, "Multipliers in spaces of differentiable functions," in: Theory of Cubature Formulas and the Application of Functional Analysis to Problems of Mathematical Physics, Tr. Sem. S. L. Sobolev, 1979, No. I, inst. Mat. Sib. Otd. Akad. Nauk SSSR, Novosibirsk (1979), ppo 39-90. D. A. Stegenga, "Multipliers of Dirichlet spaces," Illinois J. Math., 24, No. I, 1i3140 (1980). K. Hoffman, Banach Spaces of Analytic Functions [Russian translation], It, Moscow (1962). i. I. Hirschman Jr., "On multiplier transformations. If," Duke Math. J., 28, No. I, 4546 (1961). i. I. Hirschman Jr., "On multiplier transformations. III," Proe. Am. M~th. Soc., 13~ No. 5, 851-857 (1962). G. D. Taylor, "Multipliers on De," Trans. Am. Math. Soc., 123, No. 2, 229-240 (1966) o R. S. Strichartz, "Multipliers on fractional Sobolev spaces," J. Math. Mech., 16, No. 9, 1031-1060 (1967). D. R. Adams, "On the existence of capacity strong type estimates in Rn, '' Arkiv. Mat., 14, No. I, 125-140 (1976). D. J. Newman and H. S. Shapiro, "The Taylor coefficients of inner functions, ~ Michigan Math. J., 9, No. 2, 249-255 (1962). L. Carleson, On a class of Meromorphic Functions and Its Associated Exceptional Sets, Thesis, University of Upsala (1950). V. G. Maz'ya, "Summability with respect to an arbitrary measure of functions from Sobolev--Slobodetskii spaces," Investigations on Linear Operators and the Theory of Functions~ IX, Zap. Nauchn. Sere. Leningr. Otd. Mat. Inst. Steklov (LOMI), 92, 192-202 (1979). P. Ahern and D. N. Clark e "On inner functions with BP-derivative--s~" Michigan Math. J., 23, No. 1, 107-118 (1976). P. Ahern, "The mean modulus and the derivatives of inner functions, ~' Indiana Univ. Math. J., 28, No. 2, 311-348 (1979). Yu. G. Reshetnyak, "On the concept of capacity in the theory of functions with generalized derivatives," Sib. Mat. Zh., 10, No. 5, 1109-1138 (1969). V. G. Maz'ya and V. P. Khavin, "No~-inear potential theory, '~ Usp. Mat. Nauk, 27~ No. 6, 67-138 (1972). I. E. Verbitskii, "Inner functions as multipliers on the spaces Da, ~' Funkts. Anal. Prilozhen., 16, No. 3, 48-49 (1982). I. E. Ver---bitskii, "On multipliers in the spaces ~P," Funkts. Anal. Prilozhen., 14, No. 3, 67-68 (1980). I. P. Natanson, Theory of Functions of a Real Variable [in Russian], Nauka, Moscow (1974). N. K. Nikol'skii, Lectures on the Shift Operator [in Russian], Nauka, Moscow (1980).

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~-STABILITY OF CLASSES OF MAPPINGS AND SYSTEMS OF LINEAR PARTIAL DIFFERENTIAL EQUATIONS N. S. Dairbekov and A. P. Kopylov

UDC 517.54,517.55,517.95

INTRODUCTION In a recent paper [I], the second of the authors of this paper suggested the concept of G-stability (in the C-norm) of classes of mappings, going back to the Lavrent'ev--Belinskii-Reshetnyak theory of stability of conformal maps in the plane and multidimensional real spaces, and then, starting from this concept, in [2, 3] he constructed the foundations of the theory of G-stability of classes of multidimensional holomorphic mappings. In [4], Bezrukova established that the class of solutions of a Moisil--Teodoresco system is also g-stable, while it turned out that the scheme and method of investigation of the stability of classes of multidimensional mappings given in [2] also extend to the case of solutions of the latter system. In the present paper we continue the investigation of ~-stability of classes of mappings. Its basic result is the assertion that the investigation into the ~-stability of the class of solutions of a system of linear partial differential equations with constant coefficients reduces to the solution of this problem for a first-order such system, having the following special form: M

9

while

the

class

of

solutions

of t h e

latter

Og s

system is

stable

if

and only

if

this

system is

elliptic in the sense of the definition of [5]. One should note that throughout the present paper we use the notation and concepts of []-3], assuming that the reader will turn to the latter for information. I.

Systems of Partial Differential Equations and g-Stability of

Classes of Mappings.

Direction of Investigations

In [I] the following is called the main problem of the theory of ~-stability of classes of mappings. Let ~ and m be arbitrary natural numbers, l~~ a n d R m , respectively, be n-and ~ - d i m e n s i o n a l real arithmetic Euclidean space. We consider the class ~----~=m=Ig.~,A~R= - + R m} of mappings g : A c R ~ - + R m of the domain A of the space R = with values in the space Novosibirsk. Translated from Sibirskii Matematicheskii Zhurnal, Vol. 26, No. 2, pp. 7390, March-April, 1985. Original article submitted April 25, 1983.

216

0037-4466/85/2602-0216509.50

9 1985 Plenum Publishing Corporation