Removability of the wave singularities in the plane

Aequationes Mathematicae 47 (1994) 123--142 0001-9054/94/030123-20 $1.50 + 0.20/0 University of Waterloo © 1994 Birkh/iuser Verlag, Basel

Research Papers

Removability of the wave singularities in the plane

M I R O S L A V CHLEB~K A N D JOSEF KR.AL

Summary. By a measure function we mean a nondecreasing f : [0, oo[ --, [0, ~ [ which is continuous on the right; a f t is the corresponding Hausdorff measure. The total variation of a g: 1--* I~ on an interval 1 c O~ will be denoted by var{g; 1}. Given an open G c R 2 and measure functions f l , f z , we denote by aff(f t , f2; G) the class of all continuous u: G ~ R such that, for each rectangle [a, b] x [c, d] c G, the following conditions hold uniformly with respect to x e [a, b] and y E I c, d 1 as h ~, 0:

[u(x + h, y) - u(x, Y)I + var{u(x + h, .) - u(x, -); [c, dl} = (9(f, (h)),

[u(x, y + h) - u(x, y)[ + var{u(., y + h) - u(., y); [a, b]} = C(fz(h)).

Let #~(G) be the class of all continuous u: G --, ~ such that, for each (Xo, Yo) -- z0 e G, there is a square Q(zo) c G of centre z 0 for which

u(x + h, y + h) + u(x, y) = u(x + h, y) + u(x, y + h)

whenever (x, y) ~ Q(zo) and (x + h, y + h) ~ Q(zo). A relatively closed F c G is termed #~-removable for ,,~(f~ ,fz; G) provided that

(u ~ ag'( f I ,fz; G), bllG\ F ~ U/"(G\,F)) ~ gl ~ ~ ( a ) .

If n~: (x, y) ~ x and nz: (x, y) ~ y denote the projections on the coordinate axes, then the following theorem holds.

THEOREM. A relatively closed F c G is ~l¢'-removabte for W ( f l , f z ; G) i f (and also only i f provided f l (0) = 0 =f2(0)) F can be decomposed in the form F = F t u F e with af ] ' (n I F I ) = 0 = aff2(Tz2F2).

Introduction

W i t h e a c h f i n i t e c o m p a c t l y s u p p o r t e d B o r e l m e a s u r e p in t h e p l a n e ~2 w e

a s s o c i a t e t h e f u n c t i o n

v ~ : (x , y ) --, ~(1 - ~ , x[ × ] - oo, y [ ) , (1 )

AMS (1980) subject classification: Primary 39B40, 35L05; Secondary 31A15, 28A12.

Manuscript received December 13, 1990 and, in final form, October 16, 1991 (Note of Editorial Office: The proof sheets o f this paper went lost in the mail between the authors and the Editorial Office. They were reconstructed in October, 1993).

123

124 MIROSLAV CHLEBIK AND JOSEF KRAL AEQ. MATH.

arising as a convolution of It with the indicator function Z of the quadrant ]0, ~ [ x ]0, ~[ . In the sense of distribution theory, X satisfies the equation

O2Z

Ox c~y

where 6 is the Dirac unit measure concentrated at the origin; consequently, V# satisfies the equation

Ox 0y - It"

VIt will be called the wave potential of /~. (Note that the differential operator [] = ~2/Ox @ is just the wave operator in the coordinate system whose axes

coincide with the characteristic ' lines; using the change of variables = x + y , ~t - - x - y one may transform [] into the usual form 02/0~ 2 - ~2/0/~2 )

In the first part of the present paper we examine sets F c R 2 containing support of a nontrivial measure # such that its wave potential Vp satisfies on E2 a general condition of the H61der type. (Note that continuous wave potentials have been investigated in [9].) Denoting by

a, : (x , y ) ~ x

and

a2: (x, y) ~ y

the projections into the coordinate axes, we may quote as an illustration the following assertion, where ~ ~ ]0, 1], fl E ]0, 1], and where ~ is used to denote the Hausdorff measure of dimension 7. (The reader will see that 1 is a Corollary to Theorem 5 and to the last Remark in Part I.)

1. The following & a necessary and sufficient condition for an F~-set F c ~2 to

contain the support o f a non-trivial measure It whose wave potential Vg =- u satisfies,

for suitable q ~ ]0, ~ [ , the H6lder conditions

lu(x + h, y) - u(x, Y)I ~ qlhl ~,

lu(x, y + h) - u(y , Y)I ~ qlhl p,

(2)

(3)

whenever (x, y) E ~2 and h ~ R:

Vol. 47, 1994 Removability of the wave singularities in the plane 125

The set F cannot be expressed in the form

F = F1 uF2 (4)

in such a way that

Jg~(nt F~) + ~ut~t~(nzF2) = O. (5)

In the second part of this paper results concerning wave potentials are used as a tool for obtaining characterization of []-removable singularities for certain classes of functions. (Note that the relations for [] appear to be less symmetric than those for the Laplacian A discussed in [2].) In accordance with [7] we adopt the following terminology.

Let G c ~2 be an open set. Given a class ~ ' (G) of locally integrable functions in G, a relatively closed subset F in G will be termed []-removable for SC(G) if each u e off. (G) satisfying (in the sense of distribution theory) the equation E]u = 0 on G \ F must automatically satisfy the same equation on the whole G.

If ~,/~ and G have the same meaning as above, we denote by .~ff~,/~(G) the class of all u: G ~ R enjoying the following property.

For each rectangle Q = [a, b] x [c, d] ~ G there is a q e ]0, ~ [ such that (2) holds for x, x + h ~ [a, b] whenever y ~ [c, d] and, for each couple of points y, y + h E [c, d], the inequality (3) is valid whenever x ~ [a, b] and

L ]{u(x2,, y + h) - u(x2,, y)} -- {u(x2,__,, y + h) - u(xz~ ,, Y)}I <~ qJhf i = l

(3*)

w h e n e v e r a ~ x l < x ~ - . ' ~ x 2 , l < X z n ~ b . The following assertion may now serve as an illustration of results concerning

El-removability of singularities obtained in the second part of the paper. (The reader will see that 2 is a Corollary to Theorem 10.)

2. A relatively closed subset F in G is D-removable jbr J{'~I~(G) if and only if it can

be decomposed in the form (4), where the sets F1, F2 satisfy (5).

This assertion remains valid if, in the definition of ~f('~/~(G), instead of (3*) the

inequality

I{ u(x + h, y2,) - u(x, y2, ) } - { u (x + h, y2, - , ) - Y2j - , ) }l q lh [o j = l

(2*)

126 MIROSLAV CHLEB|K AND JOSEF KRAL AEQ. MATH.

is required whenever x, x + h ~ [a, b] and c <~ y , < Y2 <<." " " <<, Y2m - I < Y2m ~ d or if both (2*) and (3*) are assumed simultaneously with (2), (3). (Note that sets of []-removable singularities for continuous functions have been characterized in [4].)

Part I

3. NOTATION. The support of a measure/ t in ~2 will be denoted by spt p. By a measure function we mean a non-decreasing function f : [0, @[ ~ [0, c~[ which is continuous on the right on [0, ~ [ and strictly positive on ]0, oo[. The term interval is used to denote any (possibly void) connected subset of g~. The diameter of M c I~ will be denoted by diam M.

Given a measure function f and a bounded interval L we put

= ) ' 0 i f I = ~ f ( I > [ f (d iam I) if I ~ ~ .

For E c R and e > 0 we define

~ , r (E) = i n f ~ f ( I n >, n

where the infimum is taken over all sequences of intervals {In } such that E c Un In and diam I, ~ e whenever In ~: ~ .

The (outer) Hausdorff measure of E corresponding to the measure function f is defined by

~,V~S(E) = lim ~¢~(E) e~0

(compare chap. 2 in [8]). If there is an ct E [0, oo[ such that f ( t ) = t ° for 0 < t < then we write Jf~(-) instead of ~ I ( . ) .

4. LEMMA. Suppose that fk (k = 1, 2) are measure functions and F c R 2 is a set

which cannot be expressed in the f o rm (4) with Fl , F2 satisfying

~fYk(rtkFk) = 0, k = 1, 2. (6)

Then there is a 6 > 0 such that, f o r any two sequences o f bounded intervals {I k }~N

satisfying


the inequality

2

Z Efk(I~>>~ 6 k = l n

is valid.

Proof If there were no such 6 > 0 we could choose, for any couple of positive integers l, m, two sequences of bounded intervals I~(l, m) (k = 1, 2; n ~ t~) such that

and, simultaneously,

: 1 Y~ ~L<I~(t, m)> < - 2% k = l n m

Writing

we get from (7)

F c {el(m) × ~ } U { ~ × P2(m)},

It follows from (8) that

1 m ) > < - ,

l = l n m

and, in view of the inclusions

P'(m) = U I~(l, m), l ~ ~, n

we see that

l ,m ~ . (8)

P2(m) = ~) U 12( 1, m) (9) I = 1 n

m ~ 1 . (10)

m ~ (11)

• ,~f~(P'(m)) = O, m ~ ~l. (12)

128 MtROSLAV CHLEBiK AND JOSEF KRAL

Letting

p l = 0 P'(m), p 2 = ~ p2(m ) rn=l m=l

we have by (10) again

F = ( P 1 × R ) w ( ~ x pz),

while (11), (12) tell us that

~yff,(pk) = 0, k = 1, 2.

Defining

FI=(pIx~)~F, Fz=(RxPe)g~F

AEQ. MATH.

we observe that (4), (6) are valid, which contradicts our assumptions concerning F.

5. THEOREM. Suppose that f , , f 2 are measure functions and F c R 2 is an F~ -set

which cannot be expressed in the form (4) with Fi, F2 satisfying (6). Then there is a

finite non-trivial Borel measure p with a compact support contained in F such that its

wave potential If# satisfies the conditions

[ V/4x + h, y) - Vt4x, y)[ ~ A (h), (13)

IV#(x, y + h) - Vt~(x, Y)I <~A(h), (14)

for each (x, y) E ~2 and any h > O.

Proof Without loss of generality we may suppose that F is compact. For the sake of simplicity we assume that F c [0, 1] × [0, 1]. For each m ~ I~ we divide [0, 1] into the intervals

J,." = [(i - 1)2-% i 2 - ' ]

and define

K ~ = J ~ × J ~ , l ~ i , j ~ 2 m.

Vol. 47, 1 9 9 4 Removability of the wave singularities in the plane 129

Let

f m = {(i,j); K~ ~ F # ~} ;

c l e a r l y , f m # ~ . We are going to construct a suitable measure #m supported by [0, 1] x [0, 1] such that its density with respect to the two-dimensional Lebesgue measure remains constant in each K~. Defining

i2

pro . = U j m , l ~ i l ~ i 2 < ~ 2 m ' tit 2 i= i I

we shall require the following conditions (15)-(19) to be fulfilled by the quantities x~ = U,.(K~):

x,7 ~ o ,

x,~ = 0

i 2 2 m

2 Z i = i l j = l

2m J2

E E i = l j = j l

1 ~ i , j 62m;

whenever (i,j) ¢ j-m;

_< m x ~ f l < P i l i 2 > , l ~ i l ~ i 2 < . 2 m ;

m x~ ~A<Pj~jz>, 1 ~Jl -~J2 <<. 2m;

If (io,Jo) ~ j-,n then either there are ix, 1"2 with 1 << i~ <~ io ~ i2 <~ 2 m such that equality holds in (17) or else there are j l , j2 with 1 <~j~ ~Jo -~J2 <~ 2m such that the sign of equality holds in (18).

(15)

(16)

(17)

(18)

(19)

It follows from Lemma 6 below that one can indeed choose x~ satisfying (15)-(19), so that the measure #m is well defined. Putting

F " = U {K~; (i,j) ~ ~-'-m}

we have by (16)

spt ],l m C Fm. (20)

As a consequence of (17), (18) we get

pm [o, 1]) ~.f,<e,,,2>, •m( ili2 X ~ m

#m([O, l] x P~&) -< m

1 ~ i , ~ i 2 ~ 2 m,

1 ~ j , <~j2<2 m.

(21)

(22)

130 MIROSLAV CHLEBIK ANt) JOSEF KRtkL AEQ. MATH

With the notation

m ° m p t n L*amt = {P~,~2' 1 ~< it ~ i2 ~ 2 m, #. . (P,~: x [0. 1]) = A < i,~:>}.

£,a,~2 = {PJ.s2; I ~ j , ~J2 ~ 2% ~m([O, 1] X P;7,2) =/2(Pi'~s2>}

the condition (19) can be reformulated as follows:

Fix now k ( = 1 or 2) and consider the system £,¢''~; its elements are intervals arising as unions of the intervals J~. If some J~ is contained in three different intervals of ~,~k then at least one of these intervals is covered by the union of the remaining two and we may thus leave it out of £p,,k without changing U ~ ' ~ " After a finite number of such modifications we arrive at a subsystem

enjoying the following two properties:

~9~mk = U £a.,k, k = 1, 2; (24)

each jm is contained in at most two intervals of the system ~5 #vmk (k = 1, 2). (25)

In view of (23), (24) we have also

F~Fmc{(USPm')× [0, l]}k.){[O, 1] × (U~.<pra2)}. ( 2 6 )

It follows from (24)-(26) that each square K~ with ( i , j ) e Y " is contained in at least one and in at most four of the sets

1 7 ' x [ O , 1] ( l ~ n ~ < r , ) , [ O , l ] x I 7 2 ( l ~ n ~ < r = ) .

Hence we deduce

1

1

4

1] x [0, 1]) = ~ / x . ( K ~ ) i , j

Vol. 47, 1994

Defining

I~ l =~3" f o r n > r l , I m 2 = ~ f o r n > r 2 ,

we get for any fixed m ~ N two sequences of intervals

{lmk}.~ (k = 1, 2)

such that

Removability of the wave singularities in the plane 131

(27)

According to Lemma 4 above there is a 6 > 0 (depending on F,f~,f2 only and independent of the particular choice of the sequence (27)) such that

2 ~ fk (Imk ) ~ 6, m e N.

k = i n

Combining this with (21), (22), we get for each m ~ N

1 ~fi ~< #m([0, 1] × [0, 1]) = /~m(R 2) ~ min{f~(1),f2(1)}. (28)

Consider now an arbitrary x ~ ~ and h ~> 2 - ' . Then [x, x +h] n[0, 1] can be covered by the union of suitable three intervals Pi"~,2 with diam P~i2 ~< h. Applying (21) we obtain

/~,,([x, x + h] x ~) =/~m({[x, x + h] n[0 , 1]} × [0, 1]) ~ 3f~(h).

Similar estimates can be derived for #m(~ × [Y, Y + h]), y ~ ~. We have thus

((x, y) ~ ~2, h t>i 2 - ' ) = #,,([x, x + h] x I~) ~ 3f,(h), (29)

~m(R X [y, y +h]) ~ 3f2(h).

It follows from (28) that there is a vaguely convergent subsequence of {/~,,},,~N whose vague limit ~7 satisfies/7(R 2)//> ¼ft. Besides that, (20) implies that

spt fi c ~ Fm= F r n

132 MIROSLAV CHLEB|K AND JOSEF K R ~ L AEQ. MATH.

and (29) implies for (x, y) e ~2 and any h > 0 the inequalities

fi(]x, x + h[ x ~) ~ limsup/~m(]x, x + h[ x ~) ~ 3fl (h), m ~ c ~

fi(R x ]y, y + hD ~ limsup #m(~ x ]y, y + hD ~ 3f2(h). t n ~ o o

Since f~ ,f2 are continuous on the right, it is easy to see that the wave potential of l ~ /a = 3/1 obeys the inequalities (13), (14), so that # has all the required properties.

The following auxiliary result was needed in the above proof.

6. LEMMA. Let m e ~, (ZJ # 9-- ~ {(i,j); 1 .~ i , j <~ 2 m} and suppose that positive real numbers ci~i2 (1 ~ i l ~i2~<2"), dj-lj 2 (1 ~Jl ~j2 ~ 2 " ) have been arbitrarily prescribed. Then there exist xq ~ R (1 ~ i , j <~ 2") satisfying the following conditions (15")-(19") :

xi j~O, l ~ i , j ~ 2 m ; (15")

x e = 0 whenever (i,j) ¢ J - ; (16")

i 2 2 m

E Z X i j ~ C i l i 2 ' l ~ i l < ~ i 2 < ~ 2 m , ( 1 7 " ) i = i l ] = l

2m J2

~ x,j<~cla,j 2, l ~ j , ~ j 2 ~ 2 m ; (18") i = 1 J=Jl

I f (io,Jo) ~ 23r, then either there are il, i2 with 1 << il <~ io <~ i2 <~ 2 m such that equality holds in (17") or else there are Jl,J2 with 1 <~Jl ~Jo ~ j z ~ 2 " such that the sign of equality holds in (18"). (19")

Proof Arranging all the couples (i,j) (1 ~ i , j <~ 2 m) into a simple sequence, we may identify the set of all matrices

x = ( x , j ) 1 ~ , ~ 2 ~

I ~<j~2 m

with the space ~ 4m. The set T formed by those X = (x,;) whose entries satisfy (15*) -(18*) is clearly non-void and compact in R ~" . Defining

2 m 2 m

T(X) = Z )-', x,j (30) i = l j = l


we obtain a continuous function f o n []~ 4m which must attain its maximal value on T at a suitable X ~ T. We assert that the point X chosen in this way satisfies also the condition (19"). Indeed, let (io,Jo) E 3 and suppose that for X the following implications hold:

t2 2m

l<~il<~io~iz <~2m~ ~ E Xij<Cili2' (31) i = t l j = 1

2m J2

l <~j, ~Jo~Jz <~ 2m::~ 2 2 Xij<4",j=. (32) i = 1 J = J l

Choose e > 0 and define . ( = (20) by

= ~xij for (i,j) 4= (io,A), x o + e for (i,j) = (io,Jo).

I f e > 0 has been chosen sufficiently small then, according to (31) and (32), all the conditions (15" ) - (18" ) remain in force with x,j replaced by xij, so that )? e T. On the other hand, (30) yields f O ?) = f ( X ) + e > f ( X ) which shows that f cannot attain its maximal value on T at X.

REMARK. We shall see later (in the proof of Theorem 10) that no set F fulfilling (4) and (6) can contain support of a non-trivial Borel measure # whose wave-poten-

tial VO satisfies (13), (14) provided f 1(0) = 0 =f2(0).

Part II

7. NOTATION. Given an open set G c R 2 we shall denote by "#r(G) the class of

all the functions u: G ~ R with the following property.

For each point (Xo, Yo) = Zo ~ G there exists a square Q(zo, p) = [ x o - p , Xo + p] × [yo - p, yo + p] of side-length 2p > 0 (depending on Zo) such that

u(x + h, y + h) + u(x, y) = u(x + h, y) + u(x, y + h),

whenever (x, y) ~ Q(zo, p) and (x + h, y + h) E Q(zo, p). re(G) will stand for the class of all finite continuous real-valued functions on G. With each function u: G ~ R we shall associate the function w, of an interval in

G defined for each (possibly degenerate closed) rectangle Q = [ct,/~] × [7, 6] c G by

w~tQ) = u(~, 6) - u(~, 6) - u(~, ~,) + u(~, ~,).

134 MIROSLAV CHLEB|K AND JOSEF KRAL AIEQ. MATH.

8. REMARK. If G = R 2 is open and u ~ cg(G), then one can verify without difficulty that the following statements (i)-(i i i) are mutually equivalent (compare [1], [6], [3], [4]):

(i) u e ~¢r(G); (ii) wu(Q) = 0 for each rectangle Q = G;

(iii) Du = 0 on G in the sense of distribution theory.

We adopt the following notation. If f is a real-valued function defined on an interval J c ~, then var{f; J} will denote the total variation o f f on J.

9. LEMMA. Let G ~ ~2 be an open set, u E ~(G) and let F be a relatively closed subset o f G such that the restriction o f u to G k F belongs to 3q~(GkF).

Let

Q = [ a , b ] x [c,d] c G

and suppose that f~ ,f2 are measure functions such that u fulfils on Q at least one of the following conditions ( ~ l ) - (~3):

(~1) Uniformly, with respect to x ~ [a, b] and y ~ [c, d],

lu(x + h, y ) - u(x, Y) I = ~9(f, ([h [)) a s h ~ O,

var{u(-, y + h) - u(., y); [a, b] ) = (9(f2 (Ih I)) as h ~ O.

(33)

(34)

(,~i~2) Uniformly, with respect to x ~ [a, b] and y ~ [c, d],

lu(x,y + h) -u(x,y)l=O(fz(lh )) as h-~O,

var{u(x + h, .) - u(x, .); [c, d]} = ~ ( f , ( l h ] ) ) as h ~ 0 .

(35)

(36)

( ~ ) The estimates (33), (35) hoM uniformly with respect to (x, y) ~ Q; further-

more, wu (which is an additive function o f a two-dimensional interval in G) has bounded variation on Q.

I f F c3 Q can be decomposed into subsets F~, F 2 such that

:::'(x~ F, ) = O, (6 , )

• ,~f2Ot2Fz) = O, (62)

then

w,(Q) = o. (37)


Proof. Fix an arbi t rary ~ > 0. Using (61) (note also thatf~ ,f2 are cont inuous on the right) we get a sequence of bounded intervals I , = [a, b] relatively open in [a, b] such that

1 ~,F, cUI, , , ~f~ (I,, > < ~ ~.

n n

Since F ~ Q \ ( U . In x N) is a compac t subset o f F2, (62) implies the existence of a finite system J , consisting of relatively open sub-intervals o f [c, d] such that

1 (38)

Then F c ~ Q \ ( ~ x U J ) is a compac t subset o f Un ln x ~ so that there is a finite subsystem ,.¢ c {I, } with

Of course,

1 ~ f ~ ( I > < ~ e , I ~ J . (39)

We are now going to replace each of the systems J , J by a system of dyadic intervals. By a dyadic interval in [a, b] we mean any interval of the fo rm

[a + (i - 1)2-/(b - a), a + i2- t (b - a)] (1 e t~, 1 ~ i ~ 2t). (40)

We note that any two dyadic intervals are either distinct or one o f them is contained in the other. Given an arbi t rary non-void interval I c [a, b], on choosing 1 e I~ with

(b - a) 2 - t ~< d iam I < (b - a) 2 l ÷ 1,

we can cover I by at most three dyadic intervals o f the type (40) with diameters not exceeding d iam I. Replacing first each I ~ J by at most three dyadic intervals o f

136 MIROSLAV CHLEBiK AND JOSEF KRAL AEQ. MATH.

diameters ~ diana I and then consecutively omitting in the resulting system of dyadic intervals all intervals that are contained in another interval of the same system we finally obtain a finite system of non-overlapping dyadic intervals in [a, b]

[XI ' X2], ' " ' ' [X2r--I, X2r] ( 4 1 )

with

f l ( x 2 i - x2 i_ l) < g (42) i = l

such that

U~ ~ 0 tx2;_l,x2a. I = i

We may clearly suppose that

a ~ x l ~x2 ~" • " ~ x 2 , _ , ~x2r ~ b. (43)

In a similar way we replace the system j by a finite system of non-overlapping dyadic intervals in [c, d]

[Yl , Y2], . • • , [Y2s-l , Yzs] ( 4 4 )

with

~'~ f 2 ( Y 2 j - -Y2 j -1 ) < g (45) j = l

such that

U if" C 0 [Y2,--1' Y2j] ' j = l

Again we assume that

C ~ Yl -~ Y2 ~ ' " " ~ Y2s - 1 ~ Y2s ~ d. (46)

We have thus

~ c { 0 , . . i ~ x Ec ~,}~0=1 ,a b, x,~2. y2.j} .7,


Put x0 = a, X2r + 1 = b, Yo = c, Y2~ + ~ = d. Since the interior of any rectangle [x2i, x2~ + ~ ] x [Y2j, Yzj + ~ ] does not intersect F and the restriction of our u ~ C~(G) to G \ F belongs to ~IU(G\F) we easily observe that

Wu([X2i, x2i+l] x[y2j , y2 j+l] )=O, O ~ i ~ r , O < < . j ~ s . (48)

Assume now (~@~). Then there is a constant q ~ ]0, ~ [ (depending only on u and Q) such that

Iw~([x~,_,,x~,] ×[c,d])l<~qf~(xz,-x2,_,), l~i<<.r, (49)

L I Wu([x2'' X 2 i + I ] X [Y2j 1' ~ q A ( Y 2 , - Yz j 1), i = 0

1 ~ i ~ s (50)

Making use of (48), we get from (49), (50), (42), (45), in view of the additivity of Wu

Iw.(O) l j = l i = 0

~ q L f l (xz i - -x2 i_ , ) + q L f z ( Y 2 j - Y z j l) <2qa. i = 1 j = 1

Since e > 0 can be chosen arbitrarily small independently of q, we arrive at (37). By a similar procedure (where the role of the two variables is interchanged) we can deduce (37) from (~2).

Now assume (~3)- Let us agree to denote by [[w~ ][(R) the total variation of w,(.) on any rectangle R c Q. We shall show that our assumptions on F, u imply

1 Iwu(Q)l rlwu II(Q). (51)

According to (33), (35) we can fix a constant q e 10, ~ [ (depending on u, Q only) such that

a ~< ~ ~/~ ~ b =. Iw,([~,/3] x [c, d])l ~ qfl(/~ - a), (52)

c ~ ~ ~ 6 ~ d =~ [wu([a, bl x [7, 6])1 ~< qf2(6 - y). (53)

Supposing the contrary of (51), we fix e > 0 small enough to guarantee that

1 Iwu(a)l- q~ > ~ Ilwu I](Q).

138 MI ROSLAV CHLEB|K AND JOSEF KR,~L AEQ. MATH.

For this choice of e we then choose in the manner described above the intervals (41), (44) such that the relations (42), (45), (47) hold. Then also (48) holds again. Hence we get, by (52), (42),

I~ kl'u(tX2i, l ] X [(', d]) -~- Wu(Q) -- ~ Wu([X2i_ X2i ] X [C, d]) x2++ 17 ~=0 i = 1

i = I

>~ Iwu(a)l - q~ > ~ IIw. II(a)-

In view of (48) we have

,=oj=, w.([x.;,,+, x2,+ 1] × [Ymj-,, Y2j]) > 5 l[w'' II(Q).

In a similar way we obtain from (53), (45)

whence it follows by (48)

x [Y2:- l , Y2j])

Iw,(Q)[- q ~ f2(Y2, -Y2j-1) j = l

~> Iw.(Ol- q~ > ~ IIw./I(Q),

;=] 1 ~, Wu([X2i- I, X2i] × [Y2j, Y2j+ 1]) > 5 tiWu /I(Q).

j=O

Adding (54) and (55) we arrive at

~ Iw.([x=. x2i+ l] X [Y2j--19 yJ)l i = O j = l

+ ~ ~ Iw.([x=,_l, x=A x [y2j, y2j+ 11)1 > IIw. II(a). i = l j = O

Since there are no overlaps in the system formed by all the rectangles

{ [x~ ,_ , , x2,] x [y2j, y~j +, ] }1 . .< ,~ , O~j~s

{[x:,, x2,+ 1] x [Y2i- 1, y2A}0 ~, ~ r , l<~j<~s

(54)

(55)

(56)


the sum on the left-hand side of (56) cannot exceed Ilw, II(Q), which is a contradic- tion justifying (51). The role of Q in (51) can be replaced by any rectangle Q, c Q. Dividing Q into non-overlapping rectangles Ql . . . . . Qp and employing the additivity of [[w, II() we get

1 L wu(Q,) 1 Iwo(O,)l = Ilwo I1(o), t = l t = l

whence it follows, by taking supremum over all subdivisions {QI . . . . . Qp }, that

Hwu rl(Q) ~ 1 2 Ilwu II(a)

This is clearly possible only if [[wu II(Q) = 0 and (37) is established.

10. THEOREM. Suppose that f~,f2 are measure functions and let G c ~2 be an open set. Denote by ~ the class o f all u ~ Cg(G) such that, for each rectangle Q = [a, b] x [c, d] ~ G, at least one o f the conditions (~l)-( t~3) of Lemma 9 is satisfied. Then a relatively closed set F c G is []-removable for ~ i f (and also only i f when f l ( 0 ) = 0 =f2(0)) it can be expressed in the form (4) with F1, F2 satisfying (6).

Proof. Suppose first that F is a relatively closed subset of G which cannot be decomposed into sets Fl, F: satisfying (6). According to Theorem 5 there is a finite non-trivial Borel measure ~ with a compact support contained in F such that its wave potential V# satisfies (13), (14). We are going to verify that u - V/z satisfies, for each rectangle Q = [a, b] x [c, d], all the conditions ( ~ ) - ( ~ 3 ) listed in Lemma 9. Clearly, (13) implies (33). Fix now y ~ [c, d], h > 0 and consider an arbitrary sequence (43). It follows from the definition (1) that

{u(x2,, y + h) - u(x2i, y)} - {u(x2~_ ,, y + h) - u(x2i 1, Y)}

= ,U([X2i-- 1, X2i[ X [y, y -~- h o ~ 0,

[{u(x2,, y + h) - u (x2 , , y)} - {u(x2,_l, Y + h) - u ( x 2 , _ , , y)}[ i=1

p(] -- oo, b[ x [y, y + hD <<. [u(b, y + h) - u(b, y)[ ~f2(h).

Taking supremum over all finite sequences of the form (43) we obtain

var{u(., y + h) - u(., y); [a, b]} ~f2(h)

140 MIROSLAV CHLEB|K AND JOSEF KRAL AEQ. MATH.

and (~1) is verified. The condition (~2) can be verified symmetrically. Finally, for any choice of the sequences (43), (46) we get

Wu([X2i 1 , X2i] X [Y2j 1, Y2j]) = kt([Xei- I, x2i[ x [Y2j- I, Y2/[) //> 0,

~. IWu(tX2,_ 1, X2i ] X [Y2j-1, Y2j])I ~ #([a, b] x [c, d]), i = l j = l

so that (~3) is also valid. Since spt # c F (about spt see Part I, 3 NOTATION), we observe that the restriction of u to G \ F belongs to W~(G\F). On the other hand, p being non-trivial, we have for suitable (xo, Yo) ~ spt p and any p > 0

#([Xo - p, Xo + p[ x [Yo -- P, Yo + pD

= wu([Xo - p , Xo + p] x [yo - p , yo + p]) > o,

which shows that u cannot belong to #~(G). I f f z ( 0 ) = 0 =f2(0), then (13), (14) guarantee the continuity of Vt~ = u and we conclude that F is not []-removable for ~ff.

Conversely, it follows immediately from Lemma 9 and Remark 8 that any relatively closed F c G which can be expressed in the form (4) with FI, F2 satisfying (6) is D-removable for ~¢d. Consequently, it cannot contain support of a non-trivial Borel measure # whose wave-potential V# satisfies (13), (14) provided

A (0) = 0 =A(0) .

REMARK. For f l U ) = t =f2(t) (t e [0, oo[) the above theorem slightly extends results established in [3]. In combination with Theorem 5 it also implies the assertions 1 and 2 stated in the Introduction.

A simplification of the reasonings described in the proof of Lemma 9 permits to prove the following lemma.

11. LEMMA. Let G c R 2 be an open set, u ~ ~(G) and let F be a relatively

closed subset o f G such that the restriction of u to G \ F belongs to ~tCr(G\F). Let

Q =[a,b] x [c,d] c G . (57)

I f f l is a measure function with ~f~(rc I (Fc~Q))= 0 such that (33) holds uniformly with respect to (x, y) ~ Q then (37) is valid. Symmetrically, i f f2 is a measure function with ~,~t~f2Oz2(Fc~Q))=0 such that (35) holds uniformly with respect to ( x , y ) ~ Q then (37) is valid.


12. NOTATION. Let fk (k = 1, 2) be measure functions and suppose that G c R 2 is an open set. We shall denote by ~ ( f l , f 2 ; G) the class of all u ~ Cg(G) such that, for each rectangle (57), both conditions (33) and (35) are satisfied uniformly with respect to (x, y) e Q.

We know from Theorem 10 that the conditions (4), (6) are necessary for a relatively closed F ~ G to be D-removable for )[r(f~,f2; G), provided that f l (0) = 0 = f 2 ( 0 ) . In order to be able to prove that these conditions are also sufficient for D-removability of F we had to impose additional restrictions on the class of functions. It follows from Lemma 11 that, instead of introducing supple- mentary conditions on the class of functions considered, it is possible to impose an additional requirement on the decomposition (4) of F as shown in the following theorem.

13. THEOREM. Let f l ,f2 be measure functions and suppose that G ~ R 2 is an open set. I f F is a relatively closed subset of G for which there exist F~ -sets Fl, F2 satisfying (4) and (6), then F is [l-removable for J~r'(f~,fa; G).

Proof The following assertion based on the Baire theorem follows from results established in [5].

If relatively closed sets Hn c G (n ~ ~) are D-removable for ~ ( f ~ ,f2; G) then also each relatively closed subset F = G for which

F c U H n n

is D-removable for J~f(f~ ,f2; G). It follows from Lemma 11 that each F c G satisfying the assumptions of our

theorem can be expressed as a countable union of relatively closed subsets of G that are D-removable for ~,~((f~ ,f2; G). The above quoted assertion then implies that F

itself must also be EJ-removable for ~ ( f t , f 2 ; G).

REFERENCES

[11 BAKER, J., An analogue to the wave equation and certain related functional equations. Canad. Math. Bull. 12 (1969), 837-846.

[2] CARLESON, L., Selected problems on exceptional sets. D. Van Nostrand Co., New York, London, 1967.

[3] CHLEBIK, M. and KR~I., J., Removable singularities o f the functional wave equation. Z. Anal. Anwendungen 8(6) (1989), 495-500.

[4] DONT, M., Sets o f removable singularities o f an equation. Acta Univ. Carolin.-Math. Phys. 14 (1973), 23 -30.

142 MIROSLAV CHLEB|K AND JOSEF KR,~L AEQ. MATH.

[5] O'FAI~RELL, A. G., Removable sets for classes o f functions. Proc. Amer. Math. Soc. 41 (1973), 282-286.

[6] HARUK], H., On the functional equation f ( x + t, y) + f (x - t, y) = f (x, y + t) + f (x, y - t). Aequa- tiones Math. 5(1970), 118-119.

[7] HARVEY, R. and POLKING, J., Removable singularities o f solutions o f linear partial differential equations. Acta Math. 125 (1970), 39-56.

[8] ROGERS, C. A., Hausdorff measures. Cambridge University Press, Cambridge, 1970. [9] SCt-IULZE, B.-W., Potentiale bei der Weltengleiehung inn •2 und Charakterisierung der Mengen der

Kapazitiit Null. In Etliptische Differentialgleichungen, I (KoUoq., Berlin, 1969), (G. Anger, ed.). Akademie-Verlag, Berlin, 1970, pp. 137-157.

(Jstav aplikovanej matematiky UK, Mlynsk{: dolina, SR-842 15 Bratislava, Slovak Republic.

Matematick~ ~istav A V~R, 2itn{: 25, ~ R - I 1 5 67 Praha 1, Czech Republic.

Removability of the wave singularities in the plane

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