Reduced Hilbert transforms and singular integral …Reduced Hilbert transforms are obtained by restricting the variable x and the domain of integration in (1) to a subset R' (or two

REDUCED HILBERT TRANSFORMS AND SINGULAR

INTEGRAL EQUATIONS

BY

E L I A H U S H A M I R

in Jerusalem, Israel

I n t r o d u c t i o n

The properties of Hilbert transform on the whole real line R

(1)

oO

1 f u(Y)_dy (9~'u) (x) = ~ni J x - y

are quite well known. We mention in particular that ~ can be extended to a

bounded operator in LP(R) and that, being a convolution transform, its

Fourier transform is a multiplication operator. We havein fact

1 (2) W"(~) = - ~- sign o

Reduced Hilbert transforms are obtained by restricting the variable x and the

domain of integration in (1) to a subset R ' (or two different subsets) of R.

The case R' = R+, the positive half-line, is studied here in detail. In this case

the theory becomes quite simple after we prove in Section 1 suitable diagonali-

zation formulas. The case of a finite interval is treated briefly in the last

section.

We shall be interested in solving the following system of singular integral

equation

O0

f ~b(y) d (3) A~b + B ~ y = ~O(x), x > 0

o

where q~ and ~ are m dimensional vector functions, A and B are constant

m x m matrices.

277

278 ELIAHU SHAMIR

The system (3) will be studied in the framework of two scales of

spaces, W ~'p and H~'~, s_~ 0. For integral s, both spaces coincide with the

space of functions with derivatives up to order s in L p. For fractional s, they are

obtained by (different) interpolation methods.

Our main result is that, except for a t most m values of s (rood 1), the system

(3) admits a solution for every r in a closed, finite codimensional subspace.

The codimension increases with s, more precisely, it jumps upward at the

exceptional values of s, and the total jump is m for a unit increase in s. The

results are first obtained for s = tr, 0 _< tr < 1, and are then easily extended to

s > 1. Applications to the study of mixed boundary value problems for elliptic

partial differential equations were given in [14], [15].

Reduced Hilbert transformations in L2 were studied by Koppelman-Pincus

[5] and J. Schwartz [13]. It is Schwartz's diagonalization formula for p = 2

which we extend to 1 < p < ~ . An L p approach was given by Widom 1-17],

who obtained some of our results (the L p case for a scalar equation, m = 1).

Widom's methods are different and considerably more complicated, and their

extension to the vectorial case, m > 1, seems difficult,

This work (except for Section 6) is part of a P h . D . Thesis prepared at the

Hebrew University under the direction of Professor S. Agmon, to whcm I wish

to express my deep gratitude for valuable suggestions and encouragement.

N o t a t i o n s .

R is the real line. R+(resp. R_) is the ray x > 0 (resp. x < 0), and

Y+ (resp. Y_) its characteristic function.

C~(f~) - the class of infinitely differentiable functions with compact support

in the domain ft.

We always assume that 1 < p < o o and q=p/(p-1) ,(1/p+l/q=l) . Unless otherwise stated, tr will satisfy 0 < ~r < 1.

Fourier transformation are denoted by .~, ~--1 and also

( u " ) ( ~ ) = ( ~ u ) ( r (v * ) ( x ) = ( 9 - ~ v ) ( x ) .

Norms of functions in a domain f2 will be denoted by l[ u , ~ I[ with suitable

REDUCED HILBERT TRANSFORMS AND SINGULAR INTEGRAL EQUATIONS 279

indices. S~mi-norms will be denoted by lu, f~]. The notation II u II1 ~ I1 u 112 is

used for equivalence between the norms, i.e.

II u IlL ~ K II u 112, II u 112 ~ g II u II-

where K is independent of u. The phrase " K independent of u " will usually

be omitted in the statement of such estimates.

w D i a g o n a l i z a t i o n formulas . Hilbert transforms on R+ are defined by

(1.1) (H+ c~)(x) = lim 1 : ~k(Y) dy, x > O , ~ o ~ i x + ie- - y

o

(Hq~)(x) = l i m : . f ~(Y) dy, x > O. ~ o Z~I X -- y

I x - y l > e y>O

By Plemelj's formulas we have the relations

1 (1.2) H• = ~ - ~ T- H~b

and by M. Riesz's result, H a: and H can be extended to bounded operators

in L~ 1 < p < oo.

For a fixed p and r let

(1.3) (Wc~)(t) = et/V~b(e'), - oo < t < oo

(1.4) Mq~ = ~'U~b = (d/o~b(et))'.

Note that U is an isometry between LP(R+) and LP(R). The inverse of M is

given by

(1.5) ( M - I f ) ( x ) = ( U - l ~ - l f ) ( x ) = x - l / v ( , ~ - l f ) ( l o g x ) , x > O.

By writing explicitely by Fourier transform and using the substitutions x = e t

t = log x we obtain

280 ELIAHU SHAMIR

OD

1 / q t

j r t /q- , ,dx, oo < r < oo (l .6) (M~b)(0 = x/2--~ o

_ 1 f x l ' - l"Pf(z)dz, x > O. (1.7) ( M - ~ f ) ( x ) ~/~n _~

If the Fourier inversion formula holds for U~, then M - IM~ = ct. This is the

case i f ~ ( x ) x " e C ~ ( R + ) , 0 ~ r < I / p , and ~ satisfies a very mild growth

condit ion at + ~ (for instance ~(x) ~ x:, y < 0); for then U~ is exponential ly

decreasing at + ~ and Ma = ~rU~ is also fastly decreasing. In any case, if

ct = M - ~ M ~ then ct(x) has the following representat ion

(1.8) :~(x) = M - t M o r - 1 ? x i~-1/~ a(Odz , a = M~.

,,/ 2 rr _ o~ d

Theorem 1.1. Let - 1 + l / p < a < l / p and let a = M~ be f a s t l y decre-

asing (a e 5a). Then the transformation M diagonalizes the operators

(1.9) ( H~ ct) (x) = x - an + x~ ~(x)

and we have

(1.10) MH+ ct = p~,M~

M H ~ = (1 + Po)Met where

(I.11) p,(z) = ] - e x p ( 2 n i ( a - l / p + i z ) ) - 1 ] - 1.

P r o o f . Using (1.8) we obtain

(II~+e)(x) x - " H +x ~ = lim x-~ f ~ 1 = dtl" - - ot(z)tf '- I/i, dz. ~ o ~ i x + ie - tl x/2- ~

0 - - ~


Since a ( r ) e Lx (R), the inner integral is estimated by Kq-l / j , , and the integrand

with respect to q is estimated by K q ' - l / P / [ x + i e - q l " This expression

belongs to L~(R+), since - 1 < a - i / p < O. We can therefore change the

order of integration

(1.12)

where

-I-no

(H~+ct)(x) = lim x-~ f a(r) t ,d 'r o _

cO

= 1 f r f - l /P+i ' dq. I , 2hi x -I- ie - q

To compute I,, we integrate - z ~-~/P+g~ " ( z - i e - x ) -~ over a path con-

sisting of two circles of radii r and R around the origin and connected above

and below the positive semi axis. (We choose the branch of logz which approaches log lxl on the upper positive axis). By Cauchy's formula, this

complex integral is - (x + ie)"-a/P+~'. Letting r--* 0 and R ~ ~ , the integrals

over I z ] = r and I z ] = R tend to zero, due to the assumption on a. The in-

tegral on the upper positive axis is I,, and the lower one is I~exp2ni(tr - 1/P

+ it). Hence

I~ = (x + ie)~'-l/P+~'[exp(2ni(a - 1/p + i~)) - 1] -1

For each x > 0, l It ] is easily estimated by a constant (independent of z and e)

times x "-alp, and since - 1 < a - 1/p < 0, we can shift the limit in (1.12)

inside the integral and obtain, using (1. l 1)

cO

H,+(x) = --x-~__ f x ' - l / P + " p,,('r)a('r)d'r = M-~(p~ �9 a) = M-l(poMct). \ /2~ _ ~

Since - 1 < ~ - l / p < 0, p,(z) is a well behaved function and poMot ~5 a, like

M~. We can thus use the inversion formula and obtain MH~ + = p, Mot, which

is the desired result for H~. From (1.2) we obtain the result for H~.

282 ELIAHU SHAMIR

R e m a r k 1.1. Similar formulas can be obtained for the transforms-

I xl-'slx and x - ' J 2 Ix I ~ where Jt (resp. J2) is the Hilbert transform from

R+ to R_ (resp. f rom R_ to R+). The integrals are non-singular now, hence

the three variants J, J • coincide. In the definition of M one has only to

change U (or U -1) so that it will be an isometry between LP(R_) and LP(R).

For Ix[ - ~ J l x q one obtains the diagonalization factor

p*~(z) = [exp(2xi(o" -- l / p + iz))]-1, exp(ni(o" - I /p + iv)).

R e m a r k 1.2. For p = 2 , o~, hence also M , is an isometry and the formulas

ab3ve yield in fact the spectral decompositions of these bounded symmetric

operators (cf. [5], [13]).

R e m a r k 1.3. By using the fact that

oo

f x - l c ~ ( x ) d x = 0 ~ H + ( x - l q ~ ( x ) ) = x - X H • )

0

we can obtain formulas (1.10) in an extended range of a. For instance, if

1/p < a < 1 + I / p and f'~ x ' - 1 ~ ( x ) d x = 0 then

x - ' ~ - 1 'H + x ~- x a(x) = x -~'H + x"a(x) .

Since a - 1 is in the original range, we obtain formula (1.10) for a upon

noticing that p,_ ~ = p,.

w E s t i m a t e s in t h e E ~'p n o r m

Using the identities 2H = H + + H - and I = H - - H + (I is the identity

operator), the system (3) can be written in the form

(2.1) Ac~ =- ( C H + + D H - ) ( b = ~/,

where the given ~k and the unknown ~b are m-vectors of functions on R+,

C and D are constant m x m matrices. To avoid the trivial case where A is a

constant times the identity, and does not contain H, we assume that C + D # 0.


We introduce a norm E "'p by

(2.2) II~,R_~ll~,v= ( f Ixl-~vl~(x)lVdx) "v

If ~b=(~b t . . . . . ~b,,), then II~ll = ~ : [ l~ l l . Moreover dpEL p (or any other

space) will be an abbreviation for tkj ~ L p.

T h e o r e m 2.1. Let - 1 + 1/p < tr < l + I/p, tr r l /p and let C and D

be non-singular matrices. I f ?peCk(R+) and f~x- l~p(x)dx=O in case

a > 1/p, then

(2.3) ][ n + ~ ,R+ [I ~.v _-< g ]l ~ , g + I[~,,v

(2.4) l[ &~,R_ ][~.,p < g[] ~,R+ lily

(and similarly for Jztp, tpeC~(R_)) .

Moreover, if the eigenvalues 2~ of - C - I D are outside the ray

arg 2 = 2z(1/p - a) then

(2,5) II ~ , g + !l~,v-< KII A4,,R+ II ~,v

Proof. We note first that the condit ion ~o - , fo x 4~(x)dx = 0, imposed for

a > 1/p, implies in particular the existence of the integral, hence q5(0) = 0.

This implies in t m n that [1 ~b []~,p < oo.

Denoting a(x) = x-'~qS(x) we have [[ ~b,R+ []ro,p = [] =,R+ II,~ �9 since

H f = x - " H • ~, (2.3) is equivalent to

By Theorem 1.1.

II H~ ~(x), R + II L~ ~ K II ~(x), R + II L~

H+ ct = M - l p , M~ = U-x~- 'p ,~ ,~U~.

Since U and U -1 are isometries, H + is bounded in LP(R+) if and only if

~ - l p , . ~ - is bounded in LP(R). Or, using the conventional terminology, if

p,(T) is an Lg-multiplier (Notat ion : p, ~ d/p).

Now by M. Riesz's theorem, H + = H + is bounded in LV(R+), hence

po(T) ~ ~/p, But

284 ELIAHU SHAMIR

p,, = [exp(2nio) " exp(2rci(iT- l / p ) ) - 1] -1

has the same limit values as Po at z = + oo and the difference p , - Po dies

down exponentially. It follows that p, will be in .1-[ 9 if its denominator does

not vanish for - o o < z < 0% that is, i f a # 1/p. (One could also use here

Michlin's result for multipliers [8], [4].)

Clearly also 1 + p , e J l p if ~r# l /p , so that H~- i s bounded in L~(R+).

Thus (2.3) is proved. For (2.4) we have to show that p*ed /p , o # l ip . This is

obvious since p* dies down exponentially at z = + oo. (Since J i r is no more

singular, no wonder that we need not use the fact that Poe J /p , which is

equivalent to Riesz's theorem.)

By (2.3), A = CH + + DH- is also a bounded operator in the E *'p norm,

o # 1/p. We turn now to prove (2.5), which means that A is 1 - 1 and with

closed range in the appropriate E "Pspace. Consider the system

A r 1 6 2 + D H - d? = d/ .

Multiplying by x * and setting a = x - ~ 1 6 2 f l = x -*~,, we obtain

A,a: ~- CH + ~z + O H ; a = ft. ByTheorem 1.1 and Remark 1.3,

= M- t [Cp , , + D(I + p, ,)]- lMfl ,

Co

provided t h a t ~ x ~ ~z(x)dx=O if ~r> l /p . Now (2.5) is equivalent to

I1 a,R+ IILP <= K ]1 fl, R+ IlL" and this is true if and only if (each element of) the

matrice

(2.6) ~-1(~) = [cp,(~) + D0 + ; . (0 ) ] -1

belongs to ~r Thus our theorem will be proved after we establish the following

L e m m a 2.1. For ~r # I/p, G~ t e ,,lip i f and only if G,(z) is non singular

f o r - oo < ~ <_ oo. T h i s amounts to the non-singularity of C and D and

to the eigenvalues condition in Theorem 2.1.

P r o o f . If G,(z) is singular for some real Zo or , = + oo, then some

elements of G ; 1(0 are not bounded on the real line. Hence they are not in -~r


and a fortiori no t in dip. Since p,(oo) = - I and po( - oo) = 0, G,(O is

non-singular at + oo if and only if C and D are non-singular. I f we denote now

(2.7) E = - C - l D, /t(z) = 2r~i(tr - l ip + iz)

then

(2.8) G,"I = (e ~ ' - 1)(C +Dd') -~ = (e ~ - 1)(1 - Ee")-~C-k

I t is sufficient to examine whether - ( I - Ee u) - ~ d ip , for then

e " ( I - n e - = g [ ( I - E e - I ] e d i , ,

and also G~ 1 r ~r since G~ t C is the sum o f the last two matrices. We shall

treat the case m = 1 first. In this case C, D and E are scalars. T h e funct ion

(Be u - 1 ) - : is equal to 1 at �9 = oo and vanishes at z = - oo. These respective

values are also assumed by p,, = (e u - 1)- x , which belongs to .~(p if tr # 1/p.

The difference between the two expressions dies down exponentially, hence

the first one belongs to d ip provided that its denomina tor E d ' - 1 does no t

vanish for real ~, and in view of (2.7), this means that

a r g ( - C - t D ) # 2n(1/p - a) .

This proves the lemma for m = 1. In case m > 1, we utilize a similarity trans-

format ion which carries E to its Jo rdan ' s canonical form El . Then I - Ee ~ is

carried to B = I - E t ~ which is constructed o f diagonal blocks

I - 2e",

1 0

O, ... 0

0

0 . . . . . O, 1, 1 - 2 e ~

(2 is the generic eigenvalue o f E) The dements o f B - t will be sums o f products

o f the form

286 E L I A H U S H A M I R

(2.9) l-I(1 - 2eU)/l--[(1 - 2e~), k < m. (t,) - (m)

The denominator here, being the product of all the diagonal elements, is equal

to det(B) = det ( l - Ee u) The numerator contains k < m such elements.

A necessary condition that (the elements of) (I - Ee~')-le.../fp is therefore

that 1 - 2e ~ does not vanish for real z (hence also det ( 1 - E#') ~ 0). This means,

(as in the case m = 1), that

arg 2 # 2rr(1/p - a), 2 is any eigenvalue of E.

But this condition is also sufficient, because, after cancelling k factors, (2.9) is

I--[ (1 - 2e~) -1. No 2 is zero, since E is non singular, and this expression ( m - / O

belongs to r since it assumes at z = + oo the same values a s - p , = (1 - d ' ) -1.

w E s t i m a t e s i n W ~'p.

In this section we shall study the operators H + and A in the Sobolev spaces

W *'p, s > O, and related spaces of functions defined in R or R• The definitions

and some basic properties of the spaces will be sketched first (Cf. also [6],

O0], 05] ) .

For integral r ~ 0 W"P(R) is the completion of C~(R) with respect to the

norm

I1 ,,, R II . = Y II D'u, R l i e lalfir

For real s > 0 we define first a semi-norm of order s. For s = 0, the semi-

norm is equal to the norm. For s > 0

;; (3.1/ [ u , n ] , . , = [ [ u(x) - u(y) [P '/P [ x ~Z y-~ y-4-- f dxd , 0 < s < 1

- - 0 0 - - o 0

(3.2) [u, R],.p = ~, [D~u, R],_t,l, p , 1 =< s, I,zl = ts]

where [s] is the integral part of s.

The space W"*'(R) is defined now as the completion of C~(R) with respect

to the norm


The space W~'P(R• is defined as the class of all the restrictions to R• of

functions in W~'V(R). The norms and the semi-norms in W"~(R• are

obtained either by restricting the integrals in (3 1) to R• or by taking the

"quotient norm" . E.g. for R+

(3.3) Ilu, R+ll . =Infl[u.,Rll , , u*eW~'V(R), u*=u i n R+ . tl*

The two definitions yield equivalent norms. We also note that C~(R• is dense

in Ws'P(R•

The definitions above make it clear that in general, assertions about

W s'p, s > 1, are reduced to the case 0 < s < 1. Accordingly, we shall study

the operators H • and A in W , 'p, 0 < a < 1, and in each case indicate briefly

how the results extend to W s'p, s > 1.

For f~ = R, R• and q~ ~ C~(~), the semi-norm [~b, t)]~,p vanishes only if

q~ = 0. Hence the completion of C~(~) with respect to this semi-norm is also a

Banach space, which we denote by W"V(f~). It can be shown that II u and

[u],.p are equivalent for all the functions u supported in a fixed compact

(locally equivalent), hence a function in W "'p is locally in W "'p. Applying the

same procedure for s = r + a, we obtain the space W S'Vwhich contains all the

functions u such that D'u ~ W " P ; but functions differing in polynomial of

degree < r ( < r if a = 0) are to be identified.

The closure of C~(R• in W"P(R• [resp. in W" ~(R+)] is denoted by

Wg'P(R• [resp. W~W(R•

The results of the next theorem are contained or easily derived from [-7].

T h e o r e m 3. 1. a) W~'P(R+) = W"V(R+), a < 1/p. For tr > l /p there is

a proper inclusion.

b) The functional ~: u --* u(O), defined for continuous functions on R+, is

bounded in the II II,,p norm, > 1/p. By extension, yu = u(O) is defined and

continuous for all u ~ W "'p, tr > 1/p. We note that u(O) = 0 is equivalent to

u e Wg "p (R+) . c) For a ~ 1/p, W~'V(R+)~ E"P(R+) (cf. (2.2)), and the imbedding is

continuous.


d) For u e W ~ P(R+), let ~(x) = u(x), x > 0 and ft(x) = O, x < O. Then

~(x) e W"P(R) r u e Wo (R+),

e) I f u e W"P(R) and a ~ 1/p then

a v~ 1/p.

(3.4) [u, R]~,p ~_ K([u, R_].,. + [u, R+].,p).

R e m a r k 3.1. All the results of this theorem are true for W `'p instead of

W ,'p, and in fact in this form were (a)-(d) proved in [7]. The (stronger) results

for W "'p are obtained by a standard homogenity argument: We use the results

for u(2x) and let 2 ~ oo. The proof of (e) is obtained from (d) by considering

the map u ~ (Y_u , Y+u) and using the closed graph theorem.

The connection between W ̀ 'p and E "'p reappears in the following easily

verified formula

r

R ' ( I (3.5)[u, • 1 - y l C P - X l l u ( x ) - u ( x y ) , R • dy, O < a < l . i / 0

We also note that for a fixed y > 0

(3.6) H + (~p(xy)) = (H + ~b) (xy) ,

and similarly for J1, J2 and A. From the last two formulas, we see that every

estimate for these operators in the E ~'p norm carries over to the [ , ]~.p norm.

In particular we have:

T h e o r e m 3.2. Let 0 < a < 1 and a ~ 1/p. T h e estimates (2.3) and

(2.4) are true f o r the W ~'p norm. Moreover if the eigenvalues condition of

Theorem 2.1 is satisfied, then (2.5) is true in W~'P(R+).

We note that the condition ~ x - l dp(x)dx = 0 if tr > 1/p, which was neces- 0

sary in Theorem 2.1, becomes superfluous, since we use here the E "'Pestimates co

only for functions ~k(x) = ~b(x) - dp(xy), which satisfy f x- l~b(x)dx = O. 0

T h e o r e m 3.3. Le t 0 ~_ ~ < 1, a ~ 1/p. Then

(3.7) 1[ H+c~,R• II~., --- K II q~,R~ II


and similarly for J1, J2 and A = CH + + D H - . Moreover i f C and D are

non-singular and the eigenvalues of - C - 1 D are outside the rays

arg 2 = 2z(1/p - a) and arg 2 = 2rr/p, then

(3.8) II q~, R + II,.p < K II A ~b, R + II-

P r o o f . By homogeneity arguments we obtain that the W *'P estimates are

true if and only if they are true in LP(a = 0) and W ̀ 'p. For (3.7), the case tr = 0

is clearly true and the W*'Pestimate is given inTheorem 3.2. For (3.8) we have

to require the eigenvalues condition for both tr and 0.

R e m a r k 3.2. The estimates of the last two theorems are easily extended

to W r+~ (resp. W'+~'0, r > 0. Indeed, one has DrH+~ = H+D'd? (and

similarly for the other operators) provided that

~ ( 0 ) ---~ t# '(O) . . . . = t# ( r - 1)(0 ) = O.

These functions ~b constitute a closed subspace of a finite codimension in

W'+"P(R+). The estimates are true for this subspace. Thus the operator

H + (and the other ones) which is bounded in this subspace, must be bounded

in the whole of W'+"V(R+). On the other hand, A is a 1 - 1 operator in

W ~ P(R+), hence also in W'+"P(R+). The converse estimate (3.8) for the whole

space follows now immediately from its validity in the subspace.

We consider now in W"V(R) the operator

C 1 (3.9) adu = lim ( ~-~ ~ o

oo oo

f . , f x + -~-- 2hi x - ie - t "

I f u = 0 in R_, then affu = Au in R+.

T h e o r e m 3.4. Let a # 1/p and let C, D be non singular. I f the eigen-

values of - C -1D are outside the ray arg2 = 2n(1/p - a), then

(3.10) I'u, R]a,p ~ r ( [ u , R_'],,p + [~r R+-l,,p).

(We get the similar estimates in W ~'P if we require also the eigenralues

condition fo r a = 0.)

290 ELIAHU SHAMIR

P r o o f . By (3.4) we have

[u, R],,~ <= K([u, R_]~,,p + [u, R+]~.p), a # 1/p,

hence it is sufficient to estimate [u, R+]~.p by the right hand side of (3.10).

By Theorem 3.2 we have

(3.11) [u, R+], ,p = [Y+u,R+],,p <= KEA(Y+u),R+]e. v.

But for x > 0

.~lu = Y+..c~u = Y + d ( Y + u + Y _ u ) = A(Y+u) + Y + d Y _ u .

Using the triangle inequality

(3.12) [A(Y+u),R+],.p < [.~/u,R+]~.p + [Y+a~tY_u,R+],

[a~tu, R+]~., 4- K[u, R_]o.p .

f o u(y) The last step is true since Y+ ~r is a linear combination of - - dy, x - y - - o 0

namely of J2 u, which is bounded in the W " 'p -norm from R_ to R+. Com-

bining (3.11) and (3.12), wo obtain the desired estimate.

R e m a r k 3.3. For a unit increase in a, the ray arg 2 = 2rt(1/p - o) makes

a complete circuit. The m x m matrice E = - C - 1D has at most m eigenvalues.

It follows that the eigenvalues condition (and all the estimates which depend

on it) is satisfied for every tr except for at most m values of a(mod 1).

~4. The range of A.

We shall determine now the range Of A = CH + + DH - as an operator in

{W"P(R+)} m. We assume throughout that C and D are non-singular. C + D ~ 0

(to avoid the trivial case) and that 2j are the eigenvalues of E = - C-1D.

We recall that the diagonalization factor of x -~ ~ is given by

G~(z) = Cpa(z) + D(1 + p~(z)),

p~(T) = [exp (2~i(a - 1/p + i~)) - 11 -1.


Go(r) is a periodic, matr ix-valued analytic funct ion o f x. Its poles are the poles

o f po(z), x = i(tr - l i p + k), k integer. We note that

Go(z) = Go(x - itr), po(x) = po(x - itr).

Consider now G~ 1(0. I ts singularities are the points where G,(x) is a singular

matr ix . Since

G,(z) = I-exp (2ni(a - 1/p + iQ) - 1 ] - ~C[I - E exp (21ri(tr - l i p + ix))]

and C is non singular, the only singularities are those of the r ightmost factor ,

which is singular if one o f its eigenvalues vanishes. Thus r o is a s ingu lar i t y (of

mul t ip l ic i ty r) if and only if for some 2~ (o f multiplicity r):

1 - 2 i exp (2ni(a - 1/p - izo)) = O.

For a fixed 2~, this equa t ion has a single solut ion at each period strip. Thus,

count ing multiplicities, each period strip contains m singularities o f G~-I(o .

We have already established ( inTheorem 3.3 and L e m m a 2.1) that the range

o f A, R(A) , is closed in W "'p if and only if G j l ( Q and Gol (x ) have no real

singularities, or equivalently, if Gol(V) has no singularities on I m z = 0 and

I m �9 = - ~r. We prove now

T h e o r e m 4.1. I f Got ( z ) has l s ingulari t ies inside the s tr ip

- cr < Im x < 0 and no s ingular i ty on its boundary , then R(A) c {W~ m

is closed and o f codimension 1. In part icular, as o increases and the line

I m x = - tr passes a k -mul t ip le s ingular i ty , the codimension j u m p s by k.

P r o o f . Consider first the scalar case, m = 1. Since the range is closed,

we have to examine the possibil i ty o f solving Ark = ~b for ~b e C~ (R + ) only.

I f we could use diagonal izat ion, then, using also formulas (1.3) - (1.7):

q~(x) = ( M - 1 G o IM~b) (x) = x - 1/ , (~-- l G o 1M~k) (log x).

We shall s tudy this expression. The funct ion (M~k) (x) is analytic in Im z > - 1/p

and dies exponential ly on each line I m x = const. . I f ~b(0) = 0, M ~ is analytic

also in l m x > - 1/p - 1, for then ef/~ ') dies down at t = - ov at least as

fast as e t(1 + tip) , and M~k = ~'(e'/~


We claim tha t ~keR(A) if Go I(z)M@ has no singularities (poles) in

- tr < I m �9 < 0 Indeed, in this case Gol(Z) �9 (M~k)(z) dies down exponen-

tially on every paral lel in this strip (since M~b has this p roper ty and Go-a(z) i s

bounded) . I ts Four ie r t r ans form

( ~ - - 1 Go x M~b) (t) = O(e '('+ ~)), t - - * - - oO

= O (x ~+ ~), x --, 0 (t = log x)

f o r 5> 0 sufficiently small. Mult iplying by x-1/p wee see that ~ = O(Ix[ ~- */P+' ),

so that II R+ II and thus also [~b, R+] , .p are finite near x = 0 and

~b(0) = 0 if a > 1/p. Since Gol(z)E..Wv, we have also ripeLY(R+). Moreover ,

~b is easily seen to be a C ~ funct ion o f exponent ia l decay at x = ~ . (In part i-

cu*ar, II R§ < ). Now we can diagonalize A~b and obta in Ark = ~b,

so that ~b e R(A). Now Go l(Z) �9 (Md/)(z) will have no poles in - a < I m ~ < 0 if

(i) M e = 0 i f ~ > 1/p (i.e. ~b e Wo~'P(R+));

(ii) I f Zo is a pole o f Go l(z) in the str ip then

( M ~ ) (~o) = f ~(x)x-t/~-~~ = O.

I f we examine now what necessary condi t ions should a funct ion ~b e R(A) satisfy, we find tha t ~(0) = 0 is not one o f them. Fo r suppose tha t ~ = Aq~

does satisfy ~,(0)=0, and let (p*=~b + ~b I where ~b I is smooth , ~bx(0) = 0, and

fT-ldp~(t)dt ~0. Since A = ( D - C)I +(C + D)HandC + D O, 0

oo

r = ( A ~ * ) ( 0 ) - ( A ~ , ) ( 0 ) = (C + D) t t-14,1(t)dt ~ O. i t s

0

Thus if there is no pole To o f Gol(T) in the strip, every ~k e C~(R+) is in R(A) and the codimens ion is zero.


I f % is a pole in the strip and - t r < I m T o < 0 , then the condition

(M~b)(to) = 0 is meaningful if ~b(0)= 0 and it is clearly satisfied by

~k = Aqb ~ R(A). At the end of this section we shall show that (M@) (To) is a

continuous functional on I~o'P(R+). I f tr < 1/p, then W~'P(R+) = W"'P(R+) and

the condition (M~k)(To)= 0 characterizes the range, which is therefore of

codimension 1. I f however a > 1/p then we have:

~ ( 0 ) = 0 ~ (M~b) (To) = O.

But ~b(0) = 0 is not necessary. Hence we obtain that R(A) is a 1-codimensional

subspace containing the 2-codimensional subspace characterized by ~b(0) = 0

and (Mr = 0.

Having proved the assertion for m = 1, we pass to the case m > 1. I f all

the eigenvalues of E are distinct, then a suitable similarity transformation

carries E, I - E exp (2rti(a - 1/p + it)) and its invesrse to a diagonal form.

After this change of base we obtain m disconnected 1-dimensional equations

and the desired result follows immediately. I f some eigenvalues of E coalesce,

we consider

A;, = A + 2I = (C - ,~I)H + + (D + 2I)H- .

For [ 2 1 small, all the properties of A are preserved, (in particular the relevant

singularities remain in the same strip), except that the eigenvalues of

E x -- - (C - ,~[)-t(D + 2I) become di3tinct. Our theorem is then true for Aa, and since the index (which here is the codimension of the range) of A is

stable under small perturbations [3], R(A) and R(Aa) have the same codimension and our theorem is proved.

Remark 4.1. Consider the range in W"Pins tead of W ''p. The range is

closed if Go(z) has no singularities on Im T = - tr (i.e. G,(T) has no real sin-

gularities). The codimension is clearly the number of singularities in

- a < I m T < 0 .

R e m a r k 4.2. Consider now Ws'P(R+), s = r + a. The codirnension

here is the number of singularities of Go(T) in -- s < Im z<0 . It is sufficient to

treat the case m = 1, and we suppose first that r = 1. Then ~ e C~(R+) is in

2 9 4 ELIAIIU SHAMIR

the range for 1 + a if both ~ and ~O' belong to the range in W"P(R+). For then

3 ~b,, ~b 2 ~ W "'P(R+), A~b x = ~k', Aq5 = ~k.

But

A~b I = ~O' = (Aqg)' = A~b' + const. �9 q~(0).

However, ~b(0)=_0, since ~ ' and Aq~' vanish at x = ~ . Hence Aq~ t = Aq~'

and $1 = $ ' , so tha t q ~ Wt+~'P(R+). I t is clear now how to proceed for

higher r. (The same remark for Ws'P).

L e m m a 4.1. L~t ~(x)~L*(e, oo) for some ~ > 0 and x*7(x)~I~(O,,).

Then the functional

cO

$(x) --, f ~b(x)7(x)dx Id

0

is bounded in W~'P(R+), ~ ~ l/p.

P r o o f . We w r i t e ~ = )'x + ~ 2 , w h e r e ~ x ( x ) = 0 f o r 0 < x < e , Y z ( x ) = 0

for x => 2~, and y~, ~2x ~ i f (R+) . The funct ional ~b ~ ~ ~ b d x is bounded in 0

LP(R+) and a-for t ior i in Wg'P(R+) while

~ y ~ d x = (x-~>(~S>dx~ IIx-'~llLpll~=x~ IILq 0 0

__< g I[ ~', R + [l~,,, =< g I[ ~, R + [I, ,~,

where the last step follows f rom Theo rem 3.1 (c). As a result we obta in that if - a < I m r o < 0, then

cO

(M4,)(%) = ( ~ (x)x- 1/~-i'~ 0

is cont inuous on W~'P(R+). ( Indeed) , (x) = x -*/~-i '~ the condit ions o f

the lemma). This result was used in the p r o o f of Theorem 4.1.


w 5. E s t i m a t e s in H ~'p

In this section we shall study the operators H • A in a new scale of spaces,

H" P(R+). The results are nearly identical though the methods of proof (for

a > 0) are quite different. Let s > 0 and

s ' . = [(l +l~ Ib ~/~u ̂ (0]" �9

The space H~'P(R ") is the set of all funztions for which J~u e LP(R"). This is

a Banach space with respect to the norm

li u, R" [ I . . = II J ~u, R" I1,~.

By utilizing Michlin's multipliers theorem [8], it is easily shown (e.g. in [6])

that H~'P(R ~) = W~'P(R ") for an integral r. It is also verified that for p = 2 and

any s, H~'Z(R ~) = W~'2(R"). However in other cases the spaces H ~'p and

W~'Pare different. (Kahan~ uses laeun:try Fourier series to show this), but they

are still contiguous in the sense that for e > 0

H~+~'~(R ") = W~'P(R ") = H ~-~'r (R")

the imbMdings being continuous. (Cf. [6-I, for other properties of H ~'p, cf.

[1], [2].)

I f P u = ([ ~ [ 'u^(O) " , then

U~, R ' J ~ . = II I'u,R'[l~,

plays the r61e of a semi-norm in H ~'p and we have

II u,~R "HI~-- - I[ u, R" I1~ + Eu, R q . . .

For fl ~R" with smooth boundary we define

]]u,~I[n~.p= Inf]]ux, R"l[n~,p , u ,~HS'P(R") , u , = u i n ~ ;

and similarly for [u, ~]n~, p .

For the r~.st or this s,.~tion, II " !L,, an~ ["L- will denote the H " norm and

296 ELIAt[U SHAMIR

semi-norm. The next theorem was proved in the special case

Peetre [11].

Theorem 5.1. For u e H ~'p (R)

p = 2 by

(5i) I1 u,R II.,p II r+(/9 • i) u IlL.

where (D + i) ~ is the convolution operator whose Fourier transform is mul-

tiplication by (~ + i)'.

P r o o f . We prove (5.1) for R+ (the proof for R_ is similar). Let

Pu = (O + i) -~Y+(D + i)~u.

Then P is a bounded operator in H"P(R). Indeed:

(~ + i)"u"(r = (~ + i) r

(1 -t- I~ [ 2) ,,'z - - [ (1 + I'~ 12) "/2u ^ (~ ) ] �9

The factor in brackets belongs to ,~L t', since u e H~'P(R). The other factor

belongs to ..C/p, by Michlin's theorem. Therefore, (D + i)" maps H"P(R)

boundedly into LP(R). Also, v ~ Y + . v is bounded in LP(R), and finally

(D + i) -~ maps back boundedly into H"'P(R).

Since p2 = p, p is a projection. The subspace annihilated by P is H~'~ the

set of all the functions in H "'p which are supported in R_. This is easily derived

from Paley-Wiener theorem. Indeed if u is supported in /~_, then u^(~) is

analytic in Imr > 0 and of arbitrarily small exponential type. Since

(~ + i)" �9 u ^(~) has the same properties, the support of (/9 + i)'u is in/~_ and

Y+(D + i)r = O.

Conversely, if Pu = 0 then

Y+(D + i)~'u = (D + i)"Pu = 0

so that (D + i)'u is supported in R_, and so is u since (~ + i)-" is analytic

in Im ~ > 0. The restriction map u ~ u [~>o of H"P(R) on H~ is continuous, and

its kernel is H~f . Hence


H#"(R+) ~- H # " ( R ) / H ~ "~ PH" P(R).

In other words, for every u ~ H "'p (R)

II u, R § I1-, ~ II Pu, g I1o, �9

If we denote v = Y+(D + i) ~. u, we obtain

II u,R+ II, p "~ Ii J~(Pu),R Ilnp = II {(1 + Ir 12)~/2(r + i) -~v ^(r R II LP

.~ ][ v, R IIL, = ll Y+ (O + i) 'u IILp'

where we have used the fact that (1 + [~ [2)~/2(~ + i)-" e..C/p .

C o r o l l a r y . I f (D + iO)" is defined by ((D + iO)') ̂ = (~ + iO) ~, then

[u, R+_]~.p ..~ ]1 Y+_(D + iO)'u ]]LP" (5.2)

Indeed, for 2 > 0

J~u(Zx) = ,~[(,~-z + ~2),/2u..(~)]v(,~x )

(O + i)'u(Xx) = 2~[(~ + i /2)r

Using (5.1) for u(2x) and letting 2 -~ ~ , we obtain (5.2).

R e m a r k . The formulas of Peetre [11], [12] for the norms (and the semi-

norms)[lu, R~ [1~.2 can also be extended to any p. The proof is the same as for

n = l .

We turn now to prove the H"P version of the estimates of Section 3. However,

the logical order of the arguments will be reversed. We first prove the analogue

of Theotrem 3.4, and from it derive the other estimates.

T h e o r e m 5.2. I f the eigenvalues of - C - 1 D are outside lhe ray

arg 2 = 2rffl/p - a), then for u ~ H~'~(R)

(5.3) [u, R]~,. < K([u, R_]~,p + [ ~ u , R+].,v).

(Note that we do not require tr ~ l / p ) .

298 ELIAHU SHAMIR

P r o o f . By (5.2), we have to prove that

II r- . [I --< K(II v_(o - i O ) r I[ "~- II r+(O + e0)~'" II)

all th : norms being LP(R) norms. Or, setting v = (D - i0)%~ and noting that

lie I '(r - i~ -~ ~ or/p, we have to show that

(5.4) I[ v[I " [I I r iO)-"v ]l < g(l[ r - " [I +/] Y+(O + iO)'d(O- io) -~'vll ).

Let now ~ r + i0 )~ r - i0) - ' . Taking Fourier t ransform:

,&^(r = - CY+(r + DY_(r

~r162 = (r + i0)~(r - i 0 ) - ~ r 1 6 2 = - CY+(r + e2'#"DY_(~).

Compar ing the two transforms, we see that ,~, itself has the same form as d ,

except that e2""~ D is substituted for D. Now the required estimate (5.4),

which is

(5.5) II o II ~ K<II Y-v II + II Y+ ~ " II)

corresponds to the case a = 0 o f Theorem 3.4. Hence (5.5) is true if and only if

no eigenvalue o f - e 2'a"C-1D is on arg 2 = 2re~p, which is an equivalent way

o f stating the eigenvalues condit ion for - C -1D.

E x a m p l e . In the scalar case (m = 1) we take C = - 1, D = 1. T h e n

- C - 1D = 1 and d u = u. As a result we obtain the estimate

(5.6) [u, R]r . < K([u, R--Is,. + [u, R+'].,,p), a ~ 1/p;

(and the analogous estimate in the norms). This result is the main tool we used

in [16,] to prove a conjecture o f Lions-Magenes [7-].

We shall consider now briefly the estimate for H + and A in H~"P(R+). Firit

o f all we have, assuming the eigenvalues condit ion,

(5.7) [~b, R+]~.v < K[A~b, R + ] , , p , ~b r H "v(R +).

Indeed, for ~b ~H"'~'(R) and supported in R+, (5.7) is exactly (5.3).Thus for

a ~ 1/p we are through. I f a > 1/p, we have the estimate in a 1-codimensio~al


subspace and since A is 1 - 1 operator, the estimate extends to the whole

space. Next, to obtain the estimate

[n• R.].,v __< KEu, R+J.,v

we derive first the estimate for functions on R:

< K [ . , _-< + [u, VP,

and then argue as before. Alternatively, we can follow the arguments in the

proof of Theorem 5.2. This will show that o" r 1/p is indeed necessary here.

The estimates in the H ~'p ngrms are obtained as usual by combining the L p

with the [ , ]~.p estimates. Thus A is bounded in H"P(R+), if a j ~ 1/p. The

properties of range (A) in H ~'p are exactly as in W "'P. We have seen that the

same eigenvalues condition ensures the closeness of the range. Its codimension

is also the same in both spaces--the results of Section 4 carry over since they

were obtained by using diagonalization in L v = H ~

Finally, the extension of the results to H s,p, s]>_ 1 is again performed as in the

W s'p case.

w 6. The ca se o f a f i n i t e i n t e rva l .

We consider now the Hilbert transform on a finite interval, which can be

taken without loss of generality as 0 < x < 1,

1

1 lim f r dy, 0 < x < 1. (I• = 2-"n--~ ~ o x + ie - y 0

Again, I • is bounded in LP(0, 1). The corresponding system of singular in-

tegral equations is

(6.1) CI + r + O I - r = O.

The L 2 theory was discussed in [13], where I + is diagonalized and its spectrum

is computed. In the L p case, although I + is not diagnoalizable, x~I+x -~and

( 1 - x)'I+(1 - x ) -~ are~ where tr = 1/q - 1/p. This observat ion is sufficient

for the complete resolution of (6.1) in LP(0, 1).

For a fixed p and q~(x) defined in (0, 1), let

300 ELIAHU SHAMIR

(U~) ( t ) d/P(e-' + 1 ) 2 / P ~ b ( ~ ) , = - - o o < t < o O ;

I

- - X - X (6.2) (Mtk) (z)=(.~rU~b) (z)= /K2-,

0 < 3 < 1 .

Clearly U sets up an isometry between LP(0, 1) and LP(R). The inverse of M

is a given by

(M-~f) (x) = ( u - ' ~ - - b ( x ) = x - ' /~ (1 - x ) - ' : ' ( s ~ - ~ f ) l o g i - ~ _ x

- ~l--L~ / x - l / P ( l - x ) - ' / P f ( z ) d z , 0 < x < 1.

Note the similarity between the transformations M and U defined here and

those of Section 1. In fact, near x =!0 we have the same behaviour in both cases

Also our new M behaves relatively the same for x ~ 0 and x ~' 1.

T h e o r e m 6.1. For tr.= 1/q - 1/p we have

(6.3) x~l+x-~ = M - 1 ptMq~,

(1 - x)r - x ) - ' d p = M -1 p2Mq~, where

pl(r) = [exp(2ni(ir + l / p ) ) - 1] - i , - 0o < z < o o

P2(O = [exp (2Jzi(iz - l / p ) ) - 1] - t = pa(z).

P r o o f . The proof is essentially similar to that of Theorem 1.1. We repre-

sent ~b as M - 1 M q b = M - l f , and then compute ( x ~ I + x - ' ) M - l f . After

changing the order of integration we obtain

x ~ 1 tl - rl)-l /v- '~dtl . lim - - (z)dz ~ x + ie - '7 e ~ O % / ~ - 0

To compute the inner integral, we integrate the function

z-'/e+',(l- - , - z ) - ' : P - " / ( z - x - i O


over a closed pa th F which consists of two " s m a l l " circles o f rad ius r a round

z = 0 and z = 1, connec ted above and be low the posi t ive unit i n t e r v a l , and o f

a great circle o f rad ius R a r o u n d z . = 0. The in tegrand is h o l o m o r p h i c in the

d o m a i n bounded by F , since af ter encircl ing bo th z = 0 and z = 1 we re turn

with a fac tor

exp (2hi( - 1/q - 1 /p + i~ - i'r)) = 1.

(This is the reason for choos ing ~r = ( l / q - I /p ) . ) Let t ing r --, 0, R --* oo and

using Cauchy ' s f o rmu la we see tha t the value o f ou r integral is

(x + i e ) - 1/~+t'[exp (2n i ( i z - l / q ) ) - 11 - 1.

Let t ing e --* 0, we ob t a in (6.3). The second fo rmula is ob ta ined in the same way.

(The cor responding fo rmulas for I - can be wri t ten down immedia te ly , using

the re la t ion I - = I + + 1.)

Let now # be a CO~ funct ion such tha t p = 0 for x < 1/3 and p = 1 for

x > 2/3. Let

(6.4) ~k I = #~k, ~z = (1 - #)~, //1 = X ~ l , //2 = (1 - x)~b2 .

Then ~k = x -~ 1/1 + (1 - x ) - e ' f12 and f o r j = 1 ,2

(6.5)

where the norms are a lways in LP(0, 1). Assume for a m o m e n t tha t we can also

write ~b = x - '~1 + ( l - x ) - ' a 2 �9 Subst i tu t ing these expressions for tkand ~b in

(6.1) and equat ing the coefficients o f x -~ and o f (1 - x) -~ , we ob ta in two

systems, which we can solve by d iagona l i za t ion :

C ( x ~ x - ~ ) a l + D ( x ' I - x - ~ ' ) O t l = fll

C(1 - x )" I +(1 - x ) - a a z + D(1 - x y l - ( 1 - x) -'a2 =//z.

We shall restr ict our cons idera t ion to the scalar case, in o ther words , to the

302 ELIAHU SHAMIR

computa t ion of the spec t rum o f I § in LP(0, 1). But it will be clear that the

vectorial case is comple te ly analogous.

T h e o r e m 6.2. T h e essential spectrum o f I t in LP(0, 1) is the set of

values assumed by p l (z ) and p2(z), - ~ < z < oo. (In part icular the essential

spectrum is the same f o r p and q where 1/p + 1/q = 1.) Thi s set consists o f

two symmetr ic circular arcs, with endpoints 0 and -'~1. In case p = 2, both

arcs coincide with the real interval l" - 1, 0]. Every point 2 o f the open set

S bounded by these arcs is an eigenvalue o f mul t ipl ic i ty 2 o f I t i f p < 2,

while I t - 2 is I - 1 operator with closed range of codimension 2 i f p > 2.

Every other 2 is in the resolvent set.

P r o o f . I f p _ ~ 2 , we have or= l / q - - l / p g _ O . We first show that

( I t - 2)q~ = ~b is solvable for every ~k s C~ . To this end we use (6.4) and write

~b = x - ' f l l + (1 - x)-r . I f we have also q~ = x -r + (1 - x ) - ~ 2 then

(6.6)(I t -- 2)q~ = x r t - 2)x "cq + (1 - x) ' (1 - x ) -* ( l + -- 2)(1 - x) 'e2

= x C M - l ( p t - 2)M~1 + ( i -- x ) e M - l ( p 2 -- 2)Mc;t 2 .

Hence if we can solve in LP(O, 1) the two equat icrJs:

(6.7) M - l(pj _ 2)Mey = flj, j = 1,2,

then since tr ~ 0 if p < 2, q~ = (x-*cq +(1 - x) -~ c~2) e LP(0, 1) is the required

solution o f ( I + _ 2)tk -- ~k. But the solut ion o f (6.7) is immedia te :

(6.8) ctj = 3M- 1 (pj _ 2) - ~Mflj, j = 1, 2,

and in Section 2 it was proved that ( P i - 2) - 1 ~ . / /p i f2 ~ pj(z), - o0 <_ �9 < oo.

For such 2 we have that c t j e L p and [I~jH_<_KI[/~jI[. i f n o w

~ktk) e c ~ ( 0 , 1 ) and ~k~k)~k, then flj~k)--+fll, hence also ~ l t k ~ % and

~bt~) ~ tk = x - ' a t + ( 1 - x)-%t 2. We clearly have now ( I + - 2 ) ~ = ~b in

LP(0, 1). This shows that for p < 2 and 2 # pj(z), the range of I + - 2 is the

whole space.


The set o f values assumed by pl(T) and p2( 'C) is clearly in the essential (and

cont inuous) spectrum o f X § Now

p z ( z ) _ = [ e x p ( ~ - ~ ) e x p ( - 2 n z ) - l ] - z ,

and the set o f values o f Pz is the circular arc obtained as the image o f the posi-

t i v e a x i s under the M/Sbius t ransformat ion w = [ ( e x p ( 2 n - - i ) ) z - 1 ] - I

The endpoints are P I ( - ~ ) = O, p l ( ~ ) = - 1. Since p z ( Z ) = pl(z) , the

arc 2 = Pz(~) is the complex conjugate o f 2 = p~(z). The same two arcs are

obtained for the index q conjugate to p (but their order is reversed).

We notice now that

1 1

f (I+ -- ,~)U~ .~ - fu (x + - )~)0 0 0

so that if v .-,, rue, identities L q with (LP) *, we have (I § - 2)* = I + - ,~.

F r o m the previous result for p < 2 we obtain now for p _~ 2 that for 2 outside

the two arcs, the range o f I ~ - 2 is closed. We shall prove now that the co-

ditnension of the range is 2 if 2 is in S and zero if 2 r S. Dualizing again, we

obtain that for p < 2, 2 e S is a double eigenvalue and 2 r N is in the resolvent

(for any p in fact).

To accomplish this, we consider again (I + - 2)~b2= ~, ~ e C~ ~ Once more,

we use (6.6), and we have (6.8), with ej e L p. But now p >.2, cr = 1/q - I /p > 0 so that q~ = x - ' c q + (1 - x ) - ' e 2 is not necessarily in L p

The reasoning now is exactly analogous to the p roo f o f Theorem 4.1. In

order that x - ' e l e L p near x = 0, we have to extend the equation

M e 1 ---~ (/91 - - /~) - 1 M f l '

to the complex plane, and using the definition o f M, we see that (Px - 2) - 1Mill

has to be regular analytic in - tr < Im z < 0. In this case (remembering tha t

near x = 0 M -z here and of Section 1 behave alike) we have e 1 = O(x -~§

and cq e L p near x = 0. Since Mflz is always regular in the strip and (/91 - - ,~.)- 1

has at most one pole ro there, we have to require (Mi l l ) (%) = 0. This is a

single cont inuous linear condit ion on el (hence on ~k).

304 ELIAHU SHAMIR

Analogously, (1 - X)--~r0~2 ~ LPnear x = 1 if ( P 2 - - ~')-lMfl2 has no poles in

0 < Im z _ tr, so tha t (Mfl2) (z~) should vanish if Zo' is in this strip. This adds

another condi t ion .

The 2's for which we have to require the first condi t ion are the values of

pl(z ) for - a < Im z _< 0. For Im z = 0 we obta in the first arc of the essential

spectrum, and for Im ~ = - a, the second arc. For - tr < I m z < 0 we obta in

all the points of the domain S bounded by the arcs. The same set is covered

by ~.=p2(z), 0 < I m z < tr. Thus for every 2 ~ S , the range of I + - 2 is of

codimension 2. Every 2 ~ ~ is in the resolvent.

BIBLIOGRAPHY

1. A. P. Calderon, Intermediate spaces and interpolation, Warsaw Conference, 1960. 2. - - , Lebesgue spaces of differentiable functions, Proc. of Symposia in Pure Math.,

Vol. 4, Amer. Math. Soc., 1961, pp. 33-49. 3. I. C. Gob_berg, and M. G. Krein, Basic properties of defect numbers, root numbers

and indices of linear operators, Usp. Mat. Nauk, 12,2,1957, pp. 43-118. (A.M.S. translation: Ser. 2, vol. 13, pp. 185-264).

4. L. Htirmander, Estimates for translation invariant operators in L p spaces, Acta Math., 104, 1960, pp. 93-140.

5. W. Koppelman and J. V. Pincus, Spectral representations of finite Hilbert transformations, Math. Z., 71, 1959, pp. 399-407.

6. J. L. Lions, and E. Magenes, Problemi ai limiti non omogenei (III), Annali di Pisa, 15, 1961, pp. 39-101.

7. ~ , Probl/,'mes anx limites non homog~nes, (IV), Annali di Pisa, 15, 1961, pp. 311-326.

8. S. G. Mikhlin, Fourier integrals and multiple singular integrals. Vestnik Leningrad Univ., 1957, 7, pp, 143-155.

9. N. I. Muskhelishvili, Singular integral equations. English translation: P. Noordhof, N. V., Groningen, Holland, 1953.

10. M. Nikolski, On the imbedding, continuation and approximation theorems for differentiable functions in many variables, Usp. Mat. Nauk, 16, 5,1961, pp. 63-114.

11. J. Peetre, Another approach to elliptic boundary problems, Comm. Pure App. Math., 14, 1961, pp. 711-731.

12. - - , Mixed problems for higher order elliptic equations in two variables, Annali di Pisa, 15, 1961, pp. 337-353.

13. J. Schwartz, Some results on the spectra and spectral resolutions of a class of singular integral operators, Comm. Pure. App. Math., 15, 1962, pp. 75-90.

14. E. Shamir, Evaluation darts W s,p pour des probl~mes aux limites elliptiques mixtes dans le plan, C.R. Acad. Sci. Paris, 254, 1962, pp. 3621-3623.

15. , Mixed boundary value problems for elliptic equations in the plane - - the L j' theory, Annali diPisa, 17, 1963, pp. 117-139.


16. , Une propri6t6 des espaces H s,p, C.R. Acad. Sci. Paris, 255, 1962, pp 448--449.

17. H. Widom, Singular integral equations in L ~, Trans. Amer. Math. Soc., 97, 1960, pp. 131-159.

DEPARTMENT OF MATHEMATICS

THE HEBREW UNIVERSITY JERUSALEM, ISRAEL

AND

DEPARTMENT OF ]V[ATHEMATICS

UNIVERSITY OF CALIFORNIA

BERKELEY, CALIFORNIA) U.S.A.

(Received June 13, 1962)

Documents

Reduced Hilbert transforms and singular integral …Reduced Hilbert transforms are obtained by restricting the variable x and the domain of integration in (1) to a subset R' (or two