Math. Nachr. 165 (1994) 183-189

On Singular Integral Operators with Bounded Measurable Coefficients

By PAUL ANDREW of Launceston

(Received January 4, 1993) (Revised Version April 30, 1993)

Abstract. Necessary and sufficient analytical conditions are determined for a singular integral operator of the form aP + bQ with bounded measurable coefficients to be a @-operator on L,(T) for all 1 < p < co, where T is a closed Lyapunov curve.

0. Introduction

Suppose that T is a closed Lyapunov curve which divides the complex plane into two open regions, D+ containing the origin and D - containing the point at infinity. For each

cp E L,(T), the Cauchy principal value of - cp(s)ds is denoted by Scp whenever is exists,

r where the positive direction on T is taken so that D + is always to the left. The singular integral operator S is bounded on L,(T) for 1 < p < 00 and S2 = I .

Consider the projection operators P = -(I + S ) and A = -(I - S). If a, b E GL,(T) and

(i) ab-' t -m = cd, m integer, (ii) c*'EL:(T) , d * ' E L ; ( T ) , (iii) c[S, d] is bounded on L,(T) ,

711 s s - t

1 1 2 2

1 < p < CQ, then A = UP + bQ is a @-operator on L,(T) if and only if

where L: (T) = (cp EL,(T) : Pcp = cp} and L; (r) = {cp E L,(T) : Qcp = cp> + C

[G/K, M/P; K, 9.41. If A is a @-operator on L,(T) for all p , then the factorization in (i) is uniquely

determined up to a scalar multiple. This paper uses the algebraic conditions above to determine an equivalent analytic condition for A to be a @-operator on L,(T) for all p .

184 Math. Nachr. 165 (1994)

2. Commutators

Y where Ids1 = ePi9ds is the differential of arc measure on r, S(s) being the angle between the tangent to r at s and the x-axis.

The sharp maximal function of cp on r is

and the set

B M W ) = {cp"EL1(T): IlM#cpll, < 4 of functions with bounded mean oscillation on r is a Banach space with norm 11cpll* = I/M#cpll a3 when functions which differ a.e. by a constant are identified.

Assume that r has the parametric representation s = h ( p ) for O < p < Irl, where p is the arc length measured from some fixed point on r to s and where h'(p) satisfies a Holder condition with exponent I > 0.

As

I W I ' IUX'

for f e Ll(O,lrl) and intervals I , I' c [0, Irl), cp -+ cp 0 h is an isometric isomorphism of B M O ( r ) onto BMO(0, Irl).

A,(T) is the set of all nonnegative cp EL, (r) satisfying

Icpyll/(P-l)l c p - l / ( P - l ) l Y < k , for each arc y on r, where k , is independent of y.

Theorem. Ifc, c- EL, (r) and 1 < p < co, then both c [ S , c- '1 and c- ' [ S , c] are bounded on L,(T)for all p if and only if 1 ~ 1 " ~ A,(T)for all p and all real a, if and only if In Ic( is contained in the closure L,(r) ofL,(T) in BMO(T).

Proof. Assume that T= c [ S , c-'1 is bounded on Lp(T) for all p , and let xy be the

As characteristic function of some arc y on r with endpoint so.

Andrew, Singular Integral Operators 185

In particular, this implies c E L,( f ) for all p . As c- ' [S, c] is also bounded on ,(r) for all p , Icl: Ic-"ly is similarly bounded above for each n independently of y and c- l E L J T ) for all p . As Iclylc-'ly> 1, Icl"EA,(f) for all integers n.

An extended arc 7 on r with endpoints so, s1 is the path traced out by a point moving continuously from so to s1 in a positive direction with multiple circuits counted. cp?, Mf, BMO,(T), II . llE* and Ag(T) can be defined accordingly.

If 7 is an extended arc containing n 2 1 complete circuits of r and if cp E L (r), then

G 2 j 140 - VrI Ids1 G 2 (n + 1) J IV - 'PA Ids1 9

Y r so BMO,(T) = B M O ( r ) and the norms / I * I / $ and I / . / I * are equivalent. If cp is non- negative, cp;< ((n + l)/n) q,-, so Ag(T) = A , ( f ) as well. If 6is the periodic extension of h to R, then cp + cpo6is an isometric isomorphism of BMO,(T) onto the subspace of all a.e. periodic functions with period Irl in BMO(IR).

As 1 ~ 1 " ~ A,(T) = @(f) for all integers n, Holder's inequality implies IcI" E A f ( f ) for all real a. As cp E Af(T) if and only if (exp [In cp - (In cp)& and (exp - (l/p - 1)) [In cp)& are bounded above independently of 7 for all extended arcs on f [T; p. 2411, it follows that IcI" E A J T ) for all p and all real a. This is equivalent to In ( c I EL, (r) [G/J1.

Conversely, assume I c l " ~ A ~ ( r ) for all real a, so c, c-l e L p ( r ) for all p . For each cp~L,,(r), c p o h ELJO, Ifl) and (Sq)oh=H(cpoh)+ R(cpoh),

where

0 0

As Ico&A,, (coh)H[(c-' cp)oh] is bounded on LJO, Ifl) and, as h ' (Y)

h(Y) - h(x) 1

Y - x - 0 ly - XI-', R is bounded from L,(O, Irl) into Ls(O, Irl) whenever

1 < 1 - l/r + l/s c 1 [J; p. 2861, so c[S, c-'1 is bounded on L J T ) for all p . Similarly, c-'[S, c] is bounded on L J f ) for all p .

186 Math. Nachr. 165 (1994)

3. Analytic Conditions

Lemma. S is bounded on BMO(T).

Proof . If cpeBMO(T), then cpohEBMO(0, ITl) and (Scp)~h=H(cpoh)+ R(cp0h) for H and R as in the previous theorem.

iri If cpoh ds= 0, then Ilq~hll, < k r /IcpohJJ* by [T; V I I I , 1.51 and, as

1 R : L,(O, ITl)+L,(O, IT[) is bounded whenever - < 1 - A , IIR(cpoh)ll* G C , I l c p o ) I * for

r some constant C, . Toshow that (IH(cpoh)ll* ,< C,(Icpoh((*forsorneconstant C,,it issufficient toprove the

lemma in the case of the unit circle To, and then the general case will follow.

0

As E?(cpoh)=i(S,,cp)oh-- 1 F d s , where fif(x) is the Cauchy principal value of 2.n

1 ( cot r?) f(y) dy for 0 < x< 271, the lemma follows from IT; VIII, 3.31. 27L

0

Lemma. I fa€GL,(T) and aP + Q is a @ is a @-operator with index 0 on L,(T)for 1 < p < 00, then, for each integer n #O, there is a b e L,(r) with bn = a and IIArg bll, < k(a) n - l for some constant k(a) independent of n, for which bP + Q is a @-operator with index 0 on L,(T) for all p ,

- Proof . Let ro be the unit circle with associated projections Po and Qo. Clearly cp(t) E Lp (To) whenever q(t) E L,' (To).

If k is a conformal mapping of the unit disc onto D+ UT with restriction to the unit

circle and if a = - 0 k, then aP + Q is a @-operator with index 0 on L,(T) for all p if and

only if a E GL,(T) and aP, + Qo is a @-operator with index 0 on L,(To) for all p [ K ; 9.5, 9.61. Assume that this is the case.

a -

I4

As L,'(To)=H,, a=a+a- where a+ = I , exp (2P0 lnIct+l)ELp+(r,), a- = I 2 / E + ~ L p ( T o ) for A,,

and where Po lnIa+l EBMO(TJ [K; p. 881. Set p+ = Ai ln exp (2n-'P0 In Ia+l) and p- = Ai1"/P+, where the principal values of A:/" and A i l n are taken.

For each cp E BMO(To), ( 1 cp 11, < Ko(p ( 1 cp 11 * + 11 cp 11 ,), for some constant K O independent

of p [T; VIII, 1.51. As )I C cpk/k! 11, < C I( cp &,/k!, 1 cpk/k! -, ecp in &(To) as

N + co whenever eKop 11 cp 1) * c 1. As PolnIa+I E B M O ( ~ ~ ) , ( P o l n ~ a + ~ ) k ~ L p ( T o ) for all p and any integer k >O, so

p+ E Lp' (r,) L L: (To) when (nl is large enough. Similarly, p; E L ; (To) L L: (r,) and p?' E L ; ( T ~ ) C L L ; ( T ~ ) . As lnIa+lEL,(To), lnIP+IEL,(To) so the lemma follows

M j k S N M 5 h S N O S k S N

Andrew, Singular Integral Operators 187

for b = lal""(poR-') when p = p + p - for sufficiently large In) and also, clearly, for all n # 0.

Theorem. Zfa, b E GL,(F), then U P + bQ is a @-operator on LJT) for 1 < p < 00 ifand only ifRe Sg E L,(T) for some g E L,(f) with eg = ab-lt-"' for an integer m. Furthermore, if these conditions are satisfied then e * p g E L,' (r) and e * Og E Lp (r) for all p .

Proof. Assume that aP + bQ is a @-operator with index m in LJT) for all p , so

For each c p ~ B M o ( r ) , IIcpllp ,< K(pl(cpl(* + Ilcplll)for some constant K independent of ab-'t-"P + Q is a @-operator with index 0 on LJT) for all p .

p , so

If cp = Pu for u EL, (r), then cpk E Lp'(r) for all p and any integer k > 0, so efPu E L,' (r) whenever eKpJIPull* < 1. Similarly, ekQ"ELp(r) whenever eKp(lQull* < 1.

For each integer n # 0, ab-'t-" = u", where UP + Q is a @-operator with index 0 on L J T ) for all p and IIArgull, c k(ab-'t-"')n-'. In particular, u = u+u- for u : ' d l ( r ) and ultELp(r) for all p and lnIu+l, InIu-lEL,(T).

1 cpk/k! + e+' in LJT) as N -+ co whenever eKp llcpll. < 1. O S k S N

Select n so that eK max(llPLnull*, IIQLnull*)< 1, then as

u, exp( - PLn u) = u 1 ' exp(QL, u) E L : (T)f lL; (r) = C ,

it follows that eg= ab-lt-"', e*PgELp'(r), e*QgELp(L) and ReSgEL,(r) when g = nLn u E L , (r) for all p .

Conversely, suppose that ab- l t -" = eg for some integer m and some gEL,(T) with Re S g E L , (r).

As- g E~, (T) , Pg E B M O ( r ) and for each p there is an integer n > 0 for which w: 1 = e*n- 'Pg E L,' (r) L L: (r). Similary, w? = efn- 'Qg E Lp (r) L L; (r). As

so the theorem follows as this implies w:l E L,'(r) and w _ f ' ~ L p ( r ) for all p.

Re SgEL,(I'), InIw.1 and lnIw-lEL,(T),

4. The Index

Definition. An integer m is said to be the index of aeGL,(r) if at-"' = eg for some

As S is bounded on BMO(T) and H " ( 0 for 0 < p < 1, it can be seen that Re Sq EC(~), gEL,(T) with ReSgEL,(T).

the closure of C(r ) in BMO(r), whenever PEW. -

Theorem. Zf cp E BMO(r) , then cp E C(T) i f and only if Icp - cpyly + 0 when IyI + 0.

Proof. ~ For c p ~ B M o ( r ) and real 0, define cp" on r by cp"oh(p)=cp~E(p-o) . If c p ~ C ( r ) , then cp"ohEC[O, lrl] = WO[O, Irl) for all CJ [TI, so Icp - c p y l y -0 whenever IYI + 0.

Conversely, suppose cp E BMO(T) and Icp - qYJy + 0 whenever IyI -P 0. It is sufficient to assume that cp is real valued.

188 Math. Nachr. 165 (1994)

For q > O , restriction of q.6 to [-q,Irl+ q ) is contained in W O [ - q, Irl + q) = C [ - q, Ir( + q), so for any E > 0 there is a 0 E (0, q) and a real valued U E BMO(T) with uoh uniformly continuous on [0, Irl), IIq" - u/I * < E and

Set u = aqxy. + v for v E C(T), where M = lim uoh (0) - uo6( - 6) and y' is the positive M # ( q " - u)(h(O)) < E.

a 6-0

arc from h(0) to h(p') for some uoh(p') = uoh(0) -- and qoh(p) = 1 - p/p'.

As IqxY. - ( a q ~ ~ , ) ~ ! ~ < Iq" - qFly + M # ( q " - u)(h(O)) + 1 0 - vyly for any open arc y containing h(O), letting IyI + 0 implies la( < 2 ~ .

Select 0 < K < Irl/2 so that luoh(6,) - uo6(- S,)l< 3~ for O < 6,, 6, < K and denote the positive arc from 6( - K ) to h ( ~ ) by y". Define w on r by w = u on r\y" and

~ o ~ ( ( ~ , ) = - [ u ~ ~ ( K ) + u o ~ ( - K ) ] + - [ u o ~ ( K ) - ~ ~ ~ ( - K ) ] for - 1 < 9 < 1 , then w ~ C ( r )

and ~ ~ ~ " - ~ / ~ ~ < ~ - u ~ / ~ + ~ ~ / u - o ~ ~ , ~ 7 . 5 . As E was arbitrary and W - ~ E C ( ~ ) , it follows that q E C ( T ) .

2

1 9 2 2

Theorem. If q EL,(^) and cp = uxY,, on y' for arcs y" c y' on r with Iy"l < ly'l and u continuously differentiable on r, then Re Sq$L,(I") whenever Im a # 0, where a = u(so) for an endpoint s , ~ i n t y' of y".

Proof. It is sufficient to assume that y" is closed with initial endpoint s , ~ i n t y r . If y is an arc from so to s1 ~ i n t y", then, for SE y,

Sq(s)= -7Ln(s-so)+O(l)as s - + s o . a

711

Consider U, = {s E y :IRe S q - (Re Sq), l> A} for 1 > 0, where y c y" is any arc with a

711 initial endpoint so on which ISq(s) + Ln (s - so)] < K for some large enough K.

As

a (Re Sq) , = - (Re Sq)e-"ds = - Rey In IyI + 0(1) as IyJ + 0,

lU,l 2 I { S E Y :Is - so( < K,e-""lyl}l 2 K , e-Oa IyI ,

for some constant K , > O and small enough IyI, where o = IIma/n1-'.

IYI ' S 711

Y

This implies Re S q $ [G/J].

Corollary. Re S Ln t $ L (r) .

Proof. As Arg t is continuous on r except for one point, it can be represented in the form u( t ) xy + c ( t ) for real continuously differentiable functions u(t), c(t) on r and IYI < Irl.

Andrew, Singular Integral Operators 189

References

[G/J] J. B. GARNET and P. W. JONES, The Distance in BMO to L", Ann. of Math. 108, 1978,373-393 [G/K] I. GOHBERG and N. YA KRUPNIK, Einfuhrung in die Theorie der Eindimensionalen Singularen

Integraloperatoren, Birkhauser Verlag, Basel, Stuttgart, 1979 [J] K. JORGENS, Linear Integral Operators, Pitman, 1982 [K] N. YA KRUPNIK, Banach Algebras with Symbols and Singular Integral Operators, Birkhauser,

Verlag, Basel, Stuttgart, 1987 [M/P] S. G. MIKHLIN and S. PROSSDORF, Singular Integral Operators, Springer-Verlag, 1986 [TI A. TORCHINSKY, Real-Variable Methods in Harmonic Analysis, Academic Press, 1986

Department of Applied Computing and Mathematics University of Tasmania at Launceston GPO BOX 1214 Launceston, Tasmania 7250 Australia