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OPERATOR THEORY

TP Uđ), T88 214 © Copyright by INCREsr, 1984

ON MULTIDIMENSIONAL SINGULAR INTEGRAL OPERATORS II: THE CASE OF COMPACT MANIFOLDS

ROLAND DUDUCHAVA

INTRODUCTION

Singular integral operators (equations) Q(x, x —

(0.1) A(x) = a(x)(x) + oy) dy =f), xeM

a |x — yl"

are investigated, where M is a compact manifold with the boundary 0M #@;

f(x) € (H’?)\(M), 9 € (HP)(M), 1 < p< co, — 0 <5 < 00; proofs here are

based on [3], where the first part of the present investigations (the half-space case M = R"*t) was published

As it was already mentioned in {3] the operators (0.1) were investigated by Simonenko [6] (the case p = 2, s = 0) and by WiSik and Eskin [7] (the case p = 2, —co<s< oo)

The main tool of investigation here is the local principle (cf § 3.2°), which is a slight modification of the local principle from [5] (cf § 1.6), extended with some notions from [6]

We set forth here the numeration of sections; thus the references to §§ 1—2 will mean the reference to [3]; all notations from [3] are used without the further explanations

3 PRELIMINARIES

1° ON THE ISOMORPHISM OF SOBOLEV-SLOBODECK] SPACES Consider the operators

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where

ei(Q)=0& Fier D, €= (G4, eR’,

g6) = (L + 101)

these operators have continuous extensions

A’, 4$: H P(R") + HƯ-®)P(R") (— co <r< Co)

and these extensions are isomorphisms (A* arranges even the isometricai isomorphism

lip == A°@Pile-syp)s Obviously

AAS= MAL HL

The operators

Al, = Py, A*he,

where P,, and /,, are restricting and extending operators, also arrange isomor- phisms (cf [2, 4])

AS: HY(R"*) 3 HT 8(R™), ALAS =,

AE: HPP(RP+) > Hữ~9P(R"f), (00 < 5,6 < 00)

The following two lemmas are easy to prove (cf., for example, [2, 4])

LEMMA 3.1 The operator W° € L((H*?)*(R")) is isomorphic

we — B-*M?PB° (B=° — A=, AS, Ais)

to itself Wie L(LE(R")) operating in the space LN(R"); therefore Woe € L((H*")\(R")) if and only ïƒ a(6)c M?"X(R") (1 <p < œ, —co < s < G5)

LEMMA 3.2 The operator

(3.1) ;c ⁄((Hữ)Ẻ(R"?), (H9?) X(RP+))

is isomorphic

Wh = A Wy A’ to the operator

N/tn+ i¢ + lễ": + ] 5 ¢

(3.2) wie ⁄Ở(Œ(R"')), as) = Ww TT a(¢);

s ig — 6 — 1

hence (3.1) is valid if and only if a(¿) e M?*(R")

We remind that (C5)"*4(R") denotes the set of NxN_ matrix-functions

with entries 2(£) Cạ(R”) having continuous derivatives Dfa(é) (|ki <r) with

compact supports

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SINGULAR INTEGRAL OPERATORS Il 201 The operators

T= A-‘aAS —al, Ts = Az*aA’, — al,

~ ~ ~ 0

Tụ = Ax*a4$ — al, Ti, = Ap*aA, — aW Bis”

- , ~ —$ ~ s 1 Ty = At*a4Š — aW; › where

-

are compact

T, T,, T,€ S(LM(R")), T,, 7'e G(1J(R"°))

Proof Obviously we can suppose N = 1 and due to standard approximation consider only a € Cs°(R"); applying further Theorem 1.11 (namely the second part of it on the interpolation of compact operators) we can concentrate only on the case p = 2 (due to the boundedness of operators under the consideration in all

EL, spaces)

The inclusion T € S(L,(R")) was proved in [1], Chapter II, § 9 We consider only T!., because T)., T,, can be considered similarly and 74 = Pi47Tih4, T, =

= Pus Tihs

4413 is a pseudodifferential operator (cf [1, 4, 7]) and its symbol (ig, — |¢’] — 1)°

belongs to the class Syig(R”) (cf [7], Chapter II, § 1); the operator dA{ — Af.dI is

also pseudodifferential for all de C2(R") and its symbol belongs to Sj9'(R");

hence dA — A’dle £(A'(R"),A"-**7(R")) (here A’(R’) ef PR"); cf [7], § 6 and [3], § 18)

Let now 4(é) € Cy(R”) be such that 6(€) = 1 forall € € suppa; then a(é)b(é) = =a(€), aA’ = AK, + aA’.bl, K, = bAS, — AOI; rewrite now TL in the form

T! = Az(aK, + Kobl) + K,al, (3.4)

Ky, = as, — Asal, Kạ= AZ°A) — W9 ~s

As we already noticed K, and K, are pseudodifferential operators of order s—1 and therefore K,, K,€ £(H*(R"), H'(R")); but then (cf [4], Theorem 18.4

and [1], Chapter II, Lemma 8.1)

ak,, K,bI € S(A%(R"), L.(R")), since suppa < suppb are compact On the other hand

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and therefore (3.5) A1-°*aK\, A=ŠK;bí c S(L;(R")) Since - ,a (lũ tiết T fa = Wt $9 (SELELY, —s _ we obtain K; = Wo gs? d_) ơ g_(Â) s-,);

clearly d_,€m(R") and d_.(oo) =: 0; Theorem 1.12 yields then

(3.6) ak, € S(L2(R"));

duc to (3.4) —(3.6) the desired inclusion 7“ € S(L,(R")) is evident ey

2° LOCAL PRINCIPLE U We present here a local principle which ‘is a

slight modification of the local principles from [5,6] (cf § 1.6°) Let X,, X_ be the Banach spaces and

LY! = L(X,, X2), SS! =: S(X,, X2),

== L(Xo Xi), SY = S(X2, XG): G

L= £/S' denotes the factor-algebra with the norm

WAL SACS inf PAE TC

Tes’ 4c #⁄(X,)r #(1:)

is called a localizing class provided

S(X%) 0 S(X2) 0 4 =: GO

and for each A,, A, € A there exists A€ A such that A;A = AA; = A (j= 1,2) Two elements A, Be #' are called A-equivalent, written A 2 B, provided

inf (A — B)EL, = inf | E(A — B): = 0

Ee4 Ee4

An element A4 e Z” ¡s called /ef (righf) A-regularizable if there exist Re ¥”

and Be A such that RAB = B+-T, Te S(X,) (BAR =B+T, TE SX)

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SINGULAR INTEGRAL OPERATORS, II 203

Similarly A-invertibility, left and right A-invertibility are defined provided

T =0 (cf § 1.6°)

A system {4,}se9 of localizing classes is called a covering if from each set {Auhocas do € A, finite number of elements can be selected {Ao Seat such that

i

A=Y Aw, has the regularizer

kal

RA=I+T,, AR=I+T), Ty, T€S(X%)n SX)

THEOREM 3.4 Let {Aghwee be a covering system of localizing classes and A* B, (A,B, € £', o€Q), AB~ BAe GS’ for all BE\J Ay

woEQ

The operator A is Fredholm (has a left or a right regularizer) if and only if

B, have A,-regularizers (B, have left, have right A,-regularizers) for allw€Q Proof is completely similar to the proof of Theorem 1.1 from [5], Chapter XI (cf Theorem 1.17 above)

Consider now the particular case

X: =(Hÿ)X(M), X;¿=(H*?)X(M), 1 < p<00, ~ 0 <5 < @, where M is a r-smooth (r > s) n-dimensional compact manifold with the boundary GM #@Q; denote

LIM) = #((HŸ)Y(M), (H*?)*(M)),

(3.7)

SN(M) = S((Hÿ)*(M), (H°)*(M)) Let xe M and consider

(3.8) A, = {v,J:v,€ CM), v,(t) = 1 in some neighborhood of x € M} Clearly 4, is a localizing class in #Y(M) and {4,}xem (due to the compactness

of M) is a covering Clearly

sup |lv,J\|,,= co for s #0;

v fed,

nevertheless

su v,JI\|,) = sup inf |jvJ-+ T|!,, < 00,

(3.9) ne, lil it p ved, „ I H Pp

where the norm is taken either in (H9?)%(M) or in (H”)4(M),

Let us prove (3.9)

Due to the definition of the norms we can suppose VN = 1, M = R" (or M = R"+)

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Let Bo= A, B’= A’ or B’= A‘ for the cases of the spaces AP?(R") (M = R"), Hÿ(R"'?) or HP (R??)(M = R?”), respectively; due to

Lemma 3.3 and isomorphical properties of B° (cf § 3.1) we easily get OD sp = COBB = Ctl) = Condi, = Csup (6): = C,

where C depends only on B="

Let now M’ be another r-smooth n-dimensional (not necessarily compact) mani- fold with or without boundary; suppose there exists some r-diffeomorphism

8() :Ũ, Ù,, B(x) =}, xeM, yeM'

of neighborhoods of the points x and y The operator ` (3.10) Balt) = p(B(t)}, te U,, 0 > tẻ U, is bounded 8„ c #Ø((Hữ)Š(M), (Hữ)Š(M)) nh #((H9)Ề(M)), (H9)Š(M))

and ¡ts rcstrictions on (/ÿ)Ÿ(U,) and on (#°?)*(U,) are isomorphisms (f,, and its inverse Pz? are obviously bounded in (g’)*(U,) and (H*")*(U,) for s «+ integer;

for s # integer their boundedness follows from interpolation Theorem 1.11) Similarly to A, (x € M; cf (3.8)) the localizing class 4, () € AM’) is defined

The operators A€ £N(M) and Be LYM’) are called (A,, B, A,)-equivalen

(or quasiequivalent; cf [6]), written A ~ B © B, if there exist two neighborhoods

U,Ăc AM, Uy c M’ and a diffeomorphism ~ : U, > ,, such that

»=fQ@), BAB, 2B

All properties of quasiequivalence listed in [6], page 577 (with exception of the property b)) are valid in the considered case as well

With the help of Thecrem 3.4 we easily prove the following

THEOREM 3.5 Let Ae YN(M), BE LS(M’) and

AX Bw B;

the operator A has a left (right) A,-vegularizer if and only if B has a left (right)

A,-regularizer

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SINGULAR INTEGRAL OPERATORS II 205

LEMMA 3.6 If the operator Ae L(LN(M)) = Lo,(M) has a left (right) 4,-regularizer, it is left (right) A,-invertible

Proof Let, for definiteness, A has a left 4,-regularizer

RAv,I = vJ+T, Te S(L¥(M))

Since

inf || vp ||, = 0

ved,

for any g € LY(M), there exists such ul € A, that

(3.11) \| Tull, < 1;

let 0, € 4, be such that

0,9 = 0,0, — Ủy) then

RAv,J = + Tu) 0,1 and I + Tvp is invertible (cf (3.11)); therefore

(I+ Tuo)7*RAt,T = 0,1

and A is left A,-invertible 2

LEMMA 3.7 Let Ae L(LI(R")) (or Ae L(L}(R"*))) and

V,A=AV,, V,0(t)= et) (A> 0)

The operator A has a left (has a right) Ao-regularizer if and only if A is nor- mally solvable and dimKer A =0 (is normally solvable and dimCokerA = 0, respectively)

In particular A has a Ag-regularizer if and only if A is the invertible operator Proof Let Ahasa left Ao-regularizer; then A is left 4g-invertible (cf Lemma 3.6)

(3.12) RAtgl = tol;

we prove now that

(3.13) inf ||Ag|, > 0,

llølls=!

which is equivalent to the normal solvability and dim Ker A = 0 (ef [5])

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due to (3.12)

1 = lim ||t;@;ˆ„ = lim |, V,RV,ACi@jip <

Ane Awd

(3.15)

< RL AG lp + Alp HH | E26; — Gjlpl = 1 Rip AG; p>

where

u,(t) = rp(At) = Vir (t)

Due to (3.14) the inequality (3.15) is a contradiction; thus (3.13) holds If A has a right Ap-regularizer the conjugate operator A* will have a left 4,-re-

gularizer and the result follows from duality (A is normally solvable and

dim Coker A = 0 if and only if A* is normally solvable and dim Ker A* = 0)

A will be normally solvable and;dim Ker A = dimCoker A = 0 provided A

has both (left and right) 4o-regularizers; therefore A will be invertible ~~ a

4 SINGULAR INTEGRAL OPERATORS ON A COMPACT MANIFOLD

1° DEFINITIONS AND SIMPLEST PRCPERTIES Let M be a compact n-dimensional

r-smooth manifold with the boundary ¢4f #0; {uj}j.1 be a covering of M,

l l

Lys MucM, (J U; > 6M; B:u; > u5 be homeomorphisms on compact dc-

rel g=k+1

mains }, , up CR", ue.,, , uc R’t

The operator A €¢ LN(M), js’ <r, 1 <p < © (ef (3.7)) is called the singular

integral operator provided (cf § 3.2”):

(i) Agi — gA € S¥(M) for all ge C'(M):

(ii) A ~ B, * WS for x¢ @M,xeU, 9 M (ve B(x), a,(8) 6 (Hm,)N? X(RP)); é

(iii) A ~ Ø, Š MẠ_ for xeêMcU, (y= B(x) =(0,1s, ,3,), GS) E €(Hm,)* Ä(RẺ))

a,(¢) (xeM UCM cf Mf ) is called the symbol of the operator A

Lemma 4.1 The symbol a,(¢) € (Hm,)* *(R") (xe M) of the singular integral operator AS L3(M) is uniquelly defined

Proof We have to prove that if for a fixed x € Af

(4.1) A^f,* WỆ A^ B, Š MÃ,

where b,d¢ Hin,(R") and k=:0, &=— 1 for xéêÊÖ/, xe €Àí, respectively, then

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SINGULAR INTEGRAL OPERATORS If 207

Let first x € GM (hence & = 1 in (4.1)); from (4.1) we get

4

Wh_a= WE-WE~0 (vy=B,(x));

hence (cf Definition in § 3.2°) for any ¢ > 0 there exist an operator T, € ©,,(R“*) and a function v(t) € CS°(R"), vo(t) = 1 in some neighborhood of y € R"* such that

lIteWš + Telly <e@ (g=b— đ);

due to the isomorphism properties of A§, , and Lemmas 3.2 and 3.3 we obtain

(4.2) [eV + Thy = AL vgAAL WEAGS + ALT AS lp = []toWE + Tellp <6

where 7€ €(1?(R"+)) and

iG + lƑ| +1

B60) = ( lễ — lếi — 1 ) s(;

due to the compactness of supp tg we conclude (cf (3.3) and Theorem 1.12)

tạW (,-#-8, „e€(1?(R“')), since sim 8) — g()g,(6)| = 0; (4.1) yields then | eoW ee, Llp <é and due to (1.14)

(4.3) sup Ig(f)gs(¢)i < IIxW ee Jp = ix ee, lilp < lÍ:toff ge ll|p <8;

where x(€) is the characteristic function of the set {ế e R”: rạ(é) = l} (therefore

te = Y); from (4.3) inmediately follows b(€) = d(é)

The case x 0M (k = 0 in (4.1)) is completely similar ø LEMMA 4.2 The symbol a,(€) of the singular integral operator A€ LY(M) is a continuous function of x € 6M

Proof Let A € £3(M); by the definition of equivalence (cf § 3.2°)

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for any ¢ > 0 there exists v,J € A, (cf (3.8)) such that

1 PBi AB je — Wa) sp < 85

let x,z€U,, ,(x)€ Ú, ={ệc R*†:ø(6):= l}; IÝ be Az„œ is such that supp ở, c , and

`.9.|871A; ~ Wed sp <8;

then

SA — W* ] „ < Cỉ Bevd BAB) — WAY", ấp

+ €I, 0;[;z`A;- — W2} s < G(l + Be sp) §;

where C, depends only on s and p; similarly to (4.2) — (4.3) we obtain

(4.4) sup i[a.(¢) — a,()]g(6)! < Cid + | te, sp) &

due to (3.9), from (4.4) it follows

lim sup a(S) — a(e)} == 0, 3

z,x€u)0f

REMARK 4.3 Similarly to Lemma 3.2 one can prove that the symbol a¿(4}

is a continuous function of xe M\@M as well

Remark 4.4 If a,(€) € (Hm,)%* \(R”) is the symbol of the singular integral operator A € £N(M) and inf jdeta,(0); > 0 for all x € OM, the numbers

‘Oras

À6), tL 1—s < Red,(a,) < J — 8 Pp

ò,(4.) = 3 5

tj= l,2, ,MW,1<p< œ}

can be defined (cf §3.1°), where ,(a@,), ., 4y(a@,) are the eigenvalues of the matrix uz (1,0, ., Oa,(+-1,0, 0) (with regard of their multiplicities); partial (p.s}-

-indices zy(0', x) < < (0, x) (0' â S"-đ, x © CM) are also defined as in § 2.1”

In the scalar case N = 1 all numbers 6,(a,) and «(0’, x) = %,(0', x) are conti- nuous functions of x € 6M (since a,(¢) is continuous with respect to x ECM) and therefore x(0’, x) = const (for # > 2), x( 0’, x) = x (1) (for = 2)

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SINGULAR INTEGRAL OPERATORS II 209

Conditions (i) and (i’) are equivalent (cf Seeley’s remark in Mathematical

Reviews 31, # 3876)

2° ExAmeLe Let M c R” be a compact domain with m-smooth (m > 2) boundary 0M, let r be as in Theorem 1.1 and [s| </ < m

Suppose

(4.5) DẸO(¿, 0) e(H'?)NxN(M, S1), [kh <1, Q(é, 0) dd = 0

6"-1

and for ¢ > 0 there exists a function

qd (4.6) 2.(€, 0) = ` 9;(6)9,›(0), j=l Q)(0) dO =0, QE (CIN*M(M), Qj € (H)NXN(S"-3), gine such that

(4.7) 2 — #,|J,„ = pane MAXIDEAOCE,) — Q(E5-)Ilgra¢gn-ty < £

Consider the operator

Qe, € — 0) lễ — ml" due to Theorem 1.1 Ay € Zj(M) (cf (3.7)) The operator Ayo(é) = | o(n) dn; M

T = Aygl — gAy (geC"(M))

has a weak singular kernel and therefore is compact on L}(M); due to Theorem 1.11

(cf (3.7)) Te S¥(M)

Let „: ⁄¿ —> uạ be some homeomorphism of the neighborhoods u, < M,

ug < M,, where B(x) = 0 and

(4.8) Mem {en if xedM

R" , if xeM\eM;

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The equivalence

(4.9) Ay ~ 8, © Ay (x EM, y = B,(x))

is valid

Proof We prove first the equivalence (cf (3.8))

(4.10) Ay “= Ay,

where

(x,

A,9(é) = EE? on dy

} is — Fp:

Suppose Q = Q, (cf (4.5)), x € M\éEM and suppe, ¢ M (oJ € 4,); then in

virtue of Theorem i.1 and Lemma 3.3

vel Any — Ay] sp = -Đy [Âm — Ax sp <

4 4

_ Mp Sine „Ổn 4jsp, lap = =3 H Qj Ajo, A~*

je -

4 ~ qo x h

< ` AS, Qy AS il, ASApA-S |, [ASDA < ` "_Š 7ñ mẽ nen

jel Jol

< 3) sụp (E)1Qn(2) — Ql]! Aly < jot < Yell, jel Sup ly, (ZQ(Z) — Qa

where Q(€) == Qy(C) ~ Qr(x), ng Anp(é) = Vi 8 o(n) dy M and jj: jl, is defined by (4.7) Thus

inf J vl Aye — ALT] sp ==

tr

and (4.10) is proved for x€ M\@M, Q = Q,

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SINGULAR INTEGRAL OPERATORS II 211

For proving (4.10) we need to consider only the case xe 0M, 2 = Q,

Let supp 0, cu, (gl € Ay; cf (3.8)), vo = Beto, 2.= P10, Qa(Q=Oule)—

;;(x)) and Avg = = Bx AjoBy (cf (3.10)); applying Theorem 1.1, Lemma 3.3

and inequality (3.9) we get

q q ~

llt3L4„ — 42]Ìl„ < C¡ Y} [lee Aspor8lilap< Ca¥ [AL LOR AMeRAT|II, <

j=l jel

4 ~

<6, 3, l|Ä- sš8j.4=*\, I+ 4z4x'|l, 4i s$⁄4;°1|y = jal

€ ` supls,(E) [9a(¿) — 9nG)]I l9allu:

js rary

hence

inf \llof Ang ~~ All = v TEA,

and (4.10) is proved

Next the equivalence

(4.11) A, = B, > A, (y = B(x)

will be proved, which together with (4.10) yields (4.9)

Itsuffices to consider Q(x, €) € C(S"~") for all x € M (otherwise we can appro- ximate 2 and use the continuity property of the equivalence (4.11)); rewrite (4.11)

as (cf § 3.2)

Bye Ã,B, 2 X

in [6], Chapter I, § 4 was proved that

(4.12) HABe = A, + B+ T, where Te S(L*(M,)), Qo(t, t — th (t) de |x—/Ƒ By(t) = \ 9,(B,(x), 6)=0, D/OuŒ, ¿)c(C®9)WxX(S"~}), la < m

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3° STATEMENT OF THEOREMS

THEOREM 4.6 Let M be a compact n-dimensional r-smooth manifold with the boundary OM #0; A&€ LN(M) be a singular integral operator with the symbol

a,(é) € (HC™*+2)N*X N(R") (xe MUCM, m>n/2, —co<s<o, 1<p<o)

which is elliptic inf {a,(0)| > 0

b1

The operator Ae LN(M) is Fredholm if and only if (cf Remark 4.4) (4.13) 6,(a,) # Joo Ss for allj =1,2, ,N

p

and all partial (p, s)-indices disappear

(4.14) x(,x) = =z⁄v(0,x) =0 (0' ES", xEM)

THEOREM 4.7 Let all preliminaries of Theorem 4.6 together with (4.13) held

If #(0',x) > 0 (j= 1,2, , Ns 0’ © S’-*; x eM) but (4.14) does not hold

the operator Aé YN(M) has a left regularizer and dim Coker A = oo

If 2;(8',x) <0 (j= 1,2, .,N; 0’ € S’-?; x EM) but (4.14) does not hold, the operator A € YS(M) has a right regularizer and dim Ker A = oo

THEOREM 4.8 Let M be a compact n-dimensional r-smooth manifold with the boundary €M#@; A€ LX(M) be a singular integral operator with the symbol

(6) 6(HC"+?)NxN(R") (xe MUÊM, m>n/2, — 00 <s <0, 1 <p < co) If inf j deta) =0 (xe MUCM, 0€S"-*), the operator AE LN(M)

has no left and no right regularizer

4° PROOF OF THEOREM 4.6 Due to the definition of singular integral oper-

ators 4€ LN(M) and to Theorem 3.5, A has a regularizer (i.e A is a Fredholm

operator) if and only if W2, € P3,(R") for all xe M\¢M and W2 € £3,(R"*) for all i

xeé@M have A,-regularizers (cf (3.8); here » = f(x) for xeu;, J uj=M u eM;

gol

ef § 4.1°)

We prove now that W2 € PN(R") (x € M\GM)and Wi € £E(R"*) (xe CM) have A,-regularizers if and only if Wo ¢ LIAR") = L(LN(R")) and Wie, e £(LN(R"*)) have Ao-regularizers respectively, (g,(€) is defined by (3.3))

Let Wo, € £%,(R") has a left 4ạ-regularizer:

(4.15) RW? uyÏ = vf+T, uledA,, TEeSh(R"); then (cf § 3.1°)

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SINGULAR INTEGRAL OPERATORS I 213 where

V.y,0(Œ) = O(t FY);

obviously

V,W? V_y,= Wo? ViayV_,y€ Ag

and using Lemmas 3.1, 3.3 we get

(4.16) RoWo vol = vol + Tạ;

Ry = V,ASRA-V_,€ ZN(R"), Tye S(L9(R")); hence 4 € L(LN(R")) has a

left Ap-regularizer

Converting the argumentation from (4.16) we easily obtain (4.15)

Similarly, if Wi, € ZN(R") has a right Ap-regularizer, Wa, LA(LE(R")) will have a Ay-regularizer and vice versa

Let now Wa, € PN (R") has a left A-regularizer (x € OM, y = B,(x)), (4.17) RW; 0,1 = uT+T, vfeA,, Te SA?) R"*));

then (cf § 3.1°)

R(V,ALWLAZV_») Vy Ah agAz°V-) = V,ASagA- V_y + To,

Ryo = V, As RAZ °V ,e#Ø(17(R'")), Ty = VA TADV_, & S(LY(R"*)), since x € OM, y = B(x) = (0, ye, -, Yn) and therefore

ViWiV - >= Wis Via,V_y € Ag

(if y # 0, Viy ¢ (LI (R"*))); using Lemmas 3.2 and 3.3 we get

(4.18) RoW vol = tgÏ + Tị, T,cSŠ(17(R"*))

Converting the argumentation from (4.18) we easily obtain (4.17) Similarly is considered the case of a right 4,-regularizer of Wi

Summarizing we conclude: A€ YN(M) is a Fredholm operator if and only

if Wa, € L(Lp(R")) have Ay-regularizers for all x € M\@M and Wie, € L(L(R"™))

have ” Ag regularizers for all xe d0M

The proof of the theorem is completed now with the help of Lemma 3.7 and

Theorem 2.7

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5°, PROOFS OF THEOREMS 4.7 AND 4.8 These theorems, similarly to Theorem 4.6, are simple consequences of Theorems 2.7 — 2.8 and Lemma 3.7 if we notice

(cf (4.15) — (4.18)):

(i) if all Wo e Z(L7(R*)) (xe MỀ\£M) and all W2 „e.Z(L7(R*†)) have

a left (have a right) 4g-regularizer, 4 has a left (has a right) regularizer;

(ii) if Wi, € £(LN(R”)) (x € M\CM) or Wa, € P(LN(R"*)) (x ECM) is

not a Fredholm operator for some xe M, A will be not a Fredholm operator

as well B

This work was planned and fulfilled during the visit in the TH Darmstadt (West Germaiy) as a Fellow of AvH Foundation in 1980 198 1

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2 Dupucuava, R V., On integral equations with fixed singularities, Teubner, Leipzig, 1979

3 Duptucuava, R V., On multidimensional singular integral operators I: The half-space case, J Operator Theory, 11(1984), 41—76

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Moscow, 1973

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