J OPERATOR THEORY Copyright by INCREST, 198
11(1984), 41.76 Ẫ Copyright by Í 1984
ON MULTIDIMENSIONAL SINGULAR INTEGRAL OPERATORS I: THE HALF-SPACE CASE
ROLAND DUDUCHAVA
INTRODUCTION
Fredholm properties of a multidimensional singular integral operator
(equation)
Q(x —
(0.1) A(x) = ay 9x) + \ SA =7) g0) dy =/G), |lx—l"
Rˆ
xe R"* = R+ xR"-!, R+ = [0, 00), is investigated when 9, fe LN(R"*) (1 <p < < oo); in particular, criterion for the existence of left and right regularizers (of left and right inverses) are obtained (cf Theorems 2.7—2.8)
Those results will be used in the second part of the paper for the investigation of systems of multidimensional singular integral equations on a compact manifold with boundary in vector Sobolev-Slobodeckii spaces
Most interesting in ư 1, which deals with auxiliary propositions, is Theorem 1.4; it shows that Calderon-Zygmund (cf [4,5]) and other similar theorems on the boundedness of the operator (0.1) in L,(R"*) space is valid also for the space with
weight L,(R"t, x7); the weight x% is not pointwise here (as in Stein’s Theorem; cf [22])
Besides the interest of its own singular integral equations play an important role in mathematical physics, mechanics and boundary value problems for the partial differential equations (cf [8, 13, 18-20, 22, 24, 26, 30, 32]) It is impossible to observe here all results, obtained earlier on this subject and we refer to books {8, 13, 18, 20—22, 32] We mention only a few results closely related to our Inves-
tigations ,
Classical theory of multidimensional singular integral equations on mani- folds without boundary were developed by Tricomi, Michlin, Calderon, Zygmund, and others (cf [22]); later Simonenko [31] treated the case of a half space R"* and of a compact manifold M with boundary 6M # @, but operators were investigated
Trang 2Simonenko, Wisik and Eskin (cf [13]) treated the case of Sobolev-Slobodeckii spaces H°*(M), 0M # @ and used obtained results for the investigation of boun- dary value problems for partial differential equations
There was one more attempt to investigate operators (0.1) in L}(R"~) and
in Sobolev-Slobodeckii (4°?)4(R"*) spaces (cf [29]); but the author made mis- takes (proofs of Corollary 2.3 and Theorem 6.1 fail); nevertheless the reduction
of the matrix-case N > 1 to the scalar one N= 1 can be carried out as in [29] (obtaining more precise asymptotics (2.7) instead of Corollary 2.3) and we follow this line here; instead of Theorem 6.1 from [29] we prove below Theorem 2.12 Concerning Theorem 1.4, we do not need it here, but besides the interest of its own it can be used together with Theorem 2.12 for the investigation of the ope-
rator (0.1) in weighted spaces L'(R"*, x7) Besides Stein’s Theorem, where the weight
function âx;* is considered, recently appeared papers (cf [7,9]) dealing with much more general weight functions than ‘x,|* and |x'*; but singular integral operators have bounded and sufficiently smooth characteristics there (cf also [25])
I am indebted to Prof E Meister for stimulating conversations concerning
the paper
I am indebted to Mrs C Karl as well, whose help in preparing the manu- script was invaluable
1 AUXILIARY PROPOSITIONS 1° NOTATION
#@ — end of the proof
R’ =: Rx x R, R= (—oo, 00)
R’* =: R+ x R’-1, Rt = [0, oo) (s] integer part of a real number se R
tư " twee 1/r ; def ,u, Ậ “ “ R Ie: {° a] Ses Easy GRY K1
ẹ cử (Ga, std en) (ờ == (ờ1, ste en) € R’; ờ == (ấu ờ"))
"1<: {0:06 R”, 16! == 1} — unit sphere in R” Ali : ann DE Ee (A = (Ay, ., 4); x eR") CX,+ " xX," - & k &
XE es xtex,? x8 (we R’, k= (ky, .,k,))-
=~
F and #-1 — Fourier transforms
; 1
F Q(t) =\el@(ờ) dờ, F-W(g) = ——-\e y(n) dr,
(2)” R”
tờeR’, t-Ệ = ` hyn k=l
"
Trang 3SINGULAR INTEGRAL OPERATORS 43
F te, — partial Fourier transform
F 1,-8.0) = \ een, Edy (E= En &) ER’) - Ẫ
L(X,, X2) — space of all linear bounded operators between the Banach spaces X, and X,
def
LX) LX, X)
S(X,, X2) — space of all compact operators from #(X,, X2)
def
S(X) = SX, X)
supp a — closure of the set {€: a(ờ) 4 0} for a function a(€), € e R” (support of the function a)
Co(R”) — algebra of all infinitely differentiable functions on R” with compact
supports
P,, — restricting operator P,,đ(t) = e(t)|R"*
l,4 — extending operator (right inverse to P,,) 4.0e(t) = e(t) if te R’* and 1, g(t) = 0 if te R’\R"*t for g(t) defined on R’*
1
Xu.) = 1X,.ĩ,) = mae + sgnt) (=( ,f„)€ R”, k =1,2, n)
H(R") — algebra of homogeneous (of order 0) functions a(Agờ) = a(ờ) (A>0,
€eR’)
AHC™(R") — algebra of homogeneous (of order 0) functions, having continuous
derivatives Dfa(@) for all ||, < m on the unit sphere S’-1
(HC")X*N(R") — algebra of matrix-functions z(Š) = ||2/(@)|x„-â with entries aj, € HC"(R’)
diag (a,, ., dy) = llđ;ừ/Òllx.+ (Oj, — symbol of Kroneker)
def G a, a
diaga” == diag(a,’, a,, ,4) (a= (a, .,4y), 6 = (01, -, Oy)
Ẽ
1p
L,(M, p) — the space with norm ||đ||,, = (\ |pựi? %) , where dy is certain measure on M
LY(M, p) — the space of vector-functions p = (@1, -, Qn) € Ly(M, p) x
N 1/0
Trang 4-H°?(R") — Sobolev-Slobodeckii space, defined as the closure of C#(R") with
respect to the norm
1p i
‘one=(\ FM SFO Pa), 1 <p < 00, 5ER
R#
JTHP(R??) (HP (RẹP—)) —- subspace of H*?(R"), obtained by the closure of the
set of functions @ đ C?(R") with supp @ đ R"+ (suppg đ R"~ : : R°\R"*, res-
pectively)
H?P(R"*) — the factor-space °?(R“+) SỈ H°P(R")/HỵP(R”—)
For s <0 the space #W**{R") contains non-regular distributions, but for
H1
$ my it is a Banach algebra (Peetre) and all functions there are continuous (Sobolev); the norm in the space H*?(R”) (s > 0) can be defined also by the equality
‘Dờ + py) — DE 2 12
wie = (Vion? + | of _Dšự(x + ơ) — Dio)? ax)
‘k' : oom ly at 2A
VỊ vi
R" R” Rf
where s:=:?mr + 2, ệ < 2 < l and m is an integer (cf [12, 30])
H’?(M) -— Sobolev-Slobodeckii space on a r-smooth n-dimensional com- pact manifold without boundary (—oo <5 < oo, 1<p<oo,r>'s') defined as follows: we choose a finite covering U,, .,U,, of M (U U; =: m) and ho-
jul
meomorphisms x,: U? > U;, where Ufc R” is an open set; let ,, ,,, be the par-
„”"
tition of the unity on M, subjected to this covering (I.e 9; ;() = 1, supp ĩ, c ĩ; jot
and x7 W(x) = w,(x%(x))e C(U$)); suppose the transformation x;;: Ậ;xj7" (%;;: U2 NU? > UPN UF if UPNUP#GD) belongs to C(UPNU}) and the
Jakobian ,Dz,,(x)â # 0; the normin the space H*(M) is defined by the equality
i ian = (35 5 se i jel
the definition depends obviously on the partition of the unity, but it can be proved (cf [1, 17, 20, 34]) that different norms defined by the different partitions are equivalent
Trang 5SINGULAR INTEGRAL OPERATORS 45
It is known that H3°(M)< H°"(M), H9°(M) = H°(M) = L,(M), HẼ°?(M) =
-= L,(M) and H(M) = H°°(M) for l/fp—1<s <1/p; conjugate spaces are
(HjP(M))* = HM) (—00< s< 00, p’ = p[(p — 1) and (H*°(M))* = Hạ*?(M),
(H*?(M))* == H-‘?(M); for an integer m the space H”?(M) consists of all func-
tions g(x) having p-summable partial derivatives D‘@(x) for all |k|, <
C" H°"(X, M) — the space of functions b(x, ê) (xe Xo R/, êeM = R’,
m=0,1, , 00) which have the property: all derivatives D‘b(x, €) = 5,(x, e € H*?(M) and are uniformly continuous
lim ||2Ò(%, - } — Đ„(a, - )ll„ = 0
BON ọ
for all |k|, < mand xe X (x eX vu {oo} if X is not compact)
CH5(X, M) & COH*°(X, M)
2°, SINGULAR INTEGRAL OPERATORS Consider the (multidimensional) singular
integral operators of the type
(1.1) A(x) =\
R“
where Q(x, €) are measurable functions Q(x, AE) = Q(x, €) (A> 0; ie Q(x, -)€ e€ H(R") for all xe R”); the function Q(x, €) is called the characteristic of the
operator (1.1)
Q(x, x — y)
e(y) dy, |x — y|"
THEOREM 1.1 (cf [22]) Let r > —1/2 (r >O0 or r >(n— 1) (1/p’ — 1/2)); l<p<o, p’ =p/(p — 1) and:
a) Q(x, 6) dờ = 0,
gt-t
1 b) c= rs — EU, STs ơ — ess sup || D’Q(x, eR" | › Vila (sSĐ -)\|,r2-cn-1, < 00
The operator (L.1) is bounded in the space H°P(R") for p = 2 (for 2 < p < œ and for è < p < 2, respectively) and || A} wR") < C-C,,, where C is a constant,
independent of Q
THEOREM 1.2 (Caldờron-Zygmund; cf (4, 22]) Let Q(x, 4€) = Q(x, €)
(A >0; x,€eR") and:
a) Q(x, 6) do = 0,
git
b) Cp = SUP Q(x, On, sry < [<< p=h}
Trang 6The operator (1.1) is bounded in the space L,(R") and \|All, < C:C,, where C is a
constant, independent of Q
THEOREM 1.3 (Caldờron-Zygmund; cf [5, 22]) Let Q(x, đờ) = Q() =
=O(1') (2>0,ceR*) and:
a) 9(0) d0 := 0,
ne gh
b) l@||y „sả, < eo Bint QI], gn-2), < 00,
where Q(0) = Q(0) + O(—0)
The operator (1.1) is bounded in the space L,(R") for all < p < Now we prove the following
THEOREM 1.4 Let (1.1) be a bounded operator in the space L,({R") (1 < p < co)
and |Q(x, - ie A(R") for all xe R" If
1/p”
Kp=: sup ({ ‘Q(x, Ai” 4) : < œ,
xeR” sẽ
then the operator (1.1) is bounded in the space L,(R", |x,|°) for —I/p <2 <1 —I/p
and ||Al|, < C(|A||, + K,)
If Q(x, 6) = QE) and
K, = \ (2(0)j dờ < Ẫ,
stl
then (1.1) is @ bounded operator in L,(R", jX;!") and” ||Allpa < CC|All, -+ Ky)
Proof Assume first Q(x, €) = 2(€) and
(1.2) —-L<z<0,
P
It obviously suffices to prove the boundedness of the operator
oe _— lu) Q(x — y) I
Bo(x) = 2 \ —™ Ft p(y) dy
lxii?
rR”
in the space L,(R”), where B = —a (due to the boundedness of A in L,(R"))
Trang 7SINGULAR INTEGRAL OPERATORS 47 Introducing a constant 0 < 6 <1! and using the Hờlder inequality we get
bớ)? —_ 6p 1/p
IBp()| < -C- ({ Ile()I?i@(x — Í)|Iml #) x
lal \) be — yl" a — il? R x (ẻ Jì(x — Í)| dy y" =
be Bi" bea yal Ẽ Lal?”
R
= cal \ ee x
"` Rr” Ix — yl" ba — al?
(1.3) x ( \ Dị — xịt dy ly j Q(x = y)| dy’ y" < NV
R
iN
< (<5 lo)? Q(x — yylbv dl? any" x
Jal?) ix — yl" a — val?
R x1 x dy, \ (21, €’)| dờ’ + lạ — â|1~# |y;|Ẽ? {+ |š“12)”⁄2 —0o RU +4 dy, ( Jì(—1, #9) say la — 4/278 [yy | ad + |ê')22 x1 ga
because | |xâ|# — |yal?| < |xị — yi] and we changed the variables &’ = (&, ., €,),
t, = ee
Xị Vy
|Q(xị — YrÍ x” — y’)| = [2 (sgn (x, — Đụ), Š))
Consider now the homeomorphism of the space R’-1! on the semi-sphere
Sr = {0 = (0,0) S"-1: 66, > 0} defined by the formulas
(k = 2, .,”; due to homogeneity of |O(@| we have
1.4 =? = oe ( = ' -|/ )
Trang 8where đ == -:1 (inverse mapping is given by formulas đ, == 6,/0,, k == 2,3, - , n)
We easily obtain from (1.4)
~ “se — fre _.- tự _—Ƒ 2 2
(1.5) 6 —ư2=— 2288 WO) TO
r*)r?(6)
To change the variables in (1.3) we must find the relations between dờ’ and dờ; let’s choose for this purpose the partition of R"-1 by the curves
r(ờ') =: const, ứự — 3 coordinates of &’ = (&, .,ờ,) are fixed; E,on yet (hk == 2, .,n; —0o < ft < Co)
Using (1.5) we calculate the ratio of distances between two points across these
curves and their images on the sphere S"-1 when these points are converging to
one; we get
lim Ooo = —— for r(ờ’) = const,
Boe ig’ — Cre’)
_ 10-6 1
lim _ứ TỐ, =-~_- for € = yf;
Zag 17 — CC")
do _ la] ae i,
dờ’ r(đ") oD re’)
Using the obtained formula we get
\ iQ(e, 6) de’ \ [2(e, S| de” _
ase rE) hence (1.6) = | I(s, I8,|~18')| dỡ = '9(0)| d0 < —1 n=l Se Se < \ |O(0) đỡ = Kị 1 Ss Due to (1.3)
[Be(x)i < —( 1@(y)!?I 9x — y)i trl? dy " Si RẼ Ix — yl” xy — yyl7F
x dời yr < Ok”
yy x,['-8 Ly?" la [plese
Trang 9SINGULAR INTEGRAL OPERATORS 49 because đ dy, , (1.7) ————————- =Œixâ|ấ-%, — I — Fil” p 3 [xa | >ịm if 1—fB+ dp’ > 1 Hence
|[Be(x)|? dx < C;K†” Iyil?? lel? dy x
R” Rr"
x [xy — Wil? dxy \ {Q(x — y)| dy’ <
in
|2x|8*°P Ix — y
œ
Pịp' dp Pp dxy
< GK” \ II” leO@)“dy \ —— TT
|xul#T“? lxi —yA!~ấ
R” — 00
x (jaa) 40 = caxrr| lo)? dy,
R sink n because co dx (1.8) ——————=C.\x|-* [ql xạ — y8 7? —co
if B + dp < 1; we used also the inequality (1.6)
We will be done with the case (1.2) if we prove the compatibility of inequali- ties (cf (1.7)—(1.8))
0<d<1, 1—B+6p'>1, BP+6p <1;
but they are compatible, because B = —a and 0 < B < 1/p holds (cf (1.2))
Let now
0<az<i-L=-L (-z<-z<9)
pp Pp
consider the conjugate operator
Qơ(x — — ? WO) dy, @*(2) = BB
A*@) = NT
R”
Trang 10in the conjugate space L,.(R”, !x,!~*); in virtue of the proved part of the theorem
AẼ is bounded and hence, A is bounded in L,(R”, |x,|°)
There remains to prove only the first part of the theorem (case Q(x, ý) #
# Q(đ))
Assuming again (1.2), instead of (1.3) we write
Box)! < ~~ ( \ li?
H
gly)? yal? dy y" x
[x — yi" jx — â|7
R
' —_ , 1/
x ( \ ae "<
[x — yl? [xy — yal? yal?
R”
< „5ẻ i@@)I? Iyi|°” dy "x
7 ' “a ~
Ixy? RẼ X — y," [xy — MỊỊT?
co
{ Jamal đa vu I8G, 1, EDIE dể Ji ly âeR” R”” (+ er"?
the remainder is the same as in the considered case f2 From Theorems 1.2—1.4 it immediately follows:
COROLLARY 1.5 /f conditions of Theorem 1.2 (of Theorem 1.3} hold and —\/p <
<a<1-—1/p, the operator (1.1) is bounded in the space L,(R", |xị°) 3° INTEGRAL CONVOLUTION OPERATORS Let a(ờ)eđ L,,(R”) and
(1.9) Wig =F aFo (pe CHR");
by ,(R") denote the algebra of all (multipliers) a(ờ) for which W admits the con- tinuous extension to the space L,(R”) (1 < p < 00) By m,(R”) denote closure of
the set J M,(R") with the norm jja||} = W⁄2l|;
r€ờ(, P’)
Hm,(R") = m,(R") 1 A(R’)
If a(x, €) € m,(R") depends as well on the variable x € N, the operator will be written
as Wix,.; if suplla(x, -)||9 = supl|W2,,, ||; < 00, we write a(x, €) e L.om,(N, R”)
Trang 11SINGULAR INTEGRAL OPERATORS 51
The operator (1.1) represents the example of the operator Wix.,.) where
eO(x, t) dt
(1.10) ax, =|
It”
n
R
(cf [22, 31]) and a(x, €)e HM,(R") if (1.1) is bounded in L,(R”); a(x, €) is called the symbol of (1.1)
THEOREM 1.6 (cf [29]) Let Q(x, &) = Q(); the symbol a(ờ)€ H°(S"-}) (cf (1.10)) if and only if the characteristic Q(€)€ H*~"?.2 (S*’-}), _
COROLLARY 1.7 Let a(x, -)đ H(R") for all xe R", max sup ||D* a(x, ara") < 00 jal, <a xeR
and r >(n— l)J2, r >n/2, r>(n— l)ợp + lj2 for p=2, for 2<p<c
and for è < p < 2, respectively (p' = p|(p — 1)); then Wiix,., is a bounded oper-
ator in H°?(R") for all |s| < m
Theorem !.1 and Corollary 1.7 yield:
CoROLLARY 1.8 Let a(f) đ H(S"~4) 0 H(R") and s > n/2; then a(€) đHm,(R”)
forl<p<o, |
If s > n2, then HC(R") < Hm,(R")
Consider the operator
Z1 đef 0
(1.19) d(%, ‹ ) = Py Woe, hiss
if a(x, €)đ Lm,(R"*, R"), Wax, is a bounded operator in L,(R”*)
Obviously
AWE + Wk = Woy (k =0, 1)
It is also easy to prove the following:
PROPOSITION 1.9 If a(ờ), b()€.M,(R"), then W2W2 = W2,
Hf, additionally, a(€) has an analytic extension in the half-plane lm €, < 0 for all GER") (E = (&,, EYER") or b(E) — in the half-plane Im €, > 0 (for all
ư'c R*~!), then WiWi = W3,
Trang 12THEOREM 1.10 (Michlin-Hờrmander; cf [16, 21, 28]) If d
(1.12) ° sup \ 'x*Dka( x)? - “<< 00, kt <pasay 1 R20 R-:9-X'<4R R’
then ae m,(R")
Let M be r-smooth manifold with boundary 0M # ì and ‹„, S,, denote
the space of all bounded, all compact, operators Y,, = L(L},, Ly), Spr: = S(L5,, L,,) where either
a) Li, = HP(M), Ly =: HM) (1 < p< 00, —co <r < oo) or b) Li, = Ly = H'(R") (1 < p< 00, —00 <Fr < 00)
THEOREM 1.11 Let ạC Loy VL pr? then Aờ ê,, for
1 1-—-
Tờ mm 04.8 r = rạ(èl — ì)-rry, (0< 0<1)
P Po Pr
and
h 1 ì, :8
Alp < CHA pr gt Alyy
where ||A|l,, denotes the norm in &,,
If, additionally, Ae Spy, then AE Sy, (0 < 0 < 1)
THEOREM 1.12 Letaem,(R") (1 < p < 00), b(€)e C’(R") and jim a(ờ) ==
:= lim B(€) == 0; then bW2, Wb Ẫ S(H°(R") (€ S(L,(R") and bWi, Wibe
€ S(HP(R), H°"(R"*)) for isi <r (e S(L,(R"*)))
For s:=0, a,bđ C%(R") the theorem is well-known; the general case is treated with the help of Theorem 1.11 as the case 2 = 0 in [10], Lemma 7.1
4°, ON THE TENSOR PRODUCT OF OPERATORS If K is a finite dimensional operator
Ke)=ơ 5/0\//9eŒ dt (gE L,(R*), Wye Ly(R*) 0
j=t!
in the space L„(R*) (1< p< Ẫœ) and 4'<.ý(L,(R"-')), the tensor product
A=: K@ A’ is defined as
co
Trang 13SUNGULAR INTEGRAL OPERATORS 53 obviously A € ê(L,(R"*)) The closure of the set of such operators is denoted by
SYS SOL (R"+)); SM is an ideal inthe algebra Y, = Y(L,(R"*)) and, obvi-
ously, S, = S(L,(R"*)) c 60),
Lemma 1.13 (cf [11], Lemma 2.1) Jf Te SẼ and B; = C; @ I, where lim ||CaJ|l, = 0 for any pe L,(R*), then lim ||B;T||, = 0
jroo 17
Define the operator
Vig(t) = g(At) (0 < 4 < oo);
obviously Vz = V,,, and V,Wi = WiV, for any ae HM,(R") (k = 0, 1) Lemma 1.14 [If Te GO and V,T =TV,, then T = 0
Proof Let
Belt) = ote), peLl,(R"*),
where v(t) == | for 0 < 4, < j-* and o,(t)=0 if j-! <đ,; in virtue of Lemma 1.13
(1.13) lim || B;7||, = 0
For any đ > 0 there exists 9,(t)đ L,(R"*) such that ||l@;||, = 1, ||7@,||, >
2 |Ti,— 85 if a(t) = Ap (At) = A"?V9,(1), then ||@, 4||, = 1 and lim |[B}7@e, allp = tim |[a"?V, Vo) T@ellp =
Asoo 2Ȝ
=lim lỨ-/) Tự,lly = |TẼll; > WT, — € 5
hence ||,7||, > |èTèl, The converse mequality ||,7è|, < ||7èl, is obvious and,
therefore, ||8;T||, =: ||Tl|;; conclusion follows now from (1.13) ZB Lemma 1.15 Let ae HM,(R") and U be any neighbourhood of the point
y (0, Vax secs Yad; then
Wall, = inf Jy,W2 + TỊ, (k = 0,1), Te 6)
where x,(&) is the characteristic function of U
Proof Inequality c, = inf \|y,Wa + Til, < ||Wall, is obvious
ree)
Trang 14Assume now j|W2lj, — c„ = 4e >0 and T,đ G9) be such that c, +đ>
> ilxuWe + T,\l, > đp; without the loss of generality we can suppose that (0, 0, .,0)đ U; otherwise we can use the shift operator
B_yp(x) = @(X — ơ) = G(X, Xz — Jư; .; Xu — Vn) (i! B_yll, — > BL yWe = : WEB_y; Bz B,)
Let xz(ê) be the characteristic function of the set [—d, 6] X x [ 6, 6]
(6 > 0); then |jxs7.'|, < € for some small 6 (cf Lemma 1.13) and y5-x, = 75 Let as) be such, that |lp!!,=- 1 and ||W2I, < 1H2@n, + e: If V.g( = 9;(ấ) =: á~"/"o(2~1ỵ), then l@iy := ] and with the help of the equality
lị
limi Vata) Wp = Wl we obtain
IWSl, < IWApi, + 6 ~ limilW2xÒ) W4plụ + e = 2È
= limi, Vino) Wrelip + e= limllxsWa slp +E
€ lxsWai'p + ờs < |èxa(xuM2S + v T:)èè; + 2: S Š lèlxuW⁄2 + T:||; + 2s < Cp + 3e = | Weil, _
the obtained contradiction proves the lemma ự
LEMMA 1.16 Let a,bđ HM,(R") and y,(€) be the characteristic function of the set u, (k = 1,2,3); let (Ga, địa) X X (Gm đụ„) C ty for some ej < dy; (j = 1, 2, , n) Then
sụp â a() Đ(€)i = ẻWblls < , 2 :g = ||W2é[y =
(1.14) ,
= ` XLWfM?s, „r= âWaM), Ủy,
where
.4g= TES(L,(X)) inf ||A4 + Ti
Alt relations in (1.14) except the first one are proved similarly to Lemma 1.5;
the first relation || Welle < || We, is well known (cf [10, 16])
We need one more inequality, which is also well-known (cf [16]): if a € 47,(R”) and pe(r,r’), r’==r/(r — 1), then
Trang 15SINGULAR INTEGRAL OPERATORS 35
5° LOCAL PRINCIPLE Here some necessary information from [14], Chapter XI,
ư1 will be given
Let Z be Banach algebra with the unit element e; a set Ac Z is called a
localizing class if 0đ A and for any a, b€ A there exists ce A such that ac = be = ==ca=ch=c
Elements x, ye 2 are called A-equivalent if
inf ||(x — y)al| = inf ||a(x — y)||=0,
aca aờAa
and the notation x Ậ y is used
An element xe X is called /eft (right) A-invertible if there exist ze XZ and aờA such that zxa = a (axz =a)
A system of localizing classes {A,}.ea is called covering if from each choice of elements {a,}e0 (@,,đA,) one can find a finite number whose sum is inver-
tible in 2
THEOREM 1.17 Let {A,}yex be a covering system of localizing classes, and
A
letx ~ y, (%, VoE Z, OE Q) If x commutes with all ae \_J Ag, then it is left (right) wEQ
invertible in 2 if and only if y, are left (right) A,-invertible for ali w € Q
2 SINGULAR INTEGRAL OPERATORS ON THE HALF-SPACE
1° ON THE FACTORIZATION AND PARTIAL INDICES OF DISCONTINUOUS MATRIX-FUNCTIONS Let a(€) € (H1C™*+?)N*4(R") (m > n/2) be elliptic (nondegenerate)
inf |det a(ờ)| > 0; consider the constant matrix ges"
(2.1) a = a-(—1,0, , 0) a(+ 1,0, ., 0)
Ị
and let 4â, ., 2; be the eigenvalues of ự; with multipHcities rạ, , r; ( Yn,=N
j=l
Then the representation
đọ =gB(1)g—', detg #0 B(1) = diag [4,B"™(1), ., 2,B"(1)], 0 , v<k, 1 , v=k, Bt@) = |(b„(Ẽllf.+ - Oe = av—* (v—b!’
Trang 16The matrix-functions B’(a) have the property
B(x + B) = Bia): BB); B0) =1,
in particular
Bf(—3) == [Bf(3)]”1
Tf now Ậ, (t) = (2“)!ln( + i), we get lim [#„() — z_()]:z0, lim |z,() —
t+ - co fÈ-.co
— #_(?)] = 1; hence
lim E_())[B'„())] ` = l, lim Bœ-0))[B(+ữ))] "` = [BC]
The matrix-functions
(2.2) B„(Œ) = diag [Ba.()), , B4a.())]
are holomorphic in + ImÒ > 0 Let now
In, 1 ]
(2.3) ðj(@) = 6; =-—*, —1<Red;< (j=1,2, ,1; 1<p<oo);
2m p Pp
such a choice of 6; is obviously possible and they are defined uniquely Clearly
(2.4) bo < 1 — Ref6;-—6] Cl <j< J for some 6p > 0 Let k-\ k (2.5) 5=(6, ,6)), 5) =6 for Fr, </< 37, fH=12 5M, y=! pal
(t+ i)? = diagh( + i), ., đ 4 iPM;
clearly (đ + i)°B.(đ) = Bs(t)(t + i)° since (t + i)* is a diagonal matrix-function
having the same element inside of block of B.,(t) We set
az(ẩ) = (ối — i)-°BO"(E,) ga“ (1,0 , , 0) a(6)gB4(E:) (õi + 0P (2.6)
Trang 17SINGULAR INTEGRAL OPERATORS 37
Let us notice that if /= N (ie rp = =ry = 1) then B.() = ủ (é is the identity matrix)
LEMMA 2.1 ay(,, &’)€ C”*2(R) for all Ee R"-1 (cf (2.6)) and (cf (2.4))
DE la*(ờ:, ữ)—1„= O(g,|7*~R% +Ret—D — O(|e, [TẤT
(2.7)
ờ-> + Co, Lv= 1,2, ey N, k=0,1, Lop M+ 2,
Proof In [29], Appendix, it is proved that
Dị b(â) = O(6|T*—), k=0/1, ,m +2
-1,-1 G1 —i/
(seo = 8_(6)g~1a~1(—1,0, , 0) a(#) B,(&) — (=) )
(2.7) is now obvious since
[a„(šâ, ẩ) — 1; =éQ — )~“6(6) (ếi + DJ, = (ếi — i) 9b, (&) G+) B
By WR) (r = 0,1,2, ) denote the subalgebra of the Wiener-algebra
def
W(R) = {f() = c+ Fa(t): ge LR}, consisting of functions f(t) with the property
(1 — inkDE F(t) e WR) (k=0,1, ,?)
LEMMA 2.2 (cf [29]) W’CR) is a Banach algebra with the norm
WA” = Ill + x llq — )*DÒ/0)lly =
k=l
k=l
= |e + \ ig a+ yr lm + D#2g(0)| 44,
where ƒ= c + #g (ge Lị) and (D; + 1)“ = #~1(1 — 1)*Z
The singular integral operator
Spo(t) = + 0Πdc
Trang 18is bounded in WR) (it is decomposable) and the set of all rational functions, vanishing
at infinity and having poles off R are dense in W'(R) (it is rationally dense)
LEMMA 2.3 (cf [29]) Let r=0,1,2, and the function b(t) đ Ct*(R)
has the property
Dib(t) = O(ti-*-”), k=0,1, ,r+ 1
Then b(the WR)
CoROLLARY 2.4 (cf [29]) a„(f, ì') e (W+19)NXN(R) for all 0’ = S*~* (cf (2.6))
Lemma 2.2, Corollary 2.4 and main Theorem from [2] yield:
THEOREM 2.5 (cf [29]) The matrix-function a,(t, 6’) (cf (2.6); te R; 0’ e S"-*) can be factored on the form
t—i\*@)
(2.8) ag(t, 0") = (a5) “1, 6° diag| (“—*) Jose 00 t+i
wlere (az)°1(, 0, (2š)êNt, 6) Ẫ (WV *2)8*4(R) have analytic extensions in lower Imt <0 and upper Imt > 0 half-planes respectively for all 0’ € S"-* x(@') =
sx (9đ,(0’), ., %y(0')) is uniquelly determined x,(0') > > xy(O'); the integer
N co
(2.9) x(0) = Ÿ` x„(') = \ iengdeta,( 0)]
jel
is continuous and partial sums
y (0) (<r <N)
j=
are upper semi-continuous (i.e do not increase under small perturbations) with respect to Oe S*-*,
Integers ,(0'), ., % (0) will be called partial p-indices of a(cje
€ (HC™+2)"* N(R") (they depend on p obviously; cf (2.3))
REMARK 2.6 The index x(6’), defined by (2.9), does not depend on 06’ for n > 2 (i.e when S”~2 is connected set), because it is a continuous function admit- ting only integer values; for n == 2 x(0’) = x(+-1) and these two integers can differ
Trang 19SINGULAR INTEGRAL OPERATORS 59
In the scalar case x,(0’) = x(0’) and x,(0’) is calculated by the formula
(2.9); the integer x,(0’) and the number %,(a) (cf (2.3)) are the integer parts of the real numbers
6(a) = a \ ddfarga(d,1, ,)}— Red, forn>2 1
ừ,(2) = oo \ dd{arga(A, +1)}—Red, for n =2 Te
2° STATEMENT OF THEOREMS
THEOREM 2.7 Let a(ờ)e(HC™+?)X*\(R") be elliptic, inf|det a(@)| > 0
(0e S"~!) and xy(0') < < x,(0’) be the partial p-indices of a(€) (1 < p < @)
If (cf (2.3))
(2.10) Red; # 1/p for all j = 1,2, .,1
and
42.11) (0) = =xux() =0 (0'e S"-*),
the operator W} is invertible in LX(R"*)
If (2.10) holds, xy(0') = 0 but (2.11) does not hold the operator W} is left-in-
vertible in L}(R"*) and dim Coker Wj = oo
If (2.10) holds, %,(0’) < 0 but (2.11) does not hold the operator W3 is right-in-
yertible and dim Ker W! = oo
In all other cases (i.€ (2.10) does not hold or (2.10) holds but xy(w) <0 and
%,(@) > 0 for some w, @e S"~*) the operator W2 has no left and no right regulari- zers in L}(R"*)
THEOREM 2.8 Le ự(ẩ) 6 (HC”+3)XXX(R"); if inf|det a(0)| =0 (0đ S"-})
the operalor W2 has no left and no right regularizers in L}(R")
COROLLARY 2.9 Let conditions of Theorem 2.7 hold; Wi is a Fredholm ope- rator in L,(R"*) if and only if (2.10) and (2.11) hold
COROLLARY 2.10 Let conditions of Theorem 2.7, including (2.10), hold and
a i) | 0 +1
x = ~— -—— —-~- = 3 › +2, ee
Trang 20there exist integers x' <x" such that W) is left invertible in LX(R"*) for all x > x"
,
and is right invertible for all x < x’
Theorems will be proved below; Corollary 2.9 is obvious and we explain here only Corollary 2.10 (cf [29]): partial p-indices of a,(ờ) are x + x;(0') (j == 1,2, ., NM), where x,(6’) are partial p-indices of a(ờ); using semi-continuity
of partial sums of p-indices (cf Theorem 2.5) we conclude
—Ẫo < —x” = inf x,(6') < inf x;(0') < sup %,(6') < < sup x,(0) < —x'’ < co (ĩc S”—Ẽ),
REMARK 2.11 Conditions of Theorem 2.7 on a(ờ) đ HC™*+Ẽ(R”) can be weak- ened (we can demand less smoothness of a(ờ) with respect to ờ’e R”~'); espe-
cially can be weakened the condition in Theorem 2.8 (we can demand there a(ờ) € Hm,(R") n C(S”~9)
Condition (2.10) seems to be in fact necessary and sufficient for H⁄? being nor- mally solvable (i.e to have a closed range)
3° ON THE INVERTIBILITY OF CERTAIN OPERATORS We prove here a theorem which is a particular case of Theorem 2.7, but deals also with weighted spaces; this theorem will be used in proving Theorem 2.7 in ư 2.4°,
THEOREM 2.12 Let ig IE \? n= (B=) E= (2, 2) ER™, Wy — lễ] where 1 (2.12) z-+E-.—1<Rey<ag+-L, —L<x<1 —.L, l <p<co P P P P
The operator We, is invertible in the space L,(R"t, x{)
“
Proof Consider first the case
1
(2.13) a+——1<Rey<0
Pp
and factor function g(ờ)
8&,(6) = g-(C)+(đ),
(2.14)
Trang 21SINGULAR INTEGRAL OPERATORS 61
The operator
ly-y_ wt ot —— 0 Oo 7
W,) È WW ~ Wet Tờ, ~~
0 0 0 0 1
= Waa W hy 1 W th Wek, = Wyor — Ay,
where
0 0
Ay = Xy+ W È1,_ Why >
is inverse to Wz, on the set C7(R"*): Wi(We,) = (Wz) 'W4,9 =9 (PE
e€ Co(R"*)) and it remains to prove only the boundedness of the operator (W;,)* in the space L,(R"+, x7); obviously it is sufficient to prove the boundedness of A = xX{Ayx7°7f in L,(R"*)
We have (cf [I6])
(2.15) Weer 0) =\male~ 8) 06) 85
an
R
where g € C3(R") and m,(n) exist as distributions ? m, = F-(gz’);
Ag(t) = \ m(h, tị; EU — 2)0@(Œ)dt, tH (1), t= (VER, Rt t * + r +, m(h, tị; f)= (‡) m,(h + yý — š)m_(—n — y; Š) dy dỹ' = 1 0 Re = (Fem) (ty tt’), Ty
F(t, 113 &) = (2) ( ũi,(N + y; €9) ũ_(—n —y; ẩ9 dự, 0
m.(h, đ') = (FE 41,83") (4, đ'),
where m,, exist as distributions again
Trang 22With the help of formulas (cf [15], 3.382) = ED oy (2.17) | (B — ix)re- dx = _ 14(), x04) = „ũ + sgn2), Re >0, Rev >0, we obtain (cf (2.13) — (2.14)) I en nt dờ, (—1)” - lệ
malty, C1) = 5 \ Ge, iey- =- 2z (ễu — 16"? r(—y Pte y(n), 1+
a ‘Vi C Wag
m_(h, 4xz<“<ˆ_<È<È zi) : ừn 2z } (Gốy + lờn) ?+! —œ —— a =
_ \[ CHỈ ven _ Rr + 71) | u è -—— >! Sie Cat hence mth, t, 0) = Fs) wih, Tt!) = ta ( ny _' ews a | See eT 01929 đụ =< tị (2m11 T(È-y)T0) R1 " (—! ( " y_ me ft) —G ——\fi*Ð 7-1 + y?~!dy x (5) (2“)"~!F(—y) P() HP? -E +, -~r, h2) lẩ:
x elit ‘oot ứ; ety *t2/16] dờ’;
rR”
Trang 23SINGULAR INTEGRAL OPERATORS 63 depends only on the distance |đ’| and one can take (without loss of generality) ty = |t’| = x, tg = =t, = 0; using then spherical coordinates dờ, dờ, = == r"~3dzdr (- = Vi [ờ,|?, do — element of the sphere we easily get (4, +1, +
k=3
+ 2y = u for brevity)
\ e—*ê'+r|ế'D dờ’ = eT Eg tHE’ ger =
Ri Ri
œ
_ \: es dờ, (ew ae ate do =
co 6 “3
C -_H ‘e rề
= ar\ etl #t cos(xờ,) đế; = 9 = 2c SU QB 2cạu ( tư = —————\ r"~?K;(ứ „ấ + x*) dr = 3 2 iu +x ý 2cạu 2c,u = —-——— r"~?K) đ = wera OO Te
where co, €, are constants and K,(t) is the modified Bessel function of the
third kind
Returning back to m(t, t), t') we get
M(t, T 5=} "È `
" (2m'~(—y)TG)
x | Π+ yy? My + Oy) ly + Ty + 2y) 4
(iq + + P+ IP)”
From (2.16) we have
1/P
|4p)I < ( \ mỷn, tụ, # — Ẽ)I*ự(I &) x
Rit
(2.18)
1/P
x ( lIm(h, tụ, — t)|rrẼ ar] ;
Trang 24where
1
(2.19) 0<ð< min(- ° ah q=-?.-
P 4 p—]
‹Consider last factor in (2.18) (notation: ì = Re?)
J =: \ lm(hn, tạ, PP — 1)â1Ị°? đt <
R? +
cœ œ
<C \ tidy \ t7°4 dry \ (fq -‡- ty È- 2p) de’ 7
‘Vorm Sora? ) Watataytr 0 0 Ret ray
(2.20)
te dy ty °9-* dr, dt’
ass Cy 5 _ TT TT = nt =
(yt nyt? } (yan)re (Vi +i'2)
0 0 R1 oe „3+ dụ dz = Cf (y + tite Zed a1 + z)1~8 — 0 0
here and in what follows constants C,, C:, depend only on y and p If
(2.21) 0<dq—Bta<i],
(2.20) vield
co 3°
B-ờq—a dys
J< G#\ ———— (ty + y)1+ừ dy — C;t2+* —ˆ —~ Cyt (+ z)1†8 ;ề1+z~8
0 0
From (2.18) we obtain now
Trang 25SINGULAR INTEGRAL OPERATORS 65
if
(2.22) 0<B+6p—a<1,
integrals exist and
: d
|4ự()|” dư < €; iocorrae| ri \ l(a) Pde
Rr R? 5 RU
Finally ||Agi|, < Cglell,if (2.19) — (2.22) are valid; these inequalities are compatible, because B = Rey satisfies (2.13)
Let now
(2.23) 0<Rey<a+-L;
P
the conjugate operator We to W,, in the conjugate space L,(R”*, x;*) is invertible,
y
because
SA i, +16", \7” p
Px) “lệ Che =g.,@, ự@=—P— ’ if, — 12h 7 p—t1
and (2.23) can be rewritten as
1
—# +-— — è < Re(—y) < 0; q
invertibility of We in L,(R"*, x7*) yields the invertibility of Wy, in L,(R"*, x7) Consider now the last case: Re y = 0 but y 40; the inverse operator (W})-? =
ms a We-1 is bounded in ê,(R"*) since g.(€) = (i, + |ấ'|)#?e m,(R") (due to Theorem 1.10) The functions gz1(ờ) are homogeneous of order -+-y and Re y = 0 The operators Wes have then the form (cf [16}):
W,-19(x) = \ 9„(x — y) 00) dy - ly — xi" R”? ựx (6 = #Z,.Ò19+(0I:|~"Ẫ and therefore (cf (2.17)) Q(t) = Œ -ff”,
where Ci, are constants Hence |2.(đ)| = |Cz.| and in virtue of Theorem 1.4 the
operators W’_1 (and therefore operator (Wi )~”) are bounded in L,(R"*, x7) G
+ Lõ
Trang 264° PROOF OF THEOREM 2.7 Let ao: R?’-1— R”"-}, lo€ờ; =: be a rota- tion of R”’-! about the origin and
5,9(6) = (a6) The operator 1 ; t” We œữ) i \ wT) dt, t = (te, t), a+ 7 t— fe —œ “) , 1 ; z X:+(ấ) = X;.(ễy) = > (1 + sgn ờ3)
is bounded in L,(R"~1) and therefore 7,,(đ) € m,(R"~*) But then ìự; € m„(R"~})
(cf [16]) and 7
1H ced lip Jựơ„W2ựs|, := Wall, (de m,(R"~>)),
since o,, and conjugate o% are isomorphisms in L,(R"~)
Let 5 = L(L,(R"*)); A c Z denote the set of all operators W}, where
a-l
⁄(ư) = ME) = J z⁄.:()
faz]
ando), ., 0-1 are rotations; obviously W} = ƒ @ W?, ý H(R") and | W}i|, < co
By A,, denote the subset of A, consisting of all operators Wiđ A with ⁄(0)=
= ơ(0’) == 1 in some neighbourhood of we S*-* Clearly A, is the localizing class (cf ư 1.6°) and due to the independence of Z(ê) on đ,, AWi = WA for any de m,(R"),
AờA; it is easy also to show that the system {A}, est-2 IS a covering in 2
m vat
(U0 supp ơ 9 S"-2 = ư”-?, then g = ` ⁄Ẽ* is invertible in m,(R”) and
kol ke
> W œ = MỊ, (W})~!' = WÒÈ; cf Proposition L9), koa”
Equivalence (cf ư 1.6°)
4 1
29 WÒẼW} (wes), - a„(ư) = (Ele, ự)
is the simple consequence of the inequality (1.17) and of homogeneity of a(ờ)
(a() = al le"|-4, 8), 0 = [e721 Ze 8"),
In virtue of Theorem 1.17 the operator W) will be left (right) invertible
Trang 27SINGULAR INTEGRAL OPERATORS 67 (2.5) — (2.8) yield a,(t,@), a =a(—1,0, ,0), : ww) +8 a(t, @) = a_a=\(t, w) ( wr it — 1 (2.25) ồ„(, @) = (t + i)°ak(1, w)(t + i) BE (Ng7}, EY” = diag 1 Ƒ 1 ph” " (; + nh: lý — è ứ — è lý — è Hence (cf (2.22) — (2.23))
đu(Š) = Ao—(C)8 x(a) + 3(S) Fn +(S)s
i, +1ờ'1 \*
(2.26) 8, = diaglgy, -8, yb (0) = Lm) lếy — lễ
a,-(Ệ) = a_ọ='(& lỆ'| T1, œ), - a„+(@ = Z„(G|ẩ1~1, @)
Our next step is to prove the inclusions
(2.27) GoxlE), Goa(€) € my “™(R");
it suffices to obtain the asymptotics
(2.28) Dƒ(22,)„(t,œ) = O(/1 72), t>+o fr=3,2 N q=0,1,2, m, because then
3, sup|ýfD‡(a‡')„(6)| < eo
[kl <a and due to Theorem 1.10 (2.27) holds
We will prove that
a(t, @) € (W")N*X(R),
which by the definition of the algebra W”(R) yields (2.28)
Consider for the definiteness a(t) = @,(t, w)(we omit we S”~* for brevity); others are similar A typical entry of @., is
,
(2.29) (Z,)„() =Œ +? (a8)„@9 Inữ +, v„=0, 5} = 5}
Trang 28If Re(ð; — ð;) <0, then (@,),,đ W"+*(R); that follows from the inclusions
(t+) j °ẼTIn(+j)}e W”+1{R) (cf Lemma 2.3), (ax)„c W”+1{R) (cf, Theorem 2.5), and algebraic properties of W”+1(R)
Let j -= r; then (a,),,đ W"(R), If (a);;InŒ + )]2+e W”+1XR) for j # 4 (We remind that v„„ := 0); but we will prove more (and that will include the last case):
If Re(ð; — ð;) > 0, j #r, then
(2.30) (+i) 7 (at), 0) [Ine + DP =e WR),
for any — co < vy < co(we remind that 6; = 6, in (2.29)) The factorization (2.8) of a, easily yields
f—1
(2.31) + (! + r] 0 ƒ2„() — L] = 0) -ˆ bạŒ) [a„(Œf) — 1],
f—1
— I1
(0) == lợ + J — | ag(t), Balt) = ( is Yast
Since az € (W™+2)8**(R) clearly
(2.32) D#,@):= O(/17-9, > co, g=0,), m+1
By the same reason b; e (I?'*9X*X(R) and making use of (2.7) we casily obtain
N
Di{bo(a:, — WD] jet) = D? SY) (2) jC) — Vse(t) =
si
(2.33)
Reis, ð;?(ð7 ụ2 Dj PEW 2 (|g REC 0) 4 hy, N
= J Or,
sol
g:<0,1, ,m 4-1, where by is defined by (2.4); (2.31) — (2.33) yield
Trang 29SINGULAR INTEGRAL OPERATORS 69
If now 9,(t) =: f&Dkp(t) == Ot“) (0 < v <1, k=:0,1, ,m4+1) then
I -{- z
,„ —Ẫ<f<Cœ, zI=1] 1 z
clearly (: a=]
d 1 A(t) d
;!DƑSg0() =—ˆ “or (de 71 t—f Tỉ {eo To +T—ẻ
Ẫ -.œ Lee L+đ a LẬ ali) “i JT=C 7 hz lược L8i=1 lở|~=1 đệ — — ua J HAP), @(1) = 0] " ca) 5 — — | iF os a — Qa os os S ~ ~—— _ + tư
and we easily conclude
DES, Q(t) = Ollt|-") (jt} > 00, k=0,1, ,m) !
The operator P+ == + (1+ Sp) has the property P+a‡= a‡, P*az =0; hence
(45), () = Pơ(ay- ay) Ct) = O(|R* % _9) Ẽ
(If| + co, 0< v <5, g =0,i, ,m);
(2.30) follows now with the help of Lemma 2.3
Thus, the inclusion (2.27) is proved and the operators W, „ AT invertible (cf
Proposition 1.9)
(2.34) Wi Wia=Wiaw) =1 ok wot (+ +
Let now (2.10) and (2.11) hold; then x,(@) = 0 and in virtue of Theorem 2.12 the operator 1 gw on Wi=W,, OW, Ẽ 0W oy 8 83
(cf (2.26)) is invertible in L4(R"*) Since (cf Proposition 1.9)
1 1 ray] 1
Trang 30the operator Wi, is invertible for all w € S”~Ẽ (cf (2.34)); but then (cf (2.24)) Wt
is invertible in Lơ(R"*)
Let now (2.10) holds, x,(@) < 0 and (2.11) does not hold Since (cf Propo-
sition 1.9)
1 1 1 1 1
(2.35) Wow " Hỏ W eto) Wes Was ,
where all operators, except WwW 2 are invertible and the operator Ww wr is right-
- z(œ zl
-invertible (we remind that x,(@) < < 4(@) < 0)
W Wo Exo) Š— „(e) =t,
the operator Wa, will be right-invertible as well Due to Theorem 1.17 and (2.24) W2 is right-invertible in Lơ(R"*)
We wili finish with the case x,(@) < 0 if as soon as xy(@) <0 for some
we S"-* the operator Wj can not have a right regularizer in L4(R"*)
Because (2.10) holds and x,(w) < 0 forsomewe S"~?, it is easy to notice that
Ww Seyloy? =0
% yo) k "
a= YW ete et), t= (4, ER",
kool
where @,(t’) are arbitrary functions from L,(R"~1) Hence (cf (2.35))
Wi = b= WiaW,) 7% @ =(0,0, ,0)
and therefore dim KerW, | = co That means, among others, that Wi, cannot
have a right regularizer in L}(R”*)
Assume now W2 has the right regularizer in Z7(R”*); due to (2.24) and to Theorem 1.171
(2.36) W,W, =W+T,
where Wie A,,, Te S(L}(R"*)); (2.36) remains valid for all W> with suppx n S”~°c < suppxn S"-*, because WW; = WW; = XL Wi, == Wi Using this property
Trang 31SINGULAR INTEGRAL OPERATORS 71
and also the property oy, Wi, = We Ox for any rotation Í o, we easily conclude: there exist the operators Wr say Wy EA, Ty, -, Tyđ S(L4(R"*)) and Ẽ\ , RÒc.Zý(?((R?) such that ? W}W} R,—=W} + T, (j=12, 4), (2.37) Due to W}W} =W} M} from (2.37) it follows (2.38) W;, R =I+ T, where q q R=>%H;R, T=Yơ Tj j=l j=l
(2.38) is a contradiction, because Wi, cannot have the right regularizer in LN(R"*) Let now (2.10) holds, x,(6’) > 0 and (2.11) does not hold; the conjugate
Operator W: to the operator Wi is then right invertible in the conjugate space LP,(R?†) (p' = p/(p — 1)) and dim Ker W =: 00; but then Wiis left invertible and
dim Coker Wi = co
Let now (2.10) holds but x (w) <0 and x,(@) > 0 for some , we S"~?
As we already proved W} cannot have a right regularizer (because xy(@) < 0)
and the conjugate operator Wi cannot have right regularizer as well (because %(@) > 0) Therefore W2 has no left and no right regularizer in LY(R"*)
There remains to consider only the case when (2.10) does not hold (j = 1 for definiteness)
As we already proved, existence of a local regularizer of Wa, (i.e (2.36) is valid) immediately yields (2.38) (existence of a global regularizer); hence due to
Theorem 1.17, if Wi, has no left (or no right) regularizer for any w € S"~Ẽ the same holds for W}
Thus it suffices to make sure that W2 has no left and no right regularizer
Due to the representation (2.26)
(2.39) Wi, = Wi, Ws 8 xleaytd Wis at
Trang 32where the middle operator is diagonal one and others in the right part of (2.39) are
invertible (cf Proposition 1.9 and (2.34)) Hence it suffices to consider only the operator LG in the space L,(R"+) (r =: %,(@) for brevity and 6) =: Ip by as-
sumption)
Let first r = 0 and Ay =: Wes have a left (or a right) regularizer; due to (1.15) and stability theorems the operators A,, -: Hộ also have left (or right) regu- larizers for sufficiently small đ > 0; but the operator A_, is invertible and A.,
is left-invertible, dim Coker A,, == 00, we obtain the contradiction: if Ag : : We mm is
invertible (has two-side regularizer) the same must be A,,; if Ay has a left regula- rizer and dimCoker A) = co (has a right regularizer and dimKer Ay- : co) the same must be A_.,
Let now the integer r 4 0 and for definiteness r > 0; if By =: Ha has a
regularizer, it must be left regularizer since B, , = VN larizers and are close to the operator Bg (cf (1.15))
As it is already proved, the operator A_, = LG is invertible Consider
the factor-algebra
Ip
have only left regu-
E = L(L,(R"*))/S(L,(R"*))
The class Be 5, which contains the operator By = W} Srtay ” is left-invertible in 5 (since By has a Jeft-regularizer in 2); classes B, are also left-invertible and since
lim||By — B-,|', < lim||By — B_,|l, = 0 230 e>0 A
inverses (B_,);1 must be simultaneously bounded
sup (B_ JF", <M < 00
But then classes C, = (ấ_,)-1Ÿ, are also simultaneously bounded and therefore
convergent l\ể,— Eyllp < NCC MplB-Í — Be elloll 4, llp lim IC, — C,||, = 0 Ep a0 The limit A A Cy = lim C, e390
is inverse to the class
A ˆ A Aw A n
1 = Wi = W} 1 =limWl B_,
Trang 33SINGULAR INTEGRAL OPERATORS 73 We obtain the contradiction because Wes)p has no left and no right regularizer
3
5° PROOF OF THEOREM 2.8 Suppose detz(ìạ) = 0 for some 0ạe S”~1 but M} has the left regularizer R, in Lơ(R"*)
R,Wi=I+ 7, Te (LX(R"*)); then the class wi from the factor- -algebra
LY = #(1?(R'+))/€°\(LP(R**))(cf ư 1.49) is left invertible R,W} = ủ (since Te€(L}) c €Ẽ(L7)) Without loss of generality, we can suppose that the fũrst
column of the matrix a(ĩạ) is zero (otherwise we can consider the operator H2 = ~~ Wiel, where g is the constant matrix such that detg # 0 and the first column in a(0o)g dissapears; the operator g~+R, will be the left-regularizer for W2,)
Let now ƒV,(ê)c (Cự)X**(RÍ,
ƒ,(‡) = diag[P,(Ị),0 ,01, Ÿ,@ =J] Va(0,
kel
È ~ 0, lễ, — OR] > 2e,
VeulSx) € CO°CR), VerlSx) = {i ụ i os > k—— Yk ,
E=(€, ,6,) ER", 0) = (68, ., 8) € S77);
then
(2.40) T, = Wi, — Wor, =% Win Wr X,.€ SMLN(R"*)),
since
%- Wrdre = diap[x_ WE Xe Ẽ Ws @ @ Wee 0, , 0]
1 :
x.(ấ1) = 3 (1z signe,) Ẫ
and (cf., for example [10], Lemma 7.3)
X,- WE Nas €S(L,(R))
By assumption the first column of a(@)) dissapears; hence due to (1.15)
(2.41) lim||W,
ờ+0 telly = = 0
Let now prove
Well, = inf WS + Tilo =ilWelly>
ree? (2.42)
Trang 34Obviously
def
lWðlly — llz,, Wi,„ lip = Sax, , YOY, lle =
= -XaiMiXaiip 2 X:Wt:p > i W5i'ps where S,W? = W2S,, Swit) = W(t — A, lạ, ., th), A> 0, 1 ; ~~ Sih, = 5 [1 + sgn(n — 2)]; ‘due to Lemma 1.13
Jimny, Tlp =0 for all Te SẼ
Hence "
(Wei, = = liml'z, Wi + 4,7 We + Tiip
and therefore
[ Wally <i Wl the inverse inequality in (2.42) is trivial
From (2.40) — (2.42) we get (cf (1.14))
= = lim sửp/Ƒ (6), < lim|| Wii = v giỷp T” cÈ0
= lim|L,}W} â„<`R,!„ lim|| 2, |, = e930 8 630 8
obtained contradiction proves that Wj with a(@)) = 0 cannot have the left regu- larizer
Similar contradiction will be obtained if W3 with a(@))=-0 has the right
regularizer G
This work was planned and fulfilled during the author’s visit in the TH Darmstadt (West Ger- anany) as a Fellow of AvH Foundation in 1980/1981
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