Báo cáo toán học: "On multidimensional singular integral operators. I: The half-space case " pptx

[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 w

J OPERATOR THEORY Copyright by INCREST, 198

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 § 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-

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

in the space LX(R"*) and L(M) (cf also [13, 24, 26, 27)) Simultaneously with

Simonenko, 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 ¡x;* 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

(s] integer part of a real number se R

tư " twee 1/r ; def ,u, « “ “ R

Ie: {° a] Ses Easy GRY K1

ẹ cử (Ga, std en) (é == (é1, ste en) € R’; é == (ấu é"))

SINGULAR INTEGRAL OPERATORS 43

F te, — partial Fourier transform

F 1,-8.0) = \ een, Edy (E= En &) ER’)

supp a — closure of the set {€: a(é) 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,,¢(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,.Ú,) = mae + sgnt) (=( ,f„)€ R”, k =1,2, n)

H(R") — algebra of homogeneous (of order 0) functions a(Agé) = a(é) (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„-¡ with entries

-H°?(R") — Sobolev-Slobodeckii space, defined as the closure of C#(R") with

respect to the norm

1p i

R#

JTHP(R??) (HP (RÑP—)) —- subspace of H*?(R"), obtained by the closure of the

set of functions @ ¢ C?(R") with supp @ ¢ R"+ (suppg ¢ R"~ : : R°\R"*, res-

where s:=:?mr + 2, Ö < 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-

H*(M), H3"(M) — Sobolev-Slobodeckii spaces on a compact manifold M with boundary @M#Q, defined as in the previous case but with the help of spaces H“(R*+) and H7(R"')

SINGULAR INTEGRAL OPERATORS 45

It is known that H3°(M)< H°"(M), H9°(M) = H°(M) = L,(M), H®°?(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, £) (xe Xo R/, £eM = R’,

m=0,1, , 00) which have the property: all derivatives D‘b(x, €) = 5,(x, e

€ H*?(M) and are uniformly continuous

2°, SINGULAR INTEGRAL OPERATORS Consider the (multidimensional) singular

integral operators of the type

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)

1 b) c= rs — EU, STs ¥ — 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 Ì < p < 2, respectively) and || A} wR") < C-C,,, where C is a constant,

The 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:

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

in the space L,(R”), where B = —a (due to the boundedness of A in L,(R"))

) This theorem is a multi-dimensional analog of Babenko-Chvedelidze theorem (cf [6})

SINGULAR INTEGRAL OPERATORS 47 Introducing a constant 0 < 6 <1! and using the Hélder inequality we get

bì)? —_ 6p 1/p

IBp()| < -C- ({ Ile()I?i@(x — »)|Iml #) x

lal \) be — yl" a — il? R

< (<5 lo)? Q(x — yylbv dl? any" x

Jal?) ix — yl" a — val?

R x1

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

( ) 0 r(é’) > 6, r(é') k 2, AG r() 1 + ,

where ¢ == -:1 (inverse mapping is given by formulas ¢, == 6,/0,, k == 2,3, - , n)

We easily obtain from (1.4)

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(é’) = const,

Using the obtained formula we get

\ iQ(e, 6) de’ \ [2(e, S| de” _

SINGULAR INTEGRAL OPERATORS 49

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))

in the conjugate space L,.(R”, !x,!~*); in virtue of the proved part of the theorem

A® 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) #

ly ¡eR” 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(é)e¢ L,,(R”) and

by ,(R") denote the algebra of all (multipliers) a(é) 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€é(, 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”)

(xe N)

SINGULAR INTEGRAL OPERATORS 51

The operator (1.1) represents the example of the operator Wix.,.) where

COROLLARY 1.7 Let a(x, -)¢ 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)íp + lj2 for p=2, for 2<p<c

and for Ì < 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) ¢ H(S"~4) 0 H(R") and s > n/2; then a(€) ¢Hm,(R”)

It is also easy to prove the following:

PROPOSITION 1.9 If a(é), 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

§'c R*~!), then WiWi = W3,

We need several well-known results, which we formulate below

Let M be r-smooth manifold with boundary 0M # Ø and ‹„, S,, denote

the space of all bounded, all compact, operators Y,, = L(L},, Ly), Spr:

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(é) ==

:= lim B(€) == 0; then bW2, Wb © 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,b¢ 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)=¥ 5/0\//9eŒ dt (gE L,(R*), Wye Ly(R*)

0 j=t!

in the space L„(R*) (1< p< ©œ) and 4'<.ý(L,(R"-')), the tensor product

A=: K@ A’ is defined as

co

Aw) = (K @ 4990) = Š #0) À 9/9 (419) G09 ác (= (nD:

SUNGULAR INTEGRAL OPERATORS 53

obviously A € £(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 S® and B; = C; @ I, where lim ||CaJ|l, = 0 for any pe L,(R*), then lim ||B;T||, = 0

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-! <¢,; in virtue of Lemma 1.13

(1.13) lim || B;7||, = 0

For any ¢ > 0 there exists 9,(t)¢ 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 =

=lim lỨ-/) Tø,lly = |T®ll; > WT, — € 5

hence ||,7||, > |ÌTÌl, The converse mequality ||,7Ì|, < ||7Ìl, 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) Let for definiteness k = 0 (the case k = 1 is similar).

Assume now j|W2lj, — c„ = 4e >0 and T,¢ G9) be such that c, +¢>

> ilxuWe + T,\l, > ¢p; without the loss of generality we can suppose that (0, 0, .,0)¢ U; otherwise we can use the shift operator

B_yp(x) = @(X — ¥) = G(X, Xz — J§; .; Xu — Vn) (i! B_yll, — > BL yWe = : WEB_y; Bz B,)

Let xz(£) 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;(Ê) =: À~"/"o(2~1ÿ), then l@iy := ] and with the help of the equality

limi Vata) Wp = Wl

we obtain

IWSl, < IWApi, + 6 ~ limilW2x¿) W4plụ + e = 2¬

= limi, Vino) Wrelip + e= limllxsWa slp +E

€ lxsWai'p + és < |Ìxa(xuM2S + v T:)ÌÌ; + 2: S

Š lÌlxuW⁄2 + T:||; + 2s < Cp + 3e = | Weil, _

LEMMA 1.16 Let a,b¢ 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 ¡ a() Đ(€)i = ÍWblls < , 2 :g = ||W2Ï[y =

= ` XLWfM?s, „r= ¡WaM), 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

ki ! WFI*~^?.sup az(¿)!? oe 2ữ —?) Tả

(1.15) | Walp < Ì Walle -SUP.đ()”, ÿ = Dr - - 2° r#2, k=-0,1.

SINGULAR INTEGRAL OPERATORS 35

5° LOCAL PRINCIPLE Here some necessary information from [14], Chapter XI,

§1 will be given

Let Z be Banach algebra with the unit element e; a set Ac Z is called a

localizing class if 0¢ 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,

and the notation x « y is used

An element xe X is called /eft (right) A-invertible if there exist ze XZ and aéA such that zxa = a (axz =a)

A system of localizing classes {A,}.ea is called covering if from each choice

of elements {a,}e0 (@,,¢A,) one can find a finite number whose sum is inver-

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(é)| > 0; consider the constant matrix

The matrix-functions B’(a) have the property

The matrix-functions

are holomorphic in + Im¿ > 0

clearly (¢ + i)°B.(¢) = 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(È) = (ối — i)-°BO"(E,) ga“ (1,0 , , 0) a(6)gB4(E:) (ái + 0P

(2.6)

(= Gr ee R x R"?).

SINGULAR INTEGRAL OPERATORS 37

Let us notice that if /= N (ie rp = =ry = 1) then B.() = ï (Ï 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*(é:, ữ)—1„= O(g,|7*~R% +Ret—D — O(|e, [TẤT

(seo = 8_(6)g~1a~1(—1,0, , 0) a(#) B,(&) — (=) )

(2.7) is now obvious since

[a„(š¡, È) — 1; =ÏQ — )~“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 — )*D¿/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

is 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) ¢ 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, Ø') 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

(2.8) ag(t, 0") = (a5) “1, 6° diag| (“—*) Jose 00 t+i

wlere (az)°1(, 0, (2š)£Nt, 6) © (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 (9¢,(0’), ., %y(0')) is uniquelly determined x,(0') > > xy(O'); the integer

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

x(—1) # x(+1).