tda ~~aO$k&6d*BulfS @@dq@ZyztM mobst&%.la: . ? . . . ":pp&*;

v m * f + ~ t fhj.7 1 ~ t p ~ p l )jo 4 e f I . q ~ +ha - 8 i t ; j ~ r n n ~ y + qf q F ( $ e d * In t h i s paper we present (section 2) a survey of ream&

dsvelgpments in the search (as yet unsuccessful) fo r a dm= , , \ *J f:,:, .; :-b.% ; 7(+; t / # ' 1 '

dition ,sn q(t) wkiih is 'both necessary and suff iAiGt fo r .I

7 t o be LP. A prelimfnary inspection shows that the major%= , * : $9- c:.' , -, ,,, .$; . :.' " + > - . 4 - . * ; :?P: - % . , , . :' ' "

t y of a r i t e r i a in section 'tno'-rkil into two classes. The" '

~ r B l t : + m f thyxm~w&asbehh c.&$Q;P~s:-&~QB'B ~mtte~3.a i ia,-wFli@b the

'F&%zw@P. q(t$ 4~ 'restricted by mean$ of a so-called weight

' function (e.g. ~ ( t ) c ~ ~ ( t ) in theorem 2.3); the seoond class %!l! 1 . . ? A t . >.

cohtains' thoae ;it) which behave on a s e t of sub-intervals, , - - [P,], of [a,=). In section three we show that the first

C % t l a s is a- s t r i o t subn3et of the second.

i:. . ' . .

%@he Slm%t-point asbd lM%t - c i r c l ~ eltemat ive was- discovered

f'W4Ierma.m Way1 during t he o m e of h i s investigations i n t o

the theory of singular integral equations, and was f i r s t

%rJt;lslf$hed in h2s. remarkable memolr [G?O]. This paper also

THEDREM 2.1 (Weyl [20 I) . If q ( t ) 4 C on [ a , ~ ) f o r some

conskmt C, then T i~ LP . Note that t h i s is only a one-sided res t r ic t ion on q ( t ) .

. ,

, We sha l l see l a t e r t ha t , With s few exceptions, th is is

oha~ac te r i s t io of the functions q ( t ) f o r which the corms= _ 9 "

I .

ponding T is LP. other words T is LP provided only that

. .

- . - - , - . , . , ? .>&%ft vie. the positive values of q(t) are not too large. It ts aanh

- .. A . - fl - *f .-t =><,.:idi..- venient at this stage t o &fine the pos i t ive part of q(tf w

, . *

the weight fm@tim ~ ( t ) = eft . A-ratqer ,diffl&nt o m @ r ft,r pL:sm:i',; J '-A a, A i.'l.*,;, X' I w ere

t: 4on w a s found a l i t t l e later by P . Hartman r h

E : , . I f s - 5 - ----I <;f 4 . b ~ ~ . . & A , - - ,-. -.. , ,* ,. . L; .

THEOREM 2.4 (Hartman [ 31 ) . The formal operator T I s LP 2 .

r * I r ,

if the* exists- a positive o ~ t i n u o ~ mdnotone rrmqtion ~ ( t ) . - . . - I 7 - - .

with ' EmC:

i: rq?r ;~ ~5;:s zidaulutely Q O K ~ : ~ - : ~ ;.

b s r e q (s) is d e f i n e d by equation (2 .1) . v:: C h

I' &. m t h ~ a last msnlt, only the ave~qtp vela& of $(t)

. , 1

c", =.t*d~bea, 6 as thrller po*twi.e estimate., me 6&

Lte-rion of BrWk in. [r Erlctz%ner devlsiop~ a i r them,

RBM 2.5 YBP~II* [i] ) . The ioxdkl operator r i s LP if

,Lam axistr a positive funotian w(8) with fc r heme ~ c , y i ~ : t ~ l ; ~ t . f m d my Xnteervaq! <T :

, %ha: c_. 1 : -*- ' . ? . ' s - ' , ' " "," 71' 2' r ' I - . t . d,, " F a , ., - A

i (ii) w , w' , wn a m a l l bounded and oontinuously

8 9 : - . ! t_.>k fl& &Qdm 2@S [ a , m ) with j ( 3 ) 6 1, .Ileatiam

4(J3 is the length of the Interval 3 .

a s l igh t ly d i f fe rent point of view. In 1951 Hartman pro=

duced the following improvement :

T W ~ R E M 2.7 (Hartman [ 31 ) . ~f q ( t ) C on a sequence of

ineerv~lsr f ['an , gf 1' where (h -r a$ n 4 and b -a ae)O n n f o r a l l n, them T 2s LP.

In other words, it is suf f ic ien t t o have q+( t ) behave

on a sequence of sub-intervals whose t o t a l length is not

too > small. - . Some eleven years l a t e r , R.S. Ismagilov sub- . . *

stmt i a l l y improved t h i s r e s u l t :

j !l%EOREM 2.8 (Ismagilov[?]). The operator T is LP i f there

ex i s t s an ordered sequence of d i s jo in t sub-intervals [Pn) of

[ a , ~ ) and.numbers yn, n = 1,2, ..., with

(2 ) C (Y + W n -2 -1 1 where p = a ( P ~ ) n&l , q-. .n

.. L 1 b - - + i r I -

This l a d t theorem can be dimp'lified considerably by f i r s t

rjrL - y . ll.l. . I - Y H * R ~ 218~. The dpai-atir i is ILP it there ex i s t s an

. : - ordered siquk&e of sub-intbrvils of length i~.n with

fo r t E Pn

which, rather surprisfngly, is 5rr- faot equivalent t o the0 : @ - . . , b 7 - _

d" - a 1 . .

I I .l . I 2.8 (see [121 ) . Even more unukual is the fLmt tka t theorem *: * ,-, 1- e., - 7 ,- .;--,- r l V . - - . * , . . ? v . , t . ,. 2 . 8 ~ generblizek theorem 2.3 (see i l 3 ] ) . The moat reo&t - - i ' > < . L : 3.

i c . 2. -, G , . - -

: reault in t h i s . d i r ec t i& is given in L A ~ ' '- -

. * _ ii , .^.I - \ L I . L ^ " . i . L _ - - +

6 G , i Z L ' c ~ ~ G w : ~ t + ~ n y 7: ;. : * $ . ~ . , , c 7 ~ ~ ~ c ~ g g g ~ ~ kn" h,ml (Kn~wleb [I33 ), The operator T is LP i f them

-CPC!Y If r;!ltj & u i f f r . - i a t l y :i,rge fi.--;lt$,:2 ,..Fk: JnFi i:il : m i s t s ah ordersd saguanoa of sub-intdhiI*~ j of length

rr :+&t%h ht;9rir@: P . r'cr : z , + - , ~ : ' % *r ?it- n<:rt<n-;i:i: ibrlrAe

;>EL with ' f i x pf y-,.

Remark. In the dOoe theorem one can replace $(s) w i t h the t ,s 1 2 I function R(s) = 1, a q(r)dr 1 and thereby extend the c r i=

ter- -giv&q in [$I and [ 161 (see [ 11.1 ) .

Returnkg now to the oomment made a f t e r theorem 2.1,

we note one m a j o ~ exoeption t o the general rule given there t w.

> ,

I I

THEOREM 2.10 (~smsgllov [8] ) . The operator r is LP if there <I ,

exis t sub-intemals Pn of iength pn, and numbers % ' fo r which

nn as well. The next - - - - -, ,-- , ,

5 Ba&) a,(&) *+ dt) - qd{t) *

5 . I *

I *' I ~ ~ . ( 4 t " ~ t l ~ ~ ~ &*G) edpts ~ - d e & d aeq~gnec at ~ub-iir%erv f n

. Q) --aL'l . (p '

X I rcr so= oonatmt k, emK r i ld~ db I b T L

>

bS.s theoram fnoiudes tbe aoiterka given [ 21 p. ,1914. ' i 9 3 ( ~ 3 1 ;Ln 2 ++ -w

, .

the ,.f o l l w h g dfsoussion, we shall not consider theorem :, "* ' - P ?I-

" : ?+ t .12, M it dw$te not fal l into eiGher of the olasseq np$b

, ' I '_ _ % I ,-,: ., rr

4 . * , ' . + t 4 -;

itme&:-@ seot. one. O f the remaining oriteria. theorems $2' 2:.

z " , - ~ - < , -, *; ' ..

k9i.nurar- @ern eguibnen~p a& , &+, +be

- 6 ~ %hat theom= 2.4 and 2.6 m a y be m&ttcsn a w&& ~ $ 2 ,

pmn as well. The theorem Q equ%v@lenl t o the - v - , fog w ( a f / w i * ? , <= t w f ( s ) "fir - [ 8 ) d 3 d

%

-Y !&<8\;b(c) & 2 :Q! "- " a?-P 5 % 9 ... (2.2 . . p m ~ ~ 3.2. ~ h s opelrtorr is r S i r exists an . ,

. .' %.. . @m3arsd isequems of &ub-intamals {P f w i t h S (f )- wn lPLd

t' - Gcnceqt~::ntlg. u e k s % ; i:3;,- 9% 3> ][1

( i t ) f o r some constant K. s FLn/wn+l 4 e K n=1,2,. . .

tip). , tben exists a ccmatant C > 0 with kn q(s) 8s I

, J ' . T

,-$qr sPl 'iq)isn&ls J c P,*- for an$-- fi-.. .

PROOF OF E~UNRUWCE. ':we f i r s t show that thp existenoq or . * , 1: the sequence [Pi]' feibws from the faistence of w(s) in

fi [ theorem 2,6. l31 [ l o ] it w a s shown that for any f'unotion !i e w ( 8 ) with iw'l L M < m. there exists a pa r t i t ion (tn) of

a ,, z . ' ... - : & 'rsqaired . ~%vers@ly, given the existenoe of [ P,] in 4

theorem 3.1, we def %ne the piecewise llnear funtation w(s) to

and prooeed as in [ l o ] . In a rather similar way, the fol=

lowing is equivalent to theorem 2 .4 :

THEOREM 3.2 The operator T i s LP if there exists a partition

;t,i of' [ a . ~ ) with wn = tn+l - t and n

(1%) vn+& 4 !J, n far a l l n

(ti&) \$n $(s)da r n2 for all n.

fn conclusion, one obvious c'onsequenoe of the teahniques

r; b ] I. B R W t ; "a l~-adjqiqtpesD d 8 ~ e c W a Q Stuns- --. ---- . 0 . f ~ % p w % i f ,pp~*(Lt~+%; a&%: 8asrib.l :( %9),

' f . . 2 -.2s* *

j , % ~ f v ~ ~ i a ~ ~ h ~ ~ c & - f ; ~ : ~ ~ ~ & -e y n , + W 7 0, ? 2 " ?*< . C - . , I I - *"_

3 -?@a ,,d$q@otwn@p..$~r 80luC$.9~ts 02 .n-th order - 41f & ~ t ; W ' L a g a g ~ ~ n , $t%wJ% +'arm 347 . .'"',clB;: :. -ex. - ; .. - .-'i.: . . d & - .I

Q #5 1 2 .g 2+i. CZJlr gIptR+p.;v a a s e q t i ~ l i p e a t r p and . e& 'of *a ,po%ien~lar",~ouk$ . %&a .' f;. 23 . 1 : ($!#, . - * ? . .." - . . J t

J .

181 " ' ; ' 'On the self-adjointneras of the Sturn- Liouvil le operatora, Uspehi Mat. Nauk 18 (1963) WO. 5 (213) a I61 - 166.

. I. KNOWLES, *On a limit-pirole o r i t e r i m f o r seoond- order d i f f e r e n t i a l operatorsn, Quart J. Math. " W o r d Ser. (2) (bo appear).

;>

2 [I01 "A l imit-paint o r i t e r ion f o r a seeond order !inear d l f fe rent i a l operatorn, ( t o appear).

d: [21) "The l i m i t - p o h t and l i m i t - a i m l e olassl- a f i o a t d m of the Sturn-Liouville operator', Ph.D.

Dissertatlun, The Fllnders University of South r i ~ : Australir, 1972. 2' p [I21 9

"Mote an the s e l f -adjointaess of an n-th 7 ,

$ 7

order k i f f e r m t i a l operatorn, University of the w Witwatersrand, (unpublished preprint) t

p i c r r l , ' ,t I - . ' A L - " . ' 5 - - - * " . ,

- 7 . _. * * -*. A , i . r . , >'. "$, %& *mru~nrr- A=. I*~u= rklr*&!nu :A* -ncraplris -ncie--a -* .

rl - : - . . .,-!-* - -am. ae~l1ttoaf . o t : i r l s ~ -9 a, ,) ef - aeaaad- brar&'aam!imann . ----- . I.& L __ b&iir&*, _._ _ t o appear)

If " ~ o t e on a limit-point orlterionn, Proa. -re Math. Boa., ( t o appear)

i'] I.A.PAVLWK "~eaessary and auf lalent oonditions for h boundednesa in the spaae C (Op) for solutions of a olass of l inear d i f f e ren t id e uaticma of w o a d ordern# Dopcw. Akad. Nauk (19 8 O), 156 - 158.

i]: C .fie PUTHA& '@ks&@bls potentials and half -line speatra*, Proo, Amer, Math. Soo . 6(1955) , 243 - 246,

D.B. SEARS# ' ~ o t e on the uniqueness of orsen's funatioaer assoaiated with oertain differential equationsn, Canad. J, Math. 2 (1950), 314 - 325.

' .. - . I] M.H. STONE, se in ear ~raasformatlosra inndi lber t Bpaae".

- A m r . M a t h , gag. C013, P W . , Val, XV, NewYork, t 19 6. I *

)I E~C.~-Z&HMAR&%~ the '&iclueness o i the oremi's twret,;tOa assoaiatsd yi$h a seam8 order differen- t i a l operatorw, Cmad. 3. Math, 1 (1949), 191 - 198,

)I H . W&L, %eb& gew8hnliohe d i f ferent i a l ~ l e l a h ~ e n m l t + . . . . a p f y J , d t f i t . n rrnd Me ppeh?5rAgen sntwlaklrmg~

w 1 k;[;lrlioher funotimern , Math. h. 68 (IgIO), c $ - 222 - 269,