Boundedness of the convolution operator in L - Springer Link

,

3. 4. 5.

Z. S. Zagorskii, "On the set of points of differentiability of a continuous function," Mat. Sb., 9, 487-508 (1941). Y. Katznelson and K. Stromberg, "Everywhere differentiable, nowhere monotone, functions," Am. Math. Monthly, 81, No. 4, 349-354 (1974). C. Goffman, "Everywhere differentiable functions and the density topology," Proc. Am. Math. Soc., 51, No. i, 250 (1975). H. Whitney, "A function nonconstant on a compact set of critical points," Duke Math. J.,

!, 514-517 (1935). 6. 7.

S. Saks, Theory of the Integral, Dover, New York (1964). I. P. Natanson, Theory of Functions of a Real Variable [in Russian], Nauka, Moscow (1974).

BOUNDEDNESS OF THE CONVOLUTION OPERATOR IN Lp(Z TM) AND SMOOTHNESS OF THE SYMBOL OF THE OPERATOR UDC 517.5

S. L. Edel'shtein

Sufficient conditions for the boundedness of the convolution operator in Lp(Z TM) are found. These conditions are imposed on the symbol of the operator in terms of the spaces H a and V B (functions of bounded variation of order B). The results obtained here generalize the results of S. B. Stechkin and I. I. Hirschman [Ref. Zh. Mat. 7, No. 7821 (1960)] for the one-dimensional case.

In this article are found sufficient conditions for the boundedness of the convolution operator in Lp(Zm). These conditions are imposed on the symbol of the operator in terms of the spaces H a and V B (functions of bounded variation of order B). The results obtained here are generalizations of the results of Stechkin [I] and Hirschman [2] in the one-dimensional case. The investigated classes of symbols contain discontinuous functions and are wider than the class ~F~ (q= ~ m) for which stronger results were obtained by Birman and Solomyak [3]. For proof we will use notions of approximation and moment scales, introduced by Simonenko [4]. i. Statement of the Problem and Formulation of Results. Let f be a bounded function on [--~, ~]m and its Fourier coefficients be {at(t)}tez ~ (Z is the set of all integers), i.e., if t = (tl, ta, . 9 ., tm) , then o,(,) =

. - .

(

9 Zm)

8 -i(tlz1+tszz+ ,..

+tm z m )

,

We will denote by G(Tm) the space of finite functions over ZTM and by S(Z m) the space of all functions over ZTM. Let us define an operator Af acting from G(Z m) into S(Z TM) in the following manner: (Alx) (z) = Y',~Zr~ al (Z -- t) x (t) . We will call this operator the convolution operator generated by the function f and the function f itself will be called the symbol of the operator Af. We now formulate the basic results.

We introduce the class of operators Endp.

An oper-

ator A acting from G(Z TM) into S(Z TM) belongs to Endp if and only if it can be extended to a bounded operator A acting from Lp(Z TM) into Lp(Z TM) , and for the norm of the operator A in E n ~ we take the norm of the operator A. Let B be a Banach space and % be a mapping of [--~, ~] into B. We will say that % belongs to V~(B) (the space of functions of bounded variation of order B) if %(--~)----%(~) and Rostov State University. Translated from Matematicheskie Zametki, Vol. 22, No. 6, pp. 873-884, December, 1977. Original article submitted May 16, 1975.

978

0001-4346/77/2256-0978507.50

9 1978 Plenum Publishing Corporation

II~ liver.) = [sup ~ , 2 ~ II ~. (p~) - ~. (P~+OI1"~1~m -4- t~[-~. sup ~] II~ (t)II. < oo. The supremum in the first summand is taken over all possible partitions--u < p, ~ p ~ < . . . < Pn ~ ~. For every function ! ~ M ([--~, ~]m) ( M (X) is the space of bounded functions on X) we define a mapping %re(f) from [--~, ~] into ]kI (I--g, g]m-~) by the following rule:

{ [ ~ (01 (z)} (z~, z~, . . . , We d e n o t e Vs(C) m], where ~]m) into lows :

(C i s

z ~ - 0 -- ! (z~, z~, . . . ,

the space of complex numbers) by V[8,

z~_~, x).

1] and d e f i n e b y i n d u c t i o n

V[-8,

~ [i, oo)~ Let V [~, re-- I] be already defined. We put a function [ ~ M([--n, V[(ff,~m), re] if and only if %m(~ ~ Va n(V[~, re--i!). We define the norm as fol-

IIf IlvtCa,~=), =l = IIk= (I)II v~=(vtg. =-~)" If

~[1,

oo) ~, t h e n we a g r e e

THEOREM 1 . 1 .

If

!~V[~,

to set

FIn(~) ---- ~ ' ~ "

m! (~ ~ [ t ,

oo)"),

9 9 9 "~n.

then

At~End~

and

II a~ tl~d~ < c II/II~zm ~j, w h e r e C d o e s n o t d e p e n d on f ,

f o r a l l p ~ (2Hm (~)/[Nm (-if) -6 t], 2Urn

(6~/[n~ (~) -

This theorem was proved in the one-dimensional case by Hirshman S techkin [i].

~1).

[2], and for 8 = i by

We now introduce the space V = [~, re] ----H = ([--~, ~]m) ~ V [~, m! (~ ~ (0, i), ~ ~ [~, oo)m) with the norm ! II~tff ' m! = II / Ilvt~ ' =] + Ha([-- ~, g]m)

sup

t,, t ~ [ - n , =]m

I / (t0 - l (t=) 1/11t~ - t= II%m,

is the space of functions satisfying the H~ider condition with index ~ on

[ - ~ , ~1 ~. THEOREM 1 . 2 .

If [ E V ~ [~, ml, ~

(0, t), 13 ~ [ 2 ,

00) ~ ,

then

At~End~

and

II At where C does not depend on f, for all p E (2nm

~)/[rIm (~) +

2~], 2Hm ~ ) / [ g m

(~) -- 2~1).

This theorem was also proved for m = i in [2]. We introduce the space only if

V =,~ [~, m! (k ~ m, ~

[I, oo)m).

A function

f ~ V ~,~ [~, ml

if and

II/Uv=, ~t~. ol = IIs Ilvt6, ~ + s u P I Y (tl, to) - / (t~, t0) l / 11t, - t~ II ~2 < oo. The supremum in the last term is taken over all tI, ts ~ [--z, z!~, tI ~=ts, to ~[--z, z!m-2. THEOREM 1.3.

If

[~

V ~,~ [~, m~,

c~E (0, I), ~ ~[2,

oo)2 • [|, oo)m-~, then A! ~ E n d ~

and

II Aj II~,,d~,< C IIYllv=,2t~.~a' w h e r e C d o e s n o t d e p e n d on f ,

for all

p ~ (2II~ ( ~ ) / [ I I ~ ~ ) + 22], 2IIm (~/([I,.~ ( ~ - - 22 ]

The p l a n o f p r o o f o f T h e o r e m s 1 . 1 , 1 . 2 , and 1 . 3 i s t h e s a m e . S c a l e s o f s y m b o l s and o p erators are introduced. I t i s p r o v e d t h a t t h e o p e r a t o r p u t t i n g t h e o p e r a t o r Af i n c o r r e s p o n d e n c e w i t h t h e f u n c t i o n f i s b o u n d e d on t h e e n d s o f t h e s c a l e , and t h e p r o p e r t i e s of the i n t r o d u c e d s c a l e s and t h e i n t e r p o l a t i o n theorem are used. 2. The I n t e r p o l a t i o n Theorem. L e t B, Bo, a n d Bz b e Banach s p a c e s , w h e r e t h e c o n t i n u ousembeddings B1CB0 and B C B 0 hold. Let ?0>0, 71 > 0 , and ? , - 5 7 1 = i We will say that the space g is an approximate partition of (Bo, Bx) in the ratio Yo:Yx if the~X(%)(% ~ ~ constant C such that for each x ( ~ B ) there exists an abstract function ~ / ~ s a t i s f y i n g the following conditions: 979

a)

x(L) ~ B ,

b)

[l x - - x (~,)liB, ~

c)

II z CX) liB,-.< C (1 + x)*, II ~ Ils.

c (i + x)-~, Ilx Jim

We will say that B is a moment partition of (Bo, B:) in the ratio Yo:Yx if B, C B and there exists a constant C such that the following inequality is fulfilled for every x (~B,):

ITx lib < c If9 I1~. IIx ,~',. The d e f i n i t i o n s o f t h e a p p r o x i m a t e and t h e moment p a r t i t i o n s ko in [4], where the following theorem was proved.

were introduced

by Simonen-

!

THEOREM 2.1. Let Bo, B, B x, B', B', B: be Banach spaces, where B is an approximate partition of (Bo, B,) in the ratio y and B' is a moment partition of (B', B'x) in the ratio y', y > ,1'. Let A ~ H o m (B0, B0) be such that Air,,65Horn (B t, B~). Then A Is ~7~H o m (B.B'). 3. Spaces of Operators and Symbols. Let us denote by End~' ~ End~', where %, = {/p (p ~ [I, 2i), with the norm

~)~

the set of operators of

!l A !!~:? = tla I ,,,. -q- II A il, I t h e s p a c e V [(~, ~m), m] proximate partition o f M ([-- ~, ~]~) and r [~, m] i n t h e r a t i o 1 / ( $ m -- l ) . T h i s lemma f o l l o w s

from the following

LE~4A 3 . 3 . For every in the ratio 1/(81).

is

an a p -

temma.

8 > 1 t h e S p a c e VB(B) i s

an a p p r o x i m a t e

partition

o f (M(B), F ( B ) )

Proof. L e t F E V~ (B) . Then i t i s e a s i l y p r o v e d t h a t f o r a r b i t r a r y t ~_ [ - - ~ , ~] t h e l i m i t f r o m t h e r i g h t F + ( t ) = F ( t + 0) e x i s t s and t h a t f o r e v e r y n a t u r a l n u m b e r n t h e s e t Sn of points, for which IIF (t) -- F (t + 0) l[~>/n-V~ IIf ~v~(~), contains at most n + l elements. Let us express [--.~,x] as [J [p~,p~+~), where po = --~, and each [Pi, Pi+x) is constructed as the largest interval [Pi, d) such that !I/7+(t)-- F;.(p~)!l~ ~ n-t'~ IIFl]v$(s~ for arbitrary t ~ [p~,d). The existence of such an interval follows from the continuity of F+ from the right; it follows from the maximality of the interval that IIF+ (p~) -- F+ (Pi+l) i]~>7 n-"~ i!F !Ivs(s~. It follows from the last inequality and the inequality IIF+ IIV~(B~~ !lF live(s) that [-- ~, ~) is divided as the union of at most n + i segments. Let

F,~(t) =

and F X = Fn for L ~

980

[n, n + I).

/r (t),

for

t E Sn,

F+(p0,

for

t E [p~, p~§ \ S,,,

F (~),

for

i=1,2 t----g,

..... n+t,

The inequality IIF -- Fx 11~(~)~ (I ~- ~)-iI~IIF llggB) is valid for the function F x so constructed. This function assumes at most % + i different values, and therefore F x ~ F (B) satisfies the inequality [IFx Ilr(~)~ (i q- %)(P-I)~ IIF IIv~(~). By Ti, j we will denote the operator of permutation of the i-th and the j--th coordinates of a vector, i.e., if x ~ / ~ m (i, ] ~ m), T~,yx : y, t h e n Yk = Xk f o r k ~ i , ] , y~ -- x], a n d y j = x i . I f ~ ~ [t, c o ) a n d k i s a number, t h e n we w i l l d e n o t e t h e v e c t o r (% z , . . . , ~) ~ [t, oo) ~ b y ~ . ~_[t,

LEMMA 3.4. Let V [ ( ~ , ~m-~'), m], and

oc) ~, ~ c ~ [ t ,

oo),

] ~V[(C~,~m-'~),

m],

k 0 ,

k 0)

such that for arbitrary

0 ~ (2

-

-

~+1, 2) for all

?' ~ [1/2, (~+,II~ (-~) + 2~+')I2~+, II k (~')). The spaces of operators ( v, ~ (O~,C ~Dv,) form a moment partition in the ratio (Lemma 3.1), and the spaces of symbols form an approximate partition

(V o [(~, "~-~), m] C V~ 1(~, l~k+x~-m-'), m]

( ? 2 - I/2)/

C M ([--~, re]m)

in the ratio ~I(~- ~) (Lem~na 3.4), and the operator am acts boundedly from the extreme spaces of scales of symbols into the extreme spaces of the scales of operators (Lemmas 4.1 and the assumption of this lemma). Therefore, we can apply Theorem 2.1 to the operator ~m, and must only choose T so near to 2 and Y1 to 2~+H~(~) 2H~(~---)

that - ~T- ~ >

7~--I/2 arbitrary 71-- ?-------for ~

983

v ~ [112,(2~+I + ~+IH~ (~)12pkH~ (~)). Applying Theorem 2.1, we prove Lemma 4.6. Theorem 1.2 follows from Lemmas 5.2 and 5.3. Theorem 1.3 is proved by induction. For m = k it is valid (this follows from Theorem 1.2), the passage from mo -- i to mo is realized with the help of Lemma 4.3, which together with Lemma 4.1 enables us to conclude that ~mo acts boundedly from the extreme spaces of the approximation scale of spaces of symbols (r~,~ [~ m0] C W, ~ [(~, ~mh, too]c M ([-- ~, n]~0 rr

rcto

into the extreme spaces of the moment scale of spaces of operators ( ~ C ~ y , C 9 ~/,). In this case we can apply Theorem 2.1. That is, we have proved that if Theorem 1.3 is valid for m = mo -- i, then it is valid for m = mo also. LITERATURE CITED I. 2. 3. 4.

5. 6.

S . B . Stechkin, "On bilinear forms," Dokl. Akad. Nauk SSSR, 71, No. 2, 237-240 (1950). J . J . Hirschman, "On multiplier transformation," Duke Math. J., 26, No. 2, 221-242 (1959). M. Sh. Birman and M. Z. Solomyak, "Double Stieltjes integral operators and problems on multipliers," Dokl. Akad. Nauk SSSR, 17! , No. 6, 1251-1254 (1966). I . B . Simonenko, "The justification of the averaging method for the problems of convection in a field of rapidly oscillating forces and for other parabolic equations," Mat. SD., 87, No. 2, 236-253 (1973). A. Zygmund, Trigonometric Series, Cambridge Univ. Press, Cambridge (1959). L. H~rmander, Estimates for Operators, Invariant with Respect to Shift [Russian translation], IL, Moscow (1962).

THEORY OF SELF-CONJUGATE COBORDISMS UDC 513.836

N. Ya. Gozman

The ring of bordisms of manifolds, in the normal bundle to which is given the structure of a self-conjugate complex bundle, is studied. The results of computation of this ring up to dimension nine inclusive are given. The elements of the ring which can be represented by spheres and real projective spaces are considered. Their orders are computed and certain relations between them are found.

Introduction. duced.

In [i, 2] a K-theory based on self-conjugate vector bundles was intro-

Definition. A pair (6, X) consisting of a complex vector bundle ~ and an isomorphism X: ~ + ~ of the bundle $ and its conjugate bundle ~ is called a self-conjugate bundle or in short, an SC-bundle. The manifolds, in the normal bundle to which is given the structure of an SC-bundle, are the objects of an appropriate theory of bordisms ~SC. A classifying space BSC for the self-conjugate K-theory is constructed in the following manner (see [i]). Let ~ be a universal stable vector bundle over the space BU and 7': EU + BU be a universal p_rincipal U-bundle. Further, let the mapping g: BU § BU classify the virtual bundle ~ --~. We get the commutative diagram

Moscow District Pedagogical Institute. Translated from Matematicheskie Zametki, Vol. 22, No. 6, pp. 885-896, December, 1977. Original article submitted July 19, 1976.

984

0001-4346/77/2256-0984507.50

9 1978 Plenum Publishing Corporation