On singular integrals, multipliers, and fourier series — a local field phenomenon

Math. Ann. 265, 181-219 (1983) A m �9 Springer-Verlag 1983

On Singular Integrals, Multipliers, ~i and Fourier Series - a Local Field Phenomenon

James Daly 1'* and Keith Phillips 2'** 1 Department of Mathematics, College of the Redwoods, Eureka, CA 95501, USA 2 Department of Mathematical Sciences, New Mexico State University, Las Cruces, NM 88003, USA

I. Introduction and Notation

The work is a continuing study of integral homogeneity and the subjects in the title for spaces of distributions, functions, and operators over local fields. The principal background sources are [16, 19, 24]. The emphasis is on a unified approach in the broadest possible framework - distributions on the field K, its ring of integers P0, and the group of units D. Our main object is to obtain results on singular integrals, ~P operators, and Hardy spaces for K in terms of Fourier series on D and on P0- The topics we treat are further indicated in the following table of contents.

2. The Characters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182 3. Gamma Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 4. Additive and Multiplicative Fourier Series on D . . . . . . . . . . . . . . . . . 190 5. Integrally Homogeneous Operators . . . . . . . . . . . . . . . . . . . . . . 194 6. The Hardy Spaces ~P (0 <p_-< 1) . . . . . . . . . . . . . . . . . . . . . . . . 203 7. Continuity Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210 8. Algebras of Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214

Section 2 describes the character groups of the four principal groups involved. The gamma function relates multiplicative and additive structures and is of central importance for the subject. Section 4 treats the relationship between the character groups of D and Po, mainly in terms of the gamma function.

Sections 5 through 8 form the main content of the paper, which we briefly describe. If w is a distribution on D with total mass 0, then w extends by integral homogeneity to a homogeneous operator - "singular integral" - with kernel w/I ]. The resulting operator T on the space of test functions is a multiplier operator in that there is a unique distribution m on D for which (T~b)"= m~. This relation is described in [16]. The questions we address are:

1) How are the Fourier series of w and m on D related? On Po ? 2) Under what conditions on w and/or m is T an operator on ~2, s .~p, or

related function spaces or spaces of distributions? 3) Under what conditions does T exist and in what sense ? 4) What are the resulting algebras of operators?

* Visitor, New Mexico State University, Fall 1983 ** Visitor, University of Colorado, Boulder, CO, 1982-83

182 J. Daly and K. Phillips

We give fairly complete answers to all of these questions, and include numerous examples.

The inclusion of Sects. 2-4 make the paper reasonably self contained, at least conceptually. The history of singular integrals on local fields and the related 5p theory is mainly contained in the works [2, 3, 5-7, 14-17, 19, 23, 24] and the references contained therein.

Notation is rather voluminous, mostly as in [-16] and in the main consistent with [24]. We close this introduction with a partial glossary; other new terms are introduced and defined throughout the paper.

~, ~, 2~, ~ as in the works of Bourbaki K local field; K* - multiplicative group of K p prime element of maximal ideal m or II modular function q order of residue class field Po and P ring of integers and its maximal idea/ ~ p"Po, n~Z; A~=I + P , for n > l ; A = A 1 D, P,\P.+ l ; D= Do x* element of D for which x=pkx * (X in K*) )C central character Xv additive character 0 multiplicative character F gamma function w kernel distribution m multiplier distribution X(G) character group of G Anil annihilator ~(G) test functions for G; ~*(G) - distributions of G H mapping from 6*(0) to ~*(K) , z~, t Fourier transform on K, D, and Po, respectively

2. The Characters

We are concerned with four groups: (K, +), K*, (Po, +), and D. Let X be a fixed character of (K, +) which is trivial on P0 but nontrivial on P_ r For yz K, let L.(x) =Z(yx). The function "Y--'ZT' is a topological isomorphism of (K, +) onto X(K, +). Under this isomorphism, Anil(Pk)=P_k ' all keZ.

For K*, we have

K * = {p":n~Z} xD,

so X(K*) is T xX(D), the product of the circle group and a countable discrete group. We return to X(K*) in Sect. (2.4). It remains to describe X(Po) and X(O). We begin with X(Po). (2.1) On X(Po). We have

(i) X(K/Po) ~_ Anilr(Po) = Po

and so X(Po)_~ K/P o. An explicit isomorphism is obtained as follows. Let F be a complete set of coset representatives for Po/P. Then p-hF is complete for P_h/P_h+l (h=1,2 .. . . ), and

Singular Integrals, Multipliers, ~x and Fourier Series 183

/ ' } j = l

is complete for P_n/Po, and

(iii) V= ~) V, h=l

is complete for K/P o. The function

v + Po--* Z~ (ve V)

carries K/P o onto X(Po). For v~D_ n (h>0),)C~ is specified by h and an h-tuple from F; Zv is said to be ramified of degree h. Note that Z, Jp,@ 1; Zv[v,_,---1. Let ~ = v ~ \ v ~ _ , .

The group X(Po) is indexed on Z + by first indexing F as {v i : 0< i<=q- 1} with v0=O, letting Z(~ 1 and

(iv) Z ("J = Z~, n > 0, where v and n are related by

h h

(v) n= ~ nfl ~-1, v= ~ v,jp - j (0<nj<q) . j = l j = t

We have (vi) {X~ : v ~ E,} = {X c") : qh- 1 < n < qh}.

The resulting "order" on X(Po) is not algebraic. It has been used in [13, 26, 27]. The DirichIet kernel with respect to one of these orderings is

n - - I

(vii) D,(x)= ~ z(k)(x). k=O

A calculation gives 1

(viii) Dq~(x)= 2 ~ ) ~e,

and qk- i

(ix) ~ f*(z("l)Z(")=f.Oq~, n = O

making convergence of the blocked partial sums of Fourier series of continuous functions a near triviality.

(2.2) The groups O and X(D). There is a cyclic subgroup C = r of D of order t ) j=O q-1 for which

(i) D=C • Av The (q - 1) characters of C are given by

(ii) e~exp(Z~J~), o < j < q - 1 , �9 , ,a ~ /

and so it remains to describe XA~. The chain {A h " 1 < h} is more complicated than {P~:I <h}. All the subgroups P,/Ph+ L are the same, namely Z~ where q = f . This is the reason that Po/p (indeed, p and a) determine X(Po). An analogous treatment fer Aa is not possible, for the groups An~A,+ ~ may be different for different h.

After these prefatory remarks, we begin by letting (iii) X, = Anilx(o) [A~] (_-__ X(D/An)).

These groups satisfy

~0( ~) = o( C) = q - 1 ; o(Xn) = o(D/Ah) = ( q - 1)q n- ~ , h > 2.


We let

(iv) R h =Xh~( h_ 1, the characters ramified of degree h and note that

(V) ] R h l = q h - 2 ( q - - 1 ) 2 if h > 2 ; ]Rt l=(q-2) .

In any indexing of X(D) on Z +, let 0 0 - 1 and let R h precede Rh+ 1. Thus

R l = { O , : l < = n < q - 1 } ;

R h = {0, : ( q - 1)q h- 2 < n < ( q - 1)q h- 1 }.

There is not a canonical way to order each Rh; it depends on Ah/Ah+ 1. We normalize Haar measure 2 on (K, +) so that 2(Po)= 1. This makes ,~(D)

_ q - 1 and so the Fourier transform of 9Ef~I(D) at O~X(D) is q

q ~gOd2. (vi) g•(0)= q - 1 o

The measure q(q - 1)- t2 is required in the Parseval and Plancherel formulas; e.g.,

I J~d2 = q - 1 2fc~(O)y~x(O). o q o

The Dirichlet kernel with respect to an ordering is ~ q . 1 O k, but of more interest t / - - I k=0

is the grouped Dirichlet kernel

(vii) A , - q 2 0 . q - l x ,

Note that A, is independent of any order on X(D). As in the additive case, the fact that the X, are groups gives

(viii) A,=q"~A, if n > 0 ; A o -

and

(ix) 2 gA(0) 0 = ff@(qh~a,), Xn

| convolution on D. The kernels

q 20 (x) a k = A ~ - z t ~ _ 1 = ~ - 1 ~

are also of interest. From (viii),

q - 1

and for k > 1 :

O" k

q q on A q - 1

0"1~

- ~ q on D\A q - 1

, q k _ q k o n A k 1

ok I on A k_l\Ak on D\Ak-1.

Singular Integrals, Multipliers, ~?~ and Fourier Series 185

(2.3) Spherical Harmonics and Zonal Functions. Let ~ k = span(R~). Functions in

~k will be called spherical harmonics of order k. Since X(D)= U R k. properties k=O

(i)-(iv) below are immediate. (i) (~k : 0 < k < 00) are orthogonal in i!2(D).

(ii) ~2= ~ ~r (Hilbert space direct sum). k=O

(iii) ~ k = {Ok| ~2} (Ok| is the projection onto jcgk). (iv) a k | Y= Y for Y~ ~k.

For k > 1, define the zonal of order k on D x D by (v~ Zk(u, x) = a~(ux- i).

Then: (vi) Zk(u, X) is real valued,

(vii) IZk(u , x)l _-< Z~(u, u) = IR~I,

(rift) [1Zk(u, ")1[ 22 = IRkl,

(ix) Y(u)= S Y(x)Zk( u, x)d2(x), re ~k. D

Note (ix) is simply a restatement of (iv). An interpretation: zk(.,X).2 is a representing measure for the linear functional Y ~ Y(u) on ~k .

These properties mirror those of spherical harmonics and zonal functions in IR"; see [22]. They are introduced in [15].

(2.4) X(K*). Since K*={ph:hE2~} xD, we have X(K*) '~xX(O) . If ei'eT, the action is

(e it, O) (phx*) = eihtO(x * ) (X* E D).

Emphasizing the modular function Ixl, this can be written

(e i', O) (x) = Ix(O(x*), T = it/ln (q).

For z in the strip S=F,+i(-n/lnq, rc/lnq] let

v , ( x ) = Ixl ~ �9

Then

Y= {(y,, 0) : c~eS}

is the set of continuous (not necessarily unitary) characters of K*. Then (~, 0) is unitary if and only if Re[~) = 0; and, it is unramified if and only if 0 = 1. We let

ro = {(v,, 1 ) : ~ s } .

3. The Gamma Function - F

ILl) The Setting. The gamma function for K relates additive and multiplicative characters and Fourier transforms. The domain of F is the set of multiplicative characters y\{ 1 }. We approach F through integral homogeneity, as in [16]. For each unramified character ? of K (~, 4: m-x), there is a bijection Hr from ~*(D) to

[86 J. Daly and K. Phillips

the distributions which are integrally homogeneous of degree V; i.e., in ~'~(7). There is an injection H ( = H , , - , ) from S*(D) to ~ ( m - 1 ) for which

(i) (a, u) ~ a6 + Hu is a bijection from 112 • ~*(D) to Jct~ - 1). For y=(y, , 0)~ Y\(m- 1, 1), the distribution HrO is denoted simply y. The value F(?) is defined by the following result.

(3.2) Theorem. For y~ Y\{1, m}, there is a constant F(?) for which (i) (7/m)~=r(7)~ -1

Proof Suppose first that ? = my* with 7*~X(D) and ? * ~ 1. As a function on K* and K, ?*(pkx*)=y*(x*). Then (?*)"~W(m-1), so there is an a~07 and a u~ ~*(O) for which

(ii) (~*)" = a6 + Hu. Evaluating at ~eo, we have <Hu, ~eo> = 0 and so

<(~*L ~ o ) = a.

Since ~0 =~eo and ? * # 1, we conclude that a = 0 . (If y*= 1, then (~*)'= 1 and u = 0 . ]

Next suppose that 7 = (~, ?*) with ~ 4= 1. Then (?/m)"~ ~(?~ ~). Combining the two cases, we conclude that for (~,7")r 1), ( - 1 , 1)}, there is a u~*(D) for which

(iii) (~/m)"=Hr;,u. To prove (i), we show that u is a multiple of 7*.

For OEX(D), extend 0 to K by letting 0' = 0 on D and 0' = 0 offD. Then 0'~ ~(K) and 0' is constant on cosets of Pn if 0 is in R h. We evaluate each side of (iii) at 0'. Using the definition of Hru ['16, (2.6) or (2.8)], a calculation shows that

(iv) <Hr_ lu, 0') = q - 1 uZX(~)" q

For the left side of (ii), we use

(v) <y /m, ~> - 1 - y(o) ~ ~ m

proved in (3.9) of [16], where q~ is constant on cosets of Pl. For 4~=(0) ", l is 0 because O' is supported on Po. Calculation of the right integral for q~ = (0T gives

h

S ~(0 ' ) "d2= ~ '~(P- ' ) [! Y*(x)O-l(x)dx]'[~ ~ O(Y)Z(YP-J)dY], O# 1. p~m 1= t

(vi) ~ 7-- (O')'da = O, ?* * 1, 0 = 1 p s m

Z(O,).d,t=y,(p_l)( - 1 ] : q - - I y,____ f 1, 0 = 1 v'~m \ q ] '

which is zero unless ~* = 0. If ? * , 1, the first integral in (v) is zero ; and, if 7" = 1 and 0 # 1 then

(0')'(0) = I Od2 = O. D

Singular Integrals, Multipliers, .~ and Fourier Series 187

Thus the first term on the right in (v) is 0 unless y*=O. We have proved that u~(d)=0 unless 0=y*. Thus (iii) becomes

(y/mF = u zx(7*)H~.r ~(y*) or

( ~ , / m y = u A ( 7 " ) 7 - 1

and so (i) is proved with F(7)= ua(7*).

(3.3) Remarks. Theorem (3.2) is proved in [10] and repeated in [11]. The approach there uses homogeneity instead of integral homogeneity and depends on analytic continuation of 7 as a distribution from the case in which it is locally integrable. A similar approach is used in [20, 24]. The method presented here extends from D by homogeneity and uses Fourier series on D.

The proof given above implicitly contains formulas for F( j . To state them it is convenient to use the following lemma, whose proof is omitted.

(3.4) Lemma. The equality 0 if y e P ! 1

(i) ! )~(yx)dx = _ q- i if y~ D_ t

q-l(q - 1 ) if Y~Po holds. I f O~ Rh, h> 1, and y~ K then

(ii) S)~(yx)O(x)dx:#O r yeD_ h. D

(3.5) Corollary. I f y~-m and "~4: 1, then

(i) r ( j=?(p-h) S 7*(y)~(yp-h)dy, 7e Rh ' O

(ii) F ( J = 1 - y ( p - 1)IPl _ 1 - q ~ - 1 7* = 1 . 1 - 7 (P) 1 - q - " '

Proof. If 7 is unramified (7* = 1), the left side of (3.2, iv) is

1 (1,)zx(O) q - - 1 + 7(p_ 1 ) (q~_l) 2 1 - 7(P) q

by (3.2, vi). The right side is q - 1 F ( j . Simplification using (3.4, i) yields (i). q

If ?~Rh(h> 1), then (3.2, vi) and (3.2, iv) give

h F(7) = Z ?~(P-5 ~ 7*(Y)~(YP-6dY-

]=I D

Formula (ii) then results from (3.4, ii).

(3.6) I n t e g r a l F o r m s .

2 ( x ) ~ , ( x ) (i) F ( J = lim ~ ix~--dx (T ramified),

k ~ P~c~P-k

2(x)7(x) , Re(e) > O, ye R h. (ii) F(y)= .~ [~ ax, P - h

188

Proof Equality (3.5, i) can be written

~(x)~(x) r(T)= I Ixl

D h

J. Daly and K. Phillips

- - d x

which is equal to the integral in (i) for k > h. If Re(a) > 0, 7/mE ~ ( P _ h) so (ii) follows from (i).

(3.7) Remarks, Definition. By (3.5, ii), the proper definition of F(m) is F(m)-0 . Note that f = F ( m ) l does not hold. The correct equality is 1=6 , 6 the Dirac

0. The interpretation of (3.6)is that F(7)= [7 1(1,. measure at \m/-

The notation F (TJ* )= F(a, 7"), (a, y*)4= ( - 1 , 1), will be used when convenient.

(3.8) Functional Equations. (i) F(~, V*) = q~hF(7*) (Y~ Rh, h >= 1),

(ii) F(~, 7*) = 7 " ( - 1)F(~, y*), (iii) F(~)= 7 ( - 1)F(7), (iv) F('f)F(m/y)=?,(- 1), (v) r ( . , 7")r(1 - ~, ~*)= 1.

Proof Equality (i) is by (3.5, i). For 7* =1, (ii) follows from (3.5, ii). For 7*ER h, we have

~(Y)~(p-hy)dy = ~ y-~(-- y)%(p- hy)dy, D D

which together with (3.5, i) proves (ii). Equality (iii) is (ii) in different notation. Equality (iv) depends on

(vi) ~ = ~ ( - 1)p, as follows:

, /m =(y/mF~= [ F(7)7 T M 1] '=F(7)F ( ~ ) 7 ( - 1 ) 7 / m .

To prove (vi), use (iii) to calculate

7 ~= (~)*= F(~m)~- l m - 1 = F(ym)7-1 m - 1 = ~).

The two variable version of (iv) is

r(a, 7*)r(1 - a, ):*) = 7 ( - 1),

from which (v) follows.

(3.9) Theorem. For OeRh, h> 1,

(i) q"r(o)r(o)= 1, (ii) IF(0)l = q-h/2,

(iii) IF(~,O)[=q htR~162 l/zj, all oc.

Proof Equality (i) is a direct result of (3.8) - first (v), then (i). Equality (ii) follows, and (iii) comes from (3.8, i).

Singular Integrals, Multipliers, $3~ and Fourier Series 189

(3.10) S u m m a r y - Character Sums and Integrals. We list various formulas for easy reference. Most of the proofs have been given; we sketch those which have not appeared. Recall E h = Vh\ V h_ 1.

(i) j Z(x)dx=~{k>=o~legl, Pk

(ii) ~ X(xy)dx = 1 [(q _ l)~e ~ _ ~D-~] (Y), D q

(iii) ~ X~(y)=qh~e~(y), vEVh

(iv) ~ Z~(Y) = q " - ' [ (q-- 1)~p, - ~D~-,] (Y), vEEh

(v) ~ )~ (y )=- l , all y~D. reEl

(vi) ~ Z~(Y) = 0 / f h > 1 and ye O. VEEh

(vii) S O(x)dx = qh~x,(O), h > O. Ah

(viii) S O(x)dx=qh-l[(q - 1)r -- r ,] (O), h>O, l + D h

(ix) ~ O(y)=(q-- 1)q h-'~A,(y) (yED, h >0). OEXh

(x) ~ O(y) = ( q - 1)q h- ZE(q_ 1)~A h _ r + oh- ,3 (Y) (h > 0). OERh

(xi) ~, O(v*) = 0 if ~ :~ 1 and h > I. veE~

(xii) Y(O)=q -h ~ 0(v*)~,(p-Iv*) if VeRb, h>O, VeEh

(xiii) f , O(V*)~(p-hv*)=O if o c g h, 04: 1, h>O, vEE h

(xiv) ~r(O)O(y)= q- 1 ~,(p_~y) + _1, y~D, R~ q q

(xv) ~ r(O)O(y)= q - 1 ~(p-hy), y~D, h> 1. en q

Concerning Proofs. Formulas (i) and (ii) are immediate, and well known. Formula (iii) holds because X~--*X~(Y) is a character of X(Po/Ph). Formulas (iv)-(vi) follow from (iii). Formula (vii) is immediate and (viii) follows from it. Next, (ix) holds because O~O(y) is a character of Xh; and, (x) follows.

We prove (xi). Let OeR k and first suppose k = h. Since {v* :V~Eh} is a complete set of coset representatives for D/A h, v* ~O(v*) is a nontrivial character of D/A h. So the sum is zero. For k<h and vsE h, we may write

v*=u*(l+zp k) for U~Ek, z s P o.

Then O(v*)= O(u*) and

E 0(~*)= E E 0(~*)=o, VEEh Z U~Ek

where the sum over z is a certain finite sum. For k > h we let k = h + j and induct on j. For j = I, write

190 J. Da ly and K. Phillips

q - 1

o = ~ o(v*) = F~ o(~*) + Y~ I2 o(~* +,~,p") r e E k vsEI t 1 = 1 VEEh

= Y~ 0(~*)+ X 0(~,). Y, 0 ( ~ ? ~ * + r v~E h l = 1 V~ Eh

For each l, the set {v 7 ~v* :veE h} is a complete set for D/A h. Hence the last sum is 0(v* + Oh). Hence

v E E h

0= F, 0(v*) + u c E h

Since {v~}~[2~ is complete for D/A,

q - - 1

Y 0%). Y, o(~* + p"). l = 1 V~Eh

q - - 1

we have ~ 0(vt)=0. So /=1

0(v*)=0. The VEEh

calculation for the inductive step is substantially the same. Formulas (xii) and (xiii) follow from (3.5) and (3.4, i). We give a proof of (xiv).

By (3.5, i) we have

r(o)O(y) = [ O(xy- 1))~(p-hx)dx" D

Hence

F(O)O(y)= ~ ~ %(x)~(p-hxy)dx. R h

For h> 1, the function X~Z(p-nx) is in ~ h [this follows from formulas in (4.2)]

and so by (2.3, iv) the right side is q - 1 ~(p-hy). For h = 1, we have q

L - , = (~p-,)~(1) + Y~ ~,~_ 2(0)0 R~

= - ( q - 1) -1 + ~ 2# , (o)o . R1

Thus )~p -, + (q - 1)- 1 is in ~ 1 (but ~,-, is not). Thus we have

tr m (x)~(p - hxy)dx = ~ a t (x) [~(p-1 xy) + (q - 1)- t ]dx D O

-- I (q - 1)- lal(x)dx. D

By the definition of at, [. tr~(x)dx=O. Hence D

j" al (x)~(p-hxy)d x = ~(p- ly) + (q _ 1)-1 D

and (xiv) follows.

(3.11) Remarks. Most of the results of this section appear in either explicit or implicit form in both [11, 24]. Our context is somewhat different.

4. Additive and Multiplicative Fourier Series on D

In this section we consider Fourier series expansions in XP o and XD of functions and distributions on D. For uE ~*(D), we let ut(zo) denote the Fourier transform of

Singular Integrals, Multipliers, .~t and Fourier Series 191

the distribution u* in ~*(P0) having support D and agreeing with u on D; i.e.

(u*,r162

Let ~ be complete for P_h/Po, as in (2.1). For v~ Eh, let v* = phv (~ D). We first state a general result, whose proof is omitted.

(4.1) Proposition. Let G be a zero dimensional compact Abelian group with character 9roup X. For u~ ~*(G), the equality

(i) u= Z ut(~)x X~X

holds in ~*(G).

(4.2) Lemma. Let O~ R a, v~ E k, h > 1, k > 1.

(i)lt(z~)=blaq - 1 ; l t (1)=q - l (q_ 1),

(ii) x~(l)=b~k(q-- 1)-1; lZX(1)= 1,

(iii) Ot(Zo)=6hkF(O)O(v*),

(iv) g~(0)= q 1 6hkF(O)O(v*)" q -

The proof is by (3.4) and (3.5).

(4.3) Formulas. For uE ~*(D), u t and u a are related by

(i) ua(1)= q ut(1), q - 1

(ii) ua(0)= q v6Eh

(iii) ut(z~)=--61hq-luA(1)-+- Z ua(O)I'(O)O(v*) i f v6Eh" OeRh

Proof, Equality (i) is straight from the definitions. For (ii), let OE R h with h > 1 and write

u~(0)= q (u,O) q - 1

= q ~ v

: q[ot(1)ut(1)+q_l v*o ~ Ot(Zv)u'(Zv)]

= ~ ~. r(g)O(v*)ut(L) q - 1 E~

= q__C_ E r (O)o ( - v*)u'(xo) q - 1 E~

q q - 1

= - - Y~ r(O)o(- 1)O(v*)u*(x) Eh

= JL_ ~ rto)o(v*)u*(zv). q - 1 e~


This proves (ii). Fo r (iii), let veE h with h > 1:

u+(z~) = <u, L> = ~ (L)A(O)<u,O>

OeXD

- q-q 1[ ~:(1)uzx(1)+ o*a ~ (~)a(O)uZX(O)]

_ q - 1 ~(1)u~(1) + Z ~(O)O(-v*)u~(O) q O~Rt,

= -- 61. q - XuA(1) + ~ r(o)o( - 1)O(v*)uA(O) Rh

= - 61h q - ' uA(1) + ~ F(O)O(v*)uZX(O) Rh

= -- 61he- 'UA(1) § Z F(O)O(v*)u~(O) �9 R h

(4.4) Discussion. It is impor tant to realize that {X~ :v~ V} is not linearly independent as a set of functions on D; it is of course independent on P0. From (3.10, v), we see that l e span {Z~ : r e E 1 }, for example. The formulas in (4.3) depend on u having support D. The expansion

(i) u,-- F.u+(xv)X~ v

converges to 0 on P with u ~ defined as

(ii) u+(x~) = <u, ~DZv>" As an expansion on D, (i) is not unique because {~ ' r e V} is not linearly independent ; indeed, each {X~ : ve Vh} is not linearly independent.

Let fi denote the Fourier transform of u~ ~*(Po). A set of complex numbers {fi(Z~) : ve V} (i.e., an element of (~~ indexed by V) defines an element u of S*(Po) by

(iii) u = ~fi(~)X~. v

The restriction of u to D is of course in ~*(D), so u t is defined and satisfies the equalities in (4.3). We do not have a condition on fi to insure that u as defined by (iii) is supported on D; i.e., that fi = u*. Another perspective is that an element u in ~*(D) can be extended to ~*(Po) in many ways ; by using u* for our analysis we have extended by simply making the support D.

(4.5) The spaces 92,+ and 91". Let

92*={f : fAeV(XD)}, (i) 92+ = { f : f , eV(XPo)},

0 < r =< oo. For r >= 1, both 92 + and 92* are in E(D). The spaces 92+ and 91" are very different. We give an example. Let {?h : 1 =<h} be a sequence in XD for which ?heRk" Define u by

0 h/2 if 0=?h uA(O)= if OeXD\{~h}.~=l.

Singular Integrals, Multipliers, ~ t and Fourier Series 193

Clearly uEg.l*, but by (4.3, iii) we have

u*(zv) = Ua(7,)Yh(V*)r(~'h)

Hence

and so

and

(rE Eh).

lut(zv)l =q-h (v6Eh)

[U*(Zv)[=q-hlEhl tJ~Eh

lu t (z~) l = o o . V

So, ur Examples in the opposite direction are a little more complicated because for given u t, the condition "supp(u)CD" must be checked. We return to the matter in (5.10). For now, we have the following elementary result, obtained directly from (4.3).

(4.6) Proposition. (i) qh/2uZX~P(XD) ~ u~9.1~, (ii) qh/Zut~ll(XPo) ~ u~9~.

Proof We have

lU't(Xv)[= h~=2 ~ I'(O)O(P*)uA(O) ~ < q-h/Elghl ~ [Ua(0)[ h=2 Eh = Eh h=2 R h

and (i) follows. Similarly for (ii).

(4.7) Regularization. For u~ ~*(D), the regularization of u is 9iven by both

(i) .ix, ~)= ~ Z u~(~176 h=O Rh

(ii) u(x, n) = ~ ~ ut(x~)Zo(x). h=0 Eh

Thus, the partial sum equality (iii) u*Dq,=u*A,.

Proof. The regularization kernel Dq, is also the Dirichlet kernel, so regularization on P0 is (ii). By (4.3) of [16], (iii) holds. Hence (i) holds.

(4.8) Remarks. By (4.8), regularization corresponds to grouped partial sums of Fourier series. Hence convergence theorems about these grouped partial sums are relatively easy; see [16, p. 79] and [24, p. 175]. Actually, one can go the other way and use results on Fourier transforms to prove regularization results. The basis for this approach is Theorem (2.3), p. 331, of [17]. If u is in ~1, the grouped partial SUms converge in ~1 and pointwise a.e. If u is continuous, convergence is pointwise (see [24] and [17]).


5. Integrally Homogeneous Operators

(5.1) Introduction. For we ~ ( D ) , let m denote the unique element of ~*(D) for which

(i) (Hw)~=Hlm. Every element of ~ (m-1 ) is of the form ct5 + Hw for ce ~ and

(ii) (c6 + Hwy=c+Hlm holds. In this section, c, w, m are related by (i) and (ii). The operator T (= Tin) associated with m and w is

(iii) Tmt~ =(Hw) , r It is defined from ~ to ~ ; see Sect. 5 of [16]. By general results on convolution and Fourier transforms, the distributional equality

(iv) (Tm~b)'= $(H~m) holds. Note that T~b is a function in i3 but H~m may be only a distribution. The distribution ~(H~m) is Him multiplied by the function ~b; it has compact support because q~ has compact support. The operator T m is a multiplier operator with multiplier the distribution m. In these formulas, hypothesis' on m or w yield operator results for Tin. For openers, if met2~(D), then

(Hlm, lp) = ~mvfld2 K

from which it follows from (iv) that (v) (T.,~b)~= q~ .m,

a pointwise formula. If m e ~ o ( D ), then (vi) (Tm~b)'= ~ $(x)dm*(x),

K where m* is the integrally homogeneous extension of m to 9X(K) [16, Sect. 3].

(5.2) Theorem (Multiplicative Fourier Transform). We have m~(1)= wZX(1)=0 and (i) mZX(O)=F(O)wA(O) if 0~-1.

Proof By (3.2), (HO)~=F(O)O. In the chain

~*(D) n ~*(K) " , ~*(K)

each map is continuous, so we may compute as follows:

m =(nw) '= Y~ w~(O)(noy= y, w~(O)C(O)O. 0 0

Thus (i) holds. The equality mA(1)=w~(1)=0 holds as we ~ ( D ) .

(5.3) Theorem (Additive Fourier Transform). The equalities m*(1)= w*(1)=0 hold For veE h (h>l) , we have

O) m~(Zv)=q -h E ~v(U*)Wt(Zu), UEEh

(ii) w*(.Z~)= ~ X~(u*)m*(x.). ue Eh

Proof The equalities m*(1)=w*(1)=0 follow from (5.2) and (4.3). For veEh, we

calculate using (4.3):

Singular Integrals, Multipliers, .~1 and Fourier Series 195

mtix3 = F~,,~"io)Oiv*)r(o)

= Y~ w~(O)r(o)Oiv*)r(o) Rh

: .~[~ci-~Z q rio).o~Z wt(zo)oiu*)] rio)o(~*)rlo)

__ q

=q-h 2 2(vu*)wt(z,). u~Eh

The last step is by (3.10). Similarly,

w'/z.)= Z w~iO)O(v*)r(o) OERh

= Z m~(O)O(v *) 0

-- q Z F(O)O(v*) Z mt(z,)O(u*) q - 1 o , ~

- q - 1 q ~.r(O)O(u*v*)m'iZU)o = Z zip- hu*v*)mt(z.).

it

(5.4) Regularization Formulas. For wE ~*(D),

(i) w(y, n) = ~ qh I re(x, n)ziP- hxy)dx, h = l D

(ii) rn(y, n) = ~ ~ w(x, n)2(p- hxy)dx. h = l D

Proof. These are variations of (5.3). For (i),

w(y, ,0 = Z Z w'izo)Li- y) h = l Eh

= ~ qhmt(Z-rp-h) h = l

= ~, qh f m(x, n))~(yxp- h)dx. h = l D

Similarly for (ii).

(5 5) Discussion. The formulas in (5.2}-(5.4) yield several results immediately. First let

(i) a(o)= r(o)/tr(o)l, o~xD. Then the X(D)-expansion of m can be written

(ii) rn = ~ q- hi2 ~ a(O)wa(O)O. h = 1 Rh

196 J. D a l y a n d K . Phillips

Note that

(iii) a h | = q - ~/2 ~ a(O)wV(O)O, Rh

so regularization results give convergence to m under various hypothesis on rn; see (4.9). In addition, we see that

(iv) ~a(O)wV(O)O <Ah" ~ m continuous, A and h constants. Rh

That is, if the sums ~ are bounded by a polynomial in h, then m is continuous. Rh

Note that there is a corresponding additive result, for the grouped partial sums are the same - (4.9).

For 1 =<p_-< oo, ! p results are also immediate:

(v) w~elP(XD) =*" mZXeF(XD); Ilm~lrp<llwAIlp.

An application of Holder's inequality in (5.3, i) gives

(vi) wt elP(XPo) =*. m~ e lP(XPo) ; IIm*Hp <llw*llp.

If me P~I(D), then the integral in (5.4, i) is the same with re(x) replacing re(x, h) (h < n), and so

(vii) w(y, n)= ~'. qh S m(x)z(~-"xy)dx (mes ~) h = l D

holds for the regularization of w. This "inversion formula" for w has been known for sometime. When w is constant on cosets of Pn, the inversion is given by

(viii) w(y)= ~. qh ~ m(x)z(p-hxy)dx. h = l D

In this form, it is proved in [5]. In (5.6), we show that it holds for me 9~ § and rl ----- 00 .

If m e s ~, then T m extends to s by Fourier inversion. From the outset, a major problem in the theory of Calderon-Zygmund singular integrals has been obtaining conditions on w which insure that m is bounded; a brief discussion appears in [21, p. 39]. The problem occupies a major portion of the original paper [1] and subsequent work by Calderon and Zygmund. In the local field case, modulus of continuity conditions on w ("Dini" conditions) are found in [17, 19, 15]. It is clear from (5.2) that m and w can have very different smoothness properties. For example, an m in 9.I* can be obtained from a w whose multiplicative Fourier coefficients do not even go to zero. More on this in the "examples" section - (5.10). The additive ".t31(D) results" obtainable fairly directly from (5.2)-(5.4) are as follows. We state them as a group, for unity, in (5.6)-(5.8), then give the proofs. For yeD, Zp-~ means that ;iv for which veE h and yo-heo+Po; and, )~(ta-hyx) is its value at x.

(5.6) Theorem. megl~[ =~ we9.~ and

w(y)= ~. qJm~(~p_jr ) (p1 and a.a.y). y = t

(5.7) Theorem. wet~o t ~ lim mt(z ("~) =0 and

0= E m (zv). D j= 1 w s

Singular Integrals, Multipliers, gOa and Fourier Series 197

(5.8) Theorem. m e s ~ ~ limwt(ff~ and n ---~ oo

(i) re(y)= ~ wt(~,-,y), h = l

(ii) ~mdy= ~ q-~ ~ wt(zv). D h = ] Ve~ h

Proofs of (5.6)-(5.8). For (5.6), let w,(x)= w(x, n). By (5.4), for k<n we have

IIw.-wkll~ < ~ ~oqJ[om(x)x(P-Jxy)dxdy j = k + 1

j = k + l D j

= j=~+ 1 ~ ] Ejlv+Drn(x)zv(x)dxldy

= Z Z tmt(z~)t j = k + l Ej

Thus {%},~ x is !~l-Cauchy and so converges in it 1 and a.e. [-16, p. 79] and 1-24, p. 175].

To prove (5.7), let n > 0 and compute:

j = l Ej j = l E l p i v+Pj

= ~ qJ ]" mt(z~-,)de j = l D

The sum is the regularization of w, so converges in s Hence ~, ~mt(x,) converges to ~w. The limit follows, j= ~ ej

The partial sums for (5.8) are (5.4,ii). By regularization, they converge pointwise a.e. and i n ~ a. Equality (5.8,ii) follows by integrating; and, lira wt(zt,) ) _-0 follows from the convergence in (ii).

(5.9) Four Multiplicative Results. (i) ws~ 1 => qh/2m~SCo,

(ii) qh/2mZX~I1 =,. we~l ,

(iii) m ~ t => q-h/2WZX~CO, (iv) q-h/2WZX~p => m~t31.

Proofs. If we ~1, the regularizations of w converge to w. Hence the infinite series

(v) ~ Y r(o)-~,.~(o)o ]=1 R~

COnverges, and (i) follows. Conversely, if F-lmaeP, .the series converges ab- solutely and we ~t.

Proofs of (iii) and (iv) are similar.


(5.10) On 11 and 12. From (5.2), it is clear that (i) wa~II(XD) .r qh/2rna~ll(XD).

If B is a function on XD for which there is an e in (0,1) satisfying (ii) 13q hI2 ~ B(O) < e- l qhf2 (0~ Rh).

then (i) can be replaced by (iii) wZ~ll(XD) r Bm~eP(XD).

As an example, ifXD is indexed as in (2.2) and we let B(Oi) = i t12, then B satisfies (ii) and so

(iv) wZ~elt(XD) r {il/2mZ~(Oi)}~P(XD). If we let wZ~(Oi)=i -1 we see that w~II(XD) but m~ell(XD).

The connection for m and w in II(XPo) is not as precise because of the sums in (5.3). It is clear that

(v) mtell(XPo) ~ q-hwt~ll(XPo); (vi) wtell(XPo) =~ mt~ll(XPo).

The formula

(vii) Itm~ll2= ~ q-h~lwA(O)12 h = 1 R h

from (5.2) shows that

(viii) Slml 2dx= q - 1 ~ q_h Y, lw~(O)12 D q h= 1 Rn

(5.11) Example (Sparse w~). For each h > 1, select one yheRh and define w by

{10 if 0=Yh (i) WZ~(0)= if OeXD\{vh }.

Clearly w ~ ~ co(XD ). However mA(O)=JF(Vh) if O=Vh (ii) [o if O~gD\(Th II <h},

so mZ~eI 1. Thus we have an example of a distribution w which is not in !3 ~ but for which T,, has a continuous multiplier and so acts on ~2(K). Note, too, that w ~ l : but raAel 2.

By (5.6), m* is not in lt(XPo). Calculations for m t and w* yield (iii) w~(x~) = ~(v*)F(Yh) (V~Eh),

(iv) mt(g~) = q-h~h(-- 1)~h(V*) (rE Eh) showing that m* and w t are both in c0\lL Parseval's equality gives the forms

<w, = q--21

(v) q h=l ~e@(O) (m, r = q - 1 ~, r(7~)~(~).

q h=l An inclusion summary:

{ w~eb\co w*eco\P

(vi) m ~ e P w~eco\l 1

me 9,I~ wr 1 .

(5.12) Example - Delta fimetion. If 6" is pointmass on D centered at 1, then

(i) (J*)A(0) = q - ~ (6, 0 ) - - q q-- l" q - - i


Define w by

(ii} w = q - 1 [6* - 1], q

so tla~t

(Jill wa(O)= if 0 = !

and (ivl rna(O)=F(O), O* 1 ; mA(1)=0.

The regularizations of w and m are thus given by 1 q -

(vl w(x, n) = A, - 1 =(q - 1)q ~- l~a" - 1 ; q

(vi) = Z Z r(o)o = 1 q 1 - + - , , ~ ( p - h x ) .

h=l R~ q q h=l The additive t ransform of w is

(vii) wt(zv)= ~F(O)O(v*)= l f i l h + q - 1 ~'{v), R~ q q

and of m

mr = q- ~ ~ ~(u*)wt(z,) �9 Eh

For h> 1, we use (3.10) to obta in

~(u,)wt(x.,,)_= q-- 1 --q- 2 Eh

q - 1 2 2 . ( v * + 1) q E~

_ q-- 1 [qh- l ( ( q _ 1)~ph(~) ~ D1 - 1 ) - COb- I (v~r "~ 1))3 q

= (q - 1)qh- 2((q _ 1)~ah( -- V*) -- ~nh( -- V*)),

where Bn=Ah - 1\An" F o r h = 1,

",~ ~(u*)w' (x ' ) = .,~'Y"(u*)[ILq 4- qq 1 ~(v)]

-_- _ 1 + q - 1 [ ( q _ 1)CA,( _ V * ) - - r V*) ] . q q

Hence for v~E h we have

,viii) m*(~)= _ ~ b l h + q~qzl [(q_ l)~.~(_V,)_~h(_V,)-l.

There is just o n e v~/~ h for which --v*SAh, namely that v = - p - h m o d ( P o } . F o r - v ~Bh, the condition is v - - p - h m o d ( D _ 1); there are ( q - 1) such v in E h, Thus we "an rewrite (viii) as

0 if v r 1

!iX) m~()~v) = q - Z ~ l h + ( q - 1)2q -2 if Ve - - p - n + P 0

q-2blh~(q--1)q-2 if v e - p - n + D _ I .


Thus for each h, m~(z~)=t = 0 for the q elements of E h closest to --~p-h and mt(z~)=0 for the other v. The (distributional) Fourier series of m thus has larger and larger gaps. Of course, ~rnt(zv)X~ does not converge to a function. The regularizations of w and m can also be written using (vii) and (ix), with more complicated expressions. There is no contradiction as {X~} is not linearly dependent on D. Note that the expressions in (v) and (vi) do not give 0 on P1 ; our definition of ut for us ~*(D) yields ~ u~(X~)Zo = 0 on P r

V The action of w and of m on ~*(D) is given by Parseval's formula. Note that the

measure on D making the formula valid is q(q-1)-~2, and <u, ~b) denotes the action of u on ~b, not the inner product with respect to q(q-1)- t ,L The results:

(x) <w, 4> = q:: l E 0 . 1

(xi) <m, = q - 1 y q 0.2

An inclusion summary for the Fourier coefficients:

wAsh\Co wtsb\Co

(xii) mZXSco\l 1 m*6b\c o mr 1 wr ~ .

(5.13) Example (ws~ 2, m r Let {Ck:I <k} be a nonnegative sequence for which

(i) ~ 2 ~ C k k - 1 / 2 = O 0 C k < (~ and k= l k=l

For example, c k = [k 1/2 In(k)]-1 Let w be defined by

(ii) wZX(Ok)= ckk- 1/2F(Ok)- 1 Since Clq h- 1 < k <- C2q h- 1 for constants C I and C 2 and OkSR h, k- 1/ZF(Ok)- 1 is bounded. Hence wZ~sl: ; and, wsE 2. We have

(iii) raZe(Ok) = Ckk- 1/2,

SO both m zx and w ~ are in 12 but mZX~P. We will show that m ~ ~176 For ~bs ~(D), Parseval's equality gives

q - 1 I mc~dy = ~ Ckk-1/2~bA(0k). q o k=~

l Take ~ = ~ ~Ah" Then

4a(0)= if OsR h and h<=n.

Since Okff U R h r k<(q-1)qn- l=kn, we obtain h = l

1 ymdy=_ q ~ cgk-1/2. IAnl a . q - - l k : l


The terms on the right converge with n to oo. Hence m cannot be in ~ . In this thesis, Liu 1-15] proves there are functions w on ~2 for which mCP, ~176

under the minor restriction that char(K) :4= 2. His argument depends on w being an odd function. For char(K)=2, there are no odd functions. The above construction does not depend on the parity of q, nor on char(K). Our example- and Lifts-shows that the inequality

Ilmll~,< allwll2 fails. The norm Ilmlloo is also the !~2-operator norm of T m, so

I1Z~[I <aNwll2

fails. The corresponding inequality is valid in the Euclidean case. Daly [6] has also shown that this inequality fails for K. He uses an existential argument based on the dosed graph theorem to show that the inequality Jl Tmrtz <AJlwlr, fails for 1 < r < 0% not just r = 2. In the main the global theories of homogeniety and singular integrals for 1K II~, R", and IK run parallel. Here is an instance of individuality in the local field case.

(5.14) Example (11 - x [ - ~/2). First let

C= ~ [1- xl-1/2dx. D

A calculation (not to be given) shows that

q - 1

In this section, we study the function

(i) u={ll-xl- t l2}-C. Note u~d~(D) and for 0 ~ 1 the equality

(ii) uZX(O)= {[1 - x [ - 1/2} ~(0)

holds. We calculate these Fourier coefficients, as follows.

q - 2 5 II-xl-'/20(x)dx -- I I i - x ] - l / 20 (x )dx+ E S O(x)dx D A j = 1 ~JA

q - 2

= j Ixl- '/20(x + 1)dx + ~ O(e -s) j O(x)dx. k=l O j = l A

if0eR a, then 0 is trivial on A and the second term is -IAI. IfOeRh with h > 1 then 0 is nontrivial on A and the second term is 0. For the first term, suppose that 0~ R h. For k < h - 1, we have

For k = h - 1,

S O(x+l)dx= I O(y)dy=O. D~ Ak \Ak+ 1

S O(x+l)dx=- ~ dy=--q-(k+tl=--q-h. Dk Ak + 1

202 J. Duly and K. Phittips

For k > h - 1,

Hence

O(x + l)dx=lDk [ = q - 1 q-k. D~, q

UA(O)=__(ql/E)h-lq-h+ q - 1 ~. (q--t/E)k, h>__l. q k=h

Summing the geometric series and simplifying,

(iii) u~(O)=q-h/2=lF(O)l if OeR h. Thus u~eco\l I and ue~.~. The regularization is given using (3.10, x):

(iv) u(x, n)= q - 1 q2 ~_z~l qh/2E(q _ 1)~a h - ~1 +On-,] (X).

In terms of the action on ~(D):

(v) (u,q~> - - q - 1 ~ q-h/2~pa(O). q ~= 1 Ra

Equality (3.10, xiv) gives ut: h2 1 (vi) u ' ( x ~ ) = q - / [ q ' S x , + ~ ( P - h v * ) ] .

Thus ute Co\P. The above calculations involve only u. We make calculations for each case

w=u and m=u. First take w=u, so wE~ 1. Using a(O)=F(O)/IF(O)I, we have (vii) m~(O) = a(O)q-h,

showing that m~e r By (3.10, xiv) or (5.4), the regularization of m is

(viii) m(x,n)=q-3/2 + q - 1 ~, q_h/2~(p_hx). q h = l

The series converges uniformly and so m is a continuous function. For m*, if vEE~ then

m*tz) = Z Rh

and so (ix) m*(z,) = q- 3h/2 ~ 0 ( - v*), ve E n.

Rh

This shows that rnt~I ~. The sum can be further evaluated using (3.10, x), giving mt(z,) = 0 if v*~Ah_ 1

q--1,7- h/2 (X) rnt(;(~)= q2 ~ if v * e - - l + D h _ l

t q - - 1 2

There are ( q - 1) elements v in E h for which v*e - 1 +D h_ r Thus we have

lira*Ill = ~ [mt(x)[--=2 y' q-hi2 v h = l

=2 q l / 2 1 .

Singular Integrals, Multipliers, .~a and Fourier Series 203

It would be nice to have an explicit expression for m. One way to obtain it would be to sum the series in (viii).

We may also take m = u. Then (ix) wZX(O) = r(o)- X q- h/2 = ~(0).

We have

Y~ w~(O)O(x) = q~/~ Z r(o)o(x) Rh Ra

= qh/Z ~ F(O)O(- x) Rn

Hence the regularization of w is

(x) w(x,n)=q-1/2 + q--1 ~ qh/2Z(p_nx). q h=l

The calculation of w t :

wt(;G) =q-h/z ~ O(v*), R~

and so for V6Eh:

/ wt(zv)=0 if v*r 1

q - 1 h/z (xi) w~(L')= q2 q if V*6--1+Dh_ 1

w*(Z_ pn) = (q -- 1)~2 qh/2 q2

Thus, if the distributional multiplier m is the ~l-function u, the corresponding kernel w does not even have bounded additive Fourier coefficients. Since rnr ~, T,, does not act on ~2. One can ask if there are spaces between ~(K) and s on which T,~ acts naturally.

We summarize some of our findings as follows.

w = {11- x l - ' 2 - C} m = {11- x[- ~/2_ C}

w A e co\l ~ W't6r 1

(xii) rnZ~e co\11 mtel t w e ~ l \ ~ 2

me~

waeb\co w*r m ~ e Co\I ~ rn*e co\l 1 we ~(D)\92~(D)

6. The Hardy Spaces .~P(O < p < 1)

(6.1i Background. The spaces ,~P(K) have been developed by Chao, Janson, and Taib]eson. Taibleson presented a survey at the 1978 summer meeting in Wiiliamstown; see [23] and the references there. The spaces are also included as


spaces of homogeneous type in the comprehensive work [4]. As in the Euclidean case, there are equivalent definitions in terms of maximal functions, conjugate functions, Littlewood-Paley theory, and the atomic-molecular theory of Coiffman and Weiss [4]. In this section, we add some results based on homogeneity and on Fourier series.

For 0 < p < 1 < r < ~ and p~r, a (p,r)-atom is a function a on K satisfying (i) supp(a)CX+Pk=B, some x and k;

(ii) ][aH,< IIBll a/r- l/p;

(iii) S ad,~ = O. K

The space ~I(K) is those functions f in ~I(K) for which there is an ~l-aton~ic decomposition

(iv) f = ~ ~,a,, ~, I~.,1<~ i=1 i=1

for (1, ~)-atoms a i. For 0 < p < 1, .~P(K) is those u in ~*(K) for which there is an atomic decomposition

(v) u= ~ 2,a,, ~ 121[ p<ct3 i = I i=1

for (p, ~)-atoms a t, convergence in ~*(K). Equivalently, the a t may be taken as (p, 0-atoms for any r in [p, ~ ] , r+p. For p > 1, .~P= ~P.

We are interested in conditions on m and/or w which make T,, in ~(.~P, 2P) or in 99(~ p, ~P); we write this last space ~ ( ~ ) . The function spaces 21~ and ~.1" are defined in (4.6). We also need the function spaces on D defined by

1 El,

Since IEhl = (q--1)q h- 1, this definition is easily seen to be equivalent to

~9.1~ = { f : ~I ln(n) lf' (z~"))l < ~ }

for an indexing o fXP 0 as in (2.1). In our setting, atoms having Fourier transform equal to a multiple of ~ are

particularly useful. If & = ~D, then 1

(viii) ~ = ~p0 - q ~ e - , "

Property (i) holds for a with B = P_ l ; and, (iii) holds. The r-norms of a are:

IIc~ll o~ = q-_~l, i1~111 = 2 q - I q q

(ix) 1

'l~ll,= ~--~[1 + (q_ 1),_1] 1/'

and of course

(x) IP_ ~[=q. For each choice of (p, r), there is a constant b for which b~ is a (p, r)-atom. One simply writes down (ii) for b~ : b appears only on the left side.

The following lemma is proved by a calculation, which we do not give.


(6.2) Lemma. If f is supported on Po and 0 < p < o% then for s>=l we have

Y',lf+0~,)l , - - j" IfTPdx. h=s Eh P-s+1

(6.3) Theorem. For O < p < 1, Tme~(~ p, ~v) ~ meg.i+.

Proof For each p, there is a b=b(p) for which a=b~ is a (p, 2)-atom. Then ae.~i p and so T,,ae PP. We have

Im+(zo)P = f I(m~oFIPdx = b -p ~ I(ma)TPdx = b-" II rmall~. veV P~ K

(6.4) Theorem. For 0 < p < 1, m ~ 9 ~ + =~ Tme~(.~ p, ~P).

Proof By results in I-4], it suffices to prove that T m maps (p, oo)-atoms into ~v. Further, since T m commutes with translations the atoms may be assumed to be supported on some Pk" And since T m commutes with dilations, we can take k = 0. So let a be a (p, oo)-atom supported on Po and suppose also that a is constant on cosets of P / j > 1). We have

and so by (5.6) w is in !~(D). Referring to the proof of (3.3) of [16], we see that T=a is a convolution of functions :

(i) T~a=K,a=(~c~py),a @=~1)"

Thus T~a ~ p z and (T~a) ~ = m&. To estimate II Z~a II p, write (ii) lIT,,(a)ll~= ~ Ix*alVdx + ~ I~c,alPdx.

Po P~

For the first term,

(iii) eo~l~c*alPdx< [~o IIr ple<=ll~c*allp2<=llmll~"

For the second term,

(iv) e8 ~ Ix*alPdx= Ixlf> 1 lyl I? l a(Y)[K(x-Y)-~c(x)]dyPdx

< ~ ~ Ix(x-Y)-~:(x)tPdxdy. Iris1 Ixl>a

Let %=m~D_~, SO m = ~ m,. Each m, is in !~(K). The distributional equality

(v),n'= ~ ms s = - o 9

holds. It can be shown that l

(vi) ~o~ m; = ~G~, and so the pointwise equality

l

(vii) lira ~ m~=K

holds. Using this in (iv), we have l

(viii) ~ Ix* alPdx <= lira j / ~, I'n~( x - Y ) - m~(x)l'dxdy �9 P6 t-~oo lyl=<z Ixl>1 s = - ~


Since m~ is supported on P_ ~, m~ is constant on cosets of P,. This gives

5 l~c*al pdx< ~ I ~ ]m:(x-Y)- 'n:( x)lpdxdy ,~ lyl_=l Ixl>l s=o

f I i<(x)l,ex y s = O l y l < l Ixl > 1

=2 ~ I Im'(x)l pdx s = O p % .

=2 ~. 2 2]m'(z~ )}p $ = 0 h>s Eh

= 2 ~ s Z Im'(zJ. s = O Es

The resulting bound for II Tmal] e is independent of a and ))all ~, = 1, and so T,, maps (p, oo)-atoms boundedly to s

(6.5) Discussion. The above proof easily yields a bound for I/T,,a}J~. For v~E s and ~') = Xv, we have q~- 1 < n < q*. Hence s < 2[lnq] - 1 In(n) and we obtain

(i) 11T,~(a)Hg =< Ilm([~ + 4 ~ In(n)[m*(z~"))( p. ln~q~

The second term is also a bound for the expression

q(w,p~= f f I~(x-y)-x(x)l~dx&, I~1=<1 [xl>l

~C ~- wra- 1 For p = 1, this condition was introduced by Phillips and Taibleson in [19] and later used by Liu [15]. We will return to it in Sect. 7. For now, we have corollaries (6.6) and (6.7) below.

Since the relation between m t and m z~ is known (4.3), (6.4) has a multiplicative counterpart which we state as (6.8). The proof of (6.7) requires that me~ ~. See [15] and our Sect. 7.

Using local field analogues of the Reisz transforms, Chao [2] has shown thal for odd q the relation

holds. Hence (6.9) below is also a corollary of (6.4) Finally, (6.10) is a partial converse to (6.9).

The results (6.3) and (6.4) essentially classify bounded homogeneous multiplieI operators on bP(K). No Euclidean equivalent is known at this time.

(6.6) Corollary. I f me~91~, then tl(w, 1)< oo.

(6.7) Corollary, I f we ~lo(D ) and ~l(w, l) < o9, then T m e ~ ( b t, ~1).

(6.8) Theorem. I f

(i) ~ Im~'(0.)l ln(n)n 1re < ~ , n = 2

then T,,e ~(,~ 1, ~1).

Singular Integrals, Multipliers, .~a and Fourier Series 207

Proof. By (4.3),

[mt(zv)[ < Cq~/2 ~" Ima(O)[, Es Rs

C constant. Hence

(ii) ~ sEtm't(X~)[<=C ~ sq ~/z EImzx(O)l. s = 1 E s s = 1 R s

The right side of (ii) is bounded by the left side of (i), so the hypothesis gives

(6.9) Theorem. I f q is odd and ms P.9.I~, then T,~e ~(~v). In particular, if we ~ and t/(w, 1)< 0% then T,,e~(~I).

(6.10) Theorem. Suppose T~e~(f91). For each e > 0 , T m maps (l,2)-atoms to (2,~)- molecules if and only if \ z/

(i) ~ n 1 +~lmt(z~"))l 2 < ~ . ,=1

Proof. We must begin by describing molecules. A (2, e)-molecule M centered at x o is a function satisfying

Mole: S M d 2 = 0 , M ~ L 2, {M(x)lx-xoll /2+"}ep. 2. K

For Tme~(~l), we have msg.I~. Hence TseN(~2), and so the content of the theorem is that (i) holds if and only if "for every (1, 2)-atom a there is an x o for which

(ii) ]" [Tma I 2(x) Ix - x ol x + ~dx < co holds". First assume this last s tatement and let a be the (1, 2)-atom for which h=q-~r o. Using the fact that T,, commutes with translations, we compute :

oo> ~ ITm(a)12(x)lx-xoll+~dx Ixl > t

= ~ IT,.('Cxoa)12(x)lxll+'dx I~l > 1

=q2 f i(m~(Xo)r z ixlX+,dx Ix l>l

=q= ~ f. I(mr s = 1 D - s

=q= ~ q~"+"~Y ImP(x31 ~. s = 1 Es

The last sum is a constant times the sum in (i). Next suppose that (i) holds. By Theorem (2.2) of [24, p. 180], (i) is equivalent to

dii) ~ ~ im(x+y)_m(x)12dxlYl-2-,dy<oo lYl < 1 Ixl = 1

am~ this is the form we will use. It suffices to prove (ii) for a having suppor t in some Pk ( k < 0 and Xo=0 , for the general result then follows by translation. A

208 J, Daly and K. Phillips

calculation like the one proving (1.6) of [24, p. 220] shows that

S I T,,,al2(x) Ixl 1 + 'dx < C ~ S I(ma)(x + y) - (ma)(x)[2dx lYl- 2 -~dy. K K K

Writing the integral as a sum of differences:

II T,,al" 11 + ~IL 2 < C[ II roll o~ S~ Ih(x + y ) - fi(x)12 dx lYl- 2- ~dy] lj2

+ C[$J la(x)l 2 Im(x + y ) - re(x)[ 2 dx lYl-2-~dy] 1/2

= C[I 1 + 12].

To estimate I p use the constancy of fi on cosets of P -k :

112 = Ilmll 2 ~ [ Ifi(x+y)-fi(x)[Zdx ly[-E-~dy Pe--k K

<2[Iml[Z~ IlalL~ S Iyl-2-~dY ~'_u

= C1/[ml[ 2 [la{122q -k(1 +~) .

For I 2, use the fact that a(P_ k)= 0:

I~--(. S ...dxdy<llall~ ~ S ~.. . .dxdy. K Pe-k l=k+ l K D-~

For each l, split the y-integration into {]Yl <qt} and {]Yl ~qt}. Using homogeneity of m:

~ Im(x+y)-m(x)12dxlyl-2-~dy=q-~B, l y l<~ t D-~

where B is the integral in (iii). And,

~ [m(x+y)-m(x)12dxly[-2-'dy<Cllm[12qZ~ ly[-2-~dy lyl~q z D-~ r__<q~

<C m 2 -l, 1 ooq "

Thus

By Holder 's inequality,

/22 ~ C2 Llall~(1 + Ilmll~)q -~k.

I[&ll~q-kllaJ[ 2 .

Putting the estimates for 11 and 12 together:

(iv) I[ Z.,al.l' +'l]2 Z C(rn, e)llal[2q -k(L-~-) Since a is a (1, 2)-atom supported on Pk, we also have

(v) [[Z,.al.[' +'[12 ZC(m,e)q -k*.

(6.11) Discussion. The proof obtains the bounds (6.10,iv) and (6.10,v). Perhaps (6.10, v) written as

(i) [I T,,a{Ix[ 1 +'}11~ z C(m,e)lek[' is a nicer form.

Singular Integrals, Multipliers, .~1 and Fourier Series 209

The Euclidean equivalent of (6.10) for a homogeneous multiplier m is the necessary and sufficiently of the Hormander condition

S IDam(x)lZdx < oo, 1 <lxl<2

introduced in [12]. Proof of the sufficiently is in [25] and proof of the necessity appears in [8].

(6.12) MultipUeative Corollaries of (6.10). The equalities (4.3) yield corollaries of (6.10) in terms of m z~. The unindexed but grouped form of (6,10, i) is

(i) i qh(1 +~) E Im*(zv)l z < oo. h= 1 En

This gives :

qh, +,) E ImA(0)[ z < oo =~ h= I Rh

(ii)

(.;) Tm takes (1, 2)-atoms to -molecules.

A use of Holder's inequality gives a somewhat weaker version :

~" n2+"lm~(O,)[2 < oo =~ n = l

(iii)

T,, takes (1, 2)-atoms, to (2, 2)-molecules.

In the other direction,

Tm takes (1, 2)-atoms to 2, -molecules =~ (iv)

i qh~X ]mZ~(0)[ z < oo and ~ nqmA(O,)[ 2 < oo. h = l Rh n = l

The additive results are more precise because in the first part of the proof of (6.10) equalities hold. This is essentially due to (6.2). It is tempting to try to give "rnultiplicative" proofs to see if (ii) and (iv) can be improved. Or, to construct special functions showing that they cannot be improved. We feel that the later is more likely - that examples along the lines of (5.10) will show that (ii) and (iv) are best possible.

We conclude this section with one more corollary of (6.10).

(6.13) Corollary. Suppose that T,,G ~,(.~t), ~ > 0, and T,, maps (1, 2)-atoms to (2, e/2) molecules. Then

(i) ~ nnlm*(x(.))[ < oo ,=1

for 6<~/2.

Proof. Using Holder's inequality, the series in (i) is estimated as follows:

210 J. Daly and K. Phi!lips

Enarl-(1;~--) '+~

__< n 2~- , - , n 1 + e [mt(z(~))l 2 p l=J . n

= C. n ' + qrn'(xc"))l 2 . n 1

7. Continuity Conditions

(7.1) Introduction. In this section, we ~I(D). Let (i) Aw(x, y, j) = w(x + pSy)_ w(x), (x, y,j)e D x Pox 7] +.

Clearly, continuity conditions can be expressed in terms of Aw. Let

(ii) ~(w)= ~ S S IAw(x,y,j)ldxdy. j= 1 Po D

I t is not difficult to show that

(iii) ~(w) = ~ S w ( x - y) w(x) dxdy; lyl_-<l [x[>l x - y x

it is in this form that ~/(w) appears in (6.4); see (6.5) also. A straight forward calculation shows that

(iv) ~(w)<~ *> ~ I IlAw(x,y,j)ldxdy<oo. j = I D D

In [19], various conditions on Aw are shown to give existence of T= as an opera to r on ~P ( p > 1). It is asked if one condit ion - r / ( w ) < m - will do. In his thesis, Liu 1-15] shows that all the major ~P results follow from r/(w)< m. In this paper, we add some Fourier series and .~P results. Note that Aw is a function on D x P0 • Z+, the generic element of which we write (x, y,j). We will prove that m is bounded if ~/(w)< oo and obtain explicit formulas for Fourier coefficients and series in terms of Aw.

(7.2) Lemma. There is a constant M > 0 and a compact open set U in D for which l e U and [Z(p-ly) - I [ > M - t for ye U.

Proof. Since P_I/Po is cyclic and ;~[e-i :~1, X(p-~)~=l. The result thus follows f rom continuity of Z.

(7.3) Theorem. I f r/(w)< 0% then moP, |

Proof. We have Aw(.,y,j)et31(D). An easy calculation shows that

(i) : Aw(x, y,j)~,(x)dx = [Z~(YPS- 1]w'(xu) D

for all ue E If u~E h and j = h - 1 then

Zu(ypJ) - 1 =X(P- 1u'Y)- 1.

For ye l-u*]- 1 U, the term on the right is bounded away f rom 0 by M - ~ - by the lemma. Hence we have

(ii) [w~(x.)[ ~ M ~ lAw(x, y, h - 1)[dx D

if y~[u*] - lU and u e E h. Let zeD. There is a unique u in E h for which z6u*Ah.


t-[ence

wt(z,) = w~(x~ _ ~z) .

integrate (ii) with respect to y over [-u*]- ~ U = K,"

(ifi) I1~1 Iwt(z~-~)l < M I I [Aw(x,y , h - 1)ldxdy. Ku D

By compactness, D is a finite disjoint union of sets K u. Summing (iii) over these K,, there results

q M[. IIAw(x,y,h--1)ldxdy. (iv) Iwt(x~-~)[ < q - 1 1) o

By the inversion formula (5.4, ii),

re(z, n)= F. w t ( z , _Q . h=l

And so we have proved the bound

(v) Im(z,n)l-<M, ~ I I Idw(x,y,h-llfdxdy, h = I D D

and that re(z, n) converges boundedly to m satisfying

(vi) Ilmll~ <=M t ~ ~ ~ IAw(x,y,j)[dxdy. j = O D D

(7.4) Note. The final bound for IImll o~ is the LI(D • D • Z + ) - n o r m on Aw, and so this seems the most natural no rm to use. However , in other contexts D • P0 x Z + is more natural. A/so, j may range f rom 1 or f rom 0 depending on the si tuation (j~;g+ + or j6Z+) .

The differences Im(z,n)-m(z,k)l are small uniformly in z, so in fact the convergence of re(z, n) to m is uniform on D and so in is continuous.

(7.5) Lemma. I f w~9.~, then

(i) hw~(z.)= _ ~ ~ Aw(x,y,j)~.(x)dxdy = - ~ Aw(x, y,j)dy ~.(x)dx j=O Po D j=O

Jbr ue Eh, h > l.

Proof. For fixed y and j, Aw(x,y,j)~P~l(O). We have

A w(x, y, j)~,,(x)dx = ~ w(x + ypJ)~.(x) - wt(x.) D D

= z. (ypi)w*(z.) - wt(x.)

= [z.(yp ~) - 1] w*(z.), which can be stated

(ii) {A w(., y, j)} ~(Z,) = w*(z,) [X,(yp j) - 1 ].

The right side is 0 f o r j > h and is clearly in !i~t(D) (in y) for each j. We have

! Aw(x, y,j)~,(x)dxdy = w'~(X~) ~ [Z(uyp j ) - 1]dy. Po D Po

The integral on the right is 0 for j__> h and is - 1 for j < h. Thus (i) follows.


(7.6) Lemma. If w~d~, then h h -1 l

(i) h[trh| h-a ~ ~ ~ dwdx-(q-1) ~ ~, dwdx dy. PoLz+Dh-I j=O z+Ph j=O

Proof By (7.5,i), h-1

h[ah| ~ Z Aw Z Z.(x-z)dydx. D Po j=O u~Eh

The sum over E h is evaluated using (3.10, iv), with the result that the integral in x is

C z 1 ), s( it Aw __qh- X)dx+ Aw qh- a(q_ 1)dx ; z+Du-a j=O z + P h \ j = O /

thus (i) holds.

(7.7) Lemma. If w~Yd~ and oeE h, then

[~+D ~-~ (i) qhm~(z~)= ~ ~ Awdx-(q-1)

Po -v* h-1 j=O

Proof.

~o Awdx dy. - v*+Ph j =

By (5.3) and (7.5) h - 1

hm~(z~)= _q-h~ I 2 dw ~ ~,(x)~(v*)dydx D Po J= 0 u~Eh

h - 1

:-q-h!~o(~=oAW)(~h~,(x+v*')dydx"

Again (3.10, iv) is used on ~. The result is (i). Eu

(7.8) Theorem. If weP~, the regularizations of m are given by l h - 1

(i) A,| ~ ~, mt(z~)Zv(z)= ~ ~O]o-h ~" Aw(x,y,j)dy~[p-hxz]dx. h=l geEh h=l j=0

The functions A,| converge uniformly to m and so

(ii) re(z) = ~ ~, A w(x, y, j) dye(p- hxz)dx. h = l j = o

Proof. By the first equality in the proof of (7.7),

(iii) h ~ m~Of~,)Xv(z)=q-h~o I Aw ~, 2 Zu(x)z,(v*)xv(z)dY dx. v~Eh Po\J=O l u e E a v~Eh

The sum over v is

~, Z(vz- uv*) = 2 X~(z - u*) = qh- 1 [(q _ 1)r z_ U*) -- r ,(Z-- U*)] veE~ veEa

and so

(iv) ~ ~.(x) E ~.(v*)X,(z) u~Eh I~eEh

: qh-1 [(q__ 1 ) ~ {~(x)r ) - ~u(X)r

The set {u* :ueE h} is complete for D/P h. Hence

Y. L(x)~,~(z- u*) = Azp-"x). lteEh

Singular Integrals, Multipliers, . ~ and Fourier Series 213

For fixed z, ( z - u*) is in D h_ I for those u* which can be written

U * = z W v p h

for v~E r For such u,

Siace ~ ~ ( v ) = - 1, substitution in (i) and then in (iii) yields (i). The uniform P~E1

convergence of A. | is essentially done in (7.3) - see (7.4) also. Thus the equality (ii) results.

(7.9) Corollary. I f we 9,~ and Awe 9f(D x D x 7Z+ ), then the multiplicative Fourier series of m is

h = l Rh D j : O PO It converges uniformly on D.

Proof In (7.8, ii), evaluate ~(p-hxz) by (3.10, xiv) and rearrange the sums. The 1~q-

term for the case h = 1 does not appear because J" Aw(x, y, O)dx =0. o

(7.10) Inequalities. We have obtained either explicitly or implicitly a number of inequalities, which we want to summarize. Let

]lAwll= ~ ~ (. Idw(x,y, j ldydx. j = o D o

If the y-integration is over Po instead of D, then the integral is bounded by IIAwll. The following inequalities hold, where C denotes a constant - not the same in

all cases - and "Aw~9.1'' means s x D • 77+). (i) w e ~ =~ +lmll+~llw*lla,

(ii) Awe~ ~ => Ilmll <=CJIdwll, Off) aweYd ~ ~ IIm*JIl<CllAwll,

(iv) Awe~ 1 and veE, ~ hlw+(zv)l < h - 1

S ~ IAwldxdy~ljAw[I, j = O D D

h - 1

(v) Awe~ 1 and veE h =*. hlm+(z~)[<C ~, [. [. IAwldxdy<Clldwl[, j = O D D

(vi) me~ ' a~ =~ IIAwl[<C ~, hF~lm+(;~v)l. h ~ 1 Eh

OPt the Proofsl Inequality (i) holds by the inversion formula (5.4, ii). Inequality (ii) is proved in (7.3).

In the proof of (6.4) the inequality

t] Tm(a)llt ~ ]]m[] oo + C ]l dwll

is obtained for atoms supported on Po and constant on cosets. For one particular such atom a, the proof of (6.3) shows that

Y'. Im+(zv)l = b- 1 II Tm(a)lll, v~V


b constant. Hence (ii) above yields (iii). Note the "bootstrapping" in this argument. We first obtain m e 2 ~ by (7.3) (Liu, [15]), then use that information to obtain me~; ~.

The inequality (iv) is from (7.5) and (v) follows from it by (5.3). And (vi) is proved in (6.4).

(7.11) More Summary. Suppose that we !~ and Awe P). The following hold. (i) me!~(D),

(ii) m ~ ( D ) i.e. rnteP and lim HA,*m-m[[~ =0,

(iii) T~,e ~(i~2), (iv) T~,e M(s s if 1 < p < m,

(v) T,, is weak-type (1, 1), (vi) Tme~(~ ~, s

(vii) For f e P , p ( l < p < ~ ) , lim (~pk*f)= Tmf a.e., k ~ o o

where ~k=(W/[ [)~lq-k__< I~ I __<q~"

(viii) q-h/2mZ~ e 11 (xi) hw ~ S I ~ (ix) q-hwtel~ (xii) hmte l ~ (x) q-hwZXeP (xiii) hq-h:Zw/x~l ~

(xiv) hm A ~ 1 ~.

On the Proofs. Inclusion (i) is by Liu - our (8.3). Result (ii) follows from (8.10, iii), which is based on (vi); i.e., (6.7).

Once (i) is obtained, (iii) is immediate. From (i) and (iii) to (iv) and (v) is not trivial, but standard. It is carried out in [15], with the additional assumption that w e s ~. Section (3.5) of [-19] shows how to get by with w~s a local field adaptation of a method of Hormander [12]. Pointwise convergence- (vii) - i s also proved in [15], with the restriction w e s ~. Again, w s f ~ suffices, as in [-19].

Inclusions (viii), (ix), and (x) follow from the relations among m t, m z~, w t, w/x (in Sects. 4 and 5) and inclusion (ii). Conditions (xi) through (xiv) are of a different nature. The boundedness of hw - that is (xi) - is a direct result of Lemma (7.5), and the other inclusions follow.

(7.12) Summary-- Conditions which imply A w ~ 1. Suppose Wes t. If any of the following hold, then A w e L 1.

(i) hmte l x,

(ii) hwte l 1,

(iii) hqh/2m/x e I t, (iv) hqhwZXeP.

On Proofs. The implication hmte l 1 =~ A w e s ~ is (6.6). The other assertions follow by expressing w*, m A, w a in terms of m*.

8. Algebras of Operators

Various commutative Banach algebras of operators can be defined using smooth" ness conditions on the multiplier m or on the kernel w and appropriate normS.

Singular Integrals, Multipliers, b~ and Fourier Series 215

Principal examples are (1) me~29.I + ; and, (2) weg~*. Before discussing these we begin with a general discussion and some formulas.

(8.1) Generalities. The integrally homogeneous distributions (i) {c~+Hw :ceC, we ~(D)}

define operators (ii) cl+ T,, ( c , m ) e C •

with initial domain ~ and range ~3. These are precisely the operators on ~ which commute with translations and dilations by pJ (je Z) ; see (5.8) of [16]. It is natural to ask where the composition (clI+Tml)o(cfl+Tm2) is defined; clearly, the question reduces to that for Tin, o T,. 2. Formally, we have

(iii) Tin, ~ T,~l~b)* = (m lm2) ~ and so the questions is closely related to the existence of the product mlm 2. Since

(iv) (mira2) zx =m~*m~ (on XD), the question can be phrased in terms of the existence of the convolution of two distributions. The question can also be asked with " t" replacing "A". Continuing to operate formally and assuming m~m 2 exists, what is the corresponding w? First,

iv) Z so, of course, it need not be true that mam2~ $'~(D). We have

(vi) Tm o T,., = el + Tmo, where c is the constant in (v) and moA(1)=0. The Fourier coefficients of the w corresponding to m o are given by

(vii) wZX(O) = r(o)- ' Z r(o, lr)w oorff)w or ~ .1

(viii) w~(O) = F(O)- o , 1. The same considerations using the additive transform ? yield the following formula, where vz Eh, h> 1 :

wt(zv)=eq-2 Wtt t w~2(Z,-%*)+wt2(Zv) 2 wtt(J~,-kv *) = k = l

( q - )wlw2(X,-,~.+x)- ~ "rw'r k=h+ 1 [xcEk-h yeEk-h+ I

for which the calculation is somewhat involved. Again, the calculations are formal; convergence of the series will be discussed shortly.

(8.2) Multiplier Subalgebras of ~(~2). The multiplier subalgebra of ~(~2(K)) is (i) ~ ' = {T o :gr~O(K)} '

where

(ii) (Tgf)'=9~,, f e e 2 .

The algebra ~/t is closed and maximal Abelian and Ilgll oo is also the operator norm [{~i, ii. The subalgebra of integrally homogeneous operators is

iiii) ~/h = {cI + T~, : (c, m) E �9 • ~*(D)} ;

clearly it is closed in J/t. It is characterized as those operators in ~(~-) which commute with translations and with dilations by p-~ for j e Z. We have

IlcI+ Troll = 11c-4- m{l~ �9


Clearly {cl+ 7",, :m~E(D)} is a dosed subalgebra of M h. An element cI+T m is invertible in M h if and only if (c + m)- 1 is bounded. Thus if m is continuous and (c + m) 4: 0, then (cI + T,,,)- a exists.

We can define subsets and subalgebras of ~/h by suitably restricting w or m. If C ~ ( D ) , let

(iv) .JCh[~]={cI+T,n :msS} . F o r a subset li of ~ ( D ) , let l F = { m :wet[}. For each choice of ~, there is a bijection

(c,m)-.-,cI+ T,, or (c,w)~cI+ T m

from �9 x ~ to a subset of ~(92). A norm N on ~ yields a norm Icl + N on ~r We list some examples in Table 1. Let

(v) l iu*l[ ,= ~ hY, lu*(z~)l h= 1 Ea

for ue ~*(XPo); similarly with ~*(XD).

Table 1

N' on ~ N on ./r [~]

(1) ~3 I'.m~ll, Ic'! + IIm~ll~ (2) 9[0 ~ IIm~lll lct+ IIm*ll~ (3) ~ g [{m~(I. Ic[ + Ilmz~tl. (4"1 . ~ - Ilm~H, Icl + lira*I[. (5) ~ IIwAIla Icl + ]lwZXllx (6) 91,F Ilwtll~ Icl+llw~tl'~

For ml, m 2 in one of the listed ~, we have

(cli + T,~l)o(c2l + Tin)= [ctc2 + ~rnlm2] + T~:~2+c~ml+m o

and so

(vi) N((clI + Tm,)~ + Tin))= c,c 2 + ~m,rn21 + N'(qm2 +c2rnl +too)

We want to show that the spaces listed are Banach algebras. By (vi), it will suffice to show that

{vii) N'(mo)+ ~ rnlm 2 < g'(ml)g'(m2).

(8.3) Theorem. Each (J//~[~], N) in ( l)-(5) of Table ! is a commutative Banach algebra. The set ~/r ] is composition closed and so is a subalgebra of ~h,

Pro@. The spaces on the left are all complete in the given no rm; this is either well-known or easily established in each case. Consider J / h [ ~ ] . Since ~I~ is a Banach algebra and

(mtm2) A =m~|

Singular Integrals, Multipliers, ~ and Fourier Series 217

Jh[9/*] is composition-closed. The left side of the inequality (2, vii) is H(mlm2) A II x, so (2, vii) holds. Hence ,#t'n[9/~] is a Banach algebra. The same reasoning applies to ~h[9/~']. For ,gh[~9.l~], the equality

h(m( * m~) = m~ *(hm~) establishes composition closure and also yields a proof of (2, vii).

Formula (1,vii) shows that wA~IX(XD) if w(, w~P(XD). Hence .gh[9.l *~] is composition-closed. To show that (2, vii) holds, use (iv) to write

mlrn2[ =[(m( * m~)I(1) < ]m~t* loJ~l(1).

By (1, vii),

I~o~(0)l_-<lw~l* Iw~l(0), 04:1. Hence the left side of (2, vii) is bounded by

2 (Jw j,lw D(o), O~XD

which is bounded by the right side of (2, vii); so, (2, vii) holds and Jgh[9/~'] is a Banach algebra.

To show that Jghl-9/~-'] is composition closed, we use (1, ix) in the same way that (1, vii) was used for the multiplicative result. But in the additive case the estimates are not so transparent and we provide an outline. Call the terms on the three lines of (1,ix) Tl(v ), T~(v), Ta(v). Then

(vi) [ T~(v)] < 4q- 2[ II w~2 II, Iw~(zo)l + II w[ [Ix Iw*2(xo)l. Summing over V:

(vii) ~ i Za(v)l < 8q- : II w*~ I1~ II w*2 II ~. v~V

If ITE(V)l is summed over veEh, then each term Iw*~w*21(Xo+~) for y~E x is obtained (q- 1) times. This gives

~, ]T2(v)lN_4q-2(q-1) ~, Iwlw~l(X~) v~Er, v~Eh

and so

(viii) ~ [Z2(v) [ <4q-X ~ iwtw,210r w V o~V

For T3, note that

Ek={p-kv*+X:(V,x)~Eh• (k>h).

It follows that

~ ]T3(v)]<q-2(q_l)k~ " (~, qh-k) ~., [WIW~](Zs) h= 1 v~Eh = 1 \h~_k , s e e k

OD

Routine estimates now give

tix) Z ITs(v)l<2q -t Z Iwlwt, l(zs) �9 vcV v~V


Putting (vii), (viii), (ix) together we obtain (x) II w t II1 < 8q- 2 It w~ I1111 wtz I11 + 6q-1 II w~ wt2 lit

We have tlw w' lll z ttw ttl tiw' tt

and so cot~l 1 and J /h [9~ ~] is composition closed.

(8.4) Remarks. (i) The algebras dt'h[9.1~] and ~'h[9~-] are inverse closed. To see this note that (cI+ T,.)-1 exists if and only if (c +m)4=0. If this holds, then ( c + m ) - a e ~ (or 9,1+), by a classical theorem of Bochner. It follows that m~9~ [or 9~-), where

( c + m ) - l = d + m ' , (m')Zx(1) =0 .

(ii) We do not know if ~'n[9,I~ -"] is a Banach algebra in N. It seems likely from (1, ix) that it is not.

(iii) All of the algebras are subalgebras of ~(~2), but are not closed in the operator norm because none of the corresponding spaces are closed in ~ (D) . In the norms given, they should operate naturally on some smaller space than ~2(K). What space? In one case - Jt/h[~9/g ] - we have the answer and make it our last theorem; (8.5). Recall that J / h [ ~ 9 / + ] C ~ ( ~ l , ~ 1) by (6.4) and that ~(~1,~1) = :~(.~1) if q is odd.

(iv) An interesting choice for ~ in J//h[~] is ~=[~otC~{w:Aw eEI(D x O x 7~+)] ~. We do not know if Jr is operator closed.

(8.5) Theorem. The commutative Banach algebra ,///h[Eg.I~'] with norm N (Table t) ~s a subaloebra of ~(~91, El). The containment is continuous as the inequality

(i) IIcI+ TmIIt~,e~)<CN(cI + T m) holds. The maximal ideal space of Jgh[~9,I~] is D and if (c+m)(x)4:0, then (cI + T,)-1 ~,///n[J39A~- ].

Proof The inequality

I I T,,a II1 --< II m II | + 4(1 nq)-I ~, h Z Imt(z~)l h= 1 gh

is proved in (6.4) for atoms a; (i) follows immediately from this. To determine the maximal ideal space, consider the Banach algebras

~t = {cI + T m : N,(cl + T,,)< ov , e>0} ,

where (ii) N~(cI + T,,) 1) 1 +~lmt(zv,)l 2W2 = + lel.

pl

Note a~r is a dense subalgebra of J / / h [ ~ ] and N(c+ Tm)<N,(c+ T,,). As in (6.10) the finiteness of N, is equivalent to

(iii) [~o ~o Im(x + y)-m(x)12 dxlyl-'2+~ ~/2 < ~

A short calculation shows that if (iii) is finite and re(x) 4= 0 for all x, then (iii) is also finite for m- ~. Thus ~', is inverse closed, and if (c + T,,)- ~ exists in &(~) , then rt is in ~r For x~D, let

?~(c + Tin) = c + m(x).

Singular Integrals, Multipliers, ~ and Fourier Series 219

Then ?x is a c o n t i n u o u s m u l t i p l i c a t i v e l i n e a r f u n c t i o n a l o n ~r B y t h e i n v e r s e

c losure o f ~ r { ? x : x e D } is t h e m a x i m a l i d e a l s p a c e o f ~ r F r o m t h e d e n s e

inc lus ion o f d ~ i n Jgh[~29,I~-], i t f o l l o w s t h a t {Tx :x~D} is a l s o t h e m a x i m a l i d e a l space o f J / h [~21 t0 ] .

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Received May 3, 1983

On singular integrals, multipliers, and fourier series — a local field phenomenon

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