transactions of theamerican mathematical societyVolume 290, Number 1, July 1985
SUBSPACES OF BMO(R")
BY
MICHAEL FRAZIER
Abstract. We consider subspaces of BMO(R" ) generated by one singular integral
transform. We show that the averages along x;-lines of they'th Riesz transform of
g e BMO n Z.2(R") or g e Ve-(R") satisfy a certain strong regularity property. One
consquence of this result is that such functions satisfy a uniform doubling condition
on a.e. v-line. We give an example to show, however, that the restrictions to x-lines
of the Riesz transform of g e BMO n L2(R") do not necessarily have uniformly
bounded BMO norm. Also, for a Calderón-Zygmund singular integral operator K
with real and odd kernel, we show that rC"(BMO, ) Q Lx + K( Zf ), where Lf and
BMO, are the spaces of L°° or BMO functions of compact support, respectively, and
the closure is taken in BMO norm.
1. Introduction. Let BMO(R") be the set of complex-valued functions/ g L\oc(R")
such that
II/IIbmo = sup — [ |/- fQ\ < 00,Q \Q\ JQ
where the sup is taken over all cubes g in R" with sides parallel to the coordinate
axes, fQ = (1/1 gD/ç/, and \Q\ is the Lebesgue measure of g. For us, a singular
integral operator K with kernel ß is an operator of the form
Q(*-y)
'\x-y\-Kf(x)=limf ^f^-f(y)dy,
f-o r,x_y.>t \x y\
where ßs„-> G C00, ü(rx) = ß(x) for r > 0, and /s„iß(x) da(x) = 0 for surface
measure da on 5""1 = {x G F.": |x| = 1}. For /g l)ls¡p<xLp, Kf(x) exists a.e.
(dx), and K is a bounded operator on Lp for 1 < p < oo. Associated with K is the
Fourier multiplier m so that for/ g L2(R"), (Kf)(£) = w(£)/(|), where the Fourier
transform is defined by /(£) = jK-f(x)e2,"x'i dx; m satisfies m(r£) = m(£) for
r > 0 and | G R" - (0); and w|s„-. G C00. For/ G L°°(R"), we define
ft*)-*»/ (Bi^4-eMx..„wU)*,e-D J\x-y\>t \ \x-y\ \y\ I
where \e denotes the characteristic function of the set E. For /eLTiL00,
1 < p < oo, Kf and Kf differ only by a constant. The operator K maps Lx into
Received by the editors March 21, 1984.
1980 Mathematics Subject Classification. Primary 42B20.
Key words and phrases. BMO, singular integral operator.
©1985 American Mathematical Society
0002-9947/85 $1.00 + $.25 per page
101
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102 MICHAEL FRAZIER
BMO boundedly. Theyth Riesz transform Rrj g {1,...,«}, is the singular integral
operator with kernel ßy(x) = C„Xj/\x\ and multiplier m ■(£) = /'¿,/IH [7, p. 58].
Let H\R") be the set off g L'(R") such that
ll/IU'H!/!L> + Î \\Rjf\\Ll < ».7 = 1
In [1], Fefferman and Stein proved that (//')* = BMO (in an appropriate sense),
and obtained by duality as a corollary that BMO = L°° + L"=1 RjLx, with
(1-1) II/IIbmo ~ inf £ 11^11^:/= go + E Rjgj, modulo constants\J-0 7=1
where ~ means that the norms are equivalent. In [8], Uchiyama obtained (1.1)
constructively for / of compact support, and proved that a certain condition on the
multipliers of a set of singular integral operators Kx,...,Kn implies that BMO =
Y."=xKjLx with an appropriate equivalence of norms. This condition had been
proven necessary by Janson [5].
Using in part techniques developed by Uchiyama in [8], we consider subspaces of
BMO generated by one singular integral operator. The Riesz transforms are the most
natural to consider in this context since one may expect that functions of the form
Rjg for g G L00, or Rjg for g g BMO n L2, when restricted to lines in the Xj
direction, have regularity properties that functions in BMO(R") do not necessarily
have. Results of this type are proved in §3, Theorem 3.2 and §4, Corollary 4.3. (In
§2, we state Uchiyama's results from [8] that are needed later.) Theorem 3.2 is
obtained from Lemma 3.1, which characterizes Äy(BMO n L2) in terms of a
decomposition closely related to Uchiyama's decomposition of BMO n L2 [8, p.
224]. One direction of the characterization follows easily from Uchiyama's work; the
other direction is obtained by solving the equation RjCp = b for <p under appropriate
assumptions on b.
Corollaries of Theorem 3.2, proved in §4, state that if / = R.g, g g BMO n L2, or
/ = Rtg, g g L00, then/has a uniform doubling property on *-lines! namely
(1.2) — f f(xx,...,x„)dXj-— ( f(xx,...,x„)dXj\I\ Ji \J\ jj
< q|g||uMo
for almost all (x1,...,xj_1, xJ+1,...,x„) e R""1, whenever / and J are adjacent
intervals in R of the same length. An example of Kahane [6] from another context
shows that the uniform doubling property for a function/on ^,-lines is weaker than
the uniform BMO property on Xj-lines. Both conditions restrict the singularities of/
along Xy-lines to be at most logarithmic, but the BMO condition further controls the
packing of singularities. Kahane's example is sketched in §4; it is then modified to
show that 3G g BMO n L2(R2) such that esssup^JIFjOXx!, x2)\\BMO(x¡) = oo.
Hence (1.2) cannot be improved to obtain a uniform BMO condition on lines for the
class /^(BMO n L2).
In §5 we show that if K is a singular integral operator with a real and odd kernel
(in particular if AT is a Riesz transform), then
(1.3) #(BMO(.) <z L°° + K{Uf) ,
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SUBSPACES OF BMO(R" ) 103
where Uf and BMOc. are the spaces of functions of compact support in L00 and
BMO, respectively, and the closure is taken in BMO norm. The proof is an
adaptation of Uchiyama's main construction in [8]. This suggests that the action of
K on BMO is in some sense close to the action of K on L00.
Finally, in §6 we make some remarks about related problems which are not yet
solved.
Notation. The letter c denotes various constants depending possibly on n and, in
§5, on K. All cubes g, /, or J are assumed to be open and to have sides parallel to
the coordinate axes; xQ and /(g) denote the center and side length of g, respec-
tively. For r > 0, rg is the cube with sides parallel to the axes having center xQ and
side length rl(Q). For/ g {1,... ,n}, Dx is the derivative with respect to x¡, and for
a multi-index a = (ar,...,an), Da is the differential operator d^/dx"1 •■■ 9x°",
where \a\ = af + ■ ■ ■ + an. For/continuous on R", let
ll/llLipi = sup(|/(x)-/(y)|/|x-y|),x+y
and let CX(R") be the space of continuously differentiable functions on R". For
x, y g R*, x ■ y = £*_ixjyj.
Acknowledgements. Most of the ideas for this paper are contained in my
doctoral thesis, directed by John Garnett at UCLA. I thank him for his considerable
encouragement as well as mathematical help. I would also like to thank Akihito
Uchiyama and Alice Chang for their help at various times, and Al Baernstein for
suggestions regarding the manuscript.
2. Background. We state several lemmas from Uchiyama [8] that will be used later.
In fact, the following Lemmas U1-U6 are Lemmas 3.1-3.6, respectively, in [8].
Lemma Ul. Suppose f g BMO n L2(R") and ||/||BMO < 1- Then there exist func-
tions {b,(x)}, and {X,}¡, Xf > 0, where I is taken over all dyadic cubes in R", such
that
(2.1) /(*) = lim E aA(*),A-^oo ml.1 2-*</(/)<2*
(2.2) suppft, ç 3/, jb,(x) dx = 0, \\Dab,\\L~ a C|a|/(/)~H,
(2.3) E KU\ < c\J\ for any cube J
and
(2-4) E*,|/| = c|l/lll>all l
The last equality is not stated explicitly in [8], but follows from the definition of
the {X,} j and Plancherel's Theorem quite easily.
Lemma U2. For {X,}, satisfying (2.3), X, > 0 V/, set
vk(x)= e x,(i + 2k\x- x,\y~l,/:/</) = 2-A
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104 MICHAEL FRAZIER
and
Then \fx, y,
(2.5)
(2.6)
(2.7)
and if 1(1) = 2~p,then
(2.8)
e*(*)= E It) %-,(*)■7 = 0 V '
Vk(x) < «*(*) < c>
^(x)<c(2^-y|+l)" + 17,,(y),
e,(x)<c(2^-y|+l)" + 1£,(y),
/ E £*(*)%(*) ¿* < c|/|.
Lemma U3. Letj be a positive integer. Suppose [b,(x)}, dvadu. satisfy
(2.9) supple 2^7, fb,(x)dx = 0, Mu,i<c(2Jl(I))'\
IfX,>OVIandß>a>0, then
(2.10) E M,(*)/:«</(/)</?
<c2''" S>2,|/|1/2
Lemma U4. Suppose j, {b,}, and {X,}, are as in Lemma U3 a«i/ (2.3) ZioWj. For
a > 0, letf(x) = T.nt)<aXfb,(x). Then
(2.11) ||/||bmo < C2>".
Lemma U5. Le/ / andp,(x) e C^R") ¿>e such that
(2.12) fp,(x)dx = 0,
(2.13) |/./(x)|<(l+/(/)-1|x-x/|)"""1
(2.14) | V/>,(*)| < l(I)'l(l + l(iyl\x - x,\y~\
Then 3{ß, f(x)}f=0such that
(2.15) ||j8,JLipl < c{2H(I))'\ suppßjjQVI, fß,,fix)dx = 0
and
(2.16) p,(x)= £ 2-«"+1%J(x).7=0
If p, = (p/0,---,Pi,m) and each component of p, is real valued and satisfies the
conditions (2.12)-(2.14), and a = (a0,... ,am) g Rm + 1 with \a\ = 1 satisfies p,(x) -a
= 0 Vx, then we can take ß, y = (ßfJt.. -,ß"j) such that (2.15) is satisfied for each
component of0tJ,p,(x) = ZJ.02~J("+%j(x) andß,j(x) ■ a = 0 Vx, /.
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SUBSPACES OF BMO(R" ) 105
Lemma U6. Suppose K is a singular integral operator with multiplier m, and b,(x)
satisfies (2.2) for some cube I. Then p,(x) = Kb,(x) satisfies (2.12),
(2.17) \p,(x)\ <S C(K)(l + l(iyl\x - x^)'"'1
and
(2.18) \vpr(x)\ < C(K)l(iy\l + i{I)~l\x - x,\Y"~2,
where \C(K)\ < csup{|Z)am(£)|:|ê| = L H < C(n)}. (In fact, C(n) = n + 4 is
sufficient.) Also it is enough to assume \\Dxb,\\Upf < cl(I)'2 for j = l,...,n instead of
ll^aMI/«<cw/(/)-iai.
3. Properties of Ä,(BMO n L2). We characterize /^(BMO n L2) in terms of a
decomposition similar to the decomposition of BMO n L2 in Lemma Ul. For
convenience we take/ = 1 and write x g R" as x = (xx, x') for x' g R""1.
Lemma 3.1. Suppose f = Rxg for some g g BMO n L2 with ||g||BMO < 1. Then
there exist functions {a,(x)}r and {Xr},, X, > 0, for I taken over all dyadic cubes,
such that
O..) A»)- * I M,W,k^ccinL2 2-*</(/)<2*
(3.2) \*¿x)\<c(l*l(irl\x-.xA)?-\
(3.3) \D"a,(x)\ < ca/(/)-|a|(l + l(iyl\x - ^l)"""1"'"',
(3.4) ja,{xf,x')dxf = 0 Vx'eR"-',
(3.5) E A2|/| < cl^1 for any cube J
and
(3.6) !>2/|/|<°o.all I
Conversely, if f satisfies (3.1)-(3.6) for some {a,}, and {X¡}/, then f = Rxg for
some g g BMO n L2 with ||g||BMO < C.
Proof. The forward direction follows easily from Uchiyama's lemmas. By Lemma
Ul,
g = lim E * A
with each ¿>, satisfying (2.2), and (3.5) and (3.6) hold. Since Rx is continuous on L2,
we obtain (3.1) by setting a, = Rxb,. Then (3.2), and (3.3) for \a\ = 1, follow by
Lemma U6; for \a\ > 1, D"a, is estimated by taking the derivatives inside the
integral in the expression for a, exactly as in the proof of Lemma U6. The
cancellation property (3.4) holds because â,(£) = /¿1|¿|"1fe/(¿).
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106 MICHAEL FRAZIER
The idea of the converse is to use (3.2)-(3.4) to write a, = Rxcp, with appropriate
control of <p,. Let h,(xx, x') = j^xa,(t, x') dt. We claim
(3.7)
and
\h,(x)\^ cl(l)(l + l(iy'\x - x,\)~"
-'• H(3.8) \D«hf(x)\< cal(I)l-la\l + liiy'lx - x,\Y
To prove (3.7), we may assume x, = 0. For x such that xx < 0, let k and m be
positive integers such that -(k + 1)1(1) < xx < -kl(I) and ml(I) < \x'\ <
(m + 1)1(1). By (3.2),
¡(I)_\h,(x)\*p \a,(t,x')\dt^c¿Z
-* (l + ip2 + m2
<-,-^L=-<É/(/)(l + /(/)-l|x-xl|)-.(1 + /rC2 + W2)
If x, > 0, ///(x) = -/" a,(/, x') í/í by (3.4), so the same estimates hold. Then (3.8)
follows from (3.3) and (3.4) similarly. Note that (3.7) and (3.8) imply that h, g L2
and|Vr/,| G L1 n L2.
Define <p, = -T/,j^lRjDx h,. Then <p, G L2 since |v/t,| g L2. Note that $,(£) =
-2ir|||n/(|); therefore (ÄjtjO/)^) = -2TTi£xh,(i;) = à,(£), since £>x Ä, = a/. Hence
/?,<?, = a,. If we can show that E7 A,™, g BMO n L2, then we are done since
*i(l>/<P/) = EMi«P/ = EV, =/•
To see that E, A/rf/ g BMO n L2, we show that Dx h, satisfies the assumptions of
Lemma U5 for l=l,...,n. We have }Dxh,(x) dx = 0 since |vA,|e L1 and
lim)x|_a)A/(x) = 0 by (3.7); the estimates (2.13) and (2.14) follow from (3.8). Hence
for each /, 3{ßfJ(x)}JL0 satisfying (2.15) and Dxh,(x) = I.J=02-J,'n + 1)ßIJ(x).
Therefore, using Lemma U3,
L^,Dx,h,i
< £ 2-J(n+x
L2 7 = 0
EaA,7
<cE 2-"" + 1>2^(Ea2/|/|)y=0 \ / '
1/2
< oo,
by (3.6). Similarly, by Lemma U4,
00
LKDxh, < E 2-JiK+l) LKßuI BMO 7-0 /
2
< C.
Therefore E7 A,/^ h, g BMO n L for each /; hencen
EA/CP, = -Ea, E *A,>«/ = - E ^EA/ZV/ e BMO n Ll- °/ / 7=1 7=1 /
The cancellation property of the Riesz transforms, reflected in condition (3.4),
forces the xraverages of/ G /^(BMO n L2) to have a certain regularity. Let g be a
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SUBSPACES OF BMO(R') 107
cube in R", and write g = J X L for J ç R1 and L ç R""1. By Fubini's Theorem,
the average/y = (l/\J\)fjf(t, y ) dt is defined for almost y g L.
Theorem 3.2. Suppose f = Rxg for some g g BMO h L2(R") satisfying ||g||BMO <
1. For Q, J, and L as above, 3c > 0 and a set E ç L of (n — l)-dimensional Lebesgue
measure 0 such that
(3-9) \fjv-fj,\^c Vy,y'eL-E
and
(3.10) \f ^j^ck^ñiogJLSL ify,y'eL-El\i¿) \y - y I
and\y-y'\< \.l(Q).
Remarks. The number i is chosen because the function / -* /log(7) is increasing
on (0, ~). The proof uses only (3.1), (3.2), (3.3) for a = 1, (3.4), and the estimate
X, < c, which follows trivially from (3.5). An example of a function in BMO(R")
which does not satisfy (3.10) is/(x) = log|x'|.
Proof of Theorem 3.2. We may assume g = [-1,1]", so J = [-1,1] and L =
[-1, l]""1. Fory, y' e L fixed, let e = \y - y'\. It is sufficient to obtain |/y - /, | <
ce log} for e < i, since (3.10) implies (3.9). By Lemma 3.1, f=T.,X,a, with
(3.2)-(3.6). We also may assume that 3N so that Xr = 0 whenever /(/) > 2N or
/(/) < 2 ~N. To see this, let fk = E2-*,¡/(/)aí2* A/u/í then lim *_«,/* = fin L2 and, for
each k,fk is continuous and the sum converges absolutely. Let
n
/ = 2
where S > 0 and y = (yl7... ,y„), and similarly define A*-. The result for/A implies
that the averages (fk)Rs and (fk)R\ of / over Rsv and Rsv>, respectively, satisfy
K/a)r? _ (/a)rs,I < celogj with c independent of k or 5. Since lixrxk^,00fk = /in L2,
\fRs — fRs\ < ce logy with c independent of ô. By Lebesgue's Theorem, limÄ_0/Rs =
fj unless y is taken from a set E c L of (n — l)-dimensional Lebesgue measure 0,
and similarly for/y-; hence (3.10) follows.
For / = E, X,a, satisfying X, = 0 for /(/) > 2N or /(/) < 2_w, write / = /,+/,
+ /s + /». where
/i = E A/fl/, /2 = E A/ö/, /3 = E A/Ö,, /4 = E A,a,.HD>X /(/)<1 /(/)<* t</(/)<i
/n3(?=e> iQiQ iQ'iQ
The functions/j and/2 are estimated by Lipschitz estimates. By (3.3) and the fact
that X, < c, if / g J we have00
\fi(*,y)-fi{t.y')\<ct E M'.JO-*/('./)!A=0/(/) = 24
oo
< c E E sup I Vfl/(x)|eA=0/(/) = 2' ïG£
-
2"*< « E E-—tt < e».
k -0a<EZ" (1 +|^|)+ 2
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108 MICHAEL FRAZIER
Therefore |( /, )y - (ff)j J < ce. Similarly, if / g /,00
\fzU, y) -h{t, y')\ <cZ E M', j) - «,(*, /)|*-l /(/)=.2_i
/n3£> = 0
< ce E E SUP -A = l /(/)-2-* *eö (l + 2*|x - X,|)
/n3(?= 0
n + 2
oo jk
< ce E E -^TTk-X ,eZ" (1 + |ç|)"
< ce E 2"A = l
ce.
M»2*
Therefore |(/2)y - (/2)AJ < ce.
Using (3.4), we will obtain |(/3)y | < ce log} and K/3)/J < ce log}. Let I Q 3g be
such that /(/) = 2~k < e, with x, = (x¡, x'r), x, g R, satisfying 0 < x, < 1. Let
m g {0,1,2,...,2* - 1} and/? g Z, 0 </? < 4\/ü - 1 (2k - 1), be such that m2~k
<l-x,i<(m + 1)2"* and p2~k < \x', - y\ < (p + 1)2"*. By (3.4), (3.2), and
since \J\ = 2,
r r-x r00-Ja,(t,y)dt <2J \a,(t,y)\dt + 2j \a,(t,y)\dt\J\Jj
<cY,'0(l + Jp2+(m + s)2y + 1 (l + Jp2 + m2)"
If 7 and/? are as above except that -1 < x, < 0, and m g (0,1,... ,2k — 1} is such
that m2'k < x, + 1 < (m + 1)2"*, the same estimate is obtained by the same
methods. If /and/? are as above but 1 < |x, | < 3, say m2'k < x, — 1 < (m + 1)2"*
if x, > 1, or m2~k < -1 — x, < (m + l)2_/t if x, < -1 for some m e
{0,1,2,...,2* + 1 - 1}, then (3.4) is not used since (3.2) implies that
¿nl°Á,'y)dt <cE°(l + Jp2+(m + s)2)'
>-*< c-
(l + //?2 + m2)"
Summing over all / ç 3g with /(/) < e, we obtain
00 -,
Uh,\- L f-j L ̂ ,a,(t,y)A = [log2lAl \J\ J /(/)_2-*
/C3£?
00 1
<c E 2-* E —-—A = [log2l/e] qeZ" (1 + |?| j
WKVT2*
di
00 j
<c E ^2"* < ce log-,A = [log2lA]
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SUBSPACES OF BMO(R") 109
where we have used X, < c again, and [x] is the greatest integer in x. With the same
estimate for (f3)j-, we obtain |(/3)y - (f3)j\ < ce log}.
For/4, (3.3) and (3.4) are both used. Suppose e < /(/) - 2"* < 1 and / ç 3g. Let
y* = \(y + /)■ Let /? g Z, 0 < /? < Ayjn - 1 (2k - 1), be such that p2'k < \x', -
y*\ < (p + 1)2"*. Let m g (0,1,. ..,2k - 1} be such that m2'k < xh~ 1 <
(m + 1)2"* if 0 < xh < 1, or such that m2'k < xr¡ + 1 < (m + 1)2"* if -1 < xf¡
< 0. By (3.4),
dt
dt
— f a,(t,y')dt-— f a,(t,y)\J\ jj \J\ Jj
= ï7i"/~ ' a'(i' yï dt - Tri/"1 "'('• x!) àt\J\ ■'-90 |/| •'-00
r°° r°°+ T7ÏJ °t(t,y)dt- —f a,{t,y')
\J\ Jx \J\ Ji
\a,{t, y) - a,(t, y')\dt + 2 / \a,(t, y) - a,{t, y')\-oo •'l
Therefore, by (3.3),
CO
\(a,)jt.-{a,)jj<2^í2'k sup \va,(x)\-eí = 0 \x'-y*\<e/2
(m + s)2-k sí\xl-xl¡\^(m + s + l)2-1'
dt.
< c2-*e Ece
=°(l + Jp2+(m + s)2y (l + y!p2 + m2)
If / and p are as above but 1 < |x71 < 3, and m g {0,1,... ,2k+1 - 1} is such that
m2~k < xh - 1 < (m + 1)2'* if xh > 1 or such that m2~k < -1 - x¡ <
(m + 1)2_* if x, < -1, then (3.3) leads directly to the same estimates without use of
(3.4). Therefore, since Xf < c,
T77/ E M,fojO*-?V/ £ ̂ ,a,(t,y')dt\J\JJ „rx_->-* /K/ „r>_.,-*I-7! ̂ ,(/)»r* \J\
IQ3Q
,.z-.(l+|«|)'
/(/) = 2-IQ3Q
ce.
Therefore
|(/4)/, -(h)j,\ < «#{A:: e < 2"* < 1} = celog}.
Theorem 3.2 follows from the estimates on/j, /2, /3 and/4. D
4. Corollaries and examples. Theorem 3.2 implies that functions in Äy(BMO n L2)
satisfy a doubling condition on almost every x^-line. For convenience we take/ = 1
and write x G R" as (x,, x') for xx g R, x' g R"_1.
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110 MICHAEL FRAZIER
Corollary 4.1. Suppose f = Rxgfor some g e BMO n L2(R"). Then 3c > 0 and
a set E ç R"_1 of (n — \)-dimensional Lebesgue measure 0 such that for x' G E and
/, and I2 adjacent intervals in R with \IX\ = \I2\,
(4.1) — f f(xx, x') dxx - ■— ( f(xx, x') dx,\h\ '■ U2I J,2
< c]|g||BMO-
Proof. We can assume ||g|]BMO = 1. Assume first that Ix and I2 are dyadic
intervals. Let F be the union over all dyadic cubes of the exceptional sets obtained
for/in Theorem 3.2. Forx' <£ F, let L ç R"-1 be a closed dyadic cube containing x'
such that \IX\ = l(L), and write Qx = Ix X L, g2 = I2 X L. By Lemma 3.2, since
x' G F, \fjx - f,f\ < c for a.e. s & L - F. Therefore \fr- - /e | < c, and similarly
I//J- -/eJ < c. Since gx and g2 are adjacent cubes in R" and fe BMO(R"),
l/e, - fQl\ < cI/Ibmo < c [3, P- 223]. Therefore \fi; - ff}\ < c for x' G F. To obtaina single exceptional set E for /t, 72 not necessarily dyadic, apply this result to all
dilations and translations of/by rational coordinates. D
Example 4.2. There exists/ g BMO n L2(R") which does not satisfy (4.1). In R2,
for example, for each dyadic square /, let/, be a function satisfying 0 </,(x) < 1,
supp/, ç 37, ||/,||Lipl < cl(I)-1 and/,(x) = 1 Vx g /. Let/I, = {/dyadic: /(/) =
2"\ 0 < x,t < 1, and / n {x:x2 = 0} # 0}. Let/= E~_0E/e/<(l /,- Then it is not
difficult to show that/ G BMO n L2(R2), but
lim Í f(xx,x2)dxx - f~ f(xx, x2) dxf\ = 00.a,^0\^0 ■'1 /
For details, see [2].
Corollary 4.3. /// = Rxgfor some g g L°° such that \\g\\ L«, <\, then (3.9), (3.10)
and (4.1) hold.
Proof. The estimates hold if f = Rxgm, where gm(x) = g(x)x(x:|xKm)(x) since
gm G L2 n L°° so that /?,g„, and Ä1gm differ by a constant. The result follows since,
by a calculation in [2, p. 48], if K is any compact set in R",
lim (sup|Ä1(g-g„,)(x)|) = 0. n
A proof of Corollary 4.3 based on duality with singular measures is included in
[2].Example 4.4. If f(x) = loglog(l/|x'|) for |x'| < 1/e and/(x) = 0 otherwise, then
/G BMO(R") but/does not satisfy (3.9). Although/Í L™,f belongs to the BMO
closure of L00. Therefore L°° + RXLX, Lx + Rx(BMO n L2) and L00 + Ä^Lf ) are
not closed subspaces of BMO. This observation is relevant to §5 below. The space
L°° + RXLX is not dense in BMO in BMO norm; a proof due to Paul Koosis is
included in [2].
Example 4.5 (Kahane's Example). Kahane's example [6] can be used to show
that there exists a sequence {/„}"=i of functions on (0,1) satisfying a uniform
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SUBSPACES OF BMO(R") 111
doubling condition but such that lim„_00||/B||BMO(0il) = oo. For mx, m2,...,mn g
{1,2,3,4}, define
, m, - 1 m-, - 1 m„ - 1I = ——,-+ —-+ • '
4"
mx - 1 mn_x - 1 mn—-h • • • H-—-1--
4 4«-i 4"
and g„„...mn = /„„...„,„ X [0,4-"].Let/1(x1,x2)bedefinedby
/i = Xq, ~ Xq2 ~ Xqj + X<?4-
With sgn / = t/\t\ if / # 0 and sgnO = 0, define fn + x inductively by
4
In+1 ~ J n ' i—i am¡-m„,m,.... ,»!„ = 1
where
ami-m. = sgn/^-Xe^ „,, + XQ„n..m„2 + XQ„v..m¿ ~ XQmy.„J-
Regard/, now as restricted to (0,1). Then clearly
||/J,, = 1 V« = 1,2,3,....
If an = |supp/„|, then an decreases to 0 as n -» oo. That an decreases is clear; that
lim„ J0O an = 0 can be seen by the fact that a simple random walk in one dimension
is recurrent. Since /J f„(x) dx = 0 V«, and since
/ 1 2\1/2
II/IIbmo* sup — / |/ - fQ\Q \\Q\JQ I
we obtain lim,,_,J|/„||BMO(01) = oo since
(A -l\ ll.fl! ^ 11/11 ^ 11/J/.'(0.1) C(4-2) II/„IIbmo(o.i) > c||/J|¿2(01) > c-=— = —.
If Ix and I2 are adjacent intervals of the same length in R, then \(ff), — (/„)/,| < c,
with c independent of Ix, I2, or n, by the main estimate in Kahane's example: see [6,
p. 190], DExample 4.6. There exists G g BMO n L2(R2) such that
(4.3) esssup||Ä1G(x1,x2)||BMO(A1)= °°-
We construct RXG by symmetrizing and smoothing the functions f„(xx, x2) of
Example 4.5 in the x2 direction.
Regarding/, above as a function of (xx, x2) again, let/B(jcl5 x2) be the function
which agrees with fn(xx,xf) for x2 ^ 0 and satisfies f„(xx, -xf) = f„(xx, xf)
V(x,, x2). Define äm m similarly related to am „, . Since
ll/n+i-/Ji.^)<c2:fl,
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112 MICHAEL FRAZIER
m, ... m„ = 1
/ = lim,^^/, exists in L2(R2) and a.e. The following sequence of steps can be
carried out to show that/G BMO(R2):
(a) Since the measure of the supports of fn+x - /„ decrease rapidly,
à #-*■<«
for any 4-adic square g.
(b) If g! and g2 are 4-adic squares of the same side length with Qx n g2 =£ 0,
then \fQi — fg\ < c. For Qx and g2 horizontally adjacent, this follows from the
doubling condition for/,. For Qx and g2 vertically adjacent, this follows from the
construction.
(c) An easy observation, noted by Varopoulous, shows that (a) and (b) imply
/g BMO(R2).In this example, we wish to replace the functions äm m with versions which are
smoothed out in the x2 direction. Let \¡/ g C°°(R) be a fixed function satisfying
0 < <K*)« 1 Vx, suppt// ç (-2,2), and i>(x) = 1 if x g [-1,1]. For mx,...,mn g
{1,2,3,4}, let
?>«,..,«„(*i> *i) = X/„M „,Ui)«/'(4"x2).
Let gx = <px — <p2 - (p3 + <p4, and define g„+x(xx, x2) inductively by
4
gn + l=gn+ E b,
where
fem,...m, = Sgng„(x1,0)(-<pmi m ! + <fm¡ m¡2 + 9mi...m„3 ~ «Pm,.. .«,„4) •
Note that sgn g„(xx, 0) = sgn f„(xx, 0) for/, in Example 4.5; in particular,
supp(£m,...m< - 5Mi...mJ S /mi...mii X (x2: 4-"-1 < |x2| < 2 • 4—1).
Hence if g = limn_00g„, then g — /g L°°(R2) by the disjointness of the supports of
the different gn - f„. Therefore g g BMO n L2(R2) since/g BMO n L2(R2).
We assert that g = RXG for some G g BMO n L2; we solve for G by the method
of Lemma 3.1. Let
/xl gx(t,x2)dt-00
and
/xxf>mi...m„(t,X2)dt.
-oo
Note that hm¡Mfxx, x2) = 0 if xx € Im^Mn (set /0 = (0,1)), or if |x2| > 2 • 4—».
Further,
l-P.vA,,...„,„(*!' *2)|
= Dlj(i(4"x2))jrjX^ mi(/) - X/„„ „,„2(0 - X,„„ „,„3(0 + XV.-,(0) *
<^'(4"x2)-4"-4-"-1 < C.
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SUBSPACES OF BMO(R" ) 113
Also
supp/W..m„ C /„„...„,„ X (x2: 4-"-1 < |x2| < 2 • 4-"1},
since \p(x2) = 1 if x2 g [-1,1]. Therefore
oo 4
H = Dxh0+ E E Dxhmx_meUf.n=\ Wi,... >'"„ = 1
By Fourier transform we obtain gx = Rx(-Rxgx - R2Dxfh0) and
K „, =Rii-Ri6m „, -R2Dxhm m).
Therefore
oo 4
g = gi+E E *„,..,,,.-Ji^-Äx« - Ä2ir)./; = 1 m, ... m„ = 1
Since g, He. BMO C\L2,G= -Rxg - R2H G BMO n L2 and /vjG = g. Note that
/(x,,x2)=/„(x1, x2)for4-""1 < x2 < 4"", and hence by (4.2)
esssup||/(x1,x2)||BMO(x1) = °o-
Since RfG -fe L°°, (4.3) follows. D
5. A density theorem. Throughout this section K is a fixed singular integral
operator with kernel ß which is real and odd; it follows [7, p. 39] that the multiplier
m for K is odd and pure imaginary. We will show that
tf(BMOc) ç L00 + K(L?) ,
where the closure is taken in BMO norm. To fix notation for this section, / always
denotes a dyadic cube, / denotes a vector-valued function with two components,
/ = (/o> A). and we define
\f(x)\-(g(x) +/12(x))1/2, ll/lH/oll + ll/,11
for any function space norm || • ||, and K ■ f = f0 + Kfx. Also define
|v/(x)| = |v/o(x)| + |v/1(x)|
and
ff(x) dx = (//0(x) dx, ffx(x) dx
Our method is based closely on Uchiyama's construction in [8]. Uchiyama obtains
uniformly bounded solutions, which satisfy a certain orthogonality condition, to a
certain inversion problem. We are not able to obtain uniformly bounded solutions to
the corresponding inversion problem in our case. Hence, some additional estimates
are necessary.
First we state explicitly the assumptions we require regarding the inversion
problem. We say that a collection of real-valued functions {b,(x)}¡ satisfies (*) if
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114 MICHAEL FRAZIER
there exist constants c > 0 and / > 0 such that for all /,
(i) for any a = (a0, ax) e R2 with ax =£ 0, the equations
(5.1) Kü, = b,
and
(5.2) a-U,(x) = 0 Vx
have a solution ü, = (u0 ,ux ), u,: R" -» R2, satisfying
(5.3) \u,(x)\<c[l+ ^ )(l+l(iyl\x-x,\)
(5.4)
|«/(x)|< c
|vw/(x)|< c
-n-X
1 + /(/)"'(! + /(/)-1|x-x,|)-n-2
and
(5-5)
and
fur(x) dx = 0;
(ii) the equation K(ux ) = b, has a solution ux : R" -* R satisfying
(5.6) |Ml;(x)|^c(l + /(/)-1|x-x/|)"""1,
(5.7) |vuX/(x)| ^ c/(/)_1(l + /(/)_1|x - x,|)~"~2
and
(5.8) juXi(x) dx = 0.
Theorem 5.1. Suppose f(x) = linxk^x>Y.2-t^nJ)6i2i.Xrbl(x) (convergence in L2)
with Xj > 0 V/, w/iere //te real valued functions {b,(x)}, satisfy (*), and for some
c> 0,
(5.9) Ea2|/|<c|7| for all cubes J,IQJ
Ea2,|/|< oo,all I
limM—» oc
E a a/(/)>2M BMO
(5.10)
(5.11)
and
(5.12)
F/te/i/ g Lx + K{LX).
Before proving Theorem 5.1, we mention some consequences.
Corollary 5.2. For K as above, A:(BMO(.) £ L°° + K(LX), where the closure is
taken in BMO norm.
3 a cube S such that X, = 0 if 31 n S = 0 .
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SUBSPACES OF BMO(R" ) 115
Proof. Suppose g g BMOc. We may assume that g is real valued. By Lemma Ul,
g = E/ X,a, with (5.9) and (5.10). The functions a, satisfy supp a, C 31, \\Daar\\L*,
< c|a|/(/rH and ja,(x) dx = 0 V/. That (5.12) and the condition
limM_00||E/(/)>2" ^iai\\BMo = 0 are also obtained by the decomposition in Lemma
Ul for functions of compact support is proved in [8, pp. 232 and 234]. Set b, = Ka,;
since K is bounded on L2 and BMO, Kg = Y.X,Ka, = Y.X,b, in L2. Each b, is real
valued since the {a,}, obtained by Lemma Ul are real valued (since g is real
valued), and since K has a real-valued kernel.
We need only show that {b,}, satisfies (*). Suppose ax + 0. Then (5.1) and (5.2)
can be solved by setting
(5.13)
and
(5.14)
"o,(*)
«!,(*)
m(è)a0 - fljh,(i)
MO
(*)
(*),m(è)a0 - ax
where is the inverse Fourier transform. (This is Uchiyama's solution reduced to
our case.) Since h,(i) = m(£)â,(£), the u0 and ux are obtained from a, by the
Fourier multipliers
-axm{i) -a0m(í)«o(0 and mx(i)
m(£)a0-ax m($)a0 - ax '
respectively, which are C°° on S"^1 (since m(£) is pure imaginary and ax ¥= 0), and
homogeneous of degree 0 (since m is). In fact, since m(-£) = m(£), the same is true
for m0 and w1 and hence w0 and ux may be taken real valued. By [7, p. 75],
3a0, af g R and singular integral operators T0 and TY such that u, = a¡a, + T¡a„
/ = 0,1. An easy calculation and the fact that m is pure imaginary show that
\Dam,(i)\ < c|a|(l + Iflo/fl^WjVÍ g S""1. Also K| < H»!,!!^^.-., < Cfor /' = 0,1.
Hence by Lemma U6, the solutions u, of (5.1) and (5.2) satisfy (5.3), (5.4) and (5.5)
for / = C(n). Solving K(ux ) = br is trivial since b, = Ka,, so that ux = a, satisfies
(5.6), (5.7) and (5.8). D
Corollary 5.3. If f(x)
hold and each b, satisfies
E/ X,b,(x) in L2 with À,»0V/, and if (5.9)-(5.12)
(5.15) supp¿?,c.3/, I^^Ilípi * c/(7) 2' and jbi(x', x") dx' = ° Vx'
where x' = (xx,... ,xk) andx" = (xk + x,... ,x„), then-:-
/GL-+ T,Rj(L?)7 = 1
where the closure is taken in BMO norm.
Proof. We may assume each b, is real valued. If k = 1, we need only to show that
(*) holds for K = Rx and {b,}, under the assumption that fb,(xx, x") dxx = 0
Vx", /. For ax * 0, (5.1) and (5.2) can be solved by (5.13) and (5.14) with properties
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116 MICHAEL FRAZIER
«i,- \Ri
(5.3), (5.4) and (5.5) as above. We solve RxUf = b, as in Lemma 3.1 by setting
h,(xx,x')= J^b,(t,x')dt. Then by (5.15), supple 3/. Set
- Uxb,+ tRjDXjhtY,
then Rxux ; = b, as in Lemma 3.1. Since supp Dx h, ç 31, ¡Dxh,(x) dx = 0 and
11/^/^/7,11 Lipl < cl(I)~2,Vj,p g {1,...,«}, we obtain (5.6)-(5.8) by Lemma U6.
In general, it is sufficient to apply the case above once we note that if b, satisfies
(5.15), then b, = T.p=xb,, where each b, satisfies suppZ?, ç 31, \\DX b¡ \\LipX ̂
c/(/)-2V7, and
jb,p(xx,...,xp_x,xp,xp+x,...,xn)dxp = 0
V(xx,.. .,xp_x, xp + x,...,xn). To see this, suppose I = \~\"J=xIj and let a(xx) be a
function satisfying suppaixjc 3/,, (l/\Ix\)ja(xx) dxx = 1 and \\dpa/dx{\\L« <
cpl(iy for p = 1,2, 3. . .. Let x = (x2,. . . ,x„) and define ß(x) =
(l/\Ix\)jb,(xx, x) dxx. Set bf(x) = a(xx)ß(x) and b, = b, - bf. Then it follows
from the definitions that fbl¡(xx, x) dxx = 0 Vx, suppZ)7i ç 3/ and \\DX ¿>7 ||Lipl *S
cl(I)1 for j = 1,...,«. Similarly suppô* ç 3/, \\Dxbf\\Upf < c/(/)"'2 for / =
1,... ,n, and
I bf(xf,... ,x„) dx2 ■ ■ ■ dxk = a(xf) Iß(x) dx2 ■ ■ ■ dxk
a(Xf) r=- I b,(x) dXf ■ ■ ■ dxk = 0 Vxj.
Continuing similarly with x2 and bf, after Ac - 1 more steps we obtain b, = E* = 1 br
with the desired properties. D
Since the proof of Theorem 5.1 is an adaptation of the proof of the Main Lemma
in [8, pp. 231-238], we follow the organization and terminology of Uchiyama's proof
quite closely. Estimates which are essentially the same as Uchiyama's will only be
stated or sketched. More detail is given in [2].
Proof of Theorem 5.1. Let e > 0 be given. We will obtain g0 g L°°, gx e Lf
such that ||/- g0 - A^Hbmo < e- We may assume S = [-1,1]". Define {r\k}^_x
and {ek }"__„ as in Lemma U2 for the {X,}, given in Theorem 5.1. Let / > 0 be the
number given by ( * ). A sufficiently large positive integer M, and sufficiently large
positive numbers R and /? will be determined later, depending on e. We inductively
construct the functions {gk(x)}x=_M_x, {<pk(x)}x=_M, {0¡j(x)}?=o,«n^2»,
{jk(x)}x_Mand {ïk(x))f..M, satisfying
(5.16) supp/3;,, ç VI, \vß,j{x)\ <s cä'/'(24(/))~\ /&,,(*) dx = 6,
00
(5.17) K- E 2-jin + 1)ß,J = b, V/such that/(/) <2M,
7=0
(5.18) yk = 4>k + Ík for -M < k < oo,
(5.19)
|ít(*) - $k(y)\ < c{2°' + 2)MR{2l/p-l) + R-x/p)2k\x - y\ if \x - y\ < 2~k,
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(5.20)
(5.21)
(5.22)
(5.23)
(5.24)
(5.25)
and
(5.26)
SUBSPACES OF BMO(R" )
|^(x)|<c2<" + 2»w/í<2^-1>e,(x)7,,(x),
supple (x: |x| < 2v/ñmax(2Af~/í,l)},
supple {x: |jc|< 2/r7max(2w-*,l)},
g.w_x - (R,0),
\gk(x)\=R Vx,VA:>-M-l,
117
M
gk(x) - gk-i(x) = E A, E 2-*" + %,f(x) - <p,(x)/(/) = 2-* 7 = 0
\gk(x) - gk(y)\ < c0Rl/»ek(x)2k\x - y\ if \x-y\< 2~k.
As a consequence of the induction, we will obtain
(5-27)
and
(5.28)
E ïj ( x ) converges in L2 as k -* oo
t—M
E £(*)j = -M
< C/?"1^.
First we assume the construction and complete the proof of the theorem. By (5.25)
and (5.18), if k > -M
(5.29) gk(x) - g-M-tW = Ê 2-<" + 1> E \fít,j7=0 /: 2AI>/(/)>2-
- E 2-/<«+1> E ^A,7=A/+1 /: 2M>/(/)>2-*
- E *,- E 6-
By (5.10), (5.16) and Lemma U3, the first two terms on the right-hand side of (5.29)
converge in L2 as k -» oo. By (5.27) the last term in (5.29) converges in L2 as
k -* oo. By (5.20), (5.21), (5.9) and Lemma U2, E*=_Miy converges in L1 as A: -» oo.
Since \\gk — gj\\Lm < cR VAc, / > -M - 1, gk - g.M-X converges in L2 as k -> oo
by the argument in [8, p. 233]. Hence E*_M>pj converges in L2 as k -* oo as well.
Set
í(*) = £-a#-i(*)+ lim ,(**(*)-¿-a/-i(*))-k -» oo in L
Then |g(x)| = R Vx g R". By (5.12), (5.16), (5.21), (5.29) and (5.23),
supp(g-(/?,0))ç. {x:|x|<2Vr722M}.
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118 MICHAEL FRAZIER
Hence gx g Lx. By (5.29) and (5.17),
K-K-gk= E *A2M>/(/)>2_*
Therefore,
K'g-f- E aa/(/)>2M
/V
E 2-><-+1> E Kkjj=M+l /: 2M»/(/)»2~*:
A A \
+ I %+ E ?,/■- M y—M
E 2-'-+1> E Kkj7 = M+1 /:2M»/(/)
y—w /-M
By (5.11), lhnw_Q0||E/(/)>2« a^J^o = 0. By the boundedness of K on BMO, and
by (5.16) and Lemma U4, we obtain, as in [8, p. 237],
k- E 2-;<-1> E M/jy-A/+l /:2M>/(/)
As in [8, p. 238], from (5.19) and (5.20) we obtain
< cf2-MRl/p.
BMO
*• E Í,- M
< c2(2("+2)MÄ(2'/'-1) + /T1^).
By (5.28), ||/f • E^.m^-IIbmo < ^/T1/'. Let c4 = max(Cl, c2, c3,1). We will choose
R sufficiently large and p so that Rl/p = 8e_1c4. Choose M so that
||E/(/)>2«XAIIbmo < e/4> and so that cf2-MRl/p = Cf2-M(He-1cA)' < e/4. Now
choose R sufficiently large so that
C22<» + 2>»ä<2'//'-i>= c22<" + 2)A/(8e-1c4)2//?^1 < e/8.
Finally pick /> so that #1/'' = 8e_1c4; then c2R~1/p < e/8 and c3R~1/p < e/8. Then
||á! • g - /Hbmo *■ e> witn £o e ^°°> £i G ^°- ^OT future reference, note that our
choices allow us to assume R1/p > 1 and R2l/p •« R.
We begin the proof of the construction. Define g„M-i by (5.23). Assume
{gP}kp-=\M-i, {%)kp-J-M, {ßrjr-ox**UKn<2», {^}*:lMand [tp}kp-J_M, satisfy-
ing (5.16)-(5.26), have been constructed. Let / be a dyadic cube with /(/) = 2 k. Let
a(I) = (a0(I), 0,(1)) = gk-iix,)- Let
Ak = {/: /(/) = 2"* and|fl0(/)/fll(/)| < A1/'}.
Let/3A = {/: /(/) = 2~k and either a,(/) = 0 or ^„(.O/a^J)! > Ä1/p}.
For I e Ak, apply (*)(i), to /, b, and a = a(/) to obtain u,: R" -» R satisfying
(5.1)-(5.5). Since ^(/ya^/)! < Ä1//', by Lemma U5, 3{/J//x)}7oo=0 satisfying
supp/3^ ç 2/r, \\ß,JUpi < cR^p(Vl(I))-\ 0,j(x) dx = 0, 3(1) ■ ßfj(x) = 0 Vx,and «,(*) = E°°_o2";(" + 1)/8,,/x). Note that therefore \ßfJ(x)\ < cR'^ Vx.
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SUBSPACES OF BMO(R") 119
For / g Bk, apply (*)(ii), to solve KUf, = b, with uir satisfying (5.6)-(5.8). By
Lemma U5, l{#,,(x)}j°_o satisfying suppß,1,,,,£ VI, ||/3>^!|Lipl < c(Vl(I))~1 and
fß)j(x) dx = 0, such that
00
uu= E 2-"»+1WJ(x).7 = 0
Define ß,J(x) = (0,ß1,J). Note that \ßrj(x)\ ^ c Vx, and \a(I) ■ ßrj(x)\ =
\ax(I)ß] j(x)\ ^ cR}-1/p, since \ax(I)\ < \aa(I)\R~1/p and / g Bk, and |a0(/)| <
|gVi(*/)l-*by(5.24).Note that the ß, • defined in the above two cases satisfy (5.16) and (5.17). Define
M
hk(x)- E a,E 2-<"+I>'/3/,7(x),/:/(/) = 2"A 7 = 0
yk(x) = gk_1(x) + hk(x), gk(x) = R-f^IyaWI
and
(5.30) Vk(x) = gk-i(x) + hk(x) - gk(x)
= [gk-i(x) + hk(x)} 1 -R
\gk-i(x) + hk(x)\
yk(x)
\lk(x)\1\Ux)\-r]-
Then (5.24) and (5.25) hold. Note that hk(x) = 0 and hence, by (5.30) and (5.24),
gk(x) = 0, if |x| > 2jn~ max(2A/-*,l), by (5.12) and the fact that supply ç VI.
From \ß,j(x)\ < cRl/p we obtain
(5.31) \hk(x)\^cR</pVk(x)
as in [8, p. 235]. Similarly, from \vßtJ(x)\ < cRl/p(2Jl(I))-1, we obtain
(5.32) \hk(x)-hk(y)\<csR'^k(x)\x-y\ if |x - y| < Vk.
Therefore, if |x - y\ < 2~k, using (5.26) inductively, we obtain
\yk(x) - yk(y)\ <\gk-i(x) - gk-i(y)\ + M*i - hk(y)\
< {(c0/2)ek_f(x) + ~c5i¿x)}R">2k\x - y\
<|c0e,(x)/?'/'2*|x-y|,
if we choose c0 > |c5. Since \hk(x)\ < cR'/p «: /? = |grA:_1(x:)|, we obtain
\gk(x) - gk(y)\ < l\%(x) -%(y)\< c0ek(x)R'/p2k\x - y\
for |x — y| < 2"k. Hence (5.26) holds.
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120 MICHAEL FRAZIER
We consider now çk(x) = yk(x)[\yk(x)\ - R]/\yk(x)\. Write
\%(x)\ -R=\gk^x(x)+hk(x)\-R
_ RL + 2gk_x{x)-hk(x) + ]^(x)l2\1/2
R2 R2R
= «1 + g^M-^i-) + MúL\_R + EÁx)R2
gk-i(x)-hk(x)
R
2R¿
+ E2(x),
where the last two equalities define Ex and £2. Since |/l + a — 1 - \a\ < a2 for
\a\ < \, using (5.24) we obtain
(5.33)"
|E2<*)| < H 2̂R
,2 v 2
\gk-i(x) -hk(x)\ \hk(x)\ \ \hk(x)\-L + J-"— ^ c-
R2 R2 ^ R
Hence
- / N ?*(*) gk-i(x)-hk(x) yk(x) _Vk\x) = . , ,. -p- + 7——--¿2 = (l)k + {U)k-
\Ux)\ R \Ux)\We write
(I);gk-i(x) + hk(x) gk-x{x)-hk(x)
\gk-i(x)+hk(x)\ R
gk-i(x)-hk(x) hk(x) gk-i(x)-hk(x) gk_x(x)
R\gk-i(x) + hk(x)\
R R
, gk-i(x)-hk(x) _+-ft-Sk-x(x)
= (III),+(IV),+(V),.
[\gk-i(x)+hk(x)\
l_
R
Now
(IV),
M
M^-Ir- 5- KY,2-^gk_x(x).ß,jx)\\ /:/(/) = 2"* 7 = 0 /
( \ M
^JP- E K E 2-<«+»(g,_1(x) - gk_x(x,)) ■ ß,Jx)K
1
/:/(/) = 2-' 7 = 0
M
+ ~2 E X, I 2-><"+1)(g,_1(x) - gVt(*/))gA-i(*/) • ßrj(x)K
1
/:/(/) = 2"* 7 = 0
+ "TÏ E A, £ 2-<"+1»g,_1(x/)g,_1(x/) • &,,(*)" l:l(/) = 2-k 7 = 0
S (VI), + (VII), + (VIII),.
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and
SUBSPACES OF BMO(R" ) 121
Define $k = (II), + (III), + (V), + (VI), + (VII),, and f, = (VIII),. Then (5.18)
holds. By (5.12) and the fact that sapp'0TJ Q VI, (5.22) holds. By (5.30), $,(x) = 0
if |x| > 2{n max(2w"*, 1), and so (5.21) holds also.
We now prove (5.20). By (5.33), (5.31) and (2.5),
|(II),|=|F2(x)|<^|/;,(x)|2<cä<2^-1>1?2(x)
<c/?<2//^1»e,(x)7,,(x).
Similarly,
\(lll)k\ ^ ^\hk(x)\\\cR^p-^ek(xUk(x)
\/v\ i^l- t \ u t M \R-\gk-x(x) + hk(x)\(V)k\ < gA-i(*M*(*)|J--1-——r——
R\gk-i(x) + hk(x)\
\gk-x(x) -hk(x)\ |g,_i(x) -hk(x)\\E2\
* R%-i(x) + hk(x)\ R\gk-i(x) + hk(x)\
^^\hk(x)\\cR^p-»ek(xhk(x),
by our above estimates on |y,(x)| - R and F2. To estimate (VI), and (VII),, we
note, as in [8, p. 235], that if /(/) = 2'k and ß,j(x) ^0,0<j<M, then |x - x,\
< c2M~k, and
(5.34) \gk-i(x) - gk-Ax,)\ < c2<"+1>wÄ'/"e,„1(x)2*-1|x - x,\.
Therefore, for the same x and /,
(5.35) |(g,_,(x) - gk_f(x,)) ■ ß,Jx)\ < c2<"+2'^^e,_1(x)|/Jí,7(x)|
since |x - x,\ < c2w"* for these x and /. Therefore
M
l(VI)*| < j E A/ E 2^<" + 1)/?^2<-2)%_1(x)|/3/J(x)|/:/(/) = 2~A 7 = 0
< c2("+2>wÄ<2//"-1)e,_1(x)T,,(x),
since 1/3, ;(x)| < cfl'7'', by the method used to obtain (5.31). Similarly, using (5.34)
instead of (5.35),
M
l(VII),| < | E A, E 2-^« + 1»2'" + 1»w/?'/"e,_1(x)2^-1|x - jc,||/8,,,(jc)|
< c2<" + 2)A'/*<2//''-1,e,_1(x) £ 2-(" + 1»^ E À,
7 = 0 /:/(/) = 2_*
dis«*./)« 2^*
<c2(" + 2)WÄ<2//''-1'e,_1(x)T),(x),
as in (5.31). Hence (5.20) holds, since ek_x < \ek.
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122 MICHAEL FRAZIER
To establish (5.19), we prove corresponding estimates for <p, and f, and use (5.18).
Following [8, p. 236], for \x - y\ < 2~k, write
$k(x) - <Pk(y) = \yk(x)\'\\yk(x)\ - R)(yk(x) - y,(y))
+yk(y)RlUx)l~lUy)l\yk(x)\\yk(y)\
-(ï)*+(n)t.
Since ||y,(x)J - R\ < cR</p, |y,(x) - y,(y)| « c0ek(x)R'^2k\x - y\ and |y,(x)| >
R/2 (since \hk(x)\ •« R), we obtain
|(ï),| < cR(2l/p-1)£k(x)2k\x - y\ for \x - y\ < 2_*.
As in [8, p. 237],
|(n)*| < 2||g,_1(x) + hk(x)\ - \gk-i(y) + hk(x)\\
+ 2\\gk-i(y)+hk(x)\ -\gk-i(y) + hk(y]
= (m)k + (rv)k.
Now
(M)*-
I2 \ V2
2 v 1/2
2
L + 2gk_f(x)-hk(x) + |//,(x)|
\ Ä2 R2
_ RL 2gk_1(y)-hk(x) \hk(x)f
\ R2 R2
< ji\(gk-x(x) - gk-x{y)) -hk(x)\
< cR(2l/p-1)ek_f(x)2k\x - y\r,k(x) < cÄ(2//"-1)e,(x)2A:|x - y\,
by (5.26), (5.31) and a Lipschitz estimate for the function /(/) = (1 + /)1/2 for /
sufficiently small. Similarly,
|(IV)*| = 11 +2gk-i(y) -hk(x) \hk(x)
l2\!/2
R:
-Rll +
R-
2gk-x(y)-hk(y) | \hk(y)\2, 1/2
j£
R2 R:
j\gk-i(y)-{hk(x)-hk(y))\ + ^\\hk(x)\2 ~\hk(y)\
-|W*I+|(VI),
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SUBSPACES OF BMO(R") 123
By (5.32) and (2.5) we have |(VI),| < cR(2,/p-1)2k\x - y\. Now
l(V),l . j
+
M
E a, E 2-n"+l){gk-Ay) - gk-Ax,)) \h,j(x) - frjy))/:/(/) = 2_t 7 = 0
R
M
E A, E 2-<"+1>g,_1(x/) -(ß,Jx) - ßrj(y))/:/(/) = 2"A 7 = 0
- (VII), + (VIII),.
If ß,j(x) and ßrj(y) are not both 0, then \y - x,\ < c • 2¥~k, since |x - y| < 2""*;
so by applying (5.34) we obtain
M
|(VII),| < cRu/p-^"+2)M E a, E 2-^ + 1)|/3/J(x) - ^.(y)!/:/(/) = 2"* 7 = 0
< c/?(2//^-1)2(" + 2)%(x)2*|x -y|,
by the calculation in estimate (5.32). Since g,_1(x/) • ß,,(x) = 0 Vx whenever
I e A,., we have
Kviii)a| = £M
EX/E2-("+%(/)(/3/1,(x)-/8/1,y(y))/<eb* y-o
since ßKj = (0, #,) for / g /?,. Since |öl(/)| < R1'1^ and ||#,,l|Lipl < c(2>/(/))"1
for 7 g 73,, and by (2.5) and the proof of (5.32),
M
|(VIII),| < cR-1/" E 2-'(" + 1) E A,|x -y|7 = 0 /:/(/) = 2"A
dis«*,/) «2^*
< c/?-1/^tj,(x)2a|x - y| < c/?-1//72/i|x - y\ for |x - y| < 2"*.
Hence
I<Pa(*) - My)\ < c(2<" + 2)A/Ä<2//"-1» + R~^p)2k\x - y\
if|x-y|< 2~k.
Whenever |x — y| < 2~k, we have
M
E A,E 2^" + 1'g,_1(x/)g,_1(x/).(/i/,7(x)-/3/,/y))ImBk y-o
|f*(*)-f*(»l-¿
«Te*"1" £ X/E 2-<" + 1»|/i/1,y(x)-/3^(y)|/efi» 7=0
< cR~1/p2k\x y\,
since g,_!(x,) • ß,f(x) = 0 for / g .¿„ and for / g 73,, g,„i(*/) = (a0(/), a,(I)),
with !«!(/)[ < R1-1^ and ^ = (0, ß)/). The estimates for ¿, and fk give (5.19), by
( J. lo).
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124 MICHAEL FRAZIER
We now verify (5.27) and (5.28). We have, for k > q > -M,
1k
E I-</+i R2
M
/eu:
M
*,L2-^ + 1X-Ax,)ax(I)ßlS, 7 = 0
< cR-^p E 2-^(" + 1»7 = 0
E/Gil*,
&-i(*/)aiU)M/\y
Note that Ift-iO^a^/)/*2-1''! < c for / G U*
E f,s-q+l
< CÄ"1/P
! /3Í, so by Lemma U3
i1/2
A2,|7|
By (5.10), then, (5.27) holds. Similarly, by Lemma U4,
M
E I--M
^cR~1/pzZ 2~Ji"+1)BMO 7 = 0
E/eur_.M;
?.-i(*j)«iU)«2-1/p
A A1,,
< cR-1/p.
This establishes (5.28) and hence Theorem 5.1. D
6. Problems and remarks. The work of Janson [5] and Uchiyama [8] gives the
complete result that a collection {Kj}f=x of singular integral operators satisfy
E;_ , KjLx = BMO if and only if
'-Ai) m At) '(6.1) rank
m.
mA-a) mÂ-i)2 V| g S""
However, little is known about the subspace Y.Pj_xKjLx of BMO when (6.1) fails.
For the Riesz transforms, Corollary 4.3 implies that any f e T.p^xRjLx, p < n,
satisfies a regularity condition on appropriate /»-dimensional subspaces of R" that is
not satisfied for all BMO functions. One would hope to obtain some similar
interpretation, whenever (6.1) fails, of the fact that T.p=xKjLx # BMO. The study
of the Riesz transforms was facilitated by their cancellation property; to proceed in
the general case, one would presumably require a geometrical interpretation of the
failure of (6.1).
However, even for the Riesz transforms the results are far from complete. Our
regularity conditions hold for/g Rj(BMO fl L2) as well as RXLX; boundedness
played a role in Corollary 4.3 only in removing the L2 restriction. Further, only the
condition X, < c, rather than the full strength of the BMO packing condition (3.5),
is used in proving Theorem 3.2. Hence our regularity conclusions hold for larger
classes of functions and cannot characterize 7?,(BMO n L2) or RXLX. The decom-
position result in Lemma 3.1 is a complete, but not explicitly geometrical, char-
acterization of /?x(BMO n L2); it is not clear whether a complete geometrical
characterization is to be expected. Example 4.6 shows that we cannot expect much
stronger regularity along lines.
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SUBSPACES OF BMO(R" ) 125
Since our regularity results hold for /?x(BMO n L2) as well as RXLX, it is natural
to ask if there are additional geometrical properties that hold for RfLx but not for
/?,(BMO n L2). Corollary 5.2 suggests that any such distinction may be subtle.
Explicit examples of functions/ g Lx + /^(BMO n L2) such that/ G L°° + RfLx
would be helpful, but are not apparent since our only criterion for excluding / from
L°° + RfLx is the failure of our regularity condition, which holds for Lx +
Rx(BMO n L2) as well. Our function G g BMO n L2(R2) in Example 4.6 is
relevant: if RfG e Lx + RfLx, then a uniform BMO condition on x,-hnes does not
hold for Lx + RfLx; if not, then RfG distinguishes RX(L2 n BMO) from Lx +
RfL°°. Either case will give information as soon as it is determined which case holds.
A further question regarding each subspace Y c BMO under consideration is the
determination of dist(/, Y) = inf{||/- g||BMO: g g Y} for fe BMO. When Y =
Lx, Garnett and Jones [4] have complete results. For K as in §5, Corollary 5.2 is
obviously equivalent to the statement that
dist(/, Lx + K(LX)) = dist(/, Lx + K(BMOj).
For the Riesz transforms, our conditions on lines should play a role in this problem.
As noted above, some of our results must apply in greater generality than
presented here. Since the method of the decomposition of BMO (Lemma Ul)
appears quite general, one suspects that further information about the action of
specific operators on appropriate spaces can be obtained by similar methods.
REFERENCES
1. C. Fefferman and E. M. Stein, Hp spaces of several variables, Acta Math. 129 (1972), 137-193.
2. M. W. Frazier, Functions of hounded mean oscillation characterized by a restricted set of martingale or
Riesz transforms, Ph.D. Thesis, University of California, Los Angeles, 1983.
3. J. Garnett, Bounded analytic functions. Academic Press, New York, 1981.
4. J. Garnett and P. Jones, The distance in BMO to Í.00. Ann. of Math. (2) 108 (1978), 373-393.
5. S. Janson, Characterization of //' by singular integral transforms on martingales and R", Math. Scand.
41 (1977), 140-152.
6. J.-P. Kahane, Trois notes sur les ensembles parfaits linéaires, Enseign. Math. (2) 15 (1969), 185-192.
7. E. M. Stein, Singular integrals and differentiability properties of functions, Princeton Univ. Press,
Princeton, N. J., 1970.
8. A. Uchiyama, A constructive proof of the Fefferman-Stein decomposition o/BMO(R"), Acta Math. 148
(1982), 215-241.9. _, A constructive proof of the Fefferman-Stein decomposition of BMO on simple martingales,
Conf. on Harmonic Analysis in Honor of Antoni Zygmund, Vol. II (Beckner, et al., eds.), University of
Chicago Press, Chicago, 111., 1981, pp. 495-505.
Department of Mathematics, Washington University, St. Louis, Missouri 63130
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