Download Trace Ideal Criteria for Hankel Operators and Commutators

Trace Ideal Criteria for Hankel Operators
and Commutators
RICHARD ROCHBERG
I. Introduction and Summary
We present conditions for Hankel operators and related operators defined on
the Hardy space H2(R) to be in the Schatten ideals ':ip, p > 1. The criterion is
that the symbol have derivative in an appropriate Bergman space (i.e., that the
,syrnQQlbe in a certa.iI1~~~ovspace). Our results are analogous to and offer a new
approach to recent resultsoTv: '7. Peller involving Hankel operators defined on
the Hardy space of the circle.
Background, definitions, and the statement of the main theorem are in Section
2. The proof of the theorem is in Section 3. It is based on Fourier transform
calculations and the molecular decomposition of functions in Bergman spaces.
Variations on the main theorem are given in Section 4. These include trace ideal
criteria for various types of commutators and analogous results for operators defined using weighted projections. We also obtain conditions under which weighted
projections are trace ideal perturbations of ordinary projections. In the final section, we mention a few open questions.
II. Preliminaries
A. Function spaces. We denote by H2 the Hardy space consisting of those
functions F which are holomorphic on the upper half plane U and for which
1IFIli
= ~~~[
IF(x
+
iy)I'dx
is finite. We will identify such an F with its boundary values. The Fourier transform, P, of such a function is square integrable on the line and vanishes on the
negative half line. Thus we will regard P as an element of L2(0,00). The Fourier
transform is a unitary map of H2 onto L\0,00). We denote the inverse Fourier
transform of g by g. For further discussion of these issues, see [16].
For p, r with 0 < p < x, -1/2 < r, we define the Bergman
space Ap,r to be
,
the space of functions F holomorphic in U for which
913
Indiana University Mathematics
Journal ©, Vol. 31, No.6
(1982)
914
R.ROCHBERG
IIFII~.,~
f L IF(x
+ iy)I"y"dxdy
is finite. These spaces are discussed in detail in [5] . We now recall the decomposition theorem from [5]. Let d(',') denote the hyperbolic distance on U. Let
1) be a positive number.
A sequence of points {~J in U is called an 11-lattice if
for all i, j, i ¥ j, d (~i'~j) ;::: 1)/100 and if also for all z in U, inf
i {d(z,~J} < 11.
(For example, the set {n2m + i2m: n,m E Z} is an 11-lattice for appropriate 1).) For
z, ~ in U we denote by B (z,~) the Bergman kernel; B (z,~) = (z - ~) -z. (We are
ignoring certain inessential multiplicative constants in our definition of B and in
various Fourier transform calculations.) A special case of Theorem 2 of [5] is
>
Theorem.
Let p, d, E be given; p> 0, d> 1/2p, E
dmax(-l,p
There is an 110= 11o(p,d,E) such that if 1)
110and the points {~i} form
lattice then every function F in AP,dp-l can be written as a series
<
(2.1)
= ""'
F(z)
l-vith scalars
Ai
2: IAi I p is finite
which
satisfy
B(Z'~i)2d+E
A
L..;
I
B (~i'~i)d+E
2: /Ai Ip ::; c(1),p,d,E)IIFII~,dp-l'
then F (z) defined
- 2).
an 11-
Conversely,
if
by (2.1) is in AP,dp-l and
IIFII~,dp-l ::; c(11,p,d,E)
2: IAiIP.
Integration of order one-half (that is, the map off to (t-lI2J)v)
is a unitary map
from AZ'o to HZ (this is verified by calculating Fourier transforms). This map induces an analogous decomposition theorem for F in HZ: Given E > 0 one can
write
(2.2)
with
2: /Ai12 comparable
to IIFII;.
For later use, we record the following
t
(2.3)
(2.4)
(B (Z'~iYl,B
(z'~jr2
>
=
>0
CS1S2B (~j'~i)Sl+S2-l/Z.
For (2.3) see [16]. (2.4) follows from (2.3) and the fact that the Fourier transform
is unitary.
We use the notation B ( ., .) rather than the explicit formula for the k~rn~! to
emphasize our suspicion that most of these results will eventually be found to
have analogs for the more general Bergman spaces discussed in [5].
,
915
HANKEL OPERATORS AND COMMUTATORS
B. Hankel operators.
define Hk'
given by
Let k be a function defined on the half-line (O,x). We
the Hankel operator with symbol k, to be the linear operator on L2(O,x)
0'
(H,f)(t)
f
k (s
+
t)
f (s) ds.
The related operator on H2 which maps F to (HkF)v is also often called a Hankel
operator. General discussions of Hankel operators are given in the surveys [13]
and [1].
C. Trace ideals. Let A be a bounded linear operator on a Hilbert space. IA I
is the positive square root of the positive operator A *A. Let sp S2' .•. be the
eigenvalues of IA I. The operator A is said to be in the Schatten class ';ip' p :::> 1
if
(2.5)
IIA lip
=
(
2: sf )
]
l'p
is finite. These classes are also called trace ideals. The operators in ';i 1 are also
called trace class operators or nuclear operators. The operators in ';i 2 are also
called Hilbert-Schmidt operators. The general theory of these classes is presented
in [9] and [15].
Each ';ip is a Banach space with the norm given by (2.5) and is a two-sided
ideal in the space of all bounded operators. If we denote the operator norm by
II-II and the ';ip norm by 11'llp then
(2.6)
IIXAYllp
< IIXIIIIAllpIIYII.
For each p the set of finite rank operators is dense in ';ip and ';ip is contained in
the ideal of all compact operators. Finally, if {<pJ is any orthonormal set then
(2.7)
n
(See page 94 of [9].)
D. The main theorem.
Let Hk be the Hankel operator with symbol k. Let p ;:=: 1 be given.
There is a constant l1p which depends only on p so that if 11, n, e, and E are
constants which satisfy
Theorem 1.
1
(2.8)
and if
l1p
{~Jis
> 11, n>- ,
any l1-lattice
2
+
1
2E
p
then the following
>-
P
are equivalent
916
R. ROCHBERG
(a) Hk is in ';Jp'
(b) DTlk is in AP.npl2-J,
L
(c) k(z) =
Al(B(z,~JIB(~i,~JY3
for scalars which satisfy
If any of these hold then it is also true that
(d)
L
L
IAilP
< X.
< 00.
I<HkB(z,~;)3/4+8,B(z,-t)3/4h)B(Si,sJ-J-2eIP
If (d) holds for all TJ-lattices then (a), (b), and (c) also hold.
lip
Finally, liHkllp' liD' kllp"pIH'
and inf {
(L !,\;IP)
; all representations of k
as in (C)} are equivalent quantities with constants of equivalence which can be
chosen to depend only on the data in (2.8).
Notes. (1) Dnk denotes the nth derivative of k. We allow fractional values of
n (with Dn defined via Fourier transform). (2) The equivalence of (a) and (b) for
the analogous operators defined on e2 (Z+) is a result of Peller [12]. The equivalence of (a), (b), and (c) for p = 1 is proved in [5], earlier related results were
obtained by Howland [10].
III. Proof of Theorem 1
We write H for Hk' We suppose that the TJ-Iatticeis picked and fixed.
First note that by the decomposition theorem, (b) is satisfied for some n >
lip if and only if it is satisfied for all such n. Term-by-term integration and
differentiation yields the equivalence of a sum of the form given in (2.1) with a
representation of the type given in (c). Thus (b) is equivalent to (c).
We now show that (a) implies (d). Write h for the individual building blocks
in (2.2). That is,
/;(z)
= B(z,sJ3/4+eB(Lsi)-I/2-e.
The points obtained by reflecting the numbers {sJ in the y-axis also forms an TJlattice. Denote the corresponding functions by ft,
(Note: B (s, s) = B ( - s, - s).) Let T be the linear map from sequences to functions
on (0,00) given by T«a) = 2: aji' The representation theorem summarized in
(2.2) shows that T is a bounded map from e2 to L2(0,00). Similarly the map S
given by S«a) = 2: ajt is bounded. Let bi denote the ith standard basis element of e2; (b) j = Oij' The quantity to be estimated is
917
HANKEL OPERATORS AND COMMUTATORS
By (2.7), this is dominated by liS* HT II~.(2.6) is valid even if the three operators
are not all defined on the same space. Thus, since Sand T are bounded, we may
dominate this last quantity by IIH II~. Thus (d) is established.
We now suppose (d) holds for any "l)-lattice. We will establish (b). Write
Sj = xj + iYj" Using (2})Awe findfj(t)
= yretli2+ee-i~/.
Thus, using the definitions we can write (H fj, f /) as
c
i" i" k(s
+ t) yJ'''' t Inh SIi".
e - it" e it,i ds dt.
We replace the integration variables sand t by new variables t and w = s + t and
evaluate the t integral explicitly. This produces
i" k(w)
c yJ+2'
which by definition equals c yr2e
(3.1)
2:
wH2i
e -it,w dw
(D2+2e k)(~j)'
Thus the sum in (d) is
ID2+2e k(~)IP(y;+2e)P.
j
This sum is, roughly, a Riemann sum for
(3.2)
J J
ID"2'klpy"p+pi-lldydx
which is the integral we wish to estimate. To actually obtain the estimate we pick
a covering of U by sets Di of the sort described in Lemma 2.4 of [5]. Roughly
Di is a covering by disjoint sets which are almost balls of radius c'l1. Pick ~: in
Di at which ID2+2e kl attains its maximum. {~:} will also be a c'l1-lattice and (3.1)
for this new lattice will be a majorant for an upper sum for (3.2). Thus the integral
in (3.2) is finite.
We now show that (c) implies (a). We obtain this by complex interpolation of
operators. The most obvious limit at p = 00 of the spaces in (c) is the Bloch space.
(See Theorem 2' of [5] for a description of the Bloch space in this form.) However
the condition that k be in the Bloch space is not sufficient to insure that Hk is
bounded. On the other hand, it is not clear how to use the condition which insures
that Hk be bounded (the condition is that k is in HMO) as an endpoint for an
interpolation argument which has A 1,0 as the other endpoint. For this reason, we
consider operators a bit more general than Hankel operators. For complex a, r3
and k defined on (0,00) define Hr'~ by
(Ht' f)(t) =
r
s" t" (s
+ t) -,
Lemma 2. Suppose a is complex, Re a
H~,2+2a is a continuous map afA 1,0 to 'fl'
k(s
+ t) f
> -1/2.
(I) dt.
The map from k to
918
R. ROCHBERG
We wish to decompose k using the decomposition theorem. However
we first note that differentiation of purely imaginary order is an isomorphism of
AI,o. Hence we can obtain a decomposition (2.1) with E replaced by E + ie, e
real. 'vVedecompose k according to the decomposition theorem, that is, we write
k in the form (2.1) with p = 1, d = 1, E = - 1/2 + a. We then use (2.3) and
obtain
Proof.
Let
Ti
be the operator generated by the single term
By direct calculation, TJ is given by TJ = (f,uJ
ui(t)
= y;l2+ataei~it
and
viet)
Vi
with
= yi/2+atae-i~;t.
For a one-dimensional operator such as Ti' we can compute the g;1 norm directly,
IITill1 = Ilui11211vi112• But, .by direct calculation, Iluil12 = Ilvil12 = 0(1). Thus IITill1 =
0(1), and the operator of interest is an absolutely convergent series in g;1'
Lemma 3.
Let
tions F on U which
E
>0
be given.
satisfy IIFII =
Let BlochE be the space of holomorphic
funcsup Iy-E DEF(x + iy)1 = c < 00. [fa is a comy>O
>
plex number, Re a
0 then the map from k to H~,2a is a continuous
BlochE to the space of bounded linear maps on L2(O,00).
map from
Note.
For a real this is essentially the fact that a Hankel operator on a Bergman space is bounded if the symbol is in the Bloch space. (See Section 5 of [7].)
For more on the space BlochE see [5] and [8].
Proof.
Pickf, g in L2(O,00). We estimate I(H~,2Cif,g)l.
(H;"" f,g) ~
L" L"
= Loo
r
s" t" (s
+ t) -,-
+ t) f(t)
k(s
(u - t)"t"u-'"k(u)f(u
~Loo u-'"k(u)[(s"f)
g (s) ds dt
- t)g(t)dtdu
* (s"g)](u)du.
The Fourier transform is a unitary map from A2,r to L2((O,00),t-I-2r dt). Hence, if
we denote the inner product on A2,r by L·\ and select r = Re a-I /2 and
1m a = a we find
(H%,2af,g)
= (D -2iak, (saf)
The product (saf)A (sa g) is in A I,r with norm
A
* (SCig))),
= (D -2iak,((saf)
A
(sa g) A)r.
HANKEL OPERATORS AND COMMUTATORS
II(saf)(slY
g)111,r
< !1(saf)AII t,~lI(sag»
AII
t; =
lifll
919
~/211g 11~!2.
Thus it suffices to show that f (F) = (D -2ia k,F)r is a bounded linear functional
on A \,r. This is insured by the hypotheses on k. (See, for example, Theorem 2'
of [5] and the discussion following that theorem. The operator D -2ilY can be
"moved," f(F) = (k,D2ilYF)r')
The lemma is proved.
'vVenow complete the proof of the theorem. If p = 1 then the required result
is Lemma 2 with a = O. We now suppose p > 1. We will use complex interpolation of operators. Let w = x + iy, 0 < x < 1 be the interpolation parameter.
Let a and 13 be fixed -1/2 < a < 0 < 13. Let hew) = a + w(13 - a) and
j(w) = 2 + 2a + w(2f3 - (2 + 2a». For given k let R(w)(k) = H~(w),j(w).
Thus R
is an analytic map from functions to operators on e(O,oo). If w = iy, then
Re hew) = a, Re j(w) = 2 + 2a and we conclude, by Lemma 2, that the mapping
from k in ,1I,a to R(w)k is a continuous map into :il. For w = 1 +
iy, Re hew) = 13, Rej(w) = 213 and we conclude, now by Lemma 3, that the map
from k to R(w)k is a continuous map from Bloche to bounded operators on
L2(0,00).
The interpolation spaces between A 1,a and Blochs are AP,a 1 < P < 00. The appropriate interpolation theorem for analytic maps into :ip spaces is in Section III,
13 of [9]. (In that context, the bounded operators are the end point space, :icc')
To obtain the result of interest, we evaluate at wa = -a/(13 - a). The corresponding p is p = 1 - 0'.113 (note that we can obtain any p); h(wa) = 0, j(wa) =
21p. We conclude H~,2/p is in :ip if k is in ,1P,a. However H~,2/P = HD-2JPk' Thus
HD-2JPk
is in :ip if k is in A,P,a. This is the required conclusion.
A final note. We have been informal about the growth restrictions necessary
to apply the.interpolation theorem. What is at issue are the estimates on R(w) for
11mwi very large. The only place that 1m w affects the estimates in Lemma 2 or
Lemma 3 is through operator norm estimates of the operator Diy (and similar
operators) acting on A \,r. The required estimates for these operators follow, for
example, from the analogous estimates on the Hardy spaces H p 0 < p < I and
the inclusion theorems relating Hardy and Bergman spaces (e.g., Proposition 4.4
of [5]). The proof is complete.
IV. Variations
A. Commutators.
Roughly, commutators on the line are sums of Hankel
operators and hence Theorem 1 applies to such commutators. Let b be a function
defined on R, let P be the projection of L 2(R) onto H2(R) and let Q = I - P. Let
M be the multiplication operator Mf = bf and let M + and M _ be the analogously
defined operators using b+ = PCb) and b_ = Q(b). The commutator we consider
is [M,P]. (For more about this operator and related operators see [7], [10], [17].)
The relationship between this operator and Hankel operators is shown by the following (formal) calculation. For fin L 2, f = Pf + Qf.
920
R.ROCHBERG
[A1,P]f=
[M,P]Pf+
= QMPf-
.[M,P]Qf=
PMQf=
-[M,Q]Pf+
QM-Pf-
[M,P]Qf
PM+Qf.
The operator PM+Q is essentially the Hankel operator with symbol k(t) =
b+(t). ~'lore precisely, if we let U be the map of L2(0,::c) into L2(-oo,::c) given by
tt>
< 00
= {f(0, -t),
(Uf)(t)
then [PM+Q((Uf) v)( = Hkf. This is verified by direct calculation. Because U
is unitary onto its range, the trace ideal properties of Hk and PM +Q are the same.
Similarly QM _P is essentially the Hankel operator with symbol k(t) = b_ (-t).
Proposition 4. The commutator [M,P] is in r:Jp if and only if both b+(z) b_(i)
are in the space described by the equivalent conditions (b) and (c) of Theorem
1. Equivalently, letting b(z) be the harmonic extension of b to U, [M,P] is in
r:Jp
if and only if for some n > lip
(4.1)
f Jv
~
ayn
b(x
+
iy)
IP ynp-2
dy dx <::c.
Proof.
The first statement follows from Theorem 1 and the calculations preceding the proposition. The second statement follows from the first and the fact
that the operator P is bounded on the space of functions b which satisfy (4.1).
B. Weighted commutators. We can also obtain results on the commutator
of M with weighted projections. We will assume some familiarity with the theory
of weights as described in [2] and [14]. We will ignore the distinction between
bounded operators and densely defined operators which satisfy an a priori estimate.
We begin by recalling the A2 condition of Muckenhoupt. (We give the condition
in a form convenient for our purposes; other, simpler, forms are given in [14].)
A non-negative function V on R is said to satisfy the condition A2 if
(4.2)
sup
(xO,Yo)EV
(JR
V (x)
2
2
X
Yo + Yo d )
(x - xo)
(1 V -I (x) (x R
2
2
X
Yo + Yo d )
xo)
=c<so
If w is a positive function and w2 satisfies A2, then the operators P wand Pw-l
given by PwCf) = wP(w-If)
and Pw-1(f) = w-Ip(wf)
are bounded on L2• We
are interested in conditions which insure that the commutators [M,P w] be in :ip'
First we obtain an endpoint result similar to the a = 0 case of Lemma 2.
Lemma 5. In order for [M,P,vl to be in
and b'~(i) be in A 1.0.
Proof.
r:J
l' it is sufficient that both b':f-(z)
We consider the operator [M..,.,Pw]' The other is similar.
921
HANKEL OPERATORS AND COrvIMUTATORS
[A'I+,Pn]f(x)
. = JR b + (X)x
--
. f(y)dy.
by + (y) tv(y)
lV (X)
By the decomposition theorem "ve can write b 1- in the form shown in (c) of Theorem 1 with parameter 8 = 1/2. It suffices to obtain uniform:11 estimates for each
term in the resulting operator sum. Thus we must estimate the ':fi1 norm of the
operator Ti which sends f to
..
T
(J),)
(x
=
Yi
J(
_ -
(_~.)
x
I
Yi
_
( Y _"))
~l
\
I
-X
1
_
w(x)
-Y
(v d .
w ()f.)y
Y
This can be rewritten as
(TJ)(x)
Thus TJ = c
<J~ V) Vi
= c
J (x.
-
v'
-~J-'-f(y)dy.
~J(yw(y)
1
w(x)
with
1
Vl=-.-~
w (x) x -
In
,~
and
In
y ,Vi=W(X)~.
~i
.
X -
~i
Thus IITilli $IIUill~ IlVill~ = J U;C-'t)zdx J VJx)zdx. We now substitute the appropriate expressions for Vi and Vi and use (4.2) at the point (xo,Yo) = C't"pyJ. We
find IITilii < c. The lemma is proved.
We now interpolate between this result and Proposition 4. Let p be given
I<
P
<
ce.
6.
is in :1p'
Proposition
The following
pair
of conditions
[M,P,J
(a) b + (z) and b _ (z) satisfy condition (b) of Theorem
(b) The function w2p satisfies the condition Az.
is sufficient
to insure
that
1
We have no information to suggest (b) is actually necessary. (wz must
satisfy Az to start the analysis.)
Note.
It suffices to consider the case b = b + . If w2p satisfies (4.2) then by
the general theory of weights WZP(l+E) satisfies (4.2) for some positive E. Let
p = p + (p - l)/E. We now interpolate between the lemma for the weight
wp(l h) and Proposition 4 for p with no weight. That is, for complex ~ define
h(O = (-pl(p
- 1» (p - 1 + Ep)~ + p(1 + E). If v is real then Re h(iv) =
p(1 + E) and Re h(1 - lip + iv) = O. Also h(1 - lip) = 1. For ~ with
o $ Re ~ < 1 - lip we define R(O to be the map from functions which are analytic in U to linear operators on L z(R) given by R(O(b)(f)
= wbq;)[M,P]W~b(~)f.
Because W2p(I+E) satisfies (4.2) we may apply the previous lemma and conclude
that R(O)(b) is in:11 if b" is in AI,a. R(iv)(b) is R(O)(b) conjugated by a unitary
operator and hence is also in gl' R(I - I/fJ)(b)(f)
= [M,P](f).
By Proposition 4, this is in gp if b' is in A",p/2-1, Again, the same estimate holds for
Proof.
922
R. ROCHBERG
(1 - 1/ j5 + iv). By complex interpolation, we conclude
maps those b with b' in AP.p!2~I
to operators W[M,P]W-1
statement is the desired conclusion. The proof is complete.
R
that R
in ';fl"
0 -
1/ p )
This last
C. Other projections, other commutators.
One of the main reasons for
studying commutators is that they are, in various senses, building blocks from
which other more complicated operators are constructed. Because of this, \-ve can
use Proposition 6 as a starting point and obtain trace class criteria for other operators. We now give several examples.
\Ve will say of two operators that one is an ';f I' perturbation of the other if the
difference belongs to 'if p' Such conditions arise, for example, in the theory of
bases of Hilbert space (e.g., Section VI, 3 of [9]).
Proposition 7. Let w be a non-negative function on R. Let p be given 1 <:
p < 00. Let lvl be multiplication by log w. The following are equivalent.
(a) P w is an ~fp perturbation of P
(b) [M ,P] is in ';II'
(c) (log w)+ (z) and (log wL (2) both satisfy the conditions (b) of Theorem 1.
Proof.
The equivalence of (b) and (c) is Proposition 4. The fact that (a) implies (b) is proven the same way as the corresponding proof for bounded operators
(page 621 of [7]). We must show (c) implies (a). First note that if w satisfies
(c) then log w has vanishing mean oscillation and hence w I satisfies (4.2) for all
real t. In particular the operators P (t) = PWI are bounded for 0 <: t <: 1.
pw
-
p ~
f (:,
pet) )dt
Thus it suffices to show that (d/dt) P(t) is in ';II" (d/dt) pet) = [M,P(t)].
Because WI satisfies (4.2) for all t, Proposition 6 can be applied and we are done.
The estimates for the higher commutators now follow as in [7]. Let K 1 =
[M,P], Kn+l = [M,Kn] for n = 1,2, ..
Corollary 8. Let w, M, p be as in Proposition
for each n, n = 1, 2, ... Kn is in 'ifp'
7. If KI = U"l,P] is in
'Jp
then
The operator P w is essentially the projection P being interpreted on the weighted space L 2(w2dx). There is a closely related operator, Qw, the projection in
L2(w2dx)
interpreted as an operator on L2• That is, let Qw be the operator which
onto the closure in L2(R,w2dx)
of
is the self-adjoint projection of L2(R,w2dx)
the analytic functions. We now regard Qw as an operator defined on L\R,dx).
Q w is a non-self-adjoint projection onto analytic functions. Such operators are
considered systematically in [6]. Using the results of [6], we obtain the following
corollary of the previous proposition.
Corollary 9. Suppose w is a non-negative function on R. Let 1"1 be multiplication by log w. Let p be given, 1 ::; p < 00. If [i\lf ,P] is in ';II' then for all t
sufficiently small Q w I is an ':ip perturbation of P.
HANKEL
OPERATORS
923
AND COMMUTATORS
Our final example of an operator which can be analyzed by these methods is
a commutator of the type studied by A. P. Calderon. Let b be a function on R
and B its primitive. Let Al B be multiplication by B and let D be differentiation.
This operator is
The operator of interest is Cn defined by Cn(f) = [Mn,P]Df.
known to be bounded if b has bounded mean oscillation. One proof that C B is
bounded is based on the fact that, if we define H (i y) to be the commutator of
multiplication by Diy b with P, then C B can be represented as
•
CB(f)
(see Proposition
to CB
=c
fx-x
H(iy)(D"'f)-.
1 dv
+ y~
11 of (3]). Thus any information about the H (i y) will propogate
•
In particular all the conditions we have considered for b are preserved by differentiation of imaginary order. Hence.
Corollary 10. Let p be given 1 <p < ce. Let B be given and suppose (B ')+ (z)
and (B
(£) satisfy the conditions of (b) of Theorem 1; then C B is in ';jp'
'L
V. Comments
There are many natural variations on the issues we have considered.
tion three that seem especially intriguing.
•
'.tVe men-
A. P < 1. Theorem 1 may be true as stated for 0 < p < 1. ,The relationship
between parts (b), (c) and (d) of the theorem is valid(uslngih~
decomposition
theorem). It is still true that these conditions imply (a). The reason for this is that
if 0 < p < 1 then the estimate on individual terms can be combined in a way
similar to the case p = 1. Thus a proof of the model of Lemma 2 works essentially
without change. The new difficulty is that (2.7) is not valid for p < 1 and thus
we do not know how to draw conclusions from (a).
Intuitively, part of the content of the theorem for p = 1 is that the sets {fJ
and {fn (described in the proof) nearly diagonalize Hankel operators. A precise
quantitative form of this would be quite interesting and would probably suffice
to complete Theorem 1 for p < 1. For example, if we define P J = (f,f) fi and
QJ=
(f,fnfi
then it may be that as operators
positive C. (If this is true then H in ';jp implies
12:: PiHQi 1< CIHI for some
2: P iH Q i E
';jp for any p.)
B. Higher dimensions.
Both Hankel operators and the commutators we considered have various generalizations.
be an open convex
We could replace (O,c;cl with a more generalcol1e~ Let
We
cone in R n and let d V be volume measure on R n. Let k be defined on
r
define the generalized
to itself given by
r.
Hankel operator with symbol k as a map from L\r,dV)
924
R. ROCHBERG
)
I ,
! 1
~
(Hd)(x)
=
i
k(x
+ y)f(y)dy.
If the cone has additional structure then the techniques we have been using can
be brought to bear. It would be interesting to know about these operators. Presumably, if
is homogeneous and self dual (and hence the results of [5] are
available) then a result of the same general form as Theorem 1 is valid.
The situation with commutators for functions of several variables is l~",~~d.ear.£
For example, we do not know what techniques (invoivfng'generalized
Hankel
operators or other ideas) would be appropriate for the following natural question
in the spirit of Proposition 4.
r
".··.'~: ..c>'.",
..~.'.<.'_ .c-."
:-_c.,.,.._.:..,."'.,.". .... c.
"~.'_.'_"'_""'_. .' _"', .• :',_,",_,'
•. :<if
Question.
b betransforms
a function ononL 2R(R2. 2).LetWhat
M); be
let
aremultiplication
the conditions byonbband
which
R be
one of theLet
Riesz
are necessary and sufficient for [M);,R] to be in r::f1?
•
C. Other operators. In the last section we saw that commutators in p could
be used as starting points from which to show other operators are in p' There
are powerful general techniques for generating a very large class of naturally occurring operators from commutators ([3], [4]). It would be interesting to know
what additional conclusions could be drawn in general if the commutators so used
are assumed to be in ;j p •
(Added notes: (1) The given proof of the implication "(a) implies (b)" in Proposition 7 requires the further hypothesis that P w1+it< be an p perturbation of P for
all real a. (2) Further results on these issues can be found in [18], [19], and [20].)
r::f
r::f
r::f
REFERENCES
1. J. R. BUTZ, Compact Hankel operators
in "Contributions
to Analysis
and Geometry"
et al., Eds.), Johns Hopkins University
Press, 1982.
2. R. COIFMAN & C. FEFFERMAN, Weighted norm inequalities for maximal functions
integrals, Studia Math. LI (1974), 241-250.
(Clark
and singular
Analysis,
Proceedings
of the Seminar
held atdeElCalderon
Escorial, sur(de[,integrale
Guzman de
andCauchy,
Peral,
COIFMAN,
& Y. MEYER,
Un generalisation
du theorem
3. R. Fourier
Eds.),
Asoc.
Mat.
Espanola,
1980.
4. ---,
Au dela des operateurs pseudo-differentiels,
Asterisque,
57 (1978).
theorems for holomorphic and harmonic functions in U, Asterisque 77 (1980), 11-66.
R. COIFMAN & R. ROCHBERG, Projections in ,veighted spaces, skew projections and inversion of
Toeplitz operators, Integral Equations Operator Theory 5 (1982), 145-159.
R. COIFMAN, R. ROCHBERG & G. WEISS, Factorization theorems for Hardy spaces in several
variables, Ann. of Math. 103 (1976), 611-635.
p, DUREN, Theory of HP Spaces, Academic Press, New York, 1970.
1. C. GOHBERG & M. G. KREIN, Introduction to the Theory of Linear Non-self Adjoint Operators,
Translations
of Monographs,
18, Amer. Math. Soc., Providence,
RI, 1969.
J. S. HOWLAND, Trace class Hankel operators, Quart. J. Math. (2) 22 (1971), 147-159.
S. JANSON, Mean oscillation and commutators of singular integral operators, Ark. Mat. 16
(1978), 263-270.
5. R. R. COIFMAN & R. ROCHBERG, Representation
6.
7.
8.
9.
10.
11.
•
•
t
f'14.
f
19.
12.
HANKEL OPERATORS AND COMMUTATORS
13.
20.
16.
18.
17.
15.
925
V. V. PELLER,Nuclearity of Hankel operators, Steklov Institute of Mathematics, Leningrad,
(preprint), 1979.
S .. c. PmvER, Hankel operators on Hilbert space, Bull. London Math. Soc. 12 (1980),422-442.
D. SARASON,Function themy on the unit circle, Department of ~Iathematics, Virginia Polytechnic Institute and State University, Blacksburg, VA, 1978.
B. SIMON,Trace Ideals and Their Application, L.M.S. Lecture Note Series No. 35, Cambridge
University Press, Cambridge, 1979.
E. STEL"l& G. WEISS,Introduction to Fourier Analysis on Euclidean Spaces. Princeton University
Press, Princeton, 1971.
A. UCHIYA:YIA,
On the compactness of operators of Hankel type. Tohoku Math. J. 30 (1978),
163-171.
S. JANSON& T. WOLFF,Schatten classes and commutators of singular integral operators, (preprint), University of Chicago, 1981.
V. PELLER,Vectorial Hankel operators, commutators and related operators of the Schatten-von
Neumann class ""p' Integral Equations and Operator Theory 5 (1982), 244-272.
S. SEMMES,Ph.D. Dissertation, Washington University, St. Louis, 1983.
This work was partially supported by National Science Foundation Grant MCS-8002689.
UNIVERSITY
OF MISSOURI-ST. LOUIS, MO 63130
Received March 2, 1981
•