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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. 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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 •