Table Of ContentHANKEL OPERATORS AND THE DIXMIER TRACE
ON THE HARDY SPACE
6
1 Miroslav Engliˇs, Genkai Zhang
0
2
n Abstract. We give criteria for the membership of Hankel operators on the Hardy
a spaceonthediscintheDixmierclass,andestablishestimatesfortheirDixmiertrace.
J In contrast to the situation in the Bergman space setting, it turns out that there
4 exist Dixmier-class Hankel operators which are not measurable (i.e. their Dixmier
2 tracedependsonthechoiceoftheunderlyingBanachlimit),aswellasDixmier-class
Hankeloperatorswhichdonotbelongtothe(1,∞)Schatten-Lorentzideal. Arelated
] question concerning logarithmic interpolation of Besov spaces is also discussed.
V
C
.
h 1. Introduction
t
a
Let T be the unit circle in the complex plane C and H2 the standard Hardy
m
space of all functions in L2(T) L2 (with respect to the normalized arc-length
[ ≡
measure) whose negative Fourier coefficients vanish. For φ L (T), the Hankel
∞
1 operator H withsymbolφis theoperator fromH2 intoits or∈thogonal complement
v φ
L2 H2 defined by
8
⊖
2 H u= (I P)(φu), u H2,
φ
4 − ∈
where P : L2 H2 is the orthogonal projection. Equivalently, H is an operator
6 φ
→
0 whosematrixwithrespecttothestandardbases eikθ ofH2and e miθ of
1. L2 H2 is constant on diagonals perpendicular {to the}∞km=0ain diagonal{, t−he (k},∞mm=)1-th
⊖
0 entry being equal to the Fourier coefficient φˆ( k m 1). One can define H
φ
6 − − −
even for φ L2 as a densely defined operator, and one has H = 0 if φ H2,
1 φ
∈ ∈
v: so that Hφ effectively depends only on (I −P)φ, and thus it is enough to study
Xi Hφ only for φ = f with f ∈H2. Nehari’s theorem then asserts that Hf is bounded
if and only if f P(L (T)) = BMOA(T); similarly, H is compact if and only
r ∈ ∞ f
a if f P(C(T)) = VMOA(T). The much finer question of the membership of H
∈ f
in the Schatten classes p, 1 p < , was solved by Peller, who showed [15]
S ≤ ∞
that H p if and only if f belongs to the diagonal Besov space Bp = B1/p;
f ∈ S pp
this was later shown to prevail also for 0 < p < 1 (see e.g. [17] and the references
therein). HereBp can becharacterized asthespaceof (thenontangentialboundary
values of) all holomorphic functions f on the unit disc D which satisfy
1/p
(1) f := f(k)(z)p(1 z 2)kp 2dz <
(k),p −
k k D| | −| | ∞
(cid:16)Z (cid:17)
1991Mathematics Subject Classification. Primary 47B35;Secondary 30H25,47B10,46B70.
Keywordsandphrases. Hankeloperator,Hardyspace,Dixmiertrace,Besovspace,logarithmic
interpolation space.
Research supported by GA CˇR grant no. 201/12/0426,by RVO funding for ICˇ 67985840and
by Swedish Science Council (VR).
Typeset by AMS-TEX
1
2 M. ENGLISˇ, G. ZHANG
for some (equivalently, any) nonnegative integer k > 1/p; here dz stands for the
Lebesgue area measure. Using real interpolation, it follows more generally that H
f
belongs to the Schatten-Lorentz ideal p,q, 0 < p < , 0 < q , consisting of
S ∞ ≤ ∞
all operators T whose singular values s (T) satisfy
j
∞
(j +1)q/p 1s (T)q < , q < ,
− j
∞ ∞
(2) j=0
X
sup(j +1)1/ps (T) < , q = ,
j
∞ ∞
j
if and only if f belongs to the “Besov-Lorentz” space Bpq consisting of (the non-
tangential boundary values of) all holomorphic functions f on D satisfying
∞((1 z 2)f (z)) (t)tq/p 1dt < , q < ,
′ ∗ −
−| | ∞ ∞
(3) Z0
sup((1 z 2)f (z)) (t)t1/p < , q = ,
′ ∗
−| | ∞ ∞
t>0
atleastfor1< p < (for0 < p 1onewouldagainhavetousehigherderivatives
∞ ≤
of f as in (1)); see e.g. [11]. Here φ denotes the nonincreasing rearrangement of a
∗
function φ on D with respect to the measure (1 z 2) 2dz. For p = q, the spaces
−
Bpp = Bp agree with the Besov spaces above. T−he|re|is also an equivalent “dyadic”
description of the Besov and Besov-Lorentz spaces, which avoids the holomorphic
extension into D and which runs as follows: for n 1, introduce the trigonometric
≥
polynomials W on T by
n
∞
W (eiθ) = a ekiθ,
n nk
k=0
X
where a = 0 for k / (2n 1,2n+1), a = 1 for k = 2n, and a depends linearly
nk − nk nk
∈
on k on the intervals [2n 1,2n] and [2n,2n+1]. Setting further W (eiθ) = 1+eiθ,
− 0
we thus have for any f = ∞k=0fkekiθ on T
P
∞
(4) f = f W , where (f W )(eiθ)= a f ekiθ.
n n nk k
∗ ∗
n=0 k
X X
Then f Bs , 0 < p , 0 < q , s R, if and only if1
∈ pq ≤ ∞ ≤ ∞ ∈
(5) f dyadic,spq := 2ns f Wn Lp(T) < ,
k k { k ∗ k } lq(N) ∞
(cid:13) (cid:13)
andfor 1 = q = pthisquantityis(cid:13)(cid:13)equivalentto(1). Sim(cid:13)(cid:13)ilarly,f Bpq,0 < p < ,
s ∈ ∞
0 < q , if and only if the function φ on T N defined by
f
≤ ∞ ×
(6) φ (eiθ,n):= (f W )(eiθ)
f n
∗
1Moreprecisely Bs is the subspace of (the boundaryvalues of)holomorphic functions in the
pq
fullBesovspaceBs ,i.e.offunctionsinBs (T)whosenegativeFouriercoefficientsvanish;thefull
pq pq
BesovnorminBs beingdefineduponaddingto(5)alsothetermsn≤0(andreplacingthefactor
pq
2nsby2|n|s),whereW−n(eiθ):=Wn(e−iθ)andW0 mustbechangedtoW0(eiθ)=e−iθ+1+eiθ.
Itis morecustomary to denote Bpsq by Bpsq, andourBpsq by Aspq or(Bpsq)+, cf.[10,17]; however,
since the “full” Besov spaces Bs will not be needed anywhere in this paper, we take the liberty
pq
to use the simpler notation Bs just forthe holomorphic Besov spaces. The same also applies to
pq
the “Besov-Lorentz” spaces Bpq.
HANKEL OPERATORS AND THE DIXMIER TRACE 3
belongs to the Lorentz space Lpq(T N,dν) with respect to the measure dν given
×
by 2ndθ on T n , n N; that is, if and only if the nonincreasing rearrangement
2π ×{ } ∈
φ of φ with respect to dν satisfies
∗f f
dt 1/q
∞(t1/pφ (t))q < , q < ,
∗f t ∞ ∞
(7) (cid:16)Z0 (cid:17)
supt1/pφ (t) < , q = .
∗f ∞ ∞
t>0
Furthermore, the quantities (7) and (3) are again equivalent. We refer to Peller
[17], [16] and Krepkogorskii [11], [10] for further details on all these matters.
In addition to the Hardy space H2, there are also (big) Hankel operators on
weighted Bergman spaces A2(D) on the disc, α> 1, consisting of all functions in
α −
L2(D, α+1(1 z 2)αdz) L2 thatareholomorphiconD. Namely,forφ L (D),
π −| | ≡ α ∈ ∞
the Hankel operator H(α) :A2 L2 A2 is defined as
φ α → α⊖ α
H(α)u= (I P(α))(φu), u A2(D),
φ − ∈ α
where P(α) : L2 A2 is the orthogonal projection. Again, H(α) makes sense
α → α φ
as a densely defined operator even for any φ L2, and one has H(α) = 0 for φ
∈ α φ
holomorphic, so that H(α) in fact depends only on (I P(α))φ; furthermore, for
φ −
φ = f with f holomorphic on D, it turns out again that H(α) p if and only
f ∈ S
if f Bp, 1 < p < , while H(α) p for some 0 < p 1 only if H(α) = 0;
∈ ∞ f ∈ S ≤ f
see Arazy, Fisher and Peetre [1]. Using real interpolation, one can deduce from
this also that H(α) pq, 1 < p < , 0 < q , if and only if f Bpq (though
f ∈ S ∞ ≤ ∞ ∈
this seems not to be noted explicitly in the literature).
The Schatten-Lorentz ideals pq satisfy p1,q1 p2,q2 if p < p or if p = p ,
1 2 1 2
S S ⊂ S
q < q . A notable operator ideal lying between 1, and all p,q, p > 1, is the
1 2 ∞
S S
Dixmier ideal Dixm, consisting of all operators T whose singular values satisfy
S
n
s (T)
j=0 j
(8) sup =: T < .
Dixm
log(n+2) k k ∞
n P
Equipped with the norm (8), Dixm becomes a Banach space, and the closure
S
Dixm of the subspace of finite rank operators in Dixm consists of all T for which
Slim0 1 n s (T)= 0. Forascaling-invariSantBanach limitω onN(seethe
n logn j=0 j
→∞
next section for the definitions), one furtherdefines the Dixmier trace tr on Dixm
P ω S
by setting
n s (T)
j=0 j
tr T := ω
ω
log(n+2)
P
(cid:16) (cid:17)
for T positive, and extending to all T Dixm by linearity. The operator is
∈ S
called measurable if tr T does not depend on the choice of the Banach limit ω.
ω
Inview of the results mentioned in thelast paragraph, it is naturalto ask for which
holomorphic f on D does H(α) belong to Dixm and what is its Dixmier trace.
f S
It was shown by Rochberg and the first author [8] for α = 0, and by Tytgat [19]
4 M. ENGLISˇ, G. ZHANG
for general α, that H(α) Dixm if and only if f belongs to the Hardy 1-space H1,
′
f ∈S
and in that case the modulus H(α) = (H(α) H(α))1/2 is measurable and
∗
| f | f f
2π dθ
(9) tr H(α) = √α+1 f (eiθ) .
ω ′
| f | | |2π
Z0
The methods of [8], however, break down for A2 replaced by H2 (which in a well-
α
defined sense is the limit of A2 as α 1).
α ց −
The aim of the present paper is to characterize Hankel operators H , f H2,
f ∈
on the Hardy space that belong to Dixm, and to give estimates for the Dixmier
S
trace of H .
| f|
Ourmainresultsareasfollows. Forf H2,wedenotebyf alsotheholomorphic
∈
extension of f into D, i.e. f(z) = ∞n=0fnzn if f(eiθ)= ∞n=0fneniθ; further, let
F(t) := (1 Pz 2)2f (z) ∗(t), tP> 0,
′′
−| |
(cid:16) (cid:17)
be the nonincreasing rearrangement of (1 z 2)2f (z) with respect to the measure
′′
−| |
(1 z 2) 2dz on D, and similarly let
−
−| |
Φ(t):= (f W ) (t), t > 0,
∗
∗ ·
be the nonincreasing rearrangement of the function φ from (6) with respect to the
f
measure dν on T N.
×
Theorem 1. The following assertions are equivalent:
(i) limsup (p 1) f (z)p(1 z 2)2p 2dz < ;
(ii) limsuppց1 −1 tDF|(t′′)dt|< −; | | − ∞
t→+∞ logt R0 ∞
(iii) limsup (p 1) (f W )(eiθ)pdν(θ,n)< ;
(iv) limsuppց1 −1 RtTΦ×(Nt)|dt∗< n; | ∞
t→+∞ logt R0 ∞
(v) H Dixm.
f ∈ S R
Moreover, the quantities on the left-hand sides of (i)-(iv) are equivalent, and are
further equivalent to dist (H , Dixm).
SDixm | f| S0
Note that the integral in (i) above is just f p , which by general theory is
k k(2),p
equalto F p ;similarly,theintegralin(iii)isjust f p = Φ p .
k kLp(0, ) k kdyadic,1pp k kLp(0, )
∞ p ∞
Theorem 2. Let ω be a dilation- and power-invariant Banach limit on R , ω =
+
ω exp the corresponding translation- and dilation-invariant Banach limit on R,
◦
and tr the associated Dixmier trace on Dixm. Then the following quantities are
ω e
S
equivalent:
((iii)) ωω--lliimmr→+∞ r11 D|tfF′′((zt))|d1+t;r1(1−|z|2)r2 dz;
t + logRt 0
→ ∞
((iiivi)) ωωe--lliimmr→+∞ 1r1 TR×tNΦ|((tf)∗dtW; n)(eiθ)|1+r1 dν(θ,n);
t + logRt 0
→ ∞
(v) ter H .
ω| f| R
Furthermore, the constants in the equivalences with can be chosen independent
of ω.
HANKEL OPERATORS AND THE DIXMIER TRACE 5
Here and throughout the paper, two positive quantities X,Y are called equiv-
alent (denoted “X Y”) if there exists 0 < c < 1, independent of the variables
≍
in question, such that cX Y 1X; and we refer to Section 2 below for the
≤ ≤ c
definitions and details concerning ω, ω and tr .
ω
The first part of the next theorem is immediate from Theorem 1, which also
implies equivalence of the corresponding quotient norms2 of f with the quotient
e
norm of H in Dixm/ Dixm; for the equivalence of the norm H itself, some
f S S0 k fkDixm
extra labour seems to be needed.3
Theorem 3. For f H2, the operator H belongs to Dixm if and only if
∈ f S
1 t
f BDixm := f H2 :sup F(t)dt f <
(2),Dixm
∈ { ∈ log(2+t) ≡ k k ∞}
t>0 Z0
1 t
= f H2 :sup Φ(t)dt f < ,
dyadic,Dixm
{ ∈ log(2+t) ≡ k k ∞}
t>0 Z0
and H + f(0) f + f (0) + f(0) f .
k fkDixm | | ≍ k k(2),Dixm | ′ | | | ≍ k kdyadic,Dixm
Weremarkthat and arenormsoff andf,respectively,
(2),Dixm dyadic,Dixm ′
k·k k·k
in certain Lorentz (or Marcinkiewicz) spaces; see [2, p. 69].
Theorem 4. There exist f H2 and two dilation- and power-invariant Banach
∈
limits ω ,ω on R such that tr H = tr H ; thus H is not measurable.
1 2 + ω1| f|6 ω2| f| | f|
In [8] it was also shown that in the setting of the weighted Bergman spaces
(at least for α = 0, but the proof likely carries over to all α > 1), H Dixm
− f ∈ S
alreadyimpliesthatH evenbelongstothesmallerideal 1, Dixm ofoperators
f S ∞ ⊂ S
T with singular values s (T) = O(1); that is, there are no Hankel operators H(0),
j j f
f holomorphic, in Dixm 1, . For Hankel operators on H2, things are different.
∞
S \S
Example 5. There exists f H2 for which H Dixm 1, (in other words,
∈ f ∈ S \S ∞
f BDixm B1, ).
∞
∈ \
The equivalence (i) (v) in Theorem 1 is not new but goes back to Li and
⇔
Russo [12], and was subsequently put into a more general picture in the works of
Carey, Sukochev and coauthors [3], [4]. Combining the latter with Peller’s results
mentioned at the beginning and with standard facts from the theory of Besov
spaces yields the other parts of Theorem 1 and Theorem 2; if ω and ω are replaced
by ordinary limits, the ideas behind Theorem 2 go back at least to Connes [6,
IV.2, Proposition 4]. The proof of Theorem 3 relies on a result on logarithmic
e
§
interpolation in the context of Besov spaces, which also provides an alternative
proofoftheequivalences(v) (i) (iii)ofTheorem1andisofindependentinterest.
⇔ ⇔
2More specifically: the expressions in Theorem 1 of which limsup’s are taken are functions
belonging to L∞(1,2) in parts (i) and (iii) (as functions of p), and to L∞(0,∞) in parts (ii)
and (iv) (as functionsof t —and onehas to replace logt by log(t+2)), respectively. Theorem 1
thensaysthatthenormofthoseexpressionsintheqoutientspaceL∞/L∞(whereL∞denotesthe
0 0
subspaceoffunctionsessentiallytendingtozeroasp→1+ort→+∞,respectively)isequivalent
to the normof H in SDixm/SDixm.
f 0
3AddingkfkBMO =kHfktothequotientnormsfromthepreviousfootnoteproducesalready
norms equivalent to kH kDixm+|f(0)|, by the Closed Graph Theorem; however, that they are
f S
equivalenttotheothertwonormsmentioned inthetheorembelowseems notsostraightforward.
6 M. ENGLISˇ, G. ZHANG
Theproofs of Theorem1and Theorem2aregiven inSection 3andSection 4, re-
spectively, afterreviewingthenecessaryprerequisitesonBanachlimitsandDixmier
traces in Section 2. Interpolation of Besov spaces and the proof of Theorem 3 are
the subject of Section 5. The proof of Theorem 4 is furnished in Section 6, and
some comments and concluding remarks, including Example 5, appear in the final
Section 7.
For f a conformal map of the disc onto a Jordan domain Ω C, the Hankel
⊂
operator H is essentially the “quantum differential” dZ from IV.3 in Connes [6],
f §
whereitisalsoshownthat, uptoaconstantfactor, thefunctionalf tr (f dZ p),
ω
7→ | |
p > 1,isjusttheintegrationagainstthep-dimensionalHausdorffmeasureΛ on∂Ω.
p
Similarly, [8] (see also [19]) shows that in the weighted Bergman space setting,
1 tr H(α) equals the length of ∂Ω, i.e. Λ (∂Ω). It would be interesting to
√α+1 ω| f | 1
know if there is some kind of connection with Hausdorff measures also for tr H .
ω| f|
2. Banach limits and Dixmier traces
By a Banach limit on N, N = 0,1,2,... , we will mean a positive (i.e. tak-
{ }
ing nonnegative values on sequences whose entries are all nonnegative) continuous
linear functional on the sequence space l = l (N) which coincides with the ordi-
∞ ∞
narylimitonconvergent sequences. Similarly, byaBanach limitonR = (0,+ ),
+
∞
we will mean a positive continuous linear functional on L (R ) which coincides
∞ +
with ess-lim whenever the latter exists. Such functionals (in both cases)
t +
→ ∞
are easily constructed using the Hahn-Banach theorem. Furthermore, one can get
a Banach limit ω# on N from a Banach limit ω on R by setting
+
(10) ω#(f):= ω(f#),
where
(11) f#(x) = f for x (n,n+1], n N,f l ;
n ∞
∈ ∈ ∈
and, in fact, any Banach limit on N arises in this way (again by the Hahn-Banach
theorem).
The dilation operator D , n = 1,2,3,..., on l (N) is defined as
n ∞
D (x ,x ,...) = (x ,...,x ,x ,...,x ,...);
n 1 2 1 1 2 2
n n
similarly, the dilation operator D , a|> 0,{zon L} |(R{z) is}defined as
a ∞ +
D f(x) := f(x/a).
a
A Banach limit ω on N is called D -invariant (or scaling-invariant if n = 2)
n
if ω D =ω, and similarly ω on R is called D -invariant if ω D = ω. Clearly,
n + a a
◦ ◦
if ω is D -invariant on R , a = n N, then the ω# given by (10) will be D -
a + n
∈
invariant on N. Given an arbitrary Banach limit ω on R , its composition ω M
+
◦
with the Hardy mean
1 t dx
(12) Mf(t) := f(x)
logt x
Z1
will automatically be D -invariant for any a > 0.
a
HANKEL OPERATORS AND THE DIXMIER TRACE 7
A Banach limit ω on R is called P -invariant, for some α > 0, if ω P = ω,
+ α α
◦
where P is the “power dilation”
α
P f(x):= f(xα), x R .
α +
∈
By a Banach limit on R we will mean, by definition, a functional on L (R) of
∞
the form ω(f) = ω(f log), where ω is a Banach limit on R . Thus ω is positive,
+
◦
continuous, and ω(f) = ess-lim f(t) whenever the limit exists. Note the ω is
t +
→ ∞
P -invariant if and only if ω is D -invariant; and ω is D -invariant if and only if
α e α a e
ω T = ω, where T f(x) := f(x+c) (i.e. ω is invariant with respect to the
loga e c
◦ −
translation T by c= loga).
c e
−
Theexistenceof(alotof)BanachlimitsonRwhicharesimultaneouslydilation-,
e e e
translation- and power-invariant (i.e. ω = ω T = ω D = ω P a,α > 0
c a α
◦ ◦ ◦ ∀
c R) is a consequence of the Markov-Kakutani theorem; see [3]. The following
∀ ∈
proposition gives a simple recipe to produce translation- and dilation-invariant
e e e e
Banach limits ω on R (and, hence, dilation- and power-invariant Banach limits
ω(f)= ω(f exp) on R ).
+
◦
Proposition 6e. Let η be an arbitrary Banach limit on R . Then ω = η M ρ ,
+ +
where ρ+e :f 7→ f|R+ is the operator of restriction from R to R+, is a tra◦nsla◦tion-
and dilation-invariant Banach limit on R.
e
Proof. We already know that η M D = η M for any a > 0; since ρ commutes
a +
◦ ◦ ◦
with D , it follows immediately that
a
ω(D f)= η(MD ρ f)= η(Mρ f)= ω(f).
a a + +
For translation invariance, consider first T with c > 0. For t > 1,
e c e
1 t dx
Mρ T f(t)= f(x+c)
+ c
logt x
Z1
1 t+c dy
= f(y) .
logt y c
Z1+c −
Since 1 1 is integrable over (1 + c, ) and f is bounded, we see that the
y − y c ∞
difference of−Mρ T f(t) and
+ c
1 t+c dy
f(y)
logt y
Z1+c
t
tends to zero as t + . Similarly, replacing the limits in the last integral by
→ ∞ 1
produces an error of order O( 1 ) 0. Thus Mρ T f Mρ f 0 as t + ,
logt → + c − + → → ∞R
whenceω(T f)= ω(f),provingtheT -invarianceforc > 0. Forc < 0andassuming
c c
t > 1+c, the argument is completely analogous. (cid:3)
For eease of noteation, we will usually write ω-lim f and ω-lim f(t),
n n t +
instead of ω(f), for a Banach limit ω on N or R (or→R∞), respectively, t→o m∞ake it
+
clear which variable ω applies to.
Since the value of a Banach limit depends only on the behaviour of the sequence
or function at infinity, we will frequently also take the liberty of applying it to
8 M. ENGLISˇ, G. ZHANG
sequences or functions which are undefined or take infinite values for small values
of the argument (such as e.g. {log1n}n∈N).
For a positive operator T in Dixm and a Banach limit ω on N, one sets
S
n
s (T)
j=0 j
(13) tr T = ω- lim .
ω
n logn
→∞ P
If ω is D -invariant, one can show that tr (A+B) = tr (A)+tr (B) for any A,B
2 ω ω ω
positive. This makes it meaningful to extend tr by linearity to all of Dixm.
ω
S
We refer to [6, IV.2], [7], [3], [4] and in general to the monograph by Lord,
§
Sukochev and Zanin [13] for further details on the material in this section.
Throughout the rest of this paper, ω will be a Banach limit on R which is
+
D - and P -invariant for all α > 1; ω(f) = ω(f log) will be the corresponding
2 α
◦
Banach limit on R; ω#(f) = ω(f#) will be the Banach limit on N as in (10);
and (abusingthe notation slightly) tr will be theDixmier trace given by (13) with
ωe
ω# in the place of ω.
3. Proof of Theorem 1
The following proposition is proved in [4, Theorem 4.5] for the special case when
H isthespectralcountingfunctionofanoperator;however,theproofworkswithout
changes in general. We include the details here for the convenience of the reader.
Proposition 7. LetH be a nonnegative nonincreasing function on(0,+ ), which
∞
belongs to Lp(0,+ ) for all 1 < p < 1+δ with some δ > 0. Then the quantities
∞
H := limsup(p 1) ∞H(t)pdt
limsup
k k −
pց1 Z0
and
1 t
H := limsup H(t)dt
limlog
k k logt
t→+∞ Z0
satisfy
H H e H .
limsup limlog limsup
k k ≤ k k ≤ k k
In particular, H is finite if and only if H is.
limsup limlog
k k k k
Proof. For any C > H , let q > 0 be such that
limsup 0
k k
(p 1) ∞H(t)pdt C for 1 < p < 1+q .
0
− ≤
Z0
By H¨older’s inequality, for any 0 < q < q ,
0
t t 1 t q
H(s)ds H(s)1+qds 1+q ds 1+q
≤
Z0 (cid:16)Z0 (cid:17) (cid:16)Z0 (cid:17)
C 1+1q q tq
t1+q C .
≤ q ≤ q
(cid:16) (cid:17)
HANKEL OPERATORS AND THE DIXMIER TRACE 9
If t > e1/q0, we can take q = 1 , so that tq/q = elogt; thus
logt
1 t
H(s)ds Ce for t > e1/q0,
logt ≤
Z0
so H Ce. Hence H C H .
limlog limlog limsup
k k ≤ k k ≤ k k
Conversely, assume that
1 t
H(s)ds C t t .
0
log(t+1) ≤ ∀ ≥
Z0
In other words,
t t C
H(s)ds ds t t .
0
≤ 1+s ∀ ≥
Z0 Z0
Set
H(t), t t ,
0
≥
G(t) :=
( min H(t), 1C+t , t < t0.
(cid:16) (cid:17)
Then
t t C
G(s)ds ds t > 0,
≤ 1+s ∀
Z0 Z0
that is, G(s) C in the sense of majorization of Hardy-Littlewood; it therefore
≺ 1+s
follows (see e.g. [2, p. 88]) that for any p > 1,
C p Cp
∞G(s)pds ∞ ds = ,
≤ 1+s p 1
Z0 Z0 (cid:16) (cid:17) −
so
(14) limsup(p 1) ∞G(s)pds C.
− ≤
pց1 Z0
Since
t0 t0 C p
(p 1) G(s)pds (p 1) ds = Cp[1 (1+t )1 p] 0
0 −
− ≤ − 1+s − →
Z0 Z0 (cid:16) (cid:17)
and, by the Lebesgue Monotone Convergence Theorem,
t0 t0
H(s)pds H(s)ds Clog(t +1) <
0
→ ≤ ∞
Z0 Z0
as p 1, we get
ց
t0 t0
lim(p 1) H(s)pds = 0, lim(p 1) G(s)pds = 0.
pց1 − Z0 pց1 − Z0
Since H(t) = G(t) for t t , we thus obtain from (14)
0
≥
limsup(p 1) ∞H(s)pds C,
− ≤
pց1 Z0
implying that H H . (cid:3)
limsup limlog
k k ≤ k k
10 M. ENGLISˇ, G. ZHANG
The proof below is likewise inspired by the proof of Theorem 4.5 in [4].
Proof of Theorem 1. (i) (v) As recalled in the Introduction, it is known from
⇔
Peller [15, Theorem 4.4] that for each p > 1/2, there exists c (0,1) such that
p
∈
1
(15) c H f (0) + f H ,
pk fkp ≤ | ′ | k k(2),p ≤ c k fkp
p
where stands for the norm in p and for the Besov seminorm (1) with
p (2),p
k·k S k·k
k = 2. Now since both p and Bp, 0 < p < , form an interpolation scale under
S ∞
complex interpolation, it follows by interpolation that one can even get (15) with
c = c independent of p for 1 p 2. (See [12, p. 24] for the details; cf. also [19].)
p
≤ ≤
Consequently,
1
climsup(p 1) H p limsup(p 1) f p limsup(p 1) H p
− k fkp ≤ − k k(2),p ≤ c − k fkp
p 1 p 1 p 1
ց ց ց
for some c (0,1) independent of p.
∈
On the other hand, it is well known that the limsup on the utmost left and
right is equivalent to kHfkSDixm. Indeed, first of all, if Hf ∈/ Sp0 for some p0 > 1,
then, since p increase with p and Dixm p, both H and H
p (1,p )Sare infinite; thus we Smay as⊂sumpe>1thSat H k pfkSpDix>m 1. Bky fthkpe
∀ ∈ 0 T f ∈ S ∀
definition of the norm in p,
S
H p = ∞ s (H )p = ∞H(t)pdt
k fkp j f
j=0 Z0
X
where
(16) H = s (H ) #
{ j f }
is obtained as in (11). Therefore by the last proposition,
t
H(s)ds
limsup(p 1) H p limsup 0 limsupe(p 1) H p.
− k fkp ≤ logt ≤ − k fkp
p 1 t + R p 1
ց → ∞ ց
Furthermore, for n 1 < t n,
− ≤
(17) nj=−01sj(Hf) 0tH(s)ds nj=0sj(Hf),
logn ≤ logt ≤ log(n 1)
P R P −
whence
tH(s)ds n s (H )
limsup 0 = limsup j=0 j f .
logt logn
t + R n P
→ ∞ →∞
Combiningeverythingtogether,wethusseethatthequantityin(i)inTheorem1
is equivalent to the last limsup. However, the last limsup is finite if and only if (8)
holds, i.e. if and only if H Dixm; and it is also known that this limsup is equal
f ∈ S
to dist (H , Dixm), see [4, p. 267]. Thus indeed (i) (v) and the quantity in
SDixm f S0 ⇔
(i) is equivalent to dist (H , Dixm).
SDixm f S0
