L’INSTITUTFOURIER DE ANNALES

A L

E S

E D L ’IN IT ST T U

F O U R

ANNALES

DE

L’INSTITUT FOURIER

R. V. BESSONOV

Schatten properties of Toeplitz operators on the Paley–Wiener space Tome 68, n^o1 (2018), p. 195-215.

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SCHATTEN PROPERTIES OF TOEPLITZ OPERATORS ON THE PALEY–WIENER SPACE

by R. V. BESSONOV (*)

Abstract. — We collect several old and new descriptions of Schatten class Toeplitz operators on the Paley–Wiener space and answer a question on discrete Hilbert transform commutators posed by Richard Rochberg.

Résumé. — Nous présentons plusieurs descriptions anciennes et nouvelles des opérateurs de Toeplitz de classe de Schatten sur l’espace de Paley-Wiener et ré- pondons à une question de Richard Rochberg sur les commutateurs discrets de la transformée de Hilbert.

1. Introduction

Given a bounded function ϕ on the real line, R, consider the Toeplitz operatorTϕon the classical Paley–Wiener space PWa,

(1.1) Tϕ:f 7→Pa(ϕf), f ∈PWa.

The space PW_a could be regarded as the subspace in L²(R) of functions with Fourier spectrum in the interval [−a, a], symbolPa above denotes the orthogonal projection inL²(R) to PWa. Basic theory of Toeplitz operators on PWa can be found in paper [9] by R. Rochberg.

We are interested in description of Schatten class Toeplitz operators on PW_a in terms of their standard symbols. By the standard symbol of an operator in (1.1) we mean the entire functionϕst =F⁻¹χ2aFϕ, whereF denotes the Fourier transform on the Schwartz space of tempered distri- butions, andχ2a is the indicator function of the interval (−2a,2a). As we

Keywords:Paley–Wiener space, Schatten ideal, discrete Besov space, discrete Hilbert transform commutator.

2010Mathematics Subject Classification:47B35, 46E39.

(*) The work is supported by RFBR grant mol_a_dk 16-31-60053, by Grant MD- 5758.2015.1, and by “Native towns”, a social investment program of PJSC “Gazprom Neft”.

will see, a Toeplitz operatorT_ϕon PWa belongs to the Schatten classS^p, 0< p <∞, if and only if e^2iaxϕst belongs to a discrete oscillation Besov space introduced in 1987 by R. Rochberg [9]. Its definition we now recall.

For a measureµonRand a functionf ∈L¹_loc(µ), the oscillation of order nof f on an intervalI⊂Rwith respect toµis defined by

osc(f, I, µ, n) = inf

Pn

1 µ(I)

Z

I

|f(x)−P_n(x)|dµ(x),

where the infimum is taken over all polynomialsP_n of degree at mostn. If µ(I) = 0, we put oscI(f, I, µ, n) = 0. Define the familyIaof closed intervals

Ia,j,k= 2π

a k2^j,2π

a (k+ 1)2^j

, j, k∈Z, j>0.

Note that endpoints of intervals inIabelong to the latticeZa=_2π

ak, k∈ Z . Letpbe a positive real number, and let [¹_p] be the integer part of¹_p. The discrete oscillation Besov spaceBp(a,osc) =B^1/pp,p(Za, µ_a,osc) is defined by

Bp(a,osc) = (

f ∈L¹_loc(µ_a) :kfk^p

Bp(a,osc)=X

I∈Ia

osc

f, I, µ_a, 1

p p

<∞ )

,

whereµa= ^2π_a P

x∈Zaδx is the normalized counting measure onZa. Our main result is the following theorem.

Theorem 1.1. — Leta,pbe positive real numbers, letϕbe a bounded function on R, and let ϕ_st be the standard symbol of the Toeplitz oper- atorTϕ on PWa. Then we have Tϕ ∈ S^p(PWa) if and only ife^2iaxϕst ∈ B^p(4a,osc). Moreover, kTϕk_Sp is comparable to ke^2iaxϕstk_Bp(4a,osc) with constants depending only onp.

Theorem 1.1 complements a classical description of Toeplitz operators in S^p(PWa) given by R. Rochberg [9] for 1 6 p < ∞ and extended by V. Peller [5] to the whole range 0 < p < ∞. To formulate the result, consider a system{νj}j6−1 of infinitely smooth functions on Rsuch that suppνj ⊂[2^j−1,2^j],

06νj61, νj−1(x) =νj(x/2), X

νj= 1 on

0,1 3

.

Define ν_j(x) = ν_−j(1−x) for real x > ¹₂ and integer j > 1, put ν₀ = 1−P

j6=0νjforj= 0. Finally, letνa,j(x) =νj((x+a)/2a) for allx∈[−a, a]

andj∈Z. Observe that system{ν_a,j}provides a resolution of unity on the interval [−a, a] by functions supported on subintervalsIjwhose lengths are

comparable to the distance fromI_j to the endpoints of [−a, a]. Rochberg–

Peller theorem says that Tϕ is in S^p(PWa) for 0 < p < ∞ if and only if

aX

j∈Z

2^−|j|· kF⁻¹(ν_2a,j· Fϕ)k^p_Lp(R)<∞,

with control of the norms. R. Rochberg gives yet another characterization of Toeplitz operators in classS^p(PWa), 16p <∞, in terms of a reproducing kernel decomposition of their standard symbols, see Theorem 5.3 in [9].

Both the statement and the proof of his result forp= 1 contain errors that we correct in Section 3.

As a consequence of Theorem 1.1, we obtain the following result.

Theorem 1.2. — Leta >0. The discrete Hilbert transform commuta- tor

Cψ:f 7→ 1 π−

Z

Za

ψ(x)−ψ(t)

x−t f(t) dµa(t), f ∈L²(µa),

belongs to the trace classS¹(L²(µ_a))if and only ifψ∈B1(a,osc)∩L^∞(Za).

This answers the question posed by R. Rochberg in 1987. See Section 6 for a summary of results on discrete Hilbert transform commutators and an analogue of Theorem 1.2 for the case 0< p <1.

We would like to mention papers [11, 12] by R. Torres for readers in- terested in wavelet characterizations and interpolation theory of discrete Besov spaces. The problem of membership in Schatten classesS^p for gen- eral truncated Toeplitz operators has been recently studied by P. Lopatto and R. Rochberg [3], see also Section 4.3 in author’s paper [1].

2. Proof of Theorem 1.1 for 1< p <∞

Theorem 1.1 for 1 < p < ∞ follows from known results. Let B^p(R) = B˙^1/pp,p(R) be the standard homogeneous Besov space on the real lineR, see, e.g., Chapter 3 in [4] for definition and basic properties. Given a Toeplitz operatorT_ϕon PW_a with symbolϕ∈L^∞(R), we denote

ϕ⁻_st=F⁻¹χ_(−2a,0)Fϕ, ϕ⁺_st=F⁻¹χ_[0,2a)Fϕ,

whereχS is the indicator function of a setS. As usual, F stands for the Fourier transform on the Schwartz space of tempered distributions. The following result is a combination of Theorem 5.1 and its Corollary in [9].

Theorem (R. Rochberg). — Let 1 < p < ∞ and let a > 0. Then a Toeplitz operatorTϕonPWa belongs toSp(PWa)if and only if

ke^2iaxϕ⁻_stk_Bp(R)+ke^−2iaxϕ⁺_stk_Bp(R)<∞,

in which casekTϕk_S^pis comparable toke^2iaxϕ⁻_stk_Bp(R)+ke^−2iaxϕ⁺_stk_Bp(R) with constants depending only onp.

Denote byEathe set of tempered distributions whose Fourier transforms are supported on the interval [−a, a]. Next result is Theorem 1 in [12].

Theorem (R. Torres). — Let 1 < p < ∞ and let f be a function in E_a∩Bp(R)for somea >0. Then its restriction toZ2a belongs toBp(2a,osc) andkfk_Bp(2a,osc)is comparable tokfk_Bp(R)with constants depending only onp. Moreover, every sequence in Bp(a,osc) is the restriction toZa of a unique function (modulo polynomials) inEa∩Bp(R).

Proof of Theorem 1.1(1< p <∞). — Letϕbe a bounded function of Rand let ϕst =F⁻¹χ_(−2a,2a)Fϕbe the standard symbol of the Toeplitz operator Tϕ ∈ S^p(PWa). Then functions e^2iaxϕ⁻_st, e^−2iaxϕ⁺_st belong to E2a∩Bp(R) by R. Rochberg’s theorem above. From theorem by R. Torres we see that e^2iaxϕ⁻_st∈Bp(4a,osc) and e^−2iaxϕ⁺_st∈Bp(4a,osc) with control of the norms. Now observe that e^4iax= 1 and e^2iaxϕst= e^2iaxϕ⁻_st+ e^−2iaxϕ⁺_st onZ4a, hence e^2iaxϕ_st∈Bp(4a,osc).

Conversely, assume that the restriction of e^2iaxϕsttoZ4ais inBp(4a,osc).

Using theorem by R. Torres, find a functionf ∈ E_2a∩Bp(R) such that its restriction to Z4a agrees with e^2iaxϕst. Put f⁻ = F⁻¹χ_(−2a,0)Ff and f⁺ = F⁻¹χ_[0,2a)Ff. Observe that ˜ϕ = e^−2iaxf⁺ + e^2iaxf⁻ is an en- tire function of exponential type at most 2a coinciding with ϕ_st onZ4a. Since ϕst, ˜ϕ are the first order distributions supported on the finite in- terval [−2a,2a], we have |ϕ(x)|˜ +|ϕ(x)| 6 c+c|x| for all x ∈ R and a constant c > 0. It follows that the entire function ^ϕ−ϕ˜_z of exponen- tial type at most 2a is bounded on R and vanishes on Z4a\ {0}, hence

˜

ϕ−ϕst=psin(2az) for a polynomialpof degree at most 1. Therefore, we have Tϕ =Tϕ_st =Tϕ˜ on PWa, see Section 2.D in [9]. Since f^± ∈Bp(R), we can use R. Rochberg’s theorem and conclude that T_ϕ˜ ∈ S^p(PWa) with control of the norms: kTϕ˜k_S^p is controllable by ke^2iaxϕ˜⁻k_Bp(R)+ ke^−2iaxϕ˜⁺k_Bp(R)6c_pkfk_Bp(R)6c˜_pke^2iaxϕ_stk_Bp(4a,osc).

3. Reproducing kernel decomposition of standard symbols In this section we show that the standard symbol of a Toeplitz opera- tor on PWa from class S^p could be represented as a linear combination

of normalized reproducing kernels of PW2a with coefficients c_k such that P|ck|^p<∞. We consider only the case 0< p61. Proposition 3.1 below is a corrected version of Theorem 5.3 in [9]. In the original statement the au- thor of [9] forgot to normalize the exponentials in formula (5.6) of [9]. More importantly, he used the fact that the Fourier multiplierf 7→ F⁻¹χ_[0,1]Ff is bounded onBp(R). This is not the case forp= 1. Here is a more accurate implementation of the ideas from [9].

Let ψbe a bounded function on the real line R. Consider the standard Hardy spaceH² in the upper half-planeC⁺={λ∈C: Imλ >0} of the complex planeC. Denote by H₋² the anti-analytic subspace{f ∈L²(R) : f¯∈H²}ofL²(R). Recall that the classical Hankel operatorHψ:H²→H₋² is defined by

H_ψ:f 7→P₋(ψf), f ∈H²,

where P− denotes the orthogonal projection from L²(R) to H₋². The op- erator Hψ is completely determined by its standard anti-analytic symbol ψ_st=F⁻¹χ_(−∞,0)Fψ. The latter means thatH_ψf =H_ψstf for allf ∈H² such that sup_x∈R|xf(x)| < ∞. Take a positive number ε > 0 and define the setsU_ε⁺,U_ε⁻ by

U_ε^±={λ∈C: λ= (1 +ε)^m(εx±i); x, m∈Z}.

Forλ∈C⁺, letkλ=−_2πi¹ _z−¹_λ¯ denote the reproducing kernel ofH² atλ.

Theorem (R. Rochberg [8]). — There exists a numberε >0such that Hψ ∈ S^p(H²) if and only if ψst = P

λ∈Uε⁺cλ k_λ

kkλk², where P

|cλ|^p is fi- nite and the infimum of P|cλ|^p over all possible representations of ψst

in this form is comparable to kHψk^p_Sp with constants depending only on p∈(0,∞).

Remark that for p∈(0,1] the series definingψ_st in the theorem above converges absolutely to a bounded function on R, while for p > 1 the convergence holds only in the Besov spaceBp(R) (one need to extract con- stant terms from every summand to get the convergent series, see discussion in [8]). In order to prove an analogous result for Toeplitz operators on the Paley–Wiener space, let us consider the sets

U_ηa,ε^± =

λ∈ U_ε^±:|Imλ|> ε ηa

, Ληa,ε=U_ηa,ε⁻ ∪Zηa∪ U_ηa,ε⁺ . HereZ^ηa ={^2π_ηak, k ∈Z}. Next, for a >0 andλ∈C, denote byρa,λ the reproducing kernel of the space PWa at the pointλ. Recall that

ρa,λ:z7→ 1 π

sina(z−¯λ)

z−λ¯ , z∈C.

We are going to prove the following proposition.

Proposition 3.1. — Leta >0and letϕ∈L^∞(R). There existε >0, η >1 such that Tϕ ∈ S^p(PWa) if and only if ϕst = P

λ∈Ληa,εcλ ρ_2a,λ kρ_a,λk², where P

λ|cλ|^p is finite and the infimum of P

|cλ|^p over all possible rep- resentations of ϕst in this form is comparable to kTϕk^p_Sp with constants depending only onp∈(0,1].

We will show how to reduce Proposition 3.1 to the above theorem for Hankel operators using a splitting of the standard symbol into three pieces:

analytic, anti-analytic and a piece with “small” Fourier support.

The following two results for 0 < p 61 are consequences of Lemma 1 and Lemma 2 from [5]. The range 16p <∞has been treated earlier in [9], see also Section 2 in [10].

Lemma 3.2. — Let a > 0 and let ϕ ∈ L^∞(R). There exist bounded functionsϕ`, ϕc, and ϕr such thatTϕ=Tϕ<sub>`</sub>+Tϕ_c+Tϕ_r onPWa,

suppFϕ_`⊂[−4a,−a

2], suppFϕc⊂[−a, a], suppFϕ_r⊂[a 2,4a], and we havekTϕ_skS^p6cpkTϕkS^p for every s=`, c, r forTϕ∈ S^p(PWa).

Herecp is a constant depending only onp.

Lemma 3.3. — Let a > 0 and let ϕ∈ L^∞(R) be such thatsupp ˆϕ ⊂ [−a, a]. Then T_ϕ ∈ S^p(PW_a) if and only if ϕ ∈ L^p(R), in which case kϕk_Lp(R)is comparable tokTϕk_S^p with constants depending only onp.

Proof of Proposition 3.1. — Let ϕ ∈ L^∞(R) and let ϕst = F⁻¹χ_(−2a,2a)Fϕbe the standard symbol of the operatorT_ϕon PW_a. Then Tϕ=Tϕ_st, see Section 2.D in [9]. Suppose thatϕst=P

λ∈Ληa,εcλ ρ_2a,λ kρ_a,λk² for someε >0,η >0, and some coefficientscλ such thatP

λ∈Ληa,ε|cλ|^p<∞.

It follows from the estimate

|ρ2a,λ(z)|

kρ_a,λk² 6ce^2a|Imz|, z∈C, λ∈C,

that this series converges absolutely to an entire function of exponential type at most 2a bounded on the real line R. By triangle inequality (see, e.g., Theorem A1.1 in [6]), we have

kTϕk^p_Sp=kTϕ_stk^p_Sp6 X

λ∈Ληa,ε

|cλ|^p sup

λ∈C

kTϕ_λk^p_Sp, where we denotedϕ_λ=_kρ^ρ2a,λ

a,λk². Takeλ∈C. For everyf, g∈PW_awe have (T_ρ2a,λf, g) = (f¯g, ρ_2a,¯λ) =f(¯λ)·g(λ) = (f, ρ_a,¯λ)(ρ_a,λ, g).

It follows that the operatorT_ϕλ has rank one andkTϕλkS^p= 1. HenceT_ϕ belongs toS^p(PWa) andkTϕk^p_Sp 6P

λ|cλ|^p.

Now let ϕ be a bounded function on R such that T_ϕ ∈ S^p(PW_a). We want to show that the standard symbolϕst=F⁻¹χ_(−2a,2a)Fϕof Tϕ can be represented in the form

ϕ_st= X

λ∈Ληa,ε

c_λ ρ_2a,λ kρa,λk²

for some positive numbersε, η depending only onpand a sequence{cλ} such thatP

λ|cλ|^p is comparable tokTϕk^p_Sp. By Lemma 3.2, it suffices to consider separately the following three cases:

(1) supp ˆϕ⊂(−∞,0];

(2) supp ˆϕ⊂[−a, a];

(3) supp ˆϕ⊂[0,+∞).

Let us treat the third case first. Denote by M_e−iax the operator of multi- plication by e^−iax onL²(R). Since supp ˆϕ⊂[0,+∞), we have

H_e−2iaxϕ=M_e−iaxT_ϕP_aM_e−iax,

where H_e−2iaxϕ : H² → H₋² is the Hankel operator with symbol ψ = e^−2iaxϕ. In particular, we havekHψkS^p6kTϕkSp. By Rochberg’s Theorem above, the anti-analytic functionψst =F⁻¹χ_(−∞,0)Fe^−2iaxϕadmits the following representation:

ψst= X

λ∈U_ε⁺

cλ

k_λ kkλk², whereP

λ∈Uε⁺|cλ|^p is comparable to kHψk^p_Sp, andε > 0 does not depend onψ. This gives us decomposition forϕ_st:

ϕ_st= e^2iaxψ_st= X

λ∈U_ε⁺

c_λe^2iaxk_λ kkλk² = X

λ∈U_ε⁺

c_λP_2a(e^2iaxk_λ) kkλk² ,

whereP_2a denotes the orthogonal projection inL²(R) to PW_2a. It is easy to see that P2a(e^2iaxkλ) = e^2iaλ¯ρ_2a,λ¯ and kρ_a,¯λk² 6 2 e^2aImλ·kkλk²_L2(R), hence

ϕ_st= X

λ∈Uε⁻

cλ¯β_λ ρ_2a,λ kρa,λk²

for some complex numbersβ_λ such that sup_λ|β_λ| 62. Next, in the case where suppϕ⊂ (−∞,0] we can consider the adjoint operator T_ϕ^∗ =Tϕ^∗_st

with the standard symbol ϕ^∗_st : z 7→ ϕ_st(¯z) and conclude that in this situation

ϕst= X

λ∈Uε⁺

cλβλ¯

ρ2a,λ

kρ_a,λk².

Now let suppϕ ⊂ [−a, a]. By Lemma 3.3, we have ϕ ∈ L^p(R). In par- ticular,ϕ∈ PW2a and Plancherel–Polya theorem [7] yields the following decomposition:

ϕ=ϕst= π 2a

X

λ∈Z2a

f(λ)ρ2a,λ, X

λ∈Z2a

|f(λ)|^p6cpa^pkϕk^p_Lp(R), where the constantcp depends only on p. Put Λε =U_ε⁺∪Z2a∪ U_ε⁻. To summarize, we have proved that for every bounded functionϕon Rsuch thatT_ϕ∈ S^p(PWa) there are coefficientsc_λ,λ∈Λε, such that

(3.1) ϕ_st = X

λ∈Λε

c_λ ρ2a,λ

kρa,λk², X

λ∈Λε

|c_λ|^p6c_pkT_ϕk^p_Sp.

It remains to show that the set Λεand coefficientscλin this decomposition could be replaced by the set Ληa,ε and some new coefficientsc_λ satisfying the second estimate in (3.1). To this end, for every pointλ ∈ Λε denote byζ_λ the nearest point toλin Λ_ηa,ε ⊂Λ_ε, whereη = 2^k andk ∈Z is a positive integer number that will be specified later. Consider the function

˜

ϕ⁽¹⁾= X

λ∈Λε

c_λ ρ2a,ζ_λ

kρa,ζλk² = X

λ∈Ληa,ε

˜

c⁽¹⁾_λ ρ2a,λ

kρa,λk², c˜⁽¹⁾_λ = X

ν∈Λε, ζ_ν=λ

c_ν.

Note that ˜ϕ⁽¹⁾ has the required representation and P

|˜c⁽¹⁾_λ |^p 6 P

|cλ|^p. Moreover, we have kTϕ−T_ϕ˜(1)k^p_Sp 6 P

λ∈Λε\Ληa,ε|cλ|^p· kTϕ_λ −Tϕ_ζλk^p_Sp. On the other hand, the quasi-norm inS_p of the rank two operator

Tϕ_λ−Tϕ_ζλ = ρa,λ

kρ_a,λk⊗ ρa,λ

kρ_a,λ¯k − ρa,ζ_λ

kρ_a,ζλk ⊗ ρa,ζ_λ

kρ_a,ζλk

does not exceed 2^1p

ρ_a,ζλ

kρa,ζ_λk− ρ_a,λ kρa,λk

L²(R)

62^1p+12

1− Reρ_a,ζλ(λ) kρa,ζ_λk · kρa,λk

¹₂ .

Since|ζλ−λ|6^2πηa for allλby construction, one can choose a large number η= 2^k so thatkTϕ−Tϕ_˜k^p_Sp 6 ¹2kTϕk^p_Sp. Clearly, this choice ofη does not depend onϕanda. Iterating the process, we see that there are functions

˜

ϕ⁽ⁿ⁾= X

λ∈Ληa,ε

˜

c⁽ⁿ⁾_λ ρ2a,λ

kρ_a,λk², n= 1,2, . . .

such thatkTϕ−T_ϕ˜(1)−· · ·−T_ϕ˜(n)k^p_Sp62¹ⁿkTϕk^p_Sp,P

n,λ|˜c⁽ⁿ⁾_λ |^p6c^p_pkTϕk^p_Sp. SinceS^p(PWa) is a complete quasi-normed space and a Toeplitz operator on PWa is zero if and only if its standard symbol is zero (see Section 2.D in [9]), this gives us the required decomposition of ϕ_st with coefficients c_λ=P

n>1c˜⁽ⁿ⁾_λ , λ∈Λ_ηa,ε.

4. Interpolation of discrete Besov sequences

Denote by PW_[0,a] the Paley–Wiener space of functions inL²(R) with Fourier spectrum in the interval [0, a]. Recall that the reproducing kernel ka,λ of the space PW_[0,a] at a pointλ∈C+has the form

ka,λ(z) =− 1 2πi

1−e^ia(z−¯λ)

z−λ¯ , z∈C.

Denote byC0(Za) the set of functions onZa tending to zero at infinity. Our aim in this section is to prove the following proposition.

Proposition 4.1. — Let 0 < p 6 1, let Λ be the set Ληa,ε from Proposition 3.1, and letF =P

λ∈Λc_λkk^k_a^a,λ

2,λk² for some c_λ ∈ C such that P

λ∈Λ|cλ|^p <∞. Then the restriction ofF to Za belongs toBp(a,osc)∩ C0(Z^a). Conversely, for every functionf ∈B^p(a,osc)there exists the unique functionF as above and a polynomial q of degree at most[_p¹] such that f = q+F on Za. Moreover, the infinum of P

λ∈Λ|cλ|^p over all possi- ble representations ofF =P

λ∈Λcλ ka,λ

kka

2,λk² in this form is comparable to kfk^p

Bp(osc,a) with constants depending only onp.

The proof of Proposition 4.1 is based on the following lemma.

Lemma 4.2. — We have kka,λk_Bp(a,osc) 6 cpkk^a

2,λk² for every a > 0, 0< p61, andλ∈C, where the constantc_p depends only onp.

Proof. — At first, consider the pointsλin the support ofµa. Forλ∈Za

we have

ka,λ(x) =

(kk_a,λk², x=λ;

0, x∈suppµa\ {λ}.

TakingP_I = 0 for intervalsI∈ Ia in the definition of osc k_a,λ, I, µ_a,₁

p

, we obtain the estimate

kk_a,λk^p

Bp(a,osc)6 X

I∈Ia

1 µa(I)

Z

I

|k_a,λ(x)|dµ_a(x) ^p

=kka,λk^2pµa({λ})^p X

I∈Ia

χI(λ) µa(I)^p 6cpkk^a

2,λk^2p.

Now letλbe an arbitrary point inC\suppµa. Thenka,λ(x) =−_2πi^{1 1−e}_x−^−ia_λ¯^λ¯ for allx∈suppµa. Thus, we need to estimate an oscillation of the function x7→ _x−¹_λ¯ on the lattice Za. Divide collection Ia from Section 1 into two parts:

Ia,1={I∈ Ia: I=Ia,j,k, Reλ /∈I_a,j,k−1∪Ia,j,k∪Ia,j,k+1}, Ia,2=Ia\ Ia,1.

For an intervalI∈ I_a,1with center x_c, define the polynomialP_I of degree [¹_p] by

(4.1) 1

x−λ¯ −P_I(x) = (x−x_c)^[p1]+1 (x−λ)(¯¯ λ−xc)^[1p]+1

.

Using this polynomial, we can estimate (4.2)

osc 1

x−λ¯, I, µa, 1

p

6sup

x∈I

(x−xc)^[1p]+1 (x−λ)(¯¯ λ−xc)^[1p]+1

6 |I|^[1p]+1 dist(λ, I)^[1p]+2

,

where|I|denotes the length ofI. SinceI∈ Ia,1, we have dist(λ, I)>|I|, hence

(4.3) X

I∈Ia,1

osc 1

¯λ−x, I, µa, 1

p p

6 X

I∈Ia,1

1

|I|^p 6cp·a^p.

We also will need a more accurate estimate for the left hand side of the inequality above in the case where|Imλ|is large. For everyj>0, letI_a,1^j

be the set of intervalsI_a,j,k,k∈Z, belonging to the familyIa,1. We have X

I∈I^j_a,1

|I|^[1p]+1 dist(λ, I)^[1p]+2

!^p

= X

I∈I_a,1^j





|I|^[1p]+1

|Imλ|²+ dist(Reλ, I)²([¹_p]+2)/2





p

6c_pa 2^j

^pX

m>1







1 a

2^j 2

|Imλ|²+m²







1 2[_p¹]p+p

6c_pa 2^j

^p γ^1−[

1 p]p−2p

j ,

whereγj = max 1,₂^aj|Imλ|

. Indeed, the last inequality follows from ele- mentary estimates

∞

X

m=1

m^−1−2p<∞,

Z ∞ 1

dx

(r²+x²)^s 6c_sr^1−2s, wherer >0, and the constantc_sdepends ons >1/2. Put

N_λ=

(log₂(a|Imλ|)

, ifa|Imλ|>2, 0, ifa|Imλ|<2.

Note that ˜p=−1 + [¹_p]p+pis a positive number. It follows X

I∈Ia,1

osc 1

λ¯−x, I, µa, 1

p p

6cp

∞

X

j=0

a 2^j

^p γ^1−[

1 p]p−2p j

6cpa^−p˜|Imλ|^−p−p˜

N_λ

X

j=0

2^pj˜ +cp

∞

X

j=Nλ

a^p 2^pj 6 cp

|Imλ|^p. Combining the last estimate with (4.3), we get

X

I∈Ia,1

osc 1

¯λ−x, I, µa, 1

p p

6cpmin

a^p, 1

|Imλ|^p

.

Now consider the familyIa,2=Ia,21∪ Ia,22,

I_a,21={I∈ I_a,2:|I|6|Imλ|}, I_a,22={I∈ I_a,2:|I|>|Imλ|}.

For an intervalI∈ Ia,21 we use the polynomialPI defined by (4.1). Then formula (4.2) implies

X

I∈Ia,21

osc 1

λ¯−x, I, µ_a, 1

p ^p

6 X

I∈Ia,21

|I|^[p1]+1

|Imλ|^[p1]+2

!p

6 c_p

|Imλ|^p.

Note that if|Imλ| < ^2π_a , the set Ia,21 is empty. This shows that we can write

X

I∈I_a,21

osc 1

λ¯−x, I, µa, 1

p ^p

6cpmin

a^p, 1

|Imλ|^p

.

ForI∈ Ia,22we putP_I = 0. Denote byx₀the nearest point toλin suppµ_a, and setI⁰=I\ {x∈R:|x−Reλ|< π/a}. We have

1 µ_a(I)

Z

I

1 x−¯λ

dµa(x)6 µa({x0})

µa(I)|x0−¯λ| + 1 µ_a(I)

Z

I⁰

dx

|x−¯λ|

6 c

a|I||x0−λ|¯ + c

|I|

Z |I|

πa⁻¹

dx px²+|Imλ|²

6 c

a|I||x0−λ|¯ + c

|I|min

loga|I|

π ,log⁺ |I|

|Imλ|

.

Using estimatesP

I∈Ia,2

1

|I|^p 6c_pa^p, P

I∈Ia,2

_log

a|I|

|I|

p

6c_pa^p, and X

I∈Ia,22

1

|I|log |I|

|Imλ|

^p

6 cp

|Imλ|^p, we see that

X

I∈Ia,22

osc cp

λ¯−x, I, µa, 1

p p

6 cp

|x0−¯λ|^p +cpmin

a^p, 1

|Imλ|^p

.

Eventually, we obtain

1 x−λ¯

p

Bp(a,osc)

6 cp

|x0−λ|¯^p +cpmin

a^p, 1

|Imλ|^p

.

It follows that kka,λk^p

Bp(a,osc)6cp(1 + e^−aImλ)^pmin

a^p, 1

|Imλ|^p

+cp

1−e^−iaλ¯ x0−λ

p

6cpkk^a

2,λk^2p,

which is the desired estimate.

LetC0(R) denote the set of all continuous functions onRtending to zero at infinity. For completeness, we include the proof of the following known lemma.

Lemma 4.3. — Let0< p61,a >0. For every functionf ∈Bp(osc, a) there exists a functionF ∈Bp(R)such thatF =f onZa, and

kFk_Bp(R)6cpkfk_Bp(osc,a),

where the constantc_p depends onlyp.

Proof. — Fork∈ZputIk= [^2π_a [¹_p]k,^2π_a [¹_p](k+ 1)]. Interiors of intervals I_k are disjoint and every set I_k∩Za contains [¹_p] + 1 points. On everyI_k define the polynomial Pk of degree at most [¹_p] such that Pk(x) = f(x) for all x ∈ Ik ∩Za. Next, set F(x) = Pk(x) for x ∈ Ik. We claim that the function F is in Bp(R). To check this, let us take an interval Jj,k = [^2π_a [¹_p]k·2^j,^2π_a[¹_p](k+1)·2^j] withk, j∈Z. In the case wherej <0 we clearly have osc(F, Jj,k, m,[¹_p]) = 0 because the function F is a polynomial of degree at most [_p¹] onI. Hence, we can assume thatJ =Jj,k=I`∪. . .∪I`+N

for some`∈Zand N >1. Consider the polynomialP_J of degree at most [¹_p] such that

osc

f, J, µa, 1

p

= 1

µ_a(J) Z

J

|f(x)−PJ(x)|dµa(x).

We have 1

|J| Z

J

|F(x)−PJ(x)|dx= 1

|J|

N

X

s=0

Z

I`+s

|P`+s(x)−PJ(x)|dx

6 c_p

|J|

N

X

s=0

Z

I_`+s

|P`+s(x)−PJ(x)|dµa(x)6cposc

f, I, µa, 1

p

,

where we used the fact that Z

I_`

|P(x)|dx6cp

Z

I_`

|P(x)|dµa(x)

for every intervalI`,`∈Z, and every polynomialP of degree at most [_p¹].

It follows that kFk^p

Bp(R,m,osc)6c^p_pX

j,k

osc

f, Jj,k, µa, 1

p p

6c^p_pkfk^p

Bp(osc,a),

and henceF belongs to the spaceB^1/pp,p(R, dx,osc) =Bp(R), as required.

Proof of Proposition 4.1. — Consider a functionF of the form F =X

λ∈Λ

cλ

ka,λ

kk^a

2,λk², X

λ∈Λ

|cλ|^p<∞.

Since 0< p61 and|ka,λ(x)|6ckk^a

2,λk² for everyλ∈Candx∈R, the series above converges absolutely to a function fromC0(R) by the Lebesgue dominated convergence theorem. By Lemma 4.2, the restriction ofF toZa

(to be denoted byf) is inBp(a,osc) andkfk^p

Bp(a,osc)6cpP

λ∈Λ|cλ|^p for a constantcp depending only on p.