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Schatten–von Neumann norminequalities for two-waveletlocalization operators associated withβ-Stockwell transformsViorel Catană aa Department of Mathematics I , University Politehnica ofBucharest , Splaiul Independenţei 313, 060042 Bucharest ,RomaniaPublished online: 07 Apr 2011.

To cite this article: Viorel Catană (2012) Schatten–von Neumann norm inequalities for two-waveletlocalization operators associated with β-Stockwell transforms, Applicable Analysis: An InternationalJournal, 91:3, 503-515, DOI: 10.1080/00036811.2010.549478

To link to this article: http://dx.doi.org/10.1080/00036811.2010.549478

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Applicable AnalysisVol. 91, No. 3, March 2012, 503–515

Schatten–von Neumann norm inequalities for two-wavelet

localization operators associated with b-Stockwell transforms

Viorel Catana*

Department of Mathematics I, University Politehnica of Bucharest,Splaiul Independentei 313, 060042 Bucharest, Romania

Communicated by R.P. Gilbert

(Received 9 August 2010; final version received 10 December 2010)

In this article we define two-wavelet localization operators correspondingto an irreducible and square-integrable representation of a locally compactHausdorff group on a Hilbert space. The group structure admitting anirreducible and square-integrable representation which is related to�-Stockwell transform, that we shall use in this article �2R have beenintroduced in Boggiatto et al. [P. Boggiatto, C. Fernandez, and A. Galbis,A group representation related to the Stockwell transform, Indiana Univ.Math. J. 58(5) (2009), pp. 2277–2296]. The Schatten–von Neumann norminequalities of these two-wavelet localization operators are established. Thetraces and the trace class norm inequalities of the trace class two-waveletlocalization operators are given.

Keywords: square-integrable representation; two-wavelet localizationoperator; Schatten–von Neumann class; Stockwell transform

AMS Subject Classifications: Primary 47B10; 47G10; Secondary 22D10;43A80

1. Introduction

The aim of this article is to define two-wavelet localization operators correspondingto irreducible and square-integrable representation of a locally compact Hausdorffgroup G on Hilbert spaces H1��,0, �40, related to �-Stockwell transforms, �2R.These kind of square-integrable group representations have been found by Boggiattoet al. [1]. The two-wavelet localization operators which are constructed in this workturn out to be the same as the localization operators based on the Parseval’s formulafor the �-Stockwell transform which has been stated and proved in [1]. Based onthis fact, Schatten–von Neumann properties are established and the traces fortrace class two-wavelet localization operators are computed. The trace class norminequalities for the trace class two-wavelet localization operators are also given.As a consequence of these trace class norm inequalities, a compactness result fortwo-wavelet localization operators is given.

*Email: catana_viorel@yahoo.co.uk

ISSN 0003–6811 print/ISSN 1563–504X online

� 2012 Taylor & Francis

http://dx.doi.org/10.1080/00036811.2010.549478

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The Stockwell transform is first introduced in [2]. Some mathematical aspectsrelated to Stockwell transform can be found in [1,3–8]. For more results on the

Stockwell transform in the context of applications, see, for example, the papers[4,9–11] and the references listed there in.

In the sequel, we review some facts concerning �-Stockwell transform and werecall some main concepts and results of two-wavelet localization theory that areneeded for our investigation of Schatten–von Neumann properties for this class of

operators that will be introduced.We adopt the notations and naming of [1,12] to which we refer for more details.Let f :R!C be a measurable function and let �, k2R, k 6¼ 0. Then we set

��,kð f Þ ¼ jkj f ðkð� � �ÞÞ

and

Mk f ¼ e2�ik�f ð�Þ

to be the composition of translation and dilation operators and the modulationoperator, respectively.

Now let f2S0(R) be a tempered distribution and g2S(R) be a function in theSchwartz space. Then, for �2R, the �-Stockwell transform of f2S0(R) with respect

to the window g is given by

ðS�gf Þð�, kÞ ¼ h f,M�k��,kgi: ð1:1Þ

The brackets h f, gi denote the extension to S0(R)�S(R) of the inner product

h f, gi ¼

ZR

f ðtÞgðtÞdt ð1:2Þ

on L2(R).For a function f2L1(R) we define the Fourier transform by

f ð!Þ ¼ Fð f Þð!Þ ¼

ZR

e�2�it!f ðtÞdt: ð1:3Þ

Let us recall that the Fourier transform can be extended to a bicontinuous linearbijection F :S0(R)!S0(R).

Then it is easy to see that for f2L2(R) and g2L1(R)\L2(R) we have

S�gf ð�, kÞ ¼ Fð f ��,kgÞð�kÞ, ð1:4Þ

or more explicitly

ðS�gf Þð�, kÞ ¼ jkj

ZR

e�2�i�ktf ðtÞ gðkðt� �ÞÞdt ð1:5Þ

for all (�, k)2R�R n {0}.In the following we recall the setting for the action of the �-Stockwell transform,

which consists of triples (H, K, X ) of Hilbert spaces of tempered distributions suchthat for f2H and g2K it follows that S�g f 2X. To this end let us define for

�,�2R, �51,

L2jtþ�j� ¼ g : R! C;

ZR

j gðtÞj2jtþ �j�dt51� �

,

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H�,� ¼ f2S0ðRÞ;

ZR

j f ð!Þj2j!þ �j�d!51� �

¼ F�1ðL2j�þ!j� \ S

0ðRÞÞ,

X� ¼ F : R� R n f0g ! C;

ZR

ZR

jFð�, kÞj2d� dk

jkj�51

� �,

which are Hilbert spaces, respectively, when they are endowed with the inner

products

h f, giL2

jtþ�j2¼

ZR

f ðtÞ gðtÞ jtþ �j�dt,

h f, giH�,� ¼

ZR

f ð!Þ gð!Þ j!þ �j�d!,

hF,GiX� ¼

ZR

ZR

F ð�, kÞ Gð�, kÞd� dk

jkj�:

The following theorem which is a synthesis of Propositions 3.4 and 3.8 and

Corollary 3.10 in [1] and summarizes the main known facts about �-Stockwelltransform will be useful to us.

THEOREM 1.1 (i) For all �,�2R the �-Stockwell transform defined (by (1.1)) can be

extended as a continuous map

S� : H1��,0 �H��2,� ! X�, ð f, gÞ ! S�gf, ð1:6Þ

linear in the first argument and conjugate linear in the second one, such that

hS�g1f1,S�g2f2iX� ¼ hg1, g2iH��2,�h f1, f2iH1��,0

ð1:7Þ

for all g1, g2 in H��2,� and f1, f2 in H1��,0.(ii) For all �40 and g in H��1,�\H��2, � let us define

h ¼ F�1ðgð� � �Þj�j��1Þ, ð1:8Þ

where F�1 denotes the inverse Fourier transform. Then h and ��,kh are in H1��,0 for all

(�, k) in R�R n {0} and

h f,��,khiH1��,0¼ jkj1��e2�i��kðS�gf Þð�, kÞ a:e: ð1:9Þ

for all f in the Hilbert space H1��,0.(iii) Let �40 and let g and h be as in (ii). Then every f in H1��,0 can be recovered

from its �-Stockwell transform as

f ¼1

k gk2H��2,�

ZR2

ðS�gf Þð�, kÞhðkð� � �ÞÞe2�i��kd� dk, ð1:10Þ

where the integral is understood in a weak sense.

Remark 1.1 The statement in (iii) in the previous theorem is a reformulation of that

in (i) in the case �40.

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Remark 1.2 From (i), in the preceding theorem we get

kS�gf kX� ¼ kgkH��2�k f kH1��,0

for all f2H1��,0, g2H��2�. So, Parseval’s formula for �-Stockwell transform will

hold such that S�g is up to a constant, an isometry.

Now we recall some known facts concerning two-wavelet localization operator

theory (see, for instance, [12–14]). Let G be a locally compact and Hausdorff group

on which the left Haar measure is denoted by �. Let X be a separable and complex

Hilbert space with infinite dimension. We denote the inner product and the norm on

X by ( , ) and k k, respectively.Let B(X ) be the C�-algebra of all bounded linear operators on X and let k k�

denote the norm on B(X ). Let U(X ) be the group of unitary operators on X with

respect to the usual composition of mappings. A unitary representation

� :G!U(X ) of G on X is said to be square-integrable if there exists a nonzero

element ’ in X such that ZG

jð’,�ð gÞ’Þj2d�ð gÞ51: ð1:11Þ

We call any element ’ in X for which k’k¼ 1 and (1.11) hold an admissible

wavelet for the square-integrable representation � :G!U(X ) and we define the

constant c’ by

c’ ¼

ZG

jð’,�ð gÞ’Þj2d�ð gÞ: ð1:12Þ

We call c’ the wavelet constant associated to the admissible wavelet ’. A unitary

representation � :G!U(X ) of G on X is said to be irreducible if it has only the

trivial invariant subspaces (i.e. if M�X is a closed subspace and �(g)M�M for all g

in G, then M¼ {0} or M¼X ).It can be proved that if � :G!U(X ) is an irreducible and square-integrable

representation of G on X and if ’ is an admissible wavelet for � :G!U(X ), then

ðx, yÞ ¼1

c’

ZG

ðx,�ð gÞ’Þð�ð gÞ’, yÞd�ð gÞ ð1:13Þ

for all x and y in X. The formula (1.13) is known as the resolution of the identity

formula.Now let ’ and be two admissible wavelet for the irreducible and square-

integrable representation � :G!U(X ) of G on X, such that the two-wavelet

constant c’, defined by

c’, ¼

ZG

ð’,�ð gÞ’Þð�ð gÞ , ’Þd�ð gÞ ð1:14Þ

is nonzero. Then, we get

ðx, yÞ ¼1

c’,

ZG

ðx,�ð gÞ’Þð�ð gÞ , ’Þd�ð gÞ ð1:15Þ

for all x and y in X.

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We referred to (1.15) as the resolution of the identity formula for the irreducible

and square-integrable representation � :G!U(X ) of G on X, corresponding to the

admissible wavelets ’ and .Let F2L1(G)[L1(G) and let ’ and be admissible wavelets for the square-

integrable representation � :G!U(X ) of G on X such that c’, 6¼ 0. Then we define

the two-wavelet localization operator LF,’, :X!X by

ðLF,’, x, yÞ ¼1

c’,

ZG

Fð gÞðx,�ð gÞ’Þð�ð gÞ , yÞd�ð gÞ ð1:16Þ

for all x and y in X (see [13,14]).In the sequel we briefly review some facts concerning Schatten–von Neumann

classes (for more details, see, e.g. [13]).Let A :X!X be a bounded linear and compact operator and let jAj :X!X,

jAj ¼ (A�A)1/2, where A� is the adjoint of A. Then jAj :X!X is a bounded linear

operator, positive and compact. Let {’k : k¼ 1, 2, . . .} be an orthonormal basis for

X consisting of eigenvectors of jAj :X!X and let sk(A) be the eigenvalue of

jAj :X!X corresponding to the eigenvector ’k, k¼ 1, 2, . . . . We call sk(A),

k¼ 1, 2, . . ., the singular values of A :X!X.A compact operator A :X!X is said to be in the Schatten–von Neumann class

Sp, 1� p51 if X1k¼1

ðskðAÞÞp 51:

It is well-known that Sp, 1� p51 is a complex Banach space in which the norm

k�kSpis given by

kAkSp¼

X1k¼1

ðskðAÞÞp

( )1=p

, A2Sp:

We let S1 be the C�-algebra B(X ) of all bounded linear operators on X. Thus,

k kS1 ¼ k k�, where k k� denotes the norm in B(X ).We usually call S1 the trace class and S2 the Hilbert-Schmidt class.Now we come to the main result on the Schatten–von Neumann property of two-

wavelet localization operators which is Theorem 2.4 in [12].

THEOREM 1.2 Let F2Lp(G), 1� p�1. Then the two-wavelet localization operator

LF,’, :X!X is in Sp and

kLF,’, kSp�

1

jc’, jðc’c Þ

1=2p0kFkLp , ð1:17Þ

where p0 is the conjugate index of p (i.e. 1pþ

1p0 ¼ 1).

The following trace formula for two-wavelet localization operator is given in

Theorem 3.1 of [13].

THEOREM 1.3 Let F2L1(G) and let ’, be admissible wavelets for the square-

integrable representation � :G!U(X ) of G on X such that c’, 6¼ 0. Then the trace

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tr(LF,’, ) of the trace class localization operator LF,’, :X!X is given by

trðLF,’, Þ ¼ð , ’Þ

c’,

ZG

Fð gÞd�ð gÞ: ð1:18Þ

Now we can give a lower bound for the norm kLF,’, kS1of the trace class

two-wavelet localization operator LF,’, :X!X, with F2L1(G), by means of thefunction F’, :G!C, defined by

F’, ð gÞ ¼ ðLF,’, �ð gÞ’,�ð gÞ Þ ð1:19Þ

for all g in G. It can be proved that if F2L1(G), then F’, 2L1(G).

THEOREM 1.4 Let F2L1(G) and let ’ and be admissible wavelets for the square-integrable representation � :G!U(X ) of G on X such that c’, 6¼ 0. Then

2

c’ þ c kF’, kL1ðGÞ

� kLF,’, kS1�

1

jc’, jkFkL1ðGÞ: ð1:20Þ

The proof of the preceding theorem is similar to the proof of left-hand side inequalityin Theorem 3.3 of [12], so we omit that.

2. Unitary and square-integrable representations

Our aim is to give a group representation related to the �-Stockwell transformfollowing the point of view in [1]. To this end, let G be the set R�R n {0}� [0, 1]which becomes a group with respect to the multiplication � given by

ð�1, k1, a1Þ � ð�2, k2, a2Þ ¼ �1 þ�2k1

, k1k2, a1 þ a2 þ �1k1ð1� k2Þ

� �ð2:1Þ

for all (�1, k1, a1) and (�2, k2, a2) in G.Let us note that (0, 1, 0) is the identity element in G, and the inverse of an element

is given by

ð�, k, aÞ�1 ¼ ð��k, 1=k, �aþ �ð1� kÞÞ: ð2:2Þ

The group G is in fact a Lie group on which the Lebesgue measure is the left Haarinvariant measure. Unfortunately the right Haar invariant measure of the group G

differs from the Lebesgue measure and so the group G is not an unimodular group.Let U(H1��,0), �40 be the group all unitary operators on Hilbert space H1��,0.

Then we define the unitary representations ��,� :G!U(H1��,0) by

��,�ð�, k, aÞ f ¼ jkj�=2�1e2�iðaþ�kÞ���,k f ð2:3Þ

for all (�, k, a) in G all f in H1��,0 and �2R. Indeed, ��,�(�, k, a) :H1��,0!H1��,0

defined by (2.3) are unitary operators for all (�, k, a) in G becausek��,�ð�, k, aÞ f kH1��,0

¼ k f kH1��,0for all f in H1��,0 and ��,�(�, k, a)(H1��,0)¼H1��,0.

PROPOSITION 2.1 The unitary representation ��,� :G!U(H1��,0) is irreducible.

Proof We shall show that every vector subspace M�H1��,0, M 6¼ {0}, which isinvariant under the maps {��,�(�, k, a)}, (�, k, a)2G is necessarily dense in H1��,0.We follow the same way as in the proof of Proposition 4.11 in [1]. To this end,

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let g2M, g 6¼ 0. We prove that the linear span of {��,�(�, k, a)g}, (�, k, a)2G is a

dense subspace of H1��,0. Let us assume that f2H1��,0 is such that

ð f, ��,�ð�, k, aÞ gÞH1��,0¼ 0 ð2:4Þ

for every (�, k, a) in G. Hence, we get

ð f,��,kgÞH1��,0¼ 0 ð2:5Þ

for every (�, k)2R�R n {0}� [0, 1]. By the definition of H1��,0 there are functions

’, 2L2(R) such that

f ðuÞ ¼ jujð��1Þ=2’ðuÞ, gðuÞ ¼ jujð��1Þ=2 ðuÞ:

Now, we can write

d��,kgðuÞ ¼ e�2�i�ugu

k

� �¼ e�2�i�u

juj

jkj

� �ð��1Þ=2

u

k

� �¼juj

jkj

� �ð��1Þ=2 d��,k ðuÞ: ð2:6Þ

Then, by (2.5) and (2.6) we have

0 ¼ ð f,��,kgÞH1��,0¼

ZR

f ðuÞd��,kgðuÞjuj1��du¼

ZR

jujð��1Þ=2’ðuÞjuj

jkj

� �ð��1Þ=2 d��,k ðuÞjuj1��du¼ jkjð1��Þ=2

ZR

’ðuÞ d��,k ðuÞdu¼ jkjð1��Þ=2ð’,��,k ÞL2ðRÞ: ð2:7Þ

But the linear span of {��,k } for all (�, k)2R�R n {0} is invariant under

translations and dilations. So, spanf��,k g ¼ L2ðRÞ. Therefore, by (2.7) we have

’¼ 0 and hence the conclusion follows.

PROPOSITION 2.2 The unitary representation

��,� : G! UðH1��,0Þ, �4 0, �2R,

of the group G on H1��,0 is square-integrable.

Proof Let ’2H1��,�\H��2,�, �40, �2R and let ~’ ¼ F�1ð’ð� � �Þj�j��1Þ. Then by

(ii) in Theorem 1.1 it follows that ~’,��,k ~’2H1��,0, and moreover

ð f,��,k ~’ÞH1��,0¼ jkj1��e2�i�k�ðS�’ f Þð�, kÞ a.e. ð2:8Þ

for all f in H1��,0. Now, let us remark that

ð f, ��,�ð�, k, aÞ ~’ÞH1��,0¼ ð f, jkj�=2�1e2�iðaþ�kÞ���,k ~’ÞH1��,0

¼ jkj�=2�1e�2�iðaþ�kÞ�ð f,��,k ~’ÞH1��,0

¼ jkj��=2e�2�ia�ðS�’ f Þð�, kÞ a.e. ð2:9Þ

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Therefore, for every f in H1��,0 we getZG

ð f, ��,�ð�, k, aÞ ~’ÞH1��,0

��� ���2d� dkda ¼ ZG

jkj��jðS�’ f Þð�, kÞj2d� dkda: ð2:10Þ

So, if we take in (2.10) f ¼ ~’, it follows thatZG

ð ~’, ��,�ð�, k, aÞ ~’ÞH1��,0

��� ���2d� dkda ¼ ZR

ZR

jðS�’ ~’Þð�, kÞj2d� dk

jkj�51,

because S�’ : H1��,0 ! X�. Therefore, for every ’2H��1,�\H��2,� such that

k ~’kH1��,0¼ 1, it follows that ~’2H1��,0 is an admissible wavelet for the representation

��,� :G!U(H1��,0). Hence the proof is complete.

3. Two-wavelet localization operators for b-Stockwell transforms

Let ’, 2H��1,�\H��2,� be such that k ~’kH1��,0¼ 1, k ~ kH1��,0

¼ 1, where

~’ ¼ F�1ð’ð� � �Þj�j��1Þ, ~ ¼ F�1ð ð� � �Þj�j��1Þ, �4 0, �2R:

Let ~F 2LpðGÞ, 1� p�1. Then for all f in H1��,0 we define ~L ~F, ~’, ~ f by

ð ~L ~F, ~’, ~ f, gÞH1��,0¼

1

c ~’, ~

Z 1

0

ZR

ZR

~Fð�, k, aÞð f, ��,�ð�, k, aÞ ~’ÞH1��,0

� ð��,�ð�, k, aÞ ~ , gÞH1��,0d� dk da ð3:1Þ

for all g in H1��,0 where

c ~’, ~ ¼

Z 1

0

ZR

ZR

ð ~’, ��,�ð�, k, aÞ ~’ÞH1��,0ð��,�ð�, k, aÞ ~ , ~’ÞH1��,0

d� dk da: ð3:2Þ

In order to write (3.1), of course, we suppose that c ~’, ~ 6¼ 0.

LEMMA 3.1 Let ’ and be as above. Then

c ~’, ~ ¼ ð’, ÞH��2,� : ð3:3Þ

Proof Let us observe that by (2.9)

ð ~’, ��,�ð�, k, aÞ ~’ÞH1��,0¼ jkj��=2e�2�ia�ðS�’ ~’Þð�, kÞ, a.e. ð3:4Þ

and

ð��,�ð�, k, aÞ ~ , ~’ÞH1��,0¼ ð ~’, ��,�ð�, k, aÞ ~ Þ

H1��,0

¼ jkj��=2e2�ia�ðS� ~’Þð�, kÞ a.e. ð3:5Þ

So, by (3.3), (3.4) and (1.7) we haveZ 1

0

ZR

ZR

ð ~’, ��,�ð�, k, aÞ ~’ÞH1��,0ð��,�ð�, k, aÞ ~ , ~’ÞH1��,0

d� dkda

¼

ZR

ZR

ðS�’ ~’Þð�, kÞðS� ~’Þð�, kÞd� dk

jkj�¼ ð ~’, ~’ÞH1��,0

ð’, ÞH��2,�

¼ k ~’k2H1��,0ð’, ÞH��2,� ¼ ð’, ÞH��2,� :

Hence the lemma is proved.

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Now, by Theorem 1.2 it follows that for ~F 2LpðGÞ the two-wavelet localization

operators ~L ~F, ~’, ~ : H1��,0 !H1��,0 is in the Schatten–von Neumann class Sp. We use

this fact to prove the following result.

THEOREM 3.1 Let ’, 2H��1,�\H��2,� be such that k ~’kH1��,0¼ 1, k ~ kH1��,0

¼ 1

and ð’, ÞH��2,� 6¼ 0, where ~’ ¼ F�1ð’ð� � �Þj�j��1Þ and ~ ¼ F�1ð ð� � �Þj�j��1Þ. Let

F2Lp(R�R), 1� p�1. If we define L�F,’, f , for all f2H1��,0, by

ðL�F,’, f, gÞH1��,0¼

1

ð’, ÞH��2,�

ZR

ZR

Fð�, kÞðS�’ f Þð�, kÞðS� gÞð�, kÞ

d� dk

jkj�, ð3:6Þ

for all g2HH1��,0, then L�F,’, : H1��,0 !H1��,0 is in the Schatten–von Neumann class

Sp. Moreover,

kL�F,’, kSp�

1

jð’, ÞH��2,� jk’kH��2,�k kH��2,�

� �1=p0kFkLpðR2Þ ð3:7Þ

where p0 is the conjugate index of p (i.e. 1pþ

1p0 ¼ 1).

Proof Let us define the function ~F : G! C by ~Fð�, k, aÞ ¼ Fð�, kÞ, (�, k, a)2G.

Then ~F 2LpðGÞ. By (2.9), (3.1), (3.3) and (3.6), we get

ð ~L ~F, ~’, ~ f, gÞH1��,0¼

1

c ~’, ~

Z 1

0

ZR

ZR

~Fð�, k, aÞð f, ��,�ð�, k, aÞ ~’ÞH1��,0

� ð��,�ð�, k, aÞ ~ , gÞH1��,0d� dk da

¼1

c ~’, ~

ZR

ZR

Fð�, kÞðS�’ f Þð�, kÞðS� gÞð�, kÞ

d� dk

jkj�

¼1

ð’, ÞH��2,�

ZR

ZR

Fð�, kÞðS�’ f Þð�, kÞðS� gÞð�, kÞ

d� dk

jkj�

¼ ðL�F,’, f, gÞH1��,0ð3:8Þ

for all f and g in H1��,0. So, L�F,’, : H1��,0 !H1��,0 is the same as ~L ~F, ~’, ~ :

H1��,0!H1��,0. Therefore by Theorem 1.2 the operator L�F,’, : H1��,0 !H1��,0 is

in the Schatten–von Neumann class Sp. In addition, using the inequality (1.17) in

Theorem 1.2, we get

kL�F,’, kSp¼ k ~L ~F, ~’, ~ kSp

�1

jc ~’, ~ jðc ~’c ~ Þ

1=2p0k ~FkLpðGÞ: ð3:9Þ

But, the same reasoning as in Lemma 3.1 gives us that

c ~’ ¼

ZG

jð ~’, ��,�ð�, k, aÞ ~’Þj2H1��,0

d� dk da

¼ k’k2H��2,� � k ~’k2H1��,0¼ k’k2H��2,� ð3:10Þ

and

c ~ ¼

ZG

jð ~ , ��,�ð�, k, aÞ ~ Þj2H1��,0d� dk da ¼ k k2H��2,� : ð3:11Þ

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Finally, by using (3.3) and (3.9)–(3.11), we get

kL�F,’, kSp� jð’, ÞH��2,� j

�1ðk’kH��2,�k kH��2,� Þ1=p0kFkLpðR2Þ,

and the proof is complete.Now we give a formula for traces for trace class two-wavelet localization

operators associated with �-Stockwell transform in the following theorem.

THEOREM 3.2 Let ~’, ~ 2H1��,0 be admissible wavelets for the square-integrablerepresentation ��,� :G!U(H1��,0), where ~’ ¼ F�1ð’ð� � �Þj�j��1Þ, ~ ¼ F�1ð �ð� � �Þj�j��1Þ with ’, in H��1,�\H��2,� such that k ~’kH1��,0

¼ 1, k ~ kH1��,0¼ 1 and

ð’, ÞH��2,� 6¼ 0. Let F2L1(R�R). Then the two-wavelet localization operator L�F,’, :

H1��,0!H1��,0 is in S1 (i.e. it is a trace class operator) and his trace trðL�F,’, Þ isgiven by

trðL�F,’, Þ ¼ð ~ , ~’ÞH1��,0

ð’, ÞH��2,�

ZR

ZR

Fð�, kÞd� dk: ð3:12Þ

Proof By Theorem 1.3 and Lemma 3.1 and by the remark made in the proof ofTheorem 3.1, that is L�F,’, ¼

~L ~F, ~’, ~ , we get

trðL�F,’, Þ ¼ trð ~L ~F, ~’, ~ Þ ¼ð ~ , ~’ÞH1��,0

c ~’, ~

Z 1

0

ZR

ZR

~Fð�, k, aÞd� dkda

¼ð ~ , ~’ÞH1��,0

ð’, ÞH��2,�

ZR

ZR

Fð�, kÞd� dk,

and the proof is complete.

4. Trace class norm inequalities

In the following theorem the trace class norm inequalities are given for thetwo-wavelet localization operators associated with �-Stockwell transform L�F,’, :

H1��,0!H1��,0, �2R, �40.

THEOREM 4.1 Let ’, 2H��1,�\H��2,� be such that k ~’kH1��,0¼ 1, k ~ kH1��,0

¼ 1,ð’, ÞH��2,� 6¼ 0, where

~’ðtÞ ¼ F�1ð’ð� � �Þj�j��1Þ ð4:1Þ

and

~ ðtÞ ¼ F�1ð ð� � �Þj�j��1Þ: ð4:2Þ

Then for all functions F2L1(R�R), for �2R and �40, we have

2

jð’, ÞH��2,� jðk’k2H��2,�

þ k k2H��2,� ÞkF’, kL1ðR�RÞ

� kL�F,’, kS1�

1

jð’, ÞH��2,� jkFkL1ðR�RÞ, ð4:3Þ

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where

F’, ð�, kÞ ¼

ZR

ZR

Fð�0, k0Þð’,T ~�M��ð1� ~kÞD1��=2~k

’ÞH��1,�

�T ~�M��ð1� ~kÞD

1��=2~k

, H��1,�

d�0 dk0 ð4:4Þ

and ~� ¼ kð�0 � �Þ, ~k ¼ k0=k, a¼ a0 � aþ �(k0 � k) is such that ð�, k, aÞ�1�ð�0, k0, a0Þ ¼ ð ~�, ~k, ~aÞ, for every (�, k, a), (�0, k0, a0) in G. We note that the dilatation

operator Dtk is defined by

DtkhðxÞ ¼ jkj

thðkxÞ

for all x in R, k in R n {0}, t in [0,1) and all measurable functions h on R.

Proof In order to prove Theorem 4.1, we note that according to Theorem 3.1 we

only need to establish the lower bound for kL�F,’, kS1. To this end, let us recall that

L�F,’, ¼~L ~F, ~’, ~ , where

~Fð�, k, aÞ ¼ Fð�, kÞ, (�, k, a)2G. So, by Theorem 1.4 we have

2

c ~’ þ c ~

k ~F ~’, ~ kL1ðGÞ � k~L ~F, ~’, ~ kS1

¼ kL�F,’, kS1, ð4:5Þ

where

~F ~’, ~ ð�, k, aÞ ¼ ð~L ~F, ~’, ~ ��,�ð�, k, aÞ ~’, ��,�ð�, k, aÞ

~ ÞH1��,0

¼1

c ~’, ~

Z 1

0

ZR

ZR

~Fð�0, k0, a0Þð��,�ð�, k, aÞ ~’, ��,�ð�0, k0, a0Þ ~’ÞH1��,0

� ð��,�ð�0, k0, a0Þ ~ , ��,�ð�, k, aÞ ~ ÞH1��,0

d�0 dk0 da0

¼1

c ~’, ~

Z 1

0

ZR

ZR

Fð�0, k0Þð��,�ð�, k, aÞ ~’, ��,�ð�0, k0, a0Þ ~’ÞH1��,0

� ð��,�ð�, k, aÞ ~ , ��,�ð�0, k0, a0Þ ~ ÞH1��,0

d�0 dk0 da0: ð4:6Þ

Now we use the fact that ��,� :G!U(H1��,0) is a unitary representation of G on

H1��,0. Then for all (�, k, a) and (�0, k0, a0) in G, we have

ð��,�ð�, k, aÞ ~’, ��,�ð�0, k0, a0Þ ~’ÞH1��,0

¼ ð ~’, ��,�ðð�, k, aÞ�1� ð�0, k0, a0ÞÞ ~’ÞH1��,0

¼ ð ~’, ��,�ðð��k, 1=k, �aþ �ð1� kÞÞ � ð�0, k0, a0ÞÞ ~’ÞH1��,0

¼ ð ~’, ��,�ðkð�0 � �Þ, k0=k, a0 � aþ �ðk0 � kÞÞ ~’ÞH1��,0

¼ ð ~’, ��,�ð ~�, ~k, ~aÞ ~’ÞH1��,0

¼

ZR

b~’ðuÞð��,�ð ~�, ~k, ~aÞ ~’Þ^ðuÞjuj1�� du, ð4:7Þ

where we let

~� ¼ kð�0 � �Þ, ~k ¼ k0=k, ~a ¼ a0 � aþ �ðk0 � kÞ:

Let us remark that by (4.1) we haveb~’ðuÞ ¼ ’ðu� �Þjuj��1

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and

ð��,�ð ~�, ~k, ~aÞ ~’Þ^ðuÞ ¼ j ~kj��=2e2�ið ~aþ ~� ~kÞ�e�2�i ~�u’u

~k� �

� �juj��1

for all u2R. So, by using the preceding relations, a change of variable and thedefinitions of translation, modulation and dilatation operators, we getZ

R

b~’ðuÞð��,�ð ~�, ~k, ~aÞ ~’ÞÞ^ðuÞjuj1�� du

¼ j ~kj��=2e�2�ið ~aþ ~� ~kÞ�

ZR

’ðu� �Þ’u

~k� �

� �� e2�i ~�ujuj��1 du

¼ j ~kj��=2e�2�ið ~aþ ~� ~kÞ� � e2�i ~��ZR

’ð�Þ’�þ �ð1� ~kÞ

~k

!�

� e2�i ~��j�þ �j��1d�

¼ e�2�i ~a�ZR

’ð�ÞðT ~�M��ð1� ~kÞD1��=2~k

’Þ^

ð�Þj�þ �j��1d�

¼ e�2�i ~a�ð’,T ~�M��ð1� ~kÞD1��=2~k

’ÞH��1,� : ð4:8Þ

Similarly, it follows that

ð��,�ð�, k, aÞ ~ , ��,�ð�0, k0, a0Þ ~ ÞH1��,0

¼ e2�i ~a� ,T ~�M��ð1� ~kÞD

1��=2~k

H��1,�

¼ e2�i ~a�T ~�M��ð1� ~kÞD

1��=2~k

, H��1,�

: ð4:9Þ

By (4.6)–(4.9) we have

~F ~’, ~ ð�, k, aÞ ¼1

c ~’, ~

Z 1

0

ZR

ZR

Fð�0, k0Þð’,T ~�M��ð1� ~kÞD1��=2~k

’ÞH��1,�

� ðT ~�M��ð1� ~kÞD1��=2~k

, ÞH��1,�d�0 dk0 da0 ¼

1

c ~’, ~

F’, ð�, kÞ, ð4:10Þ

where F’, is given by (4.4).

Thus, using (3.3), (3.10), (3.11), (4.5) and (4.10), the proof is complete.Finally, we give a compactness result for the two-wavelet localization operator

L�F,’, : H1��,0!H1��,0. To this end, let us recall the following result which isProposition 2.3 of [12].

THEOREM 4.2 Let ~F 2LpðGÞ, 1� p51, and let ~’, ~ 2X be two admissible waveletsfor the unitary representation � :G!U(X ) of the group G on the Hilbert space X.Then the two-wavelet localization operator ~L ~F, ~’, ~ : X! X is compact.

THEOREM 4.3 Let ’, 2H��1,�\H��2,� be admissible wavelets, such thatk ~’kH1��,0

¼ 1, k ~ kH1��,0¼ 1, where ~’ ¼ F�1ð’ð� � �Þj � j��1Þ, ~ ¼

F�1ð ð� � �Þj � j��1Þ, �2R and �40. Then, for all F in Lp(R2), 1� p51, thetwo-wavelet localization operator L�F,’, : H1��,0!H1��,0 is compact.

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Proof It follows from Theorem 4.2 and the remark made in the proof of

Theorem 3.1 that L�F,’, ¼~L ~F, ~’, ~ , where

~Fð�, k, aÞ ¼ Fð�, kÞ for all (�, k, a) in G.

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