On the nuclear trace of Fourier integral operators

: In this paper we characterise the r -nuclearity of Fourier integral operators on Lebesgue spaces. Fourier integral operators will be considered in # n , the discrete group # n , the n-dimensional torus and symmetric spaces (compact homogeneous manifolds). We also give formulae for the nuclear trace of these operators. Explicit examples will be given on # n , the torus # n , the special unitary group SU(2)


Introduction
In this paper we characterise the r-nuclearity of Fourier integral operators on Lebesgue spaces.Fourier integral operators will be considered in # n , the discrete group # n , the n-dimensional torus and symmetric spaces (compact homogeneous manifolds).We also give formulae for the nuclear trace of these operators.Explicit examples will be given on # n , the torus # n , the special unitary group SU(2), and the projective complex plane ## 2 .Our main theorems will be applied to the characterization of r-nuclear pseudo-differential operators defined by the Weyl quantization procedure.

Outline of the paper
Let us recall that the Fourier integral operators (FIOs) on # n , are integral operators of the form where is the Fourier transform of f, or in a more general setting, linear integral operators formally defined by As it is well known, FIOs are used to express solutions to Cauchy problems of hyperbolic equations as well as for obtaining asymptotic formulas for the Weyl eigenvalue function associated to geometric operators (see Hörmander [32], [33], [34] , and Duistermaat and Hörmander [25] ).
A fundamental problem in the theory of Fourier integral operators is that of classifying the interplay between the properties of a symbol and the properties of its associated Fourier integral operator.
In this paper our main goal is to give, in terms of symbol criteria and with simple proofs, characterizations for the r-nuclearity of Fourier integral operators on Lebesgue spaces.Let us mention that this problem has been considered in the case of pseudo-differential operators by several authors.However, the obtained results belong to one of two possible approaches.e first ones are sufficient conditions on the symbol trough of summability conditions with the attempt of studying the distribution of the spectrum for the corresponding pseudo-differential operators.e second ones provide, roughly speaking, a decomposition for the symbols associated to nuclear operators, in terms of the Fourier transform, where the spatial variables and the momentum variables can be analyzed separately.Nevertheless, in both cases the results can be applied to obtain Grothendieck-Lidskii's formulae on the summability of eigenvalues when the operators are considered acting in L p spaces.
Necessary conditions for the r-nuclearity of pseudo-differential operators in the compact setting can be summarized as follows.e nuclearity and the 2/3-nuclearity of pseudo-differential operators on the circle # 1 and on the lattice # can be found in Delgado and Wong [14] .Later, the r-nuclearity of pseudo-differential operators was extensively developed on arbitrary compact Lie groups and on (closed) compact manifolds by Delgado and Ruzhansky in the works [16], [17], [18], [19], [21] , and by the author in [9] ; other conditions can be found in the works [20], [22], [23] .Finally, the subject was treated for compact manifolds with boundary by Delgado, Ruzhansky, and Tokmagambetov in [24] .
On the other hand, characterizations for nuclear operators in terms of decomposition of the symbol trough of the Fourier transform were investigated by Ghaemi, Jamalpour Birgani, and Wong in [29], [30], [36] for # 1 , #, and also for arbitrary compact and Hausdorff groups.Finally the subject has been considered for pseudo-multipliers associated to the harmonic oscillator (which can be qualified as pseudo-differential operators according to the Ruzhansky-Tokmagambetov calculus when the reference operators is the quantum harmonic oscillator) in the works of the author [3], [7], [8] .

Nuclear Fourier integral operators
In order to present our main result we recall the notion of nuclear operators.By following the classical reference Grothendieck [31] , we recall that a densely defined linear operator T : D(T) # E → F (where D(T) is the domain of T, and E, F are choose to be Banach spaces) extends to a rnuclear operator from E into F, if there exist sequences in E' (the dual space of E) and in F such that, the discrete representation holds true for all f # D(T).e class of r-nuclear operators is usually endowed with the natural semi-norm and, if r =1, n 1 (•) is a norm and we obtain the ideal of nuclear operators.In addition, when E = F is a Hilbert space and r = 1 the definition above agrees with that of trace class operators.For the case of Hilbert spaces H, the set of r-nuclear operators agrees with the Schatten-von Neumann class of order r (see Pietsch [40], [41] ).In order to characterize the r-nuclearity of Fourier integral operators on # n , we will use (same as in the references mentioned above) Delgado's characterization (see [15] ), for nuclear integral operators on Lebesgue spaces defined in σ-finite measure spaces, which in this case will be applied to L P (# n )-spaces.Consequently, we will prove that r-nuclear Fourier integral operators defined as in (1) have a nuclear trace given by In this paper our main results are the following theorems.e previous results are analogues of the main results proved in Ghaemi, Jamalpour Birgani, and Wong [29], [30], Jamalpour Birgani [36] , and Cardona and Barraza [3] .eorem 1.1, can be used for understanding the properties of the corresponding symbols in Lebesgue spaces.Moreover, we obtain the following result as a consequence of eorem 1.1.eorem 1.3.Let a(•, •) be a symbol such that Let 2 ≤ p 1 < ∞, 1 ≤ p 2 < ∞, and let F be the Fourier integral operator associated to a(•, •).
this means that and Sufficient conditions in order that pseudo-differential operators in L 2 (# n ) can be extended to (trace class) nuclear operators are well known.Let us recall that the Weyl-quantization of a distribution is the pseudo-differential operator defined by As it is well known σ = σ A (•, •) # L 1 (# 2n ), implies that A : L 2 → L 2 is class trace, and A : L 2 → L 2 is Hilbert-Schmidt if, and only if, σ A # L 2 (# 2n ).In the framework of the Weyl-Hörmander calculus of operators A associated to symbols σ in the S(m, g)-classes (see [34] ), there exist two remarkable results.e first one, due to Lars Hörmander, which asserts that σ A # S(m,g) and a σ # L 1 (# 2n ), implies that A : L 2 → L 2 is a trace class operator.e second one, due to L. Rodino and F. Nicola, expresses that σ A # S(m, g) and (the weak-L 1 space), and implies that A : L 2 → L 2 is Dixmier traceable [43] .Moreover, an open conjecture by Rodino and Nicola (see [43] ) says that gives an operator A with finite Dixmier trace.General properties for pseudo-differential operators on Schatten-von Neumann classes can be found in Buzano and To [6] .
As an application of eorem 1.1 to the Weyl quantization we present the following theorem.e proof of our main result (eorem 1.1) will be presented in Section 2 as well as the proof of eorem 1.4.e nuclearity of Fourier integral operators on the lattice # n and on compact Lie groups will be discussed in Section 3 as well as some trace formulae for FIOs on the -dimensional torus # n = # n /# n and the unitary special group SU(2).Finally, in Section 4 we consider the nuclearity of FIOs on arbitrary compact homogeneous manifolds, and we discuss the case of the complex projective space ## 2 .In this setting, we will prove analogues for the theorems 1.1 and 1.3 in every context mentioned above.

Symbol criteria for nuclear Fourier integral operators 2.1. Characterization of nuclear FIOs
In this section we prove our main result for Fourier integral operators F defined as in (1).Our criteria will be formulated in terms of the symbols a. First, let us observe that every FIO F has a integral representation with kernel K(x,y).In fact, straightforward computation shows us that where for every .In order to analyze the r-nuclearity of the Fourier integral operator F we will study its kernel K, by using as a fundamental tool the following theorem (see J. Delgado [13], [15] ).eorem 2.1.Let us consider 1 ≤ p 1 , p 2 < ∞, 0 < r ≤ 1 and let be such that Let (X 1 , μ 1 ) and (X 2 , μ 2 ) be σ-finite measure spaces.An operator T : L p1 (X 1 , μ 1 ) → L P2 (X 2 , μ 2 ) is r-nuclear if, and only if, there exist sequences (h k ) k in L P2 (μ 2 ), and (g k ) in such that for every f # L P1 (μ 1 ).In this case, if p 1 = p 2 , and μ 1 = μ 2 , (see Section 3 of [13] ) the nuclear trace of T is given by Remark 2.2.Given f # L 1 (# n ), define its Fourier transform by If we consider a function f, such that f # L 1 (# n ) with the Fourier inversion formula gives Moreover, the Hausdorff-Young inequality shows that the Fourier transform is a well defined operator on L p , 1 < p ≤ 2.
Proof of eorem 1.1.Let us assume that F is a Fourier integral operator as in (1) with associated symbol a.Let us assume that F : en there exist sequences h k in L P2 and g k in satisfying with For all z # # n , let us consider the set B(z; r), i.e., the euclidean ball centered at z with radius r > 0. Let us denote by |B(z; r)| the Lebesgue measure of B(z; r).Let us choose ξ 0 # # n and r > 0. If we define where is the characteristic function of the ball B(ξ 0 ; r), the condition 2 ≤ p 1 < ∞, together with the Hausdorff-Young inequality gives So, for every r > 0 and ξ 0 # # n , the function and we get, Taking into account that (see, e.g., Lemma 3.1 of [20] ), that and that (in view of the Lebesgue Differentiation eorem) an application of the Dominated Convergence eorem gives In fact, for a.e.w.x # # n , Since K # L 1 (# 2n ), and the function k(x, y) := |K(x, y)| is non-negative on the product space # 2n , by the Fubinni theorem applied to positive functions, the L 1 (# 2n )-norm of K can be computed from iterated integrals as By Tonelly theorem, for a.e.w.x # # n , the function Now, by the dominated convergence theorem, we have Now, from Lemma 3.4-(d) in [20] , On the other hand, if we compute from the definition (1), we have From the hypothesis that for a.e.w x # # 2 , the Lebesgue Differentiation theorem gives Consequently, we deduce the identity which in turn is equivalent to So, we have proved the first part of the theorem.Now, if we assume that the symbol a of the FIO F satisfies the decomposition formula (32)  for fixed sequences h k in L P2 and g k in satisfying ( 52 .So, if we prove that the condition (10) assures the r-nuclearity of K σ from into L P2 (# n ), we can deduce the r-nuclearity of F from L P1 (# n ) into L P2 (# n ).Here, we will be using the fact that the class of r-nuclear operators is a bilateral ideal on the set of bounded operators between Banach spaces.Now, is r-nuclear if, and only if, there exist sequences {h k }, {g k } satisfying Where for every Here, we have used the fact that for We end the proof by observing that ( 37) is in turns equivalent to (9).
Proof of eorem 1.In an analogous way we can prove that us, we finish the proof.

e nuclear trace for FIOs on #n
If we choose a r-nuclear operator T : E → E, 0 < r ≤ 1, with the Banach space E satisfying the Grothendieck approximation property (see Grothendieck [31] ), then there exist (a nuclear decomposition) sequences in E' (the dual space of E) and in E satisfying and In this case the nuclear trace of T is (a well-defined functional) given by because L p -spaces have the Grothendieck approximation property and, as consequence, we can compute the nuclear trace of every r-nuclear pseudo-multipliers.We will compute it from Delgado eorem (eorem 2.1).For doing so, let us consider a r-nuclear Fourier integral operator F : If a is the symbol associated to F, in view of (9), we have (in the sense of distributions) So, we obtain the trace formula Now, in order to determinate a relation with the eigenvalues of F, we recall that the nuclear trace of an r-nuclear operator on a Banach space coincides with the spectral trace, provided that We recall the following result (see [42] ).eorem 2.3.Let T : L P (μ) -> L P (μ) be a r-nuclear operator as in (40).if then where λ n (T), n # # is the sequence of eigenvalues of T with multiplicities taken into account.
As an immediate consequence of the preceding theorem, if the FIO F : L p (# n ) → L p (# n ) is r-nuclear, the relation implies where λ n (T), n # # is the sequence of eigenvalues of F with multiplicities taken into account.

Characterization of nuclear pseudo-differential operators defined by the Weyl quantization
As it was mentioned in the introduction, the Weyl-quantization of a distribution is the pseudo-differential operator defined by ere exist relations between pseudo-differential operators associated to the classical quantization or in a more general setting, τ-quantizations defined for every 0 < τ ≤ 1, by the integral expression (with corresponding to the Hörmander quantization), as it can be viewed in the following proposition (see Delgado [12] ).
Proposition 2.4.Let en, if, and only if, So, we have proved the first part of the characterization.On the other hand, if we assume (49), then where b(x, ξ) is defined as in (51).So, from eorem 1.1 we deduce that b(x, D x ) is r-nuclear, and from the equality a τ (x, D x ) = b(x, D x ) we deduce the r-nuclearity of a τ (x, D x ).e proof is complete.

Characterizations of Fourier integral operators on # n and arbitrary compact Lie groups 3.1. FIOs on # n
In this subsection we characterize those Fourier integral operators on # n (the set of points in # n with integral coordinates) admitting nuclear extensions on Lebesgue spaces.Now we define pseudo-differential operators and discrete Fourier integral operators on # n .e discrete Fourier transform of is defined by e Fourier inversion formula gives In this setting pseudo-differential operators on # n are defined by the integral form ese operators were introduced by Molahajloo in [38] .However, the fundamental work of Botchway L., Kibiti G., Ruzhansky M. [5] provides a symbolic calculus and other properties for these operators on In particular, Fourier integral operators on # n were defined in such reference as integral operators of the form Our main tool in the characterization of nuclear FIOs on # n is the following result, due to Jamalpour Birgani [36] .eorem 3.1.In this case, we use

FIOs on compact Lie groups
In this subsection we characterize nuclear Fourier integral operators on compact Lie groups.Although the results presented are valid for arbitrary Hausdorff and compact groups, we restrict our attention to Lie groups taking into account their differentiable structure, which in our case could give potential applications of our results to the understanding on the spectrum of certain operators associated to differential problems.
Let us consider a compact Lie group G with Lie algebra We will equip G with the Haar measure μ G .e following identities follow from the Fourier transform on G: and the Peter-Weyl eorem on G implies the Plancherel identity on L 2 (G), Notice that, since the term within the sum is the Hilbert-Schmidt norm of the matrix Any linear operator A on G mapping C ∞ (G) into D'(G) gives rise to a matrix-valued global (or full) symbol given by which can be understood from the distributional viewpoint.en it can be shown that the operator A can be expressed in terms of such a symbol as [48] , So, if is a measurable function (the phase function), and is a distribution on the Fourier integral operator F = F Φ,a associated to the symbol a(•, •) and to the phase function $ is defined by the Fourier series operator In order to present our main result for Fourier integral operators, we recall the following criterion (see Ghaemi, Jamalpour Birgani, Wong [30] ).eorem 3.5.For the proof we use the characterization of r-nuclear pseudo-differential operators mentioned above.However, this result will be generalized in the next section to arbitrary compact homogeneous manifolds.
Proof.Let us observe that the Fourier integral operator F, can be written as where So, the Fourier integral operator F coincides with the pseudo-differential operator A with symbol σ A .In view of eorem 3.5, the operator F = A : L P1 (G) → L P2 (G) is r-nuclear if, and only if, the symbol σ A (•, •) admits a decomposition of the form where and are sequences of functions satisfying Let us note that from the definition of σ A we have which is equivalent to us, we finish the proof.Remark 3.8.e nuclear trace of a r-nuclear pseudo-differential operator on G, A : L P (G) → L p (G), 1 ≤ p < ∞, can be computed according to the formula From the proof of the previous theorem, we have that F = A, where and, consequently, if F : L p (G) → L p (G), 1 ≤ p < ∞,, is r-nuclear, its nuclear trace is given by Now, we illustrate the results above with some examples.Example 3.9.(e torus).Let us consider the n-dimensional torus G = # n : = # n /# n and its unitary dual By following Ruzhansky and Turunen [48] , a Fourier integral operator F associated to the phase function and to the symbol is defined according to the rule where is the Fourier transform of f at If we identify with # n , and we define and we give the more familiar expression for F, Now, by using eorem 3.6, F is r-nuclear, 0 < r ≤ 1 if, and only if, the symbol a(•, •) admits a decomposition ohe form where and are sequences of functions satisfying e last condition have been proved for pseudo-differential operators in [29] .In this case, the nuclear trace of F can be written as Example 3.10.(e group SU(2)).Let us consider the group consinting of those orthogonal matrices A in # 2x2 , with det(A) = 1.We recall that the unitary dual of SU(2) (see [48] ) can be identified as ere are explicit formulae for t l as functions of Euler angles in terms of the so-called Legendre-Jacobi polynomials, see [48] .A Fourier integral operator F associated to the phase function and to the symbol is defined as where is the Fourier transform of f at t l .As in the case of the n-dimensional torus, if we identify with and we define we can write Now, by using eorem 3.6, F is r-nuclear, 0 < r ≤ 1 if, and only if, the symbol a (•, •) admits a decomposition of the form where and are sequences of functions satisfying e last condition have been proved for pseudo-differential operators in [30] on arbitrary Hausdorff and compact groups.In this case, in an analogous expression to the one presented above for # n , # n , and # n , the nuclear trace of F can be written as By using the diffeomorphism defined by we have where and dσ(x) denotes the surface measure on # 3 .If we consider the parametrization of S 3 defined by where then and

Nuclear Fourier integral operators on compact homogeneous manifolds
e main goal in this section is to provide a characterization for the nuclearity of Fourier integral operators on compact homogeneous manifolds Taking into account that the Peter-Weyl decompositions of L 2 (M) and L 2 (#) (where # is a Hausdorff and compact group) have an analogue structure, we classify the nuclearity of FIOs on compact homogeneous manifolds by adapting to our case the proof of eorem 2.2 in [30] , where those nuclear pseudo-differential operators in L p (#)-spaces were classified.

Global FIOs on compact homogeneous manifolds
In order to present our definition for Fourier integral operators on compact homogeneous spaces, we recall some definitions on the subject.
Compact homogeneous manifolds can be obtained if we consider the quotient space of a compact Lie groups G with one of its closed subgroups K (there exists an unique differential structure for the quotient M := G/K).Examples of compact homogeneous spaces are spheres real projective spaces ## n # SO(n + 1)/O(n), complex projective spaces ## n # SU(n + 1)/SU (1) x SU(n), and, more generally, Grassmannians Gr(r, n) # O(n)/O(n -r) x O(r).
Let us denote by the subset of of representations in G that are of class I with respect to the subgroup K. is means that if there exists at least one non trivial invariant vector a with respect to K, i.e., π(h)a = a for every h # K. Let us denote by B π to the vector space of these invariant vectors, and k π = dim B π .Now we follow the notion of Multipliers as in [1] .Let us consider the class of symbols Σ(M), for M = G/K, consisting of those matrix-valued functions Following [1] , a Fourier multiplier A on M is a bounded operator on L 2 (M) such that for some σ A # Σ(M) it satisfies where denotes the Fourier transform of the liing of f to G, given by Remark 4.1.For every symbol of a Fourier multiplier A on M, only the upper-le block in σ A (π) of the size k π x k π cannot be the trivial matrix zero.
Now, if we consider a phase function and a distribution the Fourier integral operator associated to Φ and to a (•, •) is given by We additionally require the condition σ(x,π) ij = 0 for i,j > k π for the distributional symbols considered above.Now, if we want to characterize those r-nuclear FIOs we only need to follow the proof of eorem 2.2 in [30] , where the nuclearity of pseudo-differential operators was characterized on compact and Hausdorff groups.Since the set provides an orthonormal basis of L 2 (M), we have the relation If we assume that F : L P1 (M) → L P2 (M) is r-nuclear, then we have a nuclear decomposition for its kernel, i.e., there exist sequences h k in L P2 and g k in satisfying with So, we have with 1 ≤ n,m ≤ k π , Consequently, if B t denotes the transpose of a matrix B, we obtain and by considering that Φ(x,π) # GL(d π ) for every x # M, we deduce the equivalent condition, On the other hand, if we assume that the symbol a(•, •) satisfies the condition (102) with from the definition of Fourier integral operator we can write, for φ # L P1 (M), Again, by Delgado's eorem we obtain the r-nuclearity of F. So, our adaptation of the proof of eorem 2.2 in [30] to our case of FIOs on compact manifolds leads to the following result.eorem 4.2.Let us assume M = G/K be a homogeneous manifold, 0 < r ≤ 1, 1 ≤ p 1 , p 2 < ∞, and let F be a Fourier integral operator as in (97).en, F : L P1 (M) → L P2 (M) is r-nuclear if, and only if, there exist sequences h k in L P2 and g k in satisfying with Now, we will prove that the previous (abstract) characterization can be applied in order to measure the decaying of symbols in the momentum variables.So, we will use the following formulation of Lebesgue spaces on : where φ # C ∞ (## 2 ).We additionally require the condition and for those distributional symbols considered above.As a consequence of Remark 4.5, if F : L p (## 2 ) → L p (## 2 ), 1 ≤ p < ∞, is r-nuclear, its nuclear trace is given by If is a diffeomorphism and K = SU (1) x SU (2), then where we have denoted and If we consider the parametrization of SU(3) (see, e.g., Bronzan [4] ), where and

eorem 1 . 1 .
Let 0 < r ≤ 1.Let a(•, •) be a symbol such that Let 2 ≤ p i < ∞, 1 ≤ p 2 < ∞, and let F be the Fourier integral operator associated to a(•, •).en, F : L P1 (# n ) → L P2 (# n ) is rnuclear if, and only if, the symbol a(•, •) admits a decomposition of the form where and are sequences of functions satisfying eorem 1.2.Let 0 < r ≤ 1, and let us consider a measurable function a(•, •) on # 2n .Let 1 < p 1 ≤ 2, 1 ≤ p 2 < ∞, and F be the Fourier integral operator associated to a(•, •).en, F : L P1 (# n ) → L P2 (# n ) is r-nuclear if the symbol a(•, •) admits a decomposition of the form where and are sequences of functions satisfying is theorem is sharp in the sense that the previous condition is a necessary and sufficient condition for the r-nuclearity of F when p 1 = 2.

eorem 1. 4 .
Let 0 < r ≤ 1.Let a(•, •) be a differentiable symbol.Let 2 ≤ p 1 < ∞, 1 ≤ p 2 < ∞, and let a w (x, D x ) be the Weyl quantization of the symbol a(•, •).en, a w (x, D x ) : L P1 (# n ) → L P2 (# n ) is r-nuclear if, and only if, the symbol a(•, •) admits a decomposition of the form where and are sequences of functions satisfying Remark 1.5.Let us recall that the Wigner transform of two complex functions h, g on # n , is formally defined as With a such definition in mind, if 2 ≤ p 1 < ∞, 1 ≤ p 2 < ∞, under the hypothesis of eorem 1.4, a w (x, D x ) : L P1 (# n ) → L P2 (# n ), is r-nuclear if, and only if, the symbol a(•, •) admits a decomposition (defined trough of the Wigner transform) of the type where and are sequences of functions satisfying ), then from (1) we can write (in the sense of distributions) where in the last line we have used the Fourier inversion formula.So, by Delgado eorem (eorem 2.1) we finish the proof.Proof of eorem 1.2.Let us consider the Fourier integral operator F, associated with the symbol a. e main strategy in the proof will be to analyze the natural factorization of F in terms of the Fourier transform, Clearly, if we define the operator with kernel (associated to σ = (ϕ, a)), then Taking into account the Hausdorff-Young inequality the Fourier transform extends to a bounded operator from L P1 (# n ) into

3 .
Let a(•, •) be a symbol such that Let 2 ≤ p 1 < ∞, 1 ≤ p 2 < ∞, and let F be the Fourier integral operator associated to a(•, •).If F : L P1 (# n ) → L P2 (# n ) is nuclear, then eorem 1.1 guarantees the decomposition where and are sequences of functions satisfying So, if we take the we have, Now, if we use the Hausdorff-Young inequality, we deduce that Consequently, and let a w (x, D x ) be the Weyl quantization of the symbol a(•, •).en, a w (x, D x ): L p1 (# n ) → L P2 (# n ) is r-nuclear if, and only if, the symbol a(•, •) admits a decomposition of the form where and are sequences of functions satisfying Proof.Let us assume that a τ (x, D x ) is r-nuclear from L p1 (# n ) into L P2 (# n ).By Proposition 2.4, a τ (x, D x ) = b(x, D x ), where By eorem 1.1 applied to and taking into account that b(x, D x ) is r-nuclear, there exist sequences h k in L P2 and g k in satisfying with So, we have Since we have (in the sense of distributions) and let t m be the pseudo-differential operator associated to the symbol m(•, •).en, is r-nuclear if, and only if, the symbol m(•, •) admits a decomposition of the form where and are sequences of functions satisfying As a consequence of the previous result, we give a simple proof for our characterization.eorem 3.2.Let 0 < r ≤ 1, 1 ≤ p 1 < ∞, 1 ≤ p 2 < ∞, and let be the Fourier integral operator associated to the phase function 4> and to the symbol a(•, •).en, is r-nuclear if, and only if, the symbol a(•, •) admits a decomposition of the form where and are sequences of functions satisfying Proof.Let us write the operator as where So, the discrete Fourier integral operator coincides with the discrete pseudo-differential operator t m with symbol m.By using eorem 3.5, the operator is rnuclear if, and only if, the symbol m(•, •) admits a decomposition of the form where and are sequences of functions satisfying Let us note that from the definition of m we have which, in turn, is equivalent to us, the proof is complete.Remark 3.3.e nuclear trace of a nuclear discrete pseudo-differential operator on can be computed according to the formula From the proof of the previous criterion, we have that where and, consequently, if is r-nuclear, its nuclear trace is given by Now, we present an application of the previous result.eorem 3.4.Let 2 ≤ p 1 < ∞, and 1 ≤ p 2 < ∞.If is nuclear, then this means that and e proof is only an adaptation of the proof that we have done for eorem 1.3.We only need to use a discrete Hausdorff-Young inequality.
and let A be the pseudo-differential operator associated to the symbol σ A (•, •).en, A : L P1 (G) → L P2 (G) is r-nuclear if, and only if, the symbol σ A (•, •) admits a decomposition of the form where and are sequences of functions satisfying As a consequence of the previous criterion, we give a simple proof for our characterization.eorem 3.6.Let Let 0 < r ≤ 1, 1 < p 1 < ∞, 1 ≤ p 2 < ∞, and let F be the Fourier integral operator associated to the phase function Φ and to the symbol a (•, •).en, F : L P1 (G) → L P2 (G) is r-nuclear if, and only if, the symbol a(•, •) admits a decomposition of the form where and are sequences of functions satisfying Remark 3.7.

eorem 4 . 3 .
Let us assume M # G/K be a homogeneous manifold, 2 ≤ p 1 < ∞, 1 ≤ p 2 < ∞, and let F be a Fourier integral operator as in (97) .If F : L P1 (M) → L P2 (M) is nuclear, then this means that provided that Proof.Let 2 ≤ p 1 < ∞, 1 ≤ p 2 < ∞,and let F be the Fourier integral operator associated to a(•, •).If F : L P1 (M) → L P2 (M) is nuclear, then eorem 4.2 guarantees the decomposition with So, if we take the we have, By the definition of we have We define the weights With the notations above the unitary dual of SU(3) can be identified with SU(3) = {A := \(a, b) = aa + br : a,b # N 0 , }. (116) In fact, every representation π = π λ(a,b) has highest weight λ = λ(a, b) for some (a, b) # In this case For G = SU(3) and K = SU(1) x SU(2), let us define Now, let us consider a phase function a distribution and the Fourier integral operator F associated to Φ and to a (•, •) :