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1. Introduction

In this paper we characterise the r-nuclearity of Fourier integral operators on Lebesgue spaces. Fourier integral operators will be considered in ℝⁿ, the discrete group ℤⁿ, 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 ℤⁿ, the torus 𝕋ⁿ, the special unitary group SU(2), and the projective complex plane ℂℙ². Our main theorems will be applied to the characterization of r-nuclear pseudo-differential operators defined by the Weyl quantization procedure.

1.1. Outline of the paper

Let us recall that the Fourier integral operators (FIOs) on ℝⁿ, 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]).

According to the theory of FIOs developed by Hörmander ^[32], the phase functions ϕ are positively homogeneous of order 1 and they are considered smooth at ξ ≠ 0, while the symbols are considered satisfying estimates of the form

for every compact subset K of ℝ²ⁿ. Let us observe that L^p-properties for FIOs can be found in the references Hörmander ^[32], Eskin ^[26], Seeger, Sogge and Stein ^[51], Tao ^[52], Miyachi ^[37], Peral ^[39], Asada and Fujiwara ^[2], Fujiwara ^[28], Kumano-go ^[35], Coriasco and Ruzhansky ^{[10], [11]}, Ruzhansky and Sugimoto ^{[44], [45], [46], [47]}, Ruzhansky ^[50], and Ruzhansky and Wirth ^[49].

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. The 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. The 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. The nuclearity and the 2/3-nuclearity of pseudo-differential operators on the circle 𝕊¹ and on the lattice ℤ 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 𝕊¹, ℤ, 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]}.

1.2. 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 r-nuclear 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). The class of r-nuclear operators is usually endowed with the natural semi-norm

and, if r =1, n₁(·) 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 ℝⁿ, 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(ℝⁿ)-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.

Theorem 1.1. Let 0 < r ≤ 1. Let a(·, ·) be a symbol such thatLet 2 ≤ p_i < ∞, 1 ≤ p₂ < ∞, and let F be the Fourier integral operator associated to a(·, ·). Then, F : L^P1 (ℝⁿ) → L^P2 (ℝⁿ) is r-nuclear if, and only if, the symbol a(·, ·) admits a decomposition of the form

whereandare sequences of functions satisfying

Theorem 1.2. Let 0 < r ≤ 1, and let us consider a measurable function a(·, ·) on ℝ²ⁿ. Let 1 < p₁≤ 2, 1 ≤ p₂< ∞, and F be the Fourier integral operator associated to a(·, ·). Then, F : L^P1 (ℝⁿ) → L^P2 (ℝⁿ) is r-nuclear if the symbol a(·, ·) admits a decomposition of the form

whereandare sequences of functions satisfying

This theorem is sharp in the sense that the previous condition is a necessary and sufficient condition for the r-nuclearity of F when p₁= 2.

The 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]. Theorem 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 Theorem 1.1.

Theorem 1.3. Let a(·, ·) be a symbol such thatLet 2 ≤ p₁< ∞, 1 ≤ p₂< ∞, and let F be the Fourier integral operator associated to a(·, ·). If F : L^p1 (ℝⁿ) → L^P2 (ℝⁿ) is nuclear, thenthis means that

and

Sufficient conditions in order that pseudo-differential operators in L²(ℝⁿ) 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 (·, ·) ∈ L¹(ℝ²ⁿ), implies that A : L²→ L² is class trace, and A : L²→ L² is Hilbert-Schmidt if, and only if, σ_A ∈ L²(ℝ²ⁿ). 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. The first one, due to Lars Hörmander, which asserts that σ_A ∈ S(m,g) and a σ ∈ L¹(ℝ²ⁿ), implies that A : L²→ L² is a trace class operator. The second one, due to L. Rodino and F. Nicola, expresses that σ_A ∈ S(m, g) and (the weak-L¹ space), and implies that A : L²→ L² 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 Toft ^[6].

As an application of Theorem 1.1 to the Weyl quantization we present the following theorem.

Theorem 1.4. Let 0 < r ≤ 1. Let a(·, ·) be a differentiable symbol. Let 2 ≤ p₁< ∞, 1 ≤ p₂< ∞, and let a^w(x, D_x) be the Weyl quantization of the symbol a(·, ·). Then, a^w (x, D_x) : L^P1 (ℝⁿ) → L^P2 (ℝⁿ) is r-nuclear if, and only if, the symbol a(·, ·) admits a decomposition of the form

whereandare sequences of functions satisfying

Remark 1.5. Let us recall that the Wigner transform of two complex functions h, g on ℝⁿ, is formally defined as

With a such definition in mind, if 2 ≤ p₁ < ∞, 1 ≤ p₂ < ∞, under the hypothesis of Theorem 1.4, a^w (x, D_x) : L^P1 (ℝⁿ) →L^P2 (ℝⁿ), 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

The proof of our main result (Theorem 1.1) will be presented in Section 2 as well as the proof of Theorem 1.4. The nuclearity of Fourier integral operators on the lattice ℤⁿ and on compact Lie groups will be discussed in Section 3 as well as some trace formulae for FIOs on the -dimensional torus 𝕋ⁿ = ℝⁿ/ℤⁿ 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 ℂℙ². In this setting, we will prove analogues for the theorems 1.1 and 1.3 in every context mentioned above.

2. 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]}).

Theorem 2.1. Let us consider 1 ≤ p₁, p₂< ∞, 0 < r ≤ 1 and letbe such thatLet (X₁, μ₁) and (X₂, μ₂) be σ-finite measure spaces. An operator T : L^p1 (X₁, μ₁) → L^P2 (X₂, μ₂) is r-nuclear if, and only if, there exist sequences (h_k)_kin L^P2 (μ₂), and (g_k) insuch that

for every f ∈ L^P1 (μ₁). In this case, if p₁= p₂, and μ₁= μ₂, (see Section 3 of ^[13]) the nuclear trace of T is given by

Remark 2.2. Given f ∈ L¹ (ℝⁿ), define its Fourier transform by

If we consider a function f, such that f ∈ L¹ (ℝⁿ) 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 Theorem 1.1. Let us assume that F is a Fourier integral operator as in (1) with associated symbol a. Let us assume that F : L^p1 (ℝⁿ) →L^P2 (ℝⁿ) is r-nuclear.

Then there exist sequences h_k in L^P2 and g_kin satisfying

with

For all z ∈ ℝⁿ, 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 ξ₀ ∈ ℝⁿ and r > 0. If we define where is the characteristic function of the ball B(ξ₀; r), the condition 2 ≤ p₁ < ∞, together with the Hausdorff-Young inequality gives

So, for every r > 0 and ξ₀ ∈ ℝⁿ, 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 Theorem)

an application of the Dominated Convergence Theorem gives

In fact, for a.e.w. x ∈ ℝⁿ,

Since K ∈ L¹ (ℝ²ⁿ), and the function k(x, y) := |K(x, y)| is non-negative on the product space ℝ²ⁿ, by the Fubinni theorem applied to positive functions, the L¹(ℝ²ⁿ)-norm of K can be computed from iterated integrals as

By Tonelly theorem, for a.e.w. x ∈ ℝⁿ, 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 ∈ ℝ², 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), 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 Theorem (Theorem 2.1) we finish the proof.

Proof of Theorem 1.2. Let us consider the Fourier integral operator F,

associated with the symbol a. The 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 (ℝⁿ) into . So, if we prove that the condition (10) assures the r-nuclearity of K_σ from into L^P2 (ℝⁿ), we can deduce the r-nuclearity of F from L^P1 (ℝⁿ) into L^P2 (ℝⁿ). 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 Theorem 1.3. Let a(·, ·) be a symbol such that Let 2 ≤ p₁< ∞, 1 ≤ p₂< ∞, and let F be the Fourier integral operator associated to a(·, ·). If F : L^P1 (ℝⁿ) → L^P2(ℝⁿ) is nuclear, then Theorem 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,

In an analogous way we can prove that

Thus, we finish the proof.

2.2. The nuclear trace for FIOs on ℝⁿ

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 Theorem (Theorem 2.1). For doing so, let us consider a r-nuclear Fourier integral operator F : L^p(ℝⁿ) → L^p(ℝⁿ), 2 ≤ p < ∞. 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]).

Theorem 2.3. Let T : L^P(μ) -> L^P(μ) be a r-nuclear operator as in (40). ifthen

where λ_n(T), n ∈ 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(ℝⁿ) → L^p(ℝⁿ) is r-nuclear, the relation implies

where λ_n(T), n ∈ ℕ is the sequence of eigenvalues of F with multiplicities taken into account.

2.3. 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

There 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 Then, if, and only if,

provided that 0 < τ,τ’ ≤ 1.

Theorem 2.5. Let 0 < r ≤ 1. Let a(·, ·) be a differentiable symbol. Let 2 ≤ p₁< ∞,

1 ≤ p₂ < ∞, and let a^w(x, D_x) be the Weyl quantization of the symbol a(·, ·). Then, a^w (x, D_x): L^p1 (ℝⁿ) → L^P2 (ℝⁿ) is r-nuclear if, and only if, the symbol a(·, ·) admits a decomposition of the form

whereandare sequences of functions satisfying

Proof. Let us assume that a^τ(x, D_x) is r-nuclear from L^p1(ℝⁿ) into L^P2 (ℝⁿ). By Proposition 2.4, a^τ(x, D_x) = b(x, D_x), where

By Theorem 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)

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 Theorem 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). The proof is complete.

Remark 2.6. Let us observe that from Theorem 2.5 with τ = 1/2, we deduce the Theorem 1.4 mentioned in the introduction.

3. Characterizations of Fourier integral operators on ℤⁿand arbitrary compact Lie groups

3.1. FIOs on ℤⁿ

In this subsection we characterize those Fourier integral operators on ℤⁿ (the set of points in ℝⁿ with integral coordinates) admitting nuclear extensions on Lebesgue spaces. Now we define pseudo-differential operators and discrete Fourier integral operators on ℤⁿ. The discrete Fourier transform of is defined by

The Fourier inversion formula gives

In this setting pseudo-differential operators on ℤⁿ are defined by the integral form

These 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 ℤⁿ were defined in such reference as integral operators of the form

Our main tool in the characterization of nuclear FIOs on ℤⁿ is the following result, due to Jamalpour Birgani ^[36].

Theorem 3.1. Let 0 < r ≤ 1, 1 ≤ p₁ < ∞, 1 ≤ p₂ < ∞, and let t_mbe the pseudo-differential operator associated to the symbol m(·, ·). Then,is r-nuclear if, and only if, the symbol m(·, ·) admits a decomposition of the form

whereandare sequences of functions satisfying

As a consequence of the previous result, we give a simple proof for our characterization.

Theorem 3.2. Let 0 < r ≤ 1, 1 ≤ p₁ < ∞, 1 ≤ p₂ < ∞, and letbe the Fourier integral operator associated to the phase function 4> and to the symbol a(·, ·). Then,is r-nuclear if, and only if, the symbol a(·, ·) admits a decomposition of the form

whereandare 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 Theorem 3.5, the operator is r-nuclear 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

Thus, the proof is complete.

Remark 3.3. The 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.

Theorem 3.4. Let 2 ≤ p₁ < ∞, and 1 ≤ p₂ < ∞. If is nuclear, then this means that

and

The proof is only an adaptation of the proof that we have done for Theorem 1.3. We only need to use a discrete Hausdorff-Young inequality. In this case, we use

3.2. 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. The following identities follow from the Fourier transform on G:

and the Peter-Weyl Theorem on G implies the Plancherel identity on L²(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. Then 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]).

Theorem 3.5. Let 0 < r ≤ 1, 1 < p₁< ∞, 1 ≤ p₂< ∞, and let A be the pseudo-differential operator associated to the symbol σ_A(·, ·). Then, A : L^P1 (G) → L^P2 (G) is r-nuclear if, and only if, the symbol σ_A(·, ·) admits a decomposition of the form

whereandare sequences of functions satisfying

As a consequence of the previous criterion, we give a simple proof for our characterization.

Theorem 3.6. Let Let 0 < r ≤ 1, 1 < p₁< ∞, 1 ≤ p₂ < ∞, and let F be the Fourier integral operator associated to the phase function Φ and to the symbol a (·, ·). Then, F : L^P1 (G) → L^P2 (G) is r-nuclear if, and only if, the symbol a(·, ·) admits a decomposition of the form

whereandare sequences of functions satisfying

Remark 3.7. 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 Theorem 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

whereandare sequences of functions satisfying

Let us note that from the definition of σ_A we have

which is equivalent to

Thus, we finish the proof.

Remark 3.8. The 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. (The torus). Let us consider the n-dimensional torus G = 𝕋ⁿ: = ℝⁿ/ℤⁿand its unitary dualBy following Ruzhansky and Turunen^[48], a Fourier integral operator F associated to the phase functionand to the symbolis defined according to the rule

where is the Fourier transform of f at If we identify with ℤⁿ, and we define and we give the more familiar expression for F,

Now, by using Theorem 3.6, F is r-nuclear, 0 < r ≤ 1 if, and only if, the symbol a(·, ·) admits a decomposition ofthe form

whereandare sequences of functions satisfying

The 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. (The group SU(2)). Let us consider the groupconsinting of those orthogonal matrices A in ℂ^2x2, with det(A) = 1. We recall that the unitary dual of SU(2) (see^[48]) can be identified as

There are explicit formulae for t_las functions of Euler angles in terms of the so-called Legendre-Jacobi polynomials, see^[48]. A Fourier integral operator F associated to the phase functionand to the symbolis defined as

where

is the Fourier transform of f at t_l. As in the case of the n-dimensional torus, if we identifywithand we definewe can write

Now, by using Theorem 3.6, F is r-nuclear, 0 < r ≤ 1 if, and only if, the symbol a (·, ·) admits a decomposition of the form

whereandare sequences of functions satisfying

The 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 ℝⁿ, ℤⁿ, and 𝕋ⁿ, the nuclear trace of F can be written as

By using the diffeomorphismdefined by

we have

where and dσ(x) denotes the surface measure on 𝕊³. If we consider the parametrization of S³defined bywhere

then and

4. Nuclear Fourier integral operators on compact homogeneous manifolds

The 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 L²(M) and L²(𝔾) (where 𝔾 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 Theorem 2.2 in ^[30], where those nuclear pseudo-differential operators in L^p(𝔾)-spaces were classified.

4.1. 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 ℝℙⁿ ≅ SO(n + 1)/O(n), complex projective spaces ℂℙⁿ ≅ 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. This 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 L²(M) such that for some σ_A ∈ Σ(M) it satisfies

where denotes the Fourier transform of the lifting of f to G, given by

Remark 4.1. For every symbol of a Fourier multiplier A on M, only the upper-left block in σ_A (π) of the size k_πx 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 Theorem 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 L²(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 Theorem we obtain the r-nuclearity of F. So, our adaptation of the proof of Theorem 2.2 in ^[30] to our case of FIOs on compact manifolds leads to the following result.

Theorem 4.2. Let us assume M = G/K be a homogeneous manifold, 0 < r ≤ 1, 1 ≤ p₁, p₂< ∞, and let F be a Fourier integral operator as in (97). Then, F : L^P1 (M) → L^P2 (M) is r-nuclear if, and only if, there exist sequences h_kin L^P2and g_kinsatisfying

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 :

for 1 ≤ p < ∞.

Theorem 4.3. Let us assume M ≅ G/K be a homogeneous manifold, 2 ≤ p₁< ∞, 1 ≤ p₂< ∞, and let F be a Fourier integral operator as in (97) .If F : L^P1 (M) → L^P2 (M) is nuclear, thenthis means that

provided that

Proof. Let 2 ≤ p₁< ∞, 1 ≤ p₂< ∞, and let F be the Fourier integral operator associated to a(·, ·). If F : L^P1 (M) → L^P2 (M) is nuclear, then Theorem 4.2 guarantees the decomposition

with

So, if we take the we have,

By the definition of we have

Consequently,

Now, if we use the Hausdorff-Young inequality, we deduce, Consequently,

Thus, we finish the proof.

Remark 4.4. If K = {e_G} and M = G is a compact Lie group, the condition

arises naturally in the context of pseudo-differential operators. Indeed, if we take Φ(x,ξ = ξ (x), then Φ(x, ξ)^-1 = ξ (x)*, and

Remark 4.5. As a consequence of Delgado's theorem, if F : L^p(M) - L^p(M), M ≅ G/K, 1 ≤ p < ∞, is r-nuclear, its nuclear trace is given by

Example 4.6 (The complex projective plane ℂℙ²). A point (the n-dimensional complex projective space) is a complex line through the origin in ℂⁿ⁺¹. For every n, ℂℙⁿ ≅ SU(n +1)/SU(1) x SU(n). We will use the representation theory of SU(3) in order to describe the nuclear trace of Fourier integral operators on ℂℙ² ≅ SU(3)/SU(1) x SU(2). The Lie group SU(3) (see^[27]) has dimension 8, and 3 positive square roots a, 3 and p with the property

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 ∈ N₀, }. (116)

In fact, every representation π = π_λ(a,b)has highest weight λ = λ(a, b) for some (a, b) ∈ In this caseFor G = SU(3) and K = SU(1) x SU(2), let us define

Now, let us consider a phase functiona distributionand the Fourier integral operator F associated to Φ and to a (·, ·) :

where φ ∈ C^∞ (ℂℙ²). We additionally require the conditionandfor those distributional symbols considered above. As a consequence of Remark 4.5, if F : L^p(ℂℙ²) → L^p(ℂℙ²), 1 ≤ p < ∞, is r-nuclear, its nuclear trace is given by

Ifis a diffeomorphism and K = SU(1) x SU(2), then

where we have denotedandIf we consider the parametrization of SU(3) (see, e.g., Bronzan^[4]),

whereand

then, the group measure is the determinant given by

and we have the trace formula

Acknowledgements

I would like to thanks Majid Jamalpour Birgani, who provided me a preprint of the reference ^[36].

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Notes

Author notes

^* E-mail: duvanc306@gmail.com

IR