*1. Introduction*

In this paper we characterise the *r*-nuclearity of Fourier integral operators on Lebesgue spaces. Fourier integral operators will be considered in ℝ^{n}, the discrete group ℤ^{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 ℤ^{n}, the torus 𝕋^{n}, the special unitary group SU(2), and the projective complex plane ℂℙ^{2}. 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 ℝ^{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}^{]}).

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 ℝ^{2n}. 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 𝕊^{1} 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 𝕊^{1}, 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, *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 ℝ^{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}(ℝ^{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.

Theorem 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*(·, ·)*. Then, F : L*
^{P1} (ℝ^{n}) → *L*
^{P2} (ℝ^{n}) *is r-nuclear if, and only if, the symbol a*(·, ·) *admits a decomposition of the form*

*where*
*and*
*are sequences of functions satisfying*

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

*where*
*and*
*are 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 p*
_{1}
*= 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 that*
*Let 2 ≤ p*
_{1}
*< ∞, 1 ≤ p*
_{2}
*< ∞, and let F be the Fourier integral operator associated to a*(·, ·)*. If F : L*
^{p1} (ℝ^{n}) → *L*
^{P2} (ℝ^{n}) *is nuclear, then*
*this means that*

*and*

Sufficient conditions in order that pseudo-differential operators in L^{2}(ℝ^{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}(ℝ^{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}(ℝ^{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. The first one, due to Lars Hörmander, which asserts that *σ*
_{A} ∈ *S*(m,g) and *a σ* ∈ L^{1}(ℝ^{2n}), implies that A : L^{2}
*→* L^{2} 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-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 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 ≤ *p*
_{1}
*<* ∞, 1 ≤ *p*
_{2}
*<* ∞, *and let a*
^{w}(*x, D*
_{x}) *be the Weyl quantization of the symbol a*(·, ·). *Then, a*
^{w} (x, D_{x}) : *L*
^{P1} (ℝ^{n}) → *L*
^{P2} (ℝ^{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 ℝ^{n}, is formally defined as

With a such definition in mind, if 2 ≤ *p*
_{1} < ∞, 1 ≤ p_{2} < ∞, under the hypothesis of Theorem 1.4, *a*
^{w} (x, D_{x}) : *L*
^{P1} (ℝ^{n}) **→**
*L*
^{P2} (ℝ^{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

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 ℤ^{n} and on compact Lie groups will be discussed in Section 3 as well as some trace formulae for FIOs on the -dimensional torus 𝕋^{n} = ℝ^{n}/ℤ^{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 ℂℙ^{2}. 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 ≤ 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} (ℝ^{n}), define its Fourier transform by

If we consider a function *f*, such that *f ∈* L^{1} (ℝ^{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 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} (ℝ^{n}) **→**
*L*
^{P2} (ℝ^{n}) is *r*-nuclear.

Then there exist sequences *h*
_{k} in L^{P2} and *g*
_{k}in
satisfying

with

For all z **∈** ℝ^{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} ∈ ℝ^{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} ∈ ℝ^{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 Theorem)

an application of the Dominated Convergence Theorem gives

In fact, for a.e.w. x ∈ ℝ^{n},

Since K ∈ L^{1} (ℝ^{2n}), and the function *k*(x, y) := |K(x, y)| is non-negative on the product space ℝ^{2n}, by the Fubinni theorem applied to positive functions, the L^{1}(ℝ^{2n})-norm of *K* can be computed from iterated integrals as

By Tonelly theorem, for a.e.w. x ∈ ℝ^{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 ∈ ℝ^{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), 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} (ℝ^{n}) into
. So, if we prove that the condition (10) assures the *r*-nuclearity of *K*
_{σ} from
into *L*
^{P2} (ℝ^{n}), we can deduce the *r*-nuclearity of *F* from L^{P1} (ℝ^{n}) into L^{P2} (ℝ^{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 Theorem 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*(·, ·). If F : L^{P1} (ℝ^{n}) → L^{P2}(ℝ^{n}) is nuclear, then Theorem 1.1 guarantees the decomposition

where and are sequences of functions satisfying

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* ℝ^{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 Theorem (Theorem 2.1). For doing so, let us consider a *r*-nuclear Fourier integral operator *F :* L^{p}(ℝ^{n}) → L^{p}(ℝ^{n}), 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). *if*
*then*

*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}(ℝ^{n}) → L^{p}(ℝ^{n}) 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 ≤ *p*
_{1}
*<* ∞,

1 ≤ p_{2} < ∞, *and let a*
^{w}(*x, D*
_{x}) *be the Weyl quantization of the symbol a*(·, ·)*. Then, a*
^{w} (*x, D*
_{x})*: L*
^{p1} (ℝ^{n}) → *L*
^{P2} (ℝ^{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}(ℝ^{n}) into *L*
^{P2} (ℝ^{n}). 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* ℤ^{n}
*and arbitrary compact Lie groups*

*3.1. FIOs on* ℤ^{n}

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

The Fourier inversion formula gives

In this setting pseudo-differential operators on ℤ^{n} 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 ℤ^{n} were defined in such reference as integral operators of the form

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

Theorem 3.1. *Let* 0 < r ≤ 1, 1 ≤ *p*
_{1} < ∞, 1 ≤ p_{2} < ∞, *and let* t_{m}
*be 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*

*where*
*and*
*are 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 ≤ *p*
_{1} < ∞, 1 ≤ p_{2} < ∞, *and let*
*be 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*

*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 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 ≤ p_{1} < ∞, and 1 ≤ p_{2} < ∞. *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 *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. 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 < p_{1}
*<* ∞, 1 ≤ p_{2}
*<* ∞, *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*

*where*
*and*
*are 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 < p_{1}
*<* ∞, 1 ≤ p_{2} < ∞, *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*

*where*
*and*
*are 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

*where*
*and*
*are 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 =* 𝕋^{n}: = ℝ^{n}/ℤ^{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 ℤ^{n}, 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*

*where*
*and*
*are 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 group*
*consinting 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*
_{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 Theorem 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*

*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* ℝ^{n}, ℤ^{n}, *and* 𝕋^{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* 𝕊^{3}
*. If we consider the parametrization of* S^{3}
*defined by*
*where*

**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 *L*
^{2}(*M*) and L^{2}(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 ℝℙ^{n} ≅ SO(n + 1)/O(n), complex projective spaces ℂℙ^{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*. 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 *L*
^{2}(*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*
_{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 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 *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 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 ≤ p_{1}, p_{2}
*<* ∞, *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*
_{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
*:*

for 1 ≤ *p < ∞.*

Theorem 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 Theorem 4.2 guarantees the decomposition

with

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 ℂℙ^{2}). *A point*
(*the n-dimensional complex projective space*) *is a complex line through the origin in* ℂ^{n+1}. *For every n,* ℂℙ^{n} ≅ 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* ℂℙ^{2} ≅ 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 ∈* 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* (·, ·) :

*where φ ∈* C^{∞} (ℂℙ^{2}). *We additionally require the condition*
*and*
*for those distributional symbols considered above. As a consequence of Remark 4.5, if F*: L^{p}(ℂℙ^{2}) → L^{p}(ℂℙ^{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}^{]})*,*

*then, the group measure is the determinant given by*

*and we have the trace formula*