Schatten Class and Nuclear Pseudo-Differential Operators On

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G. eg nto on go We 5 ) nti ae we paper this In G/H 21 21 20 14 7 1 be in Schatten-von Neumann class S of operators on L2(G/H) for 0 < r ; (2) to find criteria r ≤ ∞ for pseudo-differential operators from Lp1 (G/H) into Lp2 (G/H) to be r-nuclear, 0 < r 1, for ≤ 1 p1, p2 < ; and (3) applications to find a trace formula and to provide criteria for heat kernel to≤ be nuclear∞ on Lp(G/H). In order to do this, we will use the global quantization developed for compact homogeneous spaces as a non-commutative analogue of the Kohn-Nirenberg quantization of operators on Rn. Recently, several researchers started a extensive research for finding the criteria for Schatten classes and r-nuclear operators in terms of symbols with lower regularity [2, 12, 15, 40, 41]. Ruzhan- sky and Delgado [12, 14] successfully drop the regularity condition at least in their setting using the matrix-valued symbols instead of standard Kohn-Nirenberg quantization. Inspired by the work of Delgado and Ruzhansky, we present symbolic criteria for pseudo-differential operators on G/H [ to be Schatten class using the matrix-valued symbols defined on G/H G/H, a noncommutative analogue of phase space. It is well known that in the setting of Hilbert spaces× the class of r-nuclear operators agrees with the Schatten-von Neumann ideal of order r [35]. In general, for trace class operators on Hilbert spaces, the trace of an operator given by integration of its integral kernel over the diagonal is equal to the sum of its eigenvalues. However, this property fails in Banach spaces. The importance of r-nuclear operator lies in the work of Grothendieck, who proved that for 2/3-nuclear operators, the trace in Banach spaces agrees with the sum of all the eigenvalues with multiplicities counted. Therefore, the notion of r-nuclear operators becomes useful. Now, the question of finding good criteria for ensuring the r-nuclearity of operators arises but this has to be formulated in terms different from those on Hilbert spaces and has to take into account the impos- sibility of certain kernel formulations in view of Carleman’s example [6] (also see [12]). In view of this, we will establish conditions imposed on symbols instead of kernels ensuring the r-nuclearity of the corresponding operators. The initiative of finding the necessary and sufficient conditions for a pseudo-differential operator defined on a group to be r-nuclear has been started by Delgado and Wong [16]. The main ingre- dient of such a characterization is a theorem of Delgado [16]. A multilinear version of Delgado’s theorem recently proved by first author and D. Cardona to study the nuclearity of multilinear pseudo-differential operators on the lattice and the torus [4, 5]. In a seminal paper of Delgado and Ruzhansky [12], they studied the Lp-nuclearity and traces of pseudo-differential operators on compact Lie groups using the global symbolic calculus developed by Ruzhansky and Turunen [37]. Later, they with their collaborators extends these results to compact homogeneous spaces and compact manifolds [11, 12, 13, 14]. On the other hand, Wong and his collaborators extended the version of [16] in the settings of abstract compact groups with differential structure [18, 19]. Characteri- zations of nuclear operators in terms of decomposition of symbol through Fourier transform were investigated by Ghaemi, Jamalpour Birgani and Wong for S1 [18]. Later they generalized their results on nuclearity to the pseudo-differential operators for any arbitrary compact group [19]. The homogeneous spaces of abstract compact groups play an important role in mathematical physics, geometric analysis, constructive approximation and coherent state transform, see [23, 24, 25, 26, 27, 28] and the references therein. The study of pseudo-differential operators on homogeneous spaces of compact groups was started by the first author [29]. We use the operator-valued Fourier transform on homogeneous spaces of compact groups developed by Ghani Farashahi [17]. By using this Fourier transform, we introduce a global pseudo-differential calculus for homogeneous spaces of compact groups and study the Schatten class of operator on L2(G/H) and r-nuclear operators on Lp-spaces on compact homogeneous spaces. Our results can be seen as a compliment and generalization of the work of Delgado and Ruzhansky, pseudo-differential operators for compact Lie groups [12] as well as generalization of work of Ghaemi and Wong [18] on pseudo-differential operators on compact groups.

2 We begin this paper by recapitulating some basic Fourier analysis on homogeneous spaces on compact groups from [17] in Section 2 although a parallel theory of homogeneous space of compact Lie groups can be found in classical book of Vilenkin [42] and recent papers and books [1, 9, 15, 34]. Later in this section we present a global quantization (Ruzhansky and Turunen [37]) on homogeneous spaces of compact groups related to a matrix-valued symbols. In Section 3, we give symbolic criteria of r-Schatten class operators defined on L2(G/H). In Section 4, we start our investigation on r-nuclear operator. We begin this section by providing sufficient conditions for an operator be r-nuclear in term of conditions on symbol of the operator. We also present a characterization of r-nuclear pseudo-differential operator on Lp-space for homogeneous spaces of compact groups. We calculate the nuclear trace of related pseudo-differential operators. In Section 5, we find a symbol for the adjoint of r-nuclear pseudo-differential operators on homogeneous space of compact groups and give a characterization for self-adjointness. We also compute the symbol of the product of a nuclear operator and a bounded linear operator. We end this paper by presenting applications of our results in the context of heat kernels.

2. Fourier analysis and the global quantization on homogeneous spaces of compact groups We begin this section by recalling some basic and important facts of harmonic analysis on homogeneous spaces of compact (Hausdorff) groups from [17] which is almost similar to the theory given in [1] and [42] (see also [9, 15, 34]) for homogeneous spaces of compact Lie groups. Let G be a compact (Hausdorff) group with normalized Haar measure dx and let H be a closed subgroup of G with probability Haar measure dh. The left coset space G/H can be seen as a homogeneous space with respect to the action of G on G/H given by left multiplication. Let (Ω) C denote the space of continuous functions on a compact Hausdorff space Ω. Define TH : (G) (G/H) by C → C T (f)(xH)= f(xh) dh, xH G/H. H ∈ ZH Then TH is an onto map. The homogeneous space G/H has a unique normalized G-invariant positive Radon measure µ such that Weil formula

TH (f)(xH) dµ(xH)= f(x) dx ZG/H ZG 2 2 holds. The map TH can be extended to L (G/H,µ) and is a partial isometry on L (G/H) with 2 T (f),T (g) 2 = f,g 2 for all f,g L (G). h H H iL (G/H,µ) h iL (G) ∈ Let (π, π) be continuous unitary representation of compact group G on a Hilbert space π. It is well-knownH that any irreducible representation (π, ) is ﬁnite dimensional with the dimensionH Hπ dπ (say). Consider the operator valued integral

π TH := π(h) dh ZH π π deﬁned in the weak sense, i.e., TH u, v = H π(h)u, v dh, for all u, v π. Note that TH is a h i h i ∈π H bounded linear operator on π with norm bounded by one. Further, TH is a partial isometric orthogonal projection and T πHis an identityR operator if and only if π(h)= I for all h H. H ∈ [ Deﬁnition 2.1. Let H be a closed subgroup of a compact group G. Then the dual object G/H of G/H is a subset of G and given by

[ π bG/H := π G : TH =0 = π G : π(h) dh =0 . ∈ 6 ∈ H 6 n o Z b b

3 [ We would like to note here that the set G/H is the set of all type 1 representations of G with respect to H which was denoted by G0 in [34, 42]. [ H C Let π G/H. Then the functions πζ,ξ : G/H defined by ∈ c → πH (xH) := π(x)T π ζ, ξ for xH G/H, ζ,ξ h H i ∈ for ξ, ζ π, are called H -matrix elements of (π, π). If e1,e2,...,edπ is an orthonormal basis ∈ H π H H { } for π then we denote π(x)TH ei,ej by πij (xH). Now, using the orthogonality relation of matrix H h i H coefficients of G and the fact TH (πζ,ξ)= πζ,ξ we have 1 πH , ξH = δ δ δ . i,j k,l L2(G/H,µ) πξ ik jl dπ [ Let ϕ L1(G/H,µ) and π G/H. Then the group Fourier transform of ϕ at π is a bounded linear operator∈ defined by ∈

(2.1) (ϕ)(π)=ϕ ˆ(π) := ϕ(xH)Γ (xH)∗ dµ(xH) FG/H π ZG/H on the Hilbert space , where for xH G/H the notation Γ (xH) stands for the bounded linear Hπ ∈ π operator on π satisfying H π ζ, Γπ(xH)ξ = ζ, π(x)TH ξ h i h i H for all ζ, ξ π. Note that from the notation of Γπ(xH), the H-matrix coefficients πi,j (xH) are ∈ H 2 same as Γπ(xH)ij . Moreover if f L (G/H) thenϕ ˆ(π) is a Hilbert-Schmidt operator on π and satisfies the following Plancherel formula∈ as stated in next theorem. H Theorem 2.2. For ϕ L2(G/H,µ) we have ∈ 2 2 2 dπ ϕˆ(π) S2 = ϕ L (G/H,µ), [ k k k k [π]X∈G/H where . S2 stands for the Hilbert-Schmidt norm on the space of all Hilbert-Schmidt operators on . k k Hπ Theorem 2.3. For ϕ L2(G/H,µ) the following Fourier inversion formula holds ∈ π (2.2) ϕ(xH)= dπ Tr[ϕ ˆ(π)π(x)TH ] for µ a.e. xH G/H. [ − ∈ [π]X∈G/H We would like to record the following lemma whose proof is similar to [12, Lemma 2.5] by using π the fact that the operator TH is norm bounded by one. Lemma 2.4. Let G/H be a compact homogeneous space with normalized measure µ and let π [ ∈ G/H. Then for all 1 i, j d we have ≤ ≤ π 1 − q dπ 2 q , Γπ( )ij Lq(G/H) 1 ≤ ≤ ∞ − 2 k · k ≤ (dπ 1 q 2, ≤ ≤ 1 − q with the convention that for q = we have dπ =1. ∞ Given a continuous linear operator T : C(G/H) C(G/H), its matrix-valued global symbol → σ (xH, π) Cdπ×dπ is defined by T ∈ π ∗ (2.3) TH σT (xH, π)= π(x) (T Γπ)(xH), where T Γπ stands for the action of T on the matrix components of Γπ(xH).

4 Thus setting (T Γπ(xH))mn = (T (Γπmn ))(xH), we have

dπ π (TH σT (xH, π))mn := πkm(x)(T Γπ(xH))kn, Xk=1 where 1 m,n d . ≤ ≤ π Now, let the symbol σT is a matrix-valued global symbol for continuous linear operator T : C(G/H) C(G/H) as above. Then we can recover the operator T by using the Fourier inversion formula as→ follows:

π ˆ Tf(xH)= T  dπ Tr(π(x)TH f(π)) [ [π]∈G/H  X    = dπ Tr(T Γπ(xH) fˆ(π)). [ [π]X∈G/H π By using (2.3) and the relation π(x)TH =Γπ(xH), we get

(2.4) Tf(xH)= dπ Tr(Γπ(xH)σT (xH, π)fˆ(π)) [ [π]X∈G/H for all f C(G/H),µ-a.e. xH G/H and the sum is independent of the representation from each ∈ [ ∈ equivalence class [π] G/H. We will also write T = Op(σ ) for operator T given by the formula ∈ T (2.4) and will be called a pseudo-differential operator corresponding to matrix-valued symbol σT . For more details and consistent development of this quantization on compact Lie group and the corresponding symbolic calculus we refer [37] and [38]. Remark 2.5. Let H be a closed normal subgroup of the compact group G and let µ be the normalized G-invariant measure over the left quotient space G/H associated to the weil’s formula. [ Then µ is a Haar measure over the compact quotient group G/H and G/H = H⊥ := π G : π(h) = I for all h H . Moreover the Fourier transform (2.1), inverse Fourier transfo{rm∈ (2.2) ∈ } and the pseudo-differential operator given in (2.4) coincide with the classical Fourier transform,b Inverse Fourier transform and pseudo-differential operator over the compact quotient group G/H respectively.

3. r-Schatten-von Neumann class of pseudo-differential operators on L2(G/H) This section is devoted to the study of r-Schatten-von Neumann class of pseudo-diﬀerential operators on the Hilbert space L2(G/H). We begin this section with the deﬁnition of r-Schatten- von Neumann class of operators. If is a complex Hilbert space, a linear compact operator A : belongs to the r-Schatten- H H → H von Neumann class Sr( ) if H ∞ (s (A))r < , n ∞ n=1 X ∗ where sn(A) denote the singular values of A, i.e. the eigenvalues of A = √A A with multiplicities counted. | | For 1 r< , the class S ( ) is a Banach space endowed with the norm ≤ ∞ r H 1 ∞ r A = (s (A))r . k kSr n n=1 ! X

5 For 0

ξ 2 σ [ = d σ (xH, ξ)T dµ(xH) < . A L2(G/H×G/H)  ξ A H S2  k k G/H [ k k ∞ Z [ξ]∈G/H  X  The following theorem gives a characterization of Hilbert-Schmidt pseudo-differential operators on G/H. We would like to remark here that the following theorem is already proved by the first author in [29] using a different method. Theorem 3.1. Let T : L2(G/H) L2(G/H) be a continuous linear operator with the matrix- [ → valued symbol σT on G/H G/H. Then the operator T is a Hilbert-Schmidt operator if and only [ × if σ L2(G/H G/H). Moreover, we have T ∈ × T = σ [ . k kS2 k T kL2(G/H×G/H)

Proof. For all f L2(G/H), we have ∈

Tf(xH)= dξ Tr(Γξ(xH)σT (xH, ξ)fˆ(ξ)) [ [ξ]X∈G/H ∗ = dξ Tr(Γξ(xH)σT (xH, ξ)Γξ (wH) )f(wH) dµ(wH) G/H [ Z [ξ]X∈G/H = K(xH,wH)f(wH) dµ(wH), ZG/H where the kernel is given by

∗ K(xH,wH)= dξ Tr(Γξ(xH)σT (xH, ξ)Γξ(wH) ), xH,wH G/H. [ ∈ [ξ]X∈G/H We have

T 2 = K(xH,yH) 2 dµ(xH) dµ(yH) k kS2 | | ZG/H ZG/H = K(xH, xz−1H) 2 dµ(xH) dµ(zH). | | ZG/H ZG/H Now

−1 −1 ∗ K(xH, xz H)= dξ Tr Γξ(xH)σT (xH, ξ)Γξ (xz H) [ [ξ]X∈G/H ξ = dξ Tr Γξ(zH)σT (xH, ξ)TH [ [ξ]X∈G/H = −1τ(xH, )(zH), F ·

6 ξ where τ(xH, ξ)= σT (xH, ξ)TH . Therefore, using Plancherel’s formula, we have

T 2 = K(xH, xz−1H) 2dµ(xH) dµ(zH) k kS2 | | ZG/H ZG/H = −1τ(xH, )(zH) 2dµ(xH) dµ(zH) |F · | ZG/H ZG/H 2 = dξ τ(xH, ξ) S2 dµ(xH) G/H [ k k Z [ξ]X∈G/H ξ 2 = dξ σT (xH, ξ)TH S2 dµ(xH) G/H [ k k Z [ξ]X∈G/H = σ [ . k T kL2(G/H×G/H)

The following lemma is a consequence of the deﬁnition of Schatten classes (see [14]) which is needed to obtain our main result.

Lemma 3.2. Let A : be a linear compact operator. Let 0 < r,t< . Then A Sr( ) if r H → H r r t ∞ ∈ H and only if A t St( ). Moreover, A = A t . | | ∈ H k kSr k| | kSt The corollary below is the main result of this section which present a characterization of a pseudo-diﬀerential operator on L2(G/H) to be a Schatten class operator. The proof follows from Lemma 3.2 with t = 2 and Theorem 3.1. Corollary 3.3. Let T : L2(G/H) L2(G/H) be a continuous linear operator with the matrix- [ → valued symbol σ on G/H G/H. Then T S L2(G/H) if and only if T × ∈ r ξ 2 d σ r (xH, ξ)T dµ(xH) < . ξ |T | 2 H S2 G/H [ k k ∞ Z [ξ]X∈G/H

4. Characterizations and traces of r-nuclear, 0 < r 1, pseudo-differential operators on Lp(G/H) ≤ This section is devoted to study of r-nuclear operators on Banach spaces Lp(G/H). Here we present a symbolic characterization of r-nuclear operators and give a formula for the nuclear trace of such operator. We will begin this section by recalling the basic notions of nuclear operators on Banach spaces. Let 0 < r 1 and T be a bounded linear operator from a complex Banach space X into another ≤ ′ ∞ ′ complex Banach space Y such that there exist sequences xn n=1 in the dual space X of X and ∞ ∞ ′ r r { } y in Y such that x ′ y < and { n}n=1 n=1 k nkX k nkY ∞ P ∞ T x = x′ (x)y , x X. n n ∈ n=1 X Then we call T : X Y a r-nuclear operator and if X = Y, then its nuclear trace Tr(T ) is given by → ∞ ′ Tr(T )= xn (yn) . n=1 X

7 It can be proved that the definition of a nuclear operator and the definition of the trace of a ′ ∞ ∞ r-nuclear operator are independent of the choices of the sequences xn n=1 and yn n=1 . The following theorem is a characterization of r-nuclear operators on σ-finite{ measure} spaces{ } [11].

Theorem 4.1. Let 0 < r 1. Let (X1,µ1) and (X2,µ2) be σ-ﬁnite measure spaces. Then a ≤ p1 p2 bounded linear operator T : L (X1,µ1) L (X2,µ2) , 1 p1,p2 < , is r-nuclear if and ′ ∞ p → ∞≤ p ∞ only if there exist sequences g in L 1 (X ,µ ) and h in L 2 (X ,µ ) such that for all { n}n=1 1 1 { n}n=1 2 2 f Lp1 (X , µ ) ∈ 1 1

(Tf)(x)= K(x, y)f(y)dµ (y), x X , 1 ∈ 2 ZX1 where ∞ K(x, y)= h (x)g (y), x X ,y X n n ∈ 2 ∈ 1 n=1 X and ∞ r r ′ gn p hn p < . k kL 1 (X1,µ1) k kL 2 (X2,µ2) ∞ n=1 X Let 0 < r 1. Let (X,µ)bea σ-finite measure space. Let T : Lp(X,µ) Lp(X,µ), 1 p< , ′ ≤ → ∞ p ≤ ∞ be a r-nuclear operator. Then by Theorem 4.1, we can find sequences gn n=1 in L (X,µ) and h ∞ in Lp(X,µ) such that { } { n}n=1 ∞ r r ′ g p h p < k nkL (X,µ) k nkL (X,µ) ∞ n=1 X and for all f Lp(X,µ), ∈ (Tf)(x)= K(x, y)f(y) dµ(y), x X, ∈ ZX where ∞ K(x, y)= h (x)g (y), x,y X n n ∈ n=1 X and it satisfies ∞ r r ′ K(x, y) dµ(y) g p h p . | | ≤ k nkL (X,µ) k nkL (X,µ) X n=1 Z X The nuclear trace Tr(T ) of T : Lp(X,µ) Lp(X,µ) is given by → (4.1) Tr(T )= K(x, x) dµ(x). ZX Now, we present a characterization of r-nuclear pseudo-differential operators from Lp1 (G/H) into Lp2 (G/H).

p1 p2 Theorem 4.2. Let 0 < r 1 and let T : L (G/H) L (G/H), 1 p1,p2 < , be a continuous ≤ → [ ≤ ∞ linear operator with the matrix-valued symbol σT on G/H G/H. Suppose that the symbol σT satisﬁes × r 2+ p˜ 1 t ∞ ∞ r dξ σT ( , ξ) op(ℓ ,ℓ ) Lp2 (G/H) < , [ kk · k k ∞ [ξ]X∈G/H where p˜ = min 2,p . Then the operator T is r-nuclear. 1 { 1}

8 Proof. Since the operator T can be written as

∗ Tf(xH)= dξ Tr(Γξ(xH)σT (xH, ξ)Γξ(wH) )f(wH) dµ(wH), G/H [ Z [ξ]X∈G/H the kernel of T is given by ∗ K(xH,wH)= dξ Tr(Γξ(xH)σT (xH, ξ)Γξ(wH) ). [ [ξ]X∈G/H Now we write

dξ ∗ Tr(Γξ(xH)σT (xH, ξ)Γξ (wH) )= (Γξ(xH)σT (xH, ξ))ij Γξ(wH)ij , i,j=1 X ∗ and set that hξ,ij (xH)= dξ(Γξ(xH)σT (xH, ξ))ij and gξ,ij (wH) =(Γξ(wH) )ji = Γξ(wH)ij . We observe that

dξ dξ t (Γξ(xH)σT (xH, ξ))ij = Γξ(xH)ik σT (xH, ξ)kj = (σT (xH, ξ))jk Γξ(xH)ik. Xk=1 Xk=1 By taking into account that Γ (xH) 1, we get | ξ ik|≤ dξ (Γ (xH)σ (xH, ξ)) = (σ (xH, ξ))t Γ (xH) | ξ T ij | T jk ξ ik k=1 X t σ (xH, ξ) ∞ ∞ (Γ (xH ) , Γ (xH) ,..., Γ (xH) ) ∞ ≤k T kop(ℓ ,ℓ )k ξ i1 ξ i2 ξ idξ kℓ t σ (xH, ξ) ∞ ∞ . ≤k T kop(ℓ ,ℓ ) Therefore we have r r h ( ) p = d (Γ ( )σ ( , ξ)) p k ξ,ij · kL 2 (G/H) k ξ ξ · T · ij kL 2 (G/H) r t d σ ( , ξ) ∞ ∞ p2 . ≤ ξkk T · kop(ℓ ,ℓ )kL (G/H) If p′ denotes the Lebesgue conjugate of p then we have 1 + 1 = 1 whereq ˜ = max 2,p′ . By 1 1 p˜1 q˜1 1 1 r { } − ′ p˜ r 1 Lemma 2.4 we have Γξ( ) ′ d . Therefore, k · kLp1 (G/H) ≤ ξ r − ′ r r p˜1 r 2 t ′ ∞ ∞ gξ,ij ( ) p hξ,ij ( ) p2 d dξdξ σT ( , ξ) op(ℓ ,ℓ ) Lp2 (G/H) k · kL 1 (G/H)k · kL (G/H) ≤ ξ kk · k k [ξX],i,j X[ξ] 2+ r p˜1 t d σ ( , ξ) ∞ ∞ p2 < . ≤ ξ kk T · kop(ℓ ,ℓ )kL (G/H) ∞ X[ξ] Hence, by invoking Theorem 4.1 it follows that T is r-nuclear. Next theorem gives the necessary and suﬃcient conditions for an operator to be r-nuclear in terms of symbol decomposition.

p1 p2 Theorem 4.3. Let 0 < r 1 and let T : L (G/H) L (G/H), 1 p1,p2 < , be a continuous ≤ → [ ≤ ∞ linear operator with the matrix-valued symbol σT on G/H G/H. Then T is r-nuclear if and only ′ ∞ p ∞ × p if there exist sequences g L 1 (G/H) and h L 2 (G/H) such that { k}k=1 ∈ { k}k=1 ∈ ∞ r r ′ gk p hk p < k kL 1 (G/H) k kL 2 (G/H) ∞ Xk=1

9 and ∞ ξ [ T σ (xH, ξ)= ξ(x)∗ h (xH)g (ξ)∗, (xH, ξ) G/H G/H. H T k k ∈ × Xk=1 b p1 p2 Proof. Suppose that T : L (G/H) L (G/H) is r-nuclear, where 1 p1,p2 < . Then by ′ → ∞ p ∞ ≤p ∞ Theorem 4.1, there exist sequences g in L 1 (G/H) and h in L 2 (G/H) such that { k}k=1 { k}k=1 ∞ r r ′ gk p hk p < k kL 1 (G/H) k kL 2 (G/H) ∞ Xk=1 and for all f Lp1 (G/H) we have ∈

(Tf) (xH)= dπ Tr(Γπ(xH)σT (xH, π)fˆ(π)) [ [π]X∈G/H dπ = dπ (Γπ(xH)σT (xH, π))ij fˆ(π)ji [ i,j=1 [π]X∈G/H X dπ = dπ (Γπ(xH)σT (xH, π))ij Γπ(wH)ij f(wH) dµ(wH) G/H [ i,j=1 Z [π]X∈G/H X ∞ (4.2) = hk(xH)gk(wH) f(wH) dµ(wH) G/H ! Z kX=1 [ for all xH G/H. Let ξ be a ﬁxed but arbitrary element in G/H. Then for 1 m,n dξ we deﬁne the function∈ f on G/H by ≤ ≤ f(wH)=Γ (wH) , wH G/H. ξ nm ∈ Since 1 Γ (wH) Γ (wH) dµ(wH)= ξ nm π ij d ZG/H ξ if and only if π = ξ, i = n and j = m, and is zero otherwise, it follows from (4.2) that

∞ (Γξ(xH)σT (xH, ξ))nm = hk(xH) gk(wH) Γξ(wH)nm dµ(wH) k=1 ZG/H ! X∞ = hk(xH) gk(ξ) . mn kX=1 Therefore, b ∞ ξ [ T σ (xH, ξ)= ξ(x)∗ h (xH)g (ξ)∗, (xH, ξ) G/H G/H. H T k k ∈ × Xk=1 ∞ ′ ∞ b p1 p2 Conversely, suppose that there exist sequences gk k=1 in L (G/H) and hk k=1 in L (G/H) such that { } { } ∞ r r ′ gk p hk p < k kL 1 (G/H) k kL 2 (G/H) ∞ Xk=1

10 and ∞ ξ [ T σ (xH, ξ)= ξ(x)∗ h (xH)g (ξ)∗, (xH, ξ) G/H G/H. H T k k ∈ × Xk=1 Then, for all f Lp1 (G/H) b ∈ (Tf) (xH)= dξ Tr(Γξ(xH)σT (xH, ξ)fˆ(ξ)) [ [ξ]X∈G/H dξ

= dξ (Γξ(xH)σT (xH, ξ))nmfˆ(ξ)mn [ m,n=1 [ξ]X∈G/H X dξ ∞ ∗ ˆ = dξ hk(xH)gk(ξ) nm f(ξ)mn [ m,n=1 ! [ξ]X∈G/H X kX=1 b dξ ∞ = dξ hk(xH)gk(ξ)mn fˆ(ξ)mn [ m,n=1 ! [ξ]X∈G/H X kX=1 b dξ ∞ = dξ hk(xH) gk(wH)Γξ(wH)nm dµ(wH) fˆ(ξ)mn [ m,n=1 G/H ! [ξ]X∈G/H X kX=1 Z

dξ ∞ ˆ =  dξ Γξ(wH)nmf(ξ)mn hk(xH)gk(wH) dµ(wH) G/H [ Z [ξ]∈G/H m,n=1 k=1  X X  X   ∞ ˆ =  dξ Tr(Γξ(wH)f(ξ)) hk(xH)gk(wH) dµ(wH) G/H [ Z [ξ]∈G/H k=1  X  X  ∞  = hk(xH)gk(wH) f(wH) dµ(wH) G/H ! Z Xk=1 for all xH G/H. Therefore by Theorem 4.1, it follows that T is r-nuclear. ∈ In the next theorem, we will give another characterization of r-nuclear pseudo-diﬀerential operators from Lp1 (G/H) into Lp2 (G/H) in order to ﬁnd trace of r-nuclear operators from Lp(G/H) into Lp(G/H).

p1 p2 Theorem 4.4. Let 0 < r 1 and let T : L (G/H) L (G/H), 1 p1,p2 < , be a continuous ≤ → [ ≤ ∞ linear operator with the matrix-valued symbol σT on G/H G/H. Then the pseudo-diﬀerential × ∞ ′ ∞ operator T is r-nuclear if and only if there exist sequences g in Lp1 (G/H) and h in { k}k=1 { k}k=1 Lp2 (G/H) such that ∞ r r ′ gk p hk p < k kL 1 (G/H) k kL 2 (G/H) ∞ Xk=1 and ∞ ∗ dξ Tr(Γξ(xH)σT (xH, ξ)Γξ(yH) )= hk(xH)gk(yH). [ [ξ]X∈G/H kX=1

11 p1 p2 Proof. Suppose that T : L (G/H) L (G/H) is a r-nuclear operator for 1 p1,p2 < . Then ′ → ∞ p ∞ ≤p ∞ by Theorem 4.3, there exist sequences g in L 1 (G/H) and h in L 2 (G/H) such that { k}k=1 { k}k=1 ∞ r r ′ gk p hk p < k kL 1 (G/H) k kL 2 (G/H) ∞ Xk=1 and ∞ [ (Γξ(xH)σT (xH, ξ))nm = hk(xH) gk(ξ) , (xH, ξ) G/H G/H, mn ∈ × Xk=1 for all n,m with 1 n,m dξ. b Let yH G/H.≤ Then ≤ ∈ ∞

(Γξ(xH)σT (xH, ξ))nmΓξ(yH)nm = hk(xH) gk(ξ) Γξ(yH)nm mn kX=1 b ∞ = Γξ(zH)nmΓξ(yH)nm hk(xH)gk(zH)dµ(zH). G/H Z Xk=1 So we have

dξ

(Γξ(xH)σT (xH, ξ))nmΓξ(yH)nm m,n=1 X dξ ∞ = Γ (zH) Γ (yH) h (xH)g (zH)dµ(zH).  ξ nm ξ nm k k G/H m,n=1 Z X Xk=1   Therefore, for all xH, yH G/H, we get ∈ ∗ dξ Tr(Γξ(xH)σT (xH, ξ)Γξ (yH) ) [ [ξ]X∈G/H

∞ ∗ =  dξ Tr (Γξ(zH)Γξ(yH) ) hk(xH)gk(zH)dµ(zH) G/H [ Z [ξ]∈G/H k=1  X  X   ∞ ∗ =  dξTr (Γξ(yH)Γξ(zH) ) hk(xH)gk(zH)dµ(zH) G/H [ Z [ξ]∈G/H k=1  X  X   ∞ ∗ = hk(xH)  dξTr (Γξ(yH)Γξ(zH) )gk(zH) G/H [ k=1 Z [ξ]∈G/H X  X    ∞ ∗ = hk(xH)  dξTr Γξ(yH)Γξ(zH) gk(zH)  G/H [ k=1 Z [ξ]∈G/H X  X    ∞ ∞ = hk(xH) dξ Tr Γξ(yH)g(ξ)  = hk(xH) g(yH) [ k=1 [ξ]∈G/H k=1 X  X  X  b 

12 ∞ = hk(xH)gk(yH) kX=1 for all xH,yH in G/H. ′ ∞ ∞ p p Conversely, let g and h be sequences in L 1 (G/H) and L 2 (G/H) such that { k}k=1 { k}k=1 ∞ r r ′ gk p hk p < k kL 1 (G/H) k kL 2 (G/H) ∞ Xk=1 and for all xH and yH in G/H, ∞ ∗ dξ Tr(Γξ(xH)σ(xH, ξ)Γξ (yH) )= hk(xH)gk(yH). [ [ξ]X∈G/H kX=1 Then, for all f Lp1 (G/H) ∈ (Tσf) (xH)= dξ Tr(Γξ(xH)σ(xH, ξ)fˆ(ξ)) [ [ξ]X∈G/H dξ

= dξ (Γξ(xH)σ(xH, ξ))mnfˆ(ξ)nm [ m,n=1 [ξ]X∈G/H X

dξ =  dξ (Γξ(xH)σ(xH, ξ))mnΓξ(yH)mn f(yH) dµ(yH) G/H [ Z [ξ]∈G/H m,n=1  X X    ∗ =  dξ Tr(Γξ(xH)σ(xH, ξ)Γξ (yH) ) f(yH) dµ(yH) G/H [ Z [ξ]∈G/H  X   ∞  = hk(xH)gk(yH) f(yH) dµ(yH) G/H ! Z Xk=1 for all xH G/H. This completes the proof. ∈ An immediate consequence of Theorem 4.4 gives the trace of a r-nuclear pseudo-diﬀerential operator on Lp(G/H) for 1 p< . Indeed, we have the following result. ≤ ∞ Corollary 4.5. Let 0 < r 1 and let T : Lp(G/H) Lp(G/H), 1 p < , be a r-nuclear ≤ [→ ≤ ∞ operator with the matrix-valued symbol σT on G/H G/H. Then the nuclear trace of T is given by × ξ Tr (T )= dξ Tr(TH σT (xH, ξ))dµ(xH). G/H [ Z [ξ]X∈G/H

Proof. Using trace formula (4.1) and Theorem 4.4 we have ∞ Tr (T )= hk(xH)gk(xH)dµ(xH) G/H Z Xk=1

13 ∗ = dξ Tr(Γξ(xH)σT (xH, ξ)Γξ (xH) ). G/H [ Z [ξ]X∈G/H ξ Since TH is an orthogonal projection therefore,

ξ Tr (T )= dξ Tr(TH σT (xH, ξ))dµ(xH). G/H [ Z [ξ]X∈G/H

5. Adjoint and product of nuclear pseudo-differential operators In this section we give a formula for the symbols of the adjoints of r-nuclear pseudo-diﬀerential p1 p2 operators from L (G/H) into L (G/H) for 1 p1,p2 < , where G is a compact Hausdorﬀ group and H be a closed subgroup of G. ≤ ∞

p1 p2 Theorem 5.1. Let 0 < r 1 and let T : L (G/H) L (G/H), 1 p1,p2 < , be a continuous ≤ → [ ≤ ∞ ∗ linear operator with the matrix-valued symbol σT on G/H G/H. Then the adjoint T of T is also ′ ′ p p × a r-nuclear operator from L 2 (G/H) into L 1 (G/H) with symbol τ given by ∞ ξ [ T τ(xH, ξ)= ξ(x)∗ g (xH)h (ξ)∗, (xH, ξ) G/H G/H, H k k ∈ × Xk=1 ∞ ∞ ′ where g and h are two sequences incLp1 (G/H) and Lp2 (G/H) respectively such that { k}k=1 { k}k=1 ∞ r r ′ h p g p < . k kkL 2 (G/H) k kkL 1 (G/H) ∞ Xk=1

′ Proof. For f Lp1 (G/H) and g Lp2 (G/H), from the deﬁnition of the adjoint of an operator we have ∈ ∈ (Tf) (xH)g(xH) dµ(xH)= f(xH)(T ∗g) (xH) dµ(xH). ZG/H ZG/H Therefore,

dξ

dξ (Γξ(xH)σT (xH, ξ))mn G/H G/H [ m,n=1 Z Z [ξ]X∈G/H X

(5.1) Γξ(yH)mnf(yH) dµ(yH) g(xH) dµ(xH) × !

dξ

= f(xH) dξ (Γξ(xH)τ(xH, ξ))mn G/H G/H [ m,n=1 Z Z [ξ]X∈G/H X

Γξ(yH)mng(yH) dµ(yH) dµ(xH). × ! [ Now, let γ and η be elements in G/H. Then for 1 i, j dγ and 1 p, q dη, we deﬁne the functions f and g on G/H by ≤ ≤ ≤ ≤ f(xH)=Γ (xH) , xH G/H γ ij ∈

14 and g(xH)=Γ (xH) , xH G/H. η pq ∈ Therefore, from (5.1) it folllows that

(Γγ (xH)σT (xH,γ))ij Γη(xH)pq dµ(xH) ZG/H

= Γγ(xH)ij (Γη(xH)τ(xH, η))pq dµ(xH) ZG/H and so

(Γγ (xH)σT (xH,γ))ij Γη(xH)pq dµ(xH) ZG/H

= (Γη(xH)τ(xH, η))pq Γγ(xH)ij dµ(xH). ZG/H Thus, (5.2) (((Γ ( )σ ( ,γ)) )∧(η)) = (((Γ ( )τ( , η)) )∧(γ)) γ · T · ij qp η · · pq ji

[ p1 p2 for 1 i, j dγ , 1 p, q dη and all γ and η in G/H. Since T : L (G/H) L (G/H) is ≤ ≤ ≤ ≤ ∞ ′ ∞→ p1 p2 r-nuclear, by Theorem 4.3, there exist sequences gk k=1 in L (G/H) and hk k=1 in L (G/H) such that { } { } ∞ r r ′ h p g p < k kkL 2 (G/H) k kkL 1 (G/H) ∞ kX=1 [ and for all (yH,γ) G/H G/H, ∈ × ∞ (Γγ (yH)σT (yH,γ))ij = hk(yH) gk(γ) . ji kX=1 [ b So, for all (xH, η) G/H G/H ∈ × ∧ (Γη(xH)τ(xH, η))pq = dγ Tr[Γγ (xH)((Γη( )τ( , η))pq ) (γ)] [ · · [γ]X∈G/H dγ = d (Γ (xH)) (((Γ ( )τ( , η)) )∧(γ)) . γ γ ij η · · pq ji [ i,j=1 [γ]X∈G/H X [ Therefore, for all (xH, η) G/H G/H, we get from (5.2) ∈ ×

(Γη(xH)τ(xH, η))pq

dγ = d (Γ (xH)) (((Γ ( )σ ( ,γ)) )∧(η)) γ γ ij γ · T · ij qp [ i,j=1 [γ]X∈G/H X dγ

= dγ (Γγ (xH))ij (Γγ (yH)σT (yH,γ))ij Γη(yH)pq dµ(yH) [ i,j=1 G/H [γ]X∈G/H X Z

15 dγ ∞

= dγ (Γγ (xH))ij hk(yH)(gk(γ))jiΓη(yH)pq dµ(yH) [ i,j=1 G/H [γ]X∈G/H X Z kX=1 ∞ b dγ = hk(yH)Γη(yH)pq dµ(yH) dγ (Γγ(xH))ij (gk(γ))ji G/H ! [ i,j=1 kX=1 Z [γ]X∈G/H X b ∞ dγ = hk(η)qp dγ (Γγ (xH))ij (gk(γ))ji [ i,j=1 kX=1 [γ]X∈G/H X ∞ c b = hk(η)qp dγ Tr(Γγ(xH)gk(γ)) [ kX=1 [γ]X∈G/H ∞ c ∞ b ∗ = hk(η)qpgk(xH)= hk(η)pqgk(xH) kX=1 kX=1 c c for all 1 p, q dη. ≤ ≤ [ Therefore, for all (xH, η) G/H G/H, we get ∈ × ∞ ∗ Γη(xH)τ(xH, η)= hk(η) gk(xH) Xk=1 and hence c ∞ η ∗ ∗ TH τ(xH, η)= η(x) hk(η) gk(xH). kX=1 c

As an application of Theorem 4.3 and Theorem 5.1, in the next corollary we give a criterion for the self-adjointness of r-nuclear pseudo-diﬀerential operators.

Corollary 5.2. Let 0 < r 1 and let T : L2(G/H) L2(G/H) be a r-nuclear continuous linear ≤ →[ operator with the matrix-valued symbol σT on G/H G/H. Then T is self-adjoint if and only if there exist sequences g ∞ and h ∞ in L2(G/H× ) such that { k}k=1 { k}k=1 ∞ r r h 2 g 2 < , k kkL (G/H) k kkL (G/H) ∞ Xk=1 ∞ ∞ [ h (xH)g (ξ)∗ = h (ξ)∗ g (xH), (xH, ξ) G/H G/H, k k k k ∈ × kX=1 kX=1 and b c ∞ ξ [ T σ (xH, ξ)= ξ(x)∗ h (xH)g (ξ)∗, (xH, ξ) G/H G/H. H T k k ∈ × Xk=1 We can give another formula for the adjoints of rb-nuclear operators in terms of symbols. Indeed, we have the following theorem.

16 [ Theorem 5.3. Let 0 < r 1. Let σT be a matrix-valued function on G/H G/H such that the ≤ p1 p2 × corresponding pseudo-diﬀerential operator T : L (G/H) L (G/H) is r-nuclear for 1 p1,p2 < ′ ′ ∗ p →p ≤ . Then the symbol τ of the adjoint T : L 2 (G/H) L 1 (G/H) is given by ∞ → ξ ∗ ∗ TH τ(xH, ξ)= ξ(x) dη Tr[(Γη(yH)σT (yH,η)) Γη(xH)]Γξ(yH)dµ(yH) [ G/H [η]X∈G/H Z which is eventually same as

∧ ∗ ξ ∗ ∗ ∗ TH τ(xH, ξ)= ξ(x) dη Tr(σT ( , η) Γη( ) Γη(xH)) (ξ) [ · · [η]X∈G/H [ for all (xH, ξ) G/H G/H. ∈ ×

p1 p2 Proof. Suppose that T : L (G/H) L (G/H) is r-nuclear operator for 1 p1,p2 < . Then ′ → ∞ p ∞ ≤p ∞ by Theorem 4.3 , there exist sequences g in L 1 (G/H) and h in L 2 (G/H) such that { k}k=1 { k}k=1 ∞ r r ′ gk p hk p < k kL 1 (G/H) k kL 2 (G/H) ∞ Xk=1 [ and for all (yH,η) G/H G/H we have ∈ × ∞ ∗ Γη(yH)σT (yH,η)= hk(yH)gk(η) Xk=1 or b ∞ ∗ (Γη(yH)σT (yH,η)) = hk(yH)gk(η). Xk=1 [ Let (xH, ξ) G/H G/H. Then b ∈ × ∗ Tr[(Γη(yH)σT (yH,η)) Γη(xH)]Γξ(yH) dµ(yH) ZG/H ∞ = Tr hk(yH)gk(η)Γη(xH) Γξ(yH) dµ(yH) G/H " # Z kX=1 ∞ b = Tr(gk(η)Γη(xH)) hk(yH)Γξ(yH) dµ(yH) G/H Xk=1 Z ∞ b ∗ = hk(ξ) Tr[gk(η)Γη(xH)]. Xk=1 Thus by Theorem 5.1, c b

∗ dη Tr[(Γη(yH)σT (yH,η)) Γη(xH)]Γξ(yH) dµ(yH) [ G/H [η]X∈G/H Z ∞ ∗ = dη hk(ξ) Tr[gk(η)Γη(xH)] [ ! [η]X∈G/H Xk=1 c b

17 ∞ ∗ = hk(ξ) dη Tr[gk(η)Γη(xH)] [ Xk=1 [η]X∈G/H ∞ c b ∗ = hk(ξ) gk(xH)=Γξ(xH)τ(xH, ξ) Xk=1 c[ for all (xH, ξ) G/H G/H. ∈ × Another criterion for the self-adjointness of r-nuclear pseudo-differential operators on homogeneous space of compact groups is as follows. [ Corollary 5.4. Let 0 < r 1. Let σT be a matrix-valued function on G/H G/H such that T : L2(G/H) L2(G/H) is≤r-nuclear. Then T : L2(G/H) L2(G/H) is self-adjoint× if and only if → → ∧ ∗ ξ ∗ ∗ ∗ TH σT (xH, ξ)= ξ(x) dη Tr(σT ( , η) Γη( ) Γη(xH)) (ξ) [ · · [η]X∈G/H [ for all (xH, ξ) G/H G/H. ∈ × Now, we show that the product of a nuclear pseudo-differential operator on Lp(G/H) with a bounded operator again a nuclear pseudo-differential operator on Lp(G/H) for 1 p < , where G is a compact (Hausdorff) group and H is a closed subgroup of G. We present a≤ formula∞ for the symbol of product operator. The main theorem of this section is stated below. Theorem 5.5. Let T : Lp(G/H) Lp(G/H), 1 p < , be a nuclear operator with a matrix p → p ≤ ∞ valued symbol σT and let S : L (G/H) L (G/H) be a bounded linear operator with symbol σS . Then ST : Lp(G/H) Lp(G/H) is a nuclear→ operator with symbol λ given by → ∞ ξ ∗ ′ ∗ TH λ(xH, ξ)= ξ(x) hk(xH)gk(ξ) Xk=1 ′ [ ∞ ∞ b p for all (xH, ξ) G/H G/H, where gk k=1 and hk k=1 are two sequences in L (G/H) and ∞ p ∈ × { } ′ { } L (G/H) respectively such that g p h p < with k=1 k kkL (G/H) k kkL (G/H) ∞ ′ P hk(xH)= dη Tr Γη(xH)σS (xH, η)hk(η) xH G/H. [ ∈ [η]X∈G/H h i c Proof. Since T : Lp(G/H) Lp(G/H) is a nuclear pseudo-differential operator for 1 p < , → ′ ≤ ∞ from Theorem 4.3, there exist sequences g ∞ Lp (G/H) and h ∞ Lp(G/H) such that { k}k=1 ∈ { k}k=1 ∈ ∞

′ g p h p < k kkL (G/H) k kkL (G/H) ∞ Xk=1 and ∞ ξ [ T σ (xH, ξ)= ξ(x)∗ h (xH)g (ξ)∗, (xH, ξ) G/H G/H. H T k k ∈ × Xk=1 Let f LP (G/H). Then b ∈ (STf)(xH)= dη Tr(Γη(xH)σS (xH, η)Tf(η)) [ [η]X∈G/H c

18 ∗ = dη Tr Γη(xH)σS (xH, η) Tf(yH)Γη(yH) dµ(yH) [ " G/H !# [η]X∈G/H Z

= dη Tr Γη(xH)σS (xH, η) [ " G/H [η]X∈G/H Z

∗  dξ Tr(Γξ(yH)σT (yH,ξ)f(ξ)) Γη(yH) dµ(yH) × [ [ξ]∈G/H  X    b   for all xH G/H. Using the nuclearity of T , we have ∈

(STf)(xH)= dη Tr Γη(xH)σS (xH, η) dξ [ G/H [ [η]∈G/H Z [ξ]∈G/H X  X ∞ ∗ ∗ Tr hk(yH)gk(ξ) f(ξ) Γη(yH) dµ(yH) × ! # kX=1 b b ∞ ∗ = dη Tr Γη(xH)σS (xH, η) dξ Tr gk(ξ) f(ξ) [ [ [η]∈G/H [ξ]∈G/H k=1 X  X X  b b h (yH)Γ (yH)∗dµ(yH) × k η ZG/H # ∞ ∗ = dη Tr Γη(xH)σS (xH, η) dξhk(η) Tr gk(ξ) f(ξ)  [ [ [η]∈G/H [ξ]∈G/H k=1 X  X X  ∞  c b b  ∗ = dξdη Tr Γη(xH)σS (xH, η)hk(η) Tr gk(ξ) f(ξ) [ [ [η]X∈G/H kX=1 [ξ]X∈G/H h i c b b = dξ Tr Γξ(xH)λ(xH, ξ)f (ξ) , [ [ξ]X∈G/H b where ∞ ξ ∗ ∗ TH λ(xH, ξ)= ξ(x) dη Tr Γη(xH)σS (xH, η)hk(η) gk(ξ) [ kX=1 [η]X∈G/H h i ∞ c b ∗ ′ ∗ = ξ(x) hk(xH)gk(ξ) kX=1 [ for all (xH, ξ) G/H G/H, where b ∈ × ′ hk(xH)= dη Tr Γη(xH)σS (xH, η)hk(η) , xH G/H. [ ∈ [η]X∈G/H h i c

19 6. Applications to heat kernels In this section, we assume that G is a compact Lie group and H is a closed subgroup of G. Let be the Laplace-Beltrami operator (or the Casimir element of the universal enveloping algebra) LG on G. For every [ξ] G, the matrix elements of ξ are the eigenfunctions of G with same eigenvalue denoted by λ2 . Therefore,∈ L − [ξ] b ξ = λ2 ξ for all 1 i, j d . −LG ij − [ξ] ij ≤ ≤ ξ Let : C∞(G/H) C∞(G/H) be the diﬀerential operator on G/H obtained by acting −LG/H → −LG on functions that are constant on cosets of G, i.e., such that ^f = f for f C∞(G/H), −LG/H −LG ∈ where for f C∞(G/H), f˜ C∞(G) is the lifting of f given by f˜(x) = f(xH). The operator has the∈ eigenspace Γ∈ (xH) for 1 i, j d corresponding to thee common eigenvalue −LG/H ξij ≤ ≤ ξ λ2 . For more details on see [34]. We make use the symbol of the heat kernel e−tLG/H . − [ξ] −LG/H −t|ξ|2 ξ 2 Indeed, by taking in to account σ −tL (xH, ξ)= e T , where ξ = λ , we have e G/H H | | [ξ]

−tLG/H − L e f(xH)= dξTr(Γξ(xH)σe t G/H (xH, ξ)f(ξ)) [ [ξ]X∈G/H 2 b −tλ[ξ] ξ = dξTr(Γξ(xH)e TH f(ξ)) [ [ξ]X∈G/H −tλ2 b = dξe [ξ] Tr(Γξ(xH)f(ξ)). [ [ξ]X∈G/H b Now, we show that the nuclearity of heat kernel on Lp-spaces. Theorem 6.1. Let G be a compact Lie group and let H be closed subgroup of G. Then the heat kernel e−tLG/H : Lp1 (G/H) Lp2 (G/H) is nuclear for every t > 0 and all 1 p , p < . → ≤ 1 2 ∞ Moreover, if 0 < r 1, then e−tLG/H : Lp(G/H) Lp(G/H) is r-nuclear operator for every t> 0 and 1 p< . In≤ particular, on each Lp(G/H),→we have the following nuclear trace formula ≤ ∞ 2 −tLG/H −tλ[ξ] ξ Tr (e )= dξe Tr (TH ). [ [ξ]X∈G/H Proof. The kernel of e−tLG/H is given by

−tλ2 ∗ Kt(x, y)= dξe [ξ] Tr(Γξ(xH)Γ(yH) ) [ [ξ]X∈G/H −tλ2 ∗ = dξe [ξ] Tr(Γξ(xH)ξ(y) ) [ [ξ]X∈G/H with dξ ∗ Tr(Γξ(xH)ξ(y) )= Γξ(xH)ij ξ(y)ij . i,j X We set 2 −tλ[ξ] hξ,ij = dξe Γξ(xH)ij gξ,ij = ξ(y)ij . We p′ denotes the Lebesgue conjugate of p andq ˜ = max 2,p′ then by Lemma 2.4 we get 1 1 1 { 1} 1 − q˜ ′ ′ ′ 1 gξ,ij p = ξ p Γξ( )ij p d . k kL 1 (G/H) k ij kL 1 (G/H) ≤k · kL 1 (G/H) ≤ ξ

20 Also, we have

2 −tλ[ξ] h p2 = d e Γ (xH) p2 k ξ,ij kL (G/H) k ξ ξ ij kL (G/H) 2 2 −tλ[ξ] −tλ[ξ] d e Γ ( ) p2 d e . ≤k ξ k ξ · kopkL (G/H) ≤ ξ Therefore,

1 − q˜ 2 ′ 2 1 −tλ[ξ] g ( ) p h ( ) p2 d d e < , ξ,ij L 1 (G/H) ξ,ij L (G/H) ξ ξ k · k k · k ≤ [ ∞ [ξX],i,j [ξ]X∈G/H the last convergence is follows from any of the Weyl formula, see, for example [9]. Therefore, e−tLG/H is a nuclear operator. Similarly one can prove r-nuclearity of e−tLG/H . By Corollary 4.5 and by using the fact that measure µ on G/H is normalized, the nuclear trace formula of e−tLG/H given by

2 2 −tLG/H −tλ[ξ] ξ −tλ[ξ] ξ Tr(e )= dξ Tr(e TH )= dξe Tr(TH ). G/H [ [ Z [ξ]X∈G/H [ξ]X∈G/H

7. Acknowledgement Vishvesh Kumar is supported by Odysseus I Project (by FWO, Belgium) of Prof. Michael Ruzhansky. The authors thank Prof. Michael Ruzhansky for providing many valuable comments. Vishvesh Kumar thanks Duván Cardona for many fruitful discussions. Shyam Swarup Mondal thanks the Council of Scientific and Industrial Research, India, for providing financial support. He also thanks his supervisor Jitendriya Swain for his support and encouragement.

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Vishvesh Kumar, Ph. D. Department of Mathematics: Analysis, Logic and Discrete Mathematics Ghent University Krijgslaan 281, Building S8, B 9000 Ghent, Belgium . E-mail address: [email protected]

Shyam Swarup Mondal Department of Mathematics IIT Guwahati Guwahati, Assam, India. E-mail address: [email protected]