Fourier multipliers for Triebel-Lizorkin spaces on compact Lie groups

We investigate the boundedness of Fourier multipliers on a compact Lie group when acting on Triebel-Lizorkin spaces. Criteria are given in terms of the H\"ormander-Mihlin-Marcinkiewicz condition. In our analysis, we use the difference structure of the unitary dual of a compact Lie group. Our results cover the sharp H\"ormander-Mihlin theorem on Lebesgue spaces and also other historical results on the subject.


Introduction
Let G be a compact Lie group. In this work we study sufficient conditions for the boundedness of Fourier multipliers on the Triebel Lizorkin spaces F r p,q (G) in terms of the Hörmander-Mihlin condition on their symbols. The Littlewood-Paley theorem states that L p (G) ≡ F 0 p,2 (G), so that in view of the classical results of Hörmander-Mihlin type (see Hörmander [34] and Mihlin [36] for instance), Triebel-Lizorkin spaces are a good substitute of L p -spaces, when considering smoothness of distributions in different scales (see Triebel [50,51] and [37,38] for details). e −2πix·ξ f (x)dx, (1.1) it was observed by Marcinkiewicz in his classical 1939's work [35] that the condition sup ξ∈Z |σ(ξ)| + sup assures the existence of a bounded extension of A on L p (T), 1 < p < ∞. Denoting the difference operator on the lattice Z, by ∆σ := σ(· + 1) − σ, and by ∆ k , k ∈ N 0 , its successive iterations, the Marcinkiewicz condition (1.2) is satisfied by any sequence (σ(ξ)) ξ∈Z n such that |∆ k σ(ξ)| k |ξ| −k , ξ = 0, k = 0, 1, (1.3) which should be, in principle, more easier to verify that (1.2). Another generalisation of Marcinkiewicz's criterion for multipliers of the Fourier transform on R n , was done by Mihlin [36], who stated that a function σ ∈ C ∞ (R n \ {0}), satisfying estimates of the kind |∂ α ξ σ(ξ)| α |ξ| −|α| , |α| ≤ [n/2] + 1, (1.4) has a multiplier A (of the Fourier transform 1 on R n ) defined by Af (x) ≡ T σ f (x) := R n e 2πix·ξ σ(ξ) f (ξ)dξ, f ∈ C ∞ 0 (R n ), (1.5) admitting a bounded extension on L p (R n ), for 1 < p < ∞. Subsequent generalisations to Mihlin's theorem were done by Hörmander [34], Calderón and Torchinsky in [4], Taibleson and Weiss [49], Baernstein and Sawyer [3], Seeger [45,46,47] and many others. We refer the reader to Grafakos' paper [30] (and reference therein) for a complete historical revision and for recent developments about Milhin-Hörmander and Marcinkiewicz multiplier theorems on R n . Extensions of Marcinkiewicz, and Hörmander-Mihlin criteria have been proved in the context on Lie groups and several spaces of homogeneous type in the context of spectral multipliers of self-adjoint operators, e.g., sub-Laplacians, or of other operators with heat kernels satisfying Gaussian estimates, with general contexts that go beyond of the objective of this paper. In view of the extensive literature on the field, we will not review it here, but we refer the reader to [1,2,7,13,14,19,48,53] and to the extensive list of references therein.
In the framework of Fourier multipliers on compact Lie groups, by using the Calderón-Zygmund type theory in Coifman and De Guzmán [16], the L p -Fourier multipliers for SU (2) were investigated by Coifman and Weiss in their classical works [17,18]. Subelliptic Spectral multipliers for L p (SU(2)) were also considered in Cowling and Sikora [14]. Later on, criteria for the L p -boundedness of Fourier multipliers for arbitrary compact Lie groups G were given in [44], with a generalisation in [10,Section 5] to L p -subelliptic Fourier multipliers.
One of the notable questions when studying the qualitative properties for multipliers on Lie groups is to endow (the spaces of discrete functions on) the unitary dual with a difference structure. So, fixing the unitary dual G of an arbitrary compact Lie group G, when generalising the Marcinkiewicz condition on a Fourier multiplier A on G, associated to a sequence (called the symbol of A) σ : (1.6) the following questions arise: (Q1): how to define the difference operators ∆ α on G, in such a way that they generalise the usual notion of difference operators on Z ∼ = T? (Q2): which is the required order for the differences operators applied to σ in order that A admits a bounded extension on L p (G)? We note that (Q1) was satisfactorily solved in [44] by introducing a family of difference operators ∆ α := D α ξ , (defined in terms of the Fourier transform on G, as it was done in [40]) in terms of the unitary representations ξ : G → End(C ℓ ) 3 of G. About (Q2), the following Marcinkiewicz type theorem was proved in [44]. Theorem 1.1. Let us assume that G is a compact Lie group of dimension n. Let σ ∈ Σ( G) 4 be a symbol satisfying Then A ≡ T σ is of weak type (1, 1) and bounded on L p (G) for all 1 < p < ∞.

Moreover,
In the symbol condition (1.7), is the system of eigenvalues of the Bessel potential operator (1 + L G ) 1 2 associated to the Laplacian on G, which can be defined as follows. Taking an arbitrary orthonormal basis X g := {X 1 , · · · , X n } of the Lie algebra g of G, with respect to the Killing form on g, L G := n i=1 X 2 i . We refer the reader to Remark 2.1 for details about the definition of the difference operators D α = D α 1 1 · · · D αn n . They are compositions of differences operators D j of first order associated to the entries of the matrix-function ξ 0 (·) − I d ξ 0 for any choice of a unitary representation in every equivalence class [ξ 0 ] ∈ G. 2 Here, [ξ] denotes the equivalence class of a unitary, irreducible and continuous representation ξ : G → Hom(C d ξ ) on G, d ξ is the dimension of its representation space, σ(ξ) ∈ Hom(C d ξ ), and f (ξ) := G f (x)ξ(x) * dx, is the Fourier transform on the group G, of f ∈ C ∞ (G) at [ξ]. For instance, in the case of the torus G = T n , ξ(x) := e i2πx·ξ , x ∈ T n , d ξ ≡ 1, and so G ∼ = Z n . 3 We will always write d ξ := ℓ for the dimension of the representation space C ℓ . Also, I ℓ is the identity matrix of size ℓ × ℓ. Let us note that for graded Lie groups (e.g. the Heisenberg group, any stratified group and a wide class of nilpotent Lie groups where Rockland operators exist, see [26] for details) the Hörmander-Mihlin and the Marcinkiewicz conditions for L p , Triebel-Lizorkin and Hardy spaces have been investigated in [11,12,8,25].
In this work we investigate the Marcinkiewicz condition for multipliers on Triebel-Lizorkin spaces F r p,q (G) on G, extending in Theorem 1.4 to the case of compact Lie groups, the estimate of Seeger [47] for multipliers in Triebel-Lizorkin spaces F r p,q (R n ) on R n . In order to present our main result, let us define the Triebel-Lizorkin spaces F r p,q (G), as they were introduced by the second author, Nursultanov and Tikhonov in [38]. So, let us fix η ∈ C ∞ 0 (R + , [0, 1]), η = 0, so that supp(η) ⊂ [1/2, 2], and such that j∈Z η(2 −j λ) = 1, λ > 0.
(1.9) (1.10) and one can define the family of operators ψ j (B) using the functional calculus of the subelliptic Bessel potential B := (1 + L G ) 1 2 . Then, for 0 < q < ∞, and 1 < p < ∞, the Triebel-Lizorkin space F r p,q (G) consists of the distributions f ∈ D ′ (G) such that The weak-F r 1,q (G) space is defined by the distributions f ∈ D ′ (G) such that Then A ≡ T σ extends to a bounded operator from F r p,q (G) into F r p,q (G) for all 1 < p, q < ∞, and all r ∈ R. For p = 1, A admits a bounded extension from F r 1,q (G) into weak-F r 1,q (G). Moreover  [29]), we have L p (G) = F 0 p,2 (G) for 1 < p < ∞, so that we recover the L p -bound in Theorem 1.1. Because we are not assuming that the Fourier multipliers are defined by the spectral calculus, Theorem 3.1 extends the main theorem in Weiss [53] and also the historical 1939's result due to Marcinkiwicz [35] (see Remark 3.10 and Corollary 3.11). As before, we refer the reader to Remark 2.1 for details about the definition of the difference operators D α = D α 1 1 · · · D αn n .

Preliminaries
2.1. The unitary dual and the Fourier transform. First, let us record the notion of the unitary dual G of a compact Lie group G. So, let us assume that ξ is a continuous, unitary and irreducible representation of G, this means that, • ξ ∈ Hom(G, U(H ξ )), for some finite-dimensional vector space H ξ ∼ = C d ξ , i.e. ξ(xy) = ξ(x)ξ(y) and for the adjoint of ξ(x), ξ(x) * = ξ(x −1 ), for every x, y ∈ G.
Let Rep(G) be the set of unitary, continuous and irreducible representations of G. The relation, for every x ∈ G, is an equivalence relation and the unitary dual of G, denoted by G is defined via G := Rep(G)/∼.
By a suitable changes of basis, we always can assume that every ξ is matrix-valued and that H ξ = C d ξ . If a representation ξ is unitary, then is a subgroup (of the group of matrices C d ξ ×d ξ ) which is isomorphic to the original group G. Thus the homomorphism ξ allows us to represent the compact Lie group G as a group of matrices. This is the motivation for the term 'representation'. Here, as usually, 2.2. Difference operators. Difference operators on compact Lie groups were introduced in [40] to endow the unitary dual of a compact Lie group with a difference structure. In terms of them, Hörmander classes of pseudo-differential operators on a compact Lie group can be characterised, see [41,42,43]. Same as in [44], where differences operators were used to study L p -multipliers we will extend that analysis to the case of Triebel-Lizorkin spaces (see [37,38] for instance).
We will denote by Σ( G) the space of matrix-valued functions, By following [44], a difference operator Q ξ of order k, can be applied to a symbol 1) where Q ξ is associated with a smooth function q vanishing of order k at the identity e = e G . We will denote by diff k ( G) the set of all difference operators of order k. For a fixed smooth function q, the associated difference operator will be denoted by ∆ q := Q ξ . We will choose an admissible collection of difference operators (see e.g. [44]),

We say that this admissible collection is strongly admissible if
Remark 2.1. A special type of difference operators can be defined by using the unitary representations of G. Indeed, if ξ 0 is a fixed irreducible and unitary representation of G, consider the matrix (2.2) Then, we associated to the function q ij (g) : 3) If the representation is fixed we omit the index ξ 0 so that, from a sequence D 1 = D ξ 0 ,j 1 ,i 1 , · · · , D n = D ξ 0 ,jn,in of operators of this type we define D α = D α 1 1 · · · D αn n , where α ∈ N n . Remark 2.2 (Leibniz rule for difference operators). The difference structure on the unitary dual G, induced by the difference operators acting on the momentum variable [ξ] ∈ G, implies the following Leibniz rule for a 1 , a 2 ∈ C ∞ (G, S ′ ( G)). For details we refer the reader to [40]. Remark 2.3 (Difference operators of fractional order and Sobolev spaces on the unitary dual). In the spirit of the Sobolev spaces on the unitary dual of a graded Lie group [24], Sobolev spaces also can be defined for the unitary dual of a compact Lie group. They can be defined as follows: Let ∆ q 1 , be a difference operator of first order associated to a smooth and non-negative function q 1 ≥ 0. For s ∈ R, the Sobolev So, for every s ∈ R, the difference operator ∆ s q 1 := ∆ q s 1 of fractional order s, can be defined in terms of the Fourier transform, via: 2.3. Calderón-Zygmund type estimates for multipliers. In order to provide L p -estimates for multipliers in the subelliptic context, we will use the techniques developed by the second author and J. Wirth in [44], where a special case (compatible with the notion of difference operators and the difference structure that they provide for the unitary dual) of a statement of Coifman and de Guzmán ( [16], Theorem 2) was established. We record it as follows (see [44, p. 630]).
The family {ψ r } r>0 that appears in Theorem 2.4 is defined by where the functions in the net {φ r } r>0 , satisfy, among other things, the following properties (see [44,Lemma 3.3]): where Ad : G → U(g), and ∆ 0 is the system of positive roots. It can be decomposed into irreducible representations as, where e G is the trivial representation. With the consideration on the centre Z(G) = {e G }, it can be shown (see Lemma 3.1 of [44]) that • ρ 2 (x) 0 and ρ(x) = 0 if and only if x = e G .
• ∆ ρ 2 ∈ diff 2 ( G). If G is not semi-simple, we refer the reader to [44,Remark 3.2] for the modifications in the definition of ρ, in this particular case. The construction of the functions φ r , is as follows. By choosingφ ∈ C ∞ 0 (R) such thatφ ≥ 0,φ(0) ≡ 1, and ∂ ℓ tφ (0) = 0, for ℓ ≥ 1, we define with the normalisation condition used to define c r . An equivalent statement to the Hörmander-Mihlin Theorem 2.4 will be given in terms of the family of functions ϕ r −1 ≡ ψ r −n , r > 0, (see Theorem 2.6) that provide a continuous dyadic decomposition.
2.4. L p (G)-boundedness of Fourier multipliers. In this subsection we recall the Hörmander-Mihlin theorem and the Marcinkiewicz-Mihlin theorem for L p -multipliers on compact Lie groups [44].
Remark 2.5. Let A be a Fourier multiplier with symbol σ. Let us observe that that Theorem 2.4 can be re-written in terms of the Sobolev spacesL 2 s ( G) on the unitary dual, showing that, (2.8) is indeed, an analogue on compact Lie groups of the Hörmander-Mihlin theorem in [34]. So, with the notation in Theorem 2.4, and making use of the Plancherel theorem, we have for the continuous dyadic approximation of the identity in (2.9). Theorem 2.6 (Hörmander-Mihlin Theorem for L p (G)). Let G be a compact Lie group and let s > n 2 . Let σ ∈ Σ( G) be a symbol satisfying Then A ≡ T σ is of weak type (1, 1) and bounded on L p (G) for all 1 < p < ∞.
We present Theorem 1.1 in the following form.
Theorem 2.7 (Marcinkiewicz-Mihlin Theorem). Let G be a compact Lie group and let κ ∈ 2N be such that κ > n 2 . Let σ ∈ Σ( G) be a symbol satisfying Then A ≡ T σ is of weak type (1, 1) and bounded on L p (G) for all 1 < p < ∞.
Remark 2.8. Several properties for the Sobolev spaces L 2 s ( G) on the unitary dual were established in [23]. In particular, the Hörmander-Mihlin Theorem 2.6 was reobtained, by observing that the L p (G)-boundedness and the weak (1,1) type of a Fourier multiplier, also can be obtained if one verifies the following condition: for any η = 0, η ∈ C ∞ 0 (R + 0 ), and s > n/2. Both Hörmander-Mihlin conditions ((2.12) or (2.13)) allow us to write Theorem 2.7 in the form of Theorem 1.1.
Remark 2.9. The boundedness of pseudo-differential operators on compact Lie groups, including oscillating Fourier multipliers, on L p , subelliptic Sobolev and Besov spaces can be found in [8,15] and [5,6,12]. The boundedness of Fourier multipliers on L p -spaces, Triebel-Lizorkin spaces and Hardy spaces on graded Lie groups can be found in [25,8] and [33], respectively.
Theorem 2.10. Let G be a compact Lie group. Then we have the following properties: ( for all r ∈ R, and all 1 < p < ∞, where L p r (G) are the standard Sobolev spaces on G. First, let us deduce the proof of Theorem 1.4 from the following result. The distribution ϑ t := η(tB)δ denotes the right-convolution kernel of the operator η(tB).

3)
with s > n 2 . Then A ≡ T σ extends to a bounded operator from F r p,q (G) into F r p,q (G) for all 1 < p, q < ∞, and all r ∈ R. For p = 1, A admits a bounded extension from Remark 3.2. Note that the Fourier transform of the distribution ϑ t := η(tB)δ is given by and that A :

G) is bounded and it satisfies the (Coifman-Weiss type) condition
uniformly in r > 0, the Plancherel Theorem makes the hypothesis in Theorem 3.1 equivalent to (3.6). Moreover, (3.4) is equivalent to (3.7) in view of (3.5).
Four our further analysis we will use the following auxiliary estimate.

8)
provided that s > n/2, s ∈ N. Moreover, the right-convolution kernel κ ℓ of Aψ ℓ , satisfies the uniform estimate in ℓ ∈ Z, Proof. That (3.9) is a consequence of (3.  (3.10) in the sense that if the right-hand side is finite then the left-hand side is finite and the inequality holds. So, applying (3.10) to τ = ψ ℓ ( ξ )I d ξ , we have Observe that, . Also, in view of Lemma 5.4 of [23], for any positive and smooth function f ∈ C ∞ 0 (R), supported in [0, a] with a > 0, one has which applied to f = ψ and t = 2 −ℓ , gives The analysis above shows that Thus, the proof is complete.

Proof of Theorem 1.4. In view of Lemma 3.3, we have that
In view of the Hörmander-Mihlin Theorem 3.1 applied to s := [ n 2 ]+1, A ≡ T σ extends to a bounded operator from F r p,q (G) into F r p,q (G) for all 1 < p, q < ∞, and all r ∈ R. For p = 1, A admits a bounded extension from F r 1,q (G) into weak-F r 1,q (G). Proof of Theorem 3.1. By observing that (1+B) r 2 : F r p,q (G) → F 0 p,q (G) and (1+B) − r 2 : F 0 p,q (G) → F r p,q (G) are both isomorphisms, it is enough to prove that A admits a bounded extension from F 0 p,q (G) into F 0 p,q (G). For this, let us define the vector-valued operator W : Observe that W is well-defined (bounded from L 2 (G, ℓ 2 (N 0 )) into L 2 (G, ℓ 2 (N 0 ))), because A admits a bounded extension on L 2 (G) and also, in view of the following estimate . So, observe that, in order to prove Theorem 3.1 it is enough to prove the following two lemmas and the estimate (3.4).
Proof. Indeed, by the Marcinkiewicz interpolation, these two lemmas are enough to show that W : L p (G, ℓ q (N 0 )) → L p (G, ℓ q (N 0 )) admits a bounded extension for all 1 < p ≤ q < ∞. The case 1 < q ≤ p < ∞ follows from the fact that L p ′ (G, ℓ q ′ (N 0 )) is the dual of L p (G, ℓ q (N 0 )) and also that Lemma 3.4 and Lemma 3.5 hold if we change A by its standard L 2 -adjoint.
Remark 3.7 (Lemmas 3.4 and 3.5 imply Theorem 3.1). By defining ψ −1 = ψ 0 , we have . Also note that from Lemma 3.5, we have . By observing that Lemmas 3.4 and 3.5 are enough for proving Theorem 3.1, we will proceed with their proofs. For this, let us recall that on any amenable topological group, and also in several spaces of homogeneous type one has the Calderón-Decomposition lemma. We fix it in the following remark.
Remark 3.8. For any non-negative function f ∈ L 1 (G), one has its Calderón-Zygmund decomposition. Indeed, by following Hebish [31], (whose construction remains valid for any amenable group, in particular, compact Lie groups) one can obtain a suitable Now, for every j ∈ N 0 , let us define R j by R j := sup{R > 0 : B(z j , R) ⊂ I j , for some z j ∈ I j }, (3.13) where B(z j , R) = {x ∈ I j : |z −1 j x| < R}. Every I j is bounded, and one can assume that I j ⊂ B(z j , 2R j ), where z j ∈ I j . Let us note that by assuming f (e G ) > t, (this can be done, just by re-defining f ∈ L 1 (G) at the identity element e G of G) we should have that e G ∈ j I j , (3.14) because f (x) ≤ t, for a.e. x ∈ G \ ∪ j≥0 I j .
Let us define, for every x ∈ I j , , (3.15) and for x ∈ G \ ∪ j≥0 I j , g(x) = f (x), b(x) = 0. Then, one has the decomposition f = g + b, with b having null average on I j .

Proof of Lemma 3.4.
Let us fix f ∈ L 1 (G) to be a non-negative function, and let us consider its Calderón-Zygmund decomposition f = g + b as in Remark 3.8. It is enough to demonstrate that the linear operators W ℓ , ℓ ∈ N 0 , are uniformly bounded on L q (G), 1 < q < ∞. This fact is straightforward if q = 2, so it is suffices (by the duality argument) that the operators W ℓ are uniformly bounded from L 1 (G) into L 1,∞ (G). So, we will prove the existence of C > 0, independent of f ∈ L 1 (G), and ℓ ∈ N 0 , such that Let us remark that for every x ∈ I j , By the Minkowski inequality, we have In view of the Chebyshev inequality, we get in view of the L 2 (G)-boundedness of A and the fact that the operators ψ ℓ (B) are L 2 (G)-bounded uniformly in ℓ ∈ N 0 . Additionally, note that the estimate Taking into account that b ≡ 0 on G \ ∪ j I j , we have that Let us assume that I * j is an open set, such that I j ⊂ I * j , and |I * j | = K|I j | for some K > 0, and dist(∂I * j , ∂I j ) ≥ 4c dist(∂I j , e G ), where c > 0 and e G is the identity element of G. So, by the Minkowski inequality we have Consequently, we deduce the estimates Now, using the Chebyshev inequality to estimate the right hand side above we obtain From now, let us denote by κ ℓ the right convolution kernel of W ℓ := Aψ ℓ (B). Observe that By using that the average of b k on I k is zero, Assuming the following uniform estimate we have So, if we prove the uniform estimate (3.18), we obtain the weak (1,1) inequality for f ∈ L 1 (G), f ≥ 0. For the proof of (3.18) let us use the estimates of the Calderón-Zygmund kernel of every operator W ℓ . Before continuing with the proof let us use the following geometrical property of the open sets I j , j ≥ 0: Lemma 3.9. Let us consider (3.14). Then, with the notation above, for any x ∈ G \ ∪ j I * j , and for all z ∈ I k , we have |z| |x|, i.e. for some c > 0, 4c|z| ≤ |x|.
We will postpone the proof of Lemma 3.9 for a moment to continue with the proof of Lemma 3.4, in order to use the estimate 4c|z| ≤ |x|, for x ∈ G \ ∪ j I * j , z ∈ I k , together with (3.9) in Lemma 3.3 for ℓ ≥ 0, as follows, The estimate above implies that A similar analysis, by splitting a complex function f ∈ L 1 (G) into its real and imaginary parts allows to conclude the weak (1,1) inequality (3.16) to complex functions. Thus, the proof of Lemma 3.4 would be complete if we prove Lemma 3.9. That lemma was proved in [8,Page 17] for graded Lie groups and the same proof applies for any amenable group (see Hebish [31]), we will present the proof here for the case of a compact Lie group G for completeness.
Proof of Lemma 3.9. In view of the estimate dist(∂I * j , ∂I j ) ≥ 4cdist(∂I j , e G ), we will prove that for x ∈ G \ ∪ j I * j , and all z ∈ I k , 4c|z| = 4c × dist(z, e G ) dist(∂I * k , ∂I k ) ≤ |x|. Indeed, fix ε > 0, and let us take w ∈ ∂I k , and w ′ ∈ ∂I * k such that d(w, w ′ ) ≤ dist(∂I k , ∂I * k ) + ε. Then, from the triangle inequality, we have (3.20) where in the last line we have assumed that diam(I k ) ≍ dist(∂I k , ∂I * k ), (with constants of proportionality independent in k) and that dist(∂I k , ∂I * k ) is proportional to R k in view of the relation |I * k | = K|I k |. Assuming (3.20), one has that for all ε > 0, d(z, e G ) dist(∂I k , ∂I * k ) + ε, which implies that d(z, e G ) dist(∂I k , ∂I * k ). (3.21) To show that the proportionality constant in (3.21) is uniform in k, let us recall the definition of the radii R ′ k s in (3.13), that B(z k , R k ) ⊂ I k ⊂ B(z k , 2R k ), and that B(z k , R k /C) ⊂ I * k ⊂ B(z k , CR k ) for some C > 2 independent of k, where for any k, z k ∈ I k . From this remark observe that: On the other hand, by observing that in every step above we can replace I * k := B(z k , CR k ), in view of the inclusion To show that dist(∂I * k , ∂I k ) ≤ |x|, observe that from Remark 3.14, e G ∈ ∪ j I j , and because of x ∈ G \ ∪ j I j , dist(∂I * k , ∂I k ) diam(∪ j I j ) d(x, e G ) = |x|. So, we have guaranteed the existence of a positive constant, which we again denote by c > 0, such that, {x ∈ G : x ∈ G \ ∪ j I * j } ⊂ {x ∈ G : for all z ∈ I k , 4c|z| ≤ |x|}.
Proof of Lemma 3.5. Now, we claim that extends to a bounded operator. For the proof of (3.22), we need to show that there exists a constant C > 0 independent of {f ℓ } ∈ L 1 (G, ℓ r (N 0 )) and t > 0, such that So, fix {f ℓ } ∈ L 1 (G, ℓ r (N 0 )) and t > 0, and let h(x) := ( ∞ ℓ=0 |f ℓ (x)| r ) 1 r . Apply the Calderón-Zygmund decomposition Lemma to h ∈ L 1 (G) in Remark 3.8 in order to obtain a disjoint collection , for all j. Now, we will define a suitable decomposition of f ℓ , for every ℓ ≥ 0. Recall that every I j is bounded, and that I j ⊂ B(z j , 2R j ), where z j ∈ I j (see Remark 3.8). Let us define, for every ℓ, and x ∈ I j , . (3.24) and for x ∈ G \ ∪ j≥0 I j , g ℓ (x) = f ℓ (x), b ℓ (x) = 0.
(3.25) So, for a.e. x ∈ G, f ℓ (x) = g ℓ (x)+b ℓ (x). Note that for every 1 < r < ∞, {g ℓ } r L r (ℓ r ) ≤ t r−1 {f ℓ } L 1 (ℓ r ) . Indeed, for x ∈ I j , Minkowski integral inequality gives, and from the fact that h(x) ≤ t, for a.e. x ∈ G \ ∪ j≥0 I j , we have Now, by using the Minkowski and the Chebyshev inequality, we obtain In view of Lemma 3.4, W : L r (G, ℓ r (N 0 )) → L r (G, ℓ r (N 0 )), extends to a bounded operator and Now, we only need to prove that   Taking into account that b ℓ ≡ 0 on G \ ∪ j I j , we have that Let us assume that I * j is a open set, such that |I * j | = K|I j | for some K > 0, and dist(∂I * j , ∂I j ) ≥ cdist(∂I j , e G ), where e G is the identity element of G. So, by the Minkowski inequality we have, Taking into account that Observe that the Chebyshev inequality implies Now, if κ ℓ is the right convolution Calderón-Zygmund kernel of W ℓ , and by using that I k b k,ℓ (y)dy = 0, we have that Now, we will proceed as follows. By using that |b ℓ,k (y)| r ≤ ∞ ℓ ′ =0 |b ℓ ′ ,k (y)| r , by an application of the Minkowski integral inequality, we have Consequently, we deduce, Let us recall that for x ∈ G \ ∪ j I * j , and y ∈ I k , in view of Lemma 3.9, we have that 4c|y| ≤ |x|. So, {x ∈ G : x ∈ G \ ∪ j I * j } ⊂ {x ∈ G : for all z ∈ I k , 4c|z| ≤ |x|}. Now, in view of (3.9) in Lemma 3.3 with ℓ ≥ 0, we deduce (3.19) and as a consequence we get Thus, we have proved the estimate    x ∈ G : Thus, the proof of the weak (1,1) inequality is complete and we have that W : L 1 (G, ℓ r (N 0 )) → L 1,∞ (G, ℓ r (N 0 )), 1 < r < ∞,  It was proved in [23], that for 0 ≤ s 0 ≤ n, a symbol satisfying (3.31) also satisfies the Hörmander-Mihlin condition (3.3) for any s > n/2. In view of Theorem 3.1, a Fourier multiplier A ≡ T σ satisfying (3.31) extends to a bounded operator from F r p,q (G) into F r p,q (G) for all 1 < p, q < ∞, and all r ∈ R, and for p = 1, A admits a bounded extension from F r 1,q (G) into weak-F r 1,q (G). In the case of the torus G = T, 1 < p, q < ∞, and n = 1 = s 0 , (3.31) is just (3.30), and A ≡ T σ extends to a bounded operator from F r p,q (T) into F r p,q (T) for all 1 < p, q < ∞, and all r ∈ R, and for p = 1, A admits a bounded extension from F r 1,q (T) into weak-F r 1,q (T). In view of the Littlewood-Paley theorem, for 1 < p < ∞, q = 2 and r = 0, the previous estimate recovers the classical Marcinkiewicz estimate [35]. We summarise this discussion in the following corollary.
Corollary 3.11. Let us assume that G is a compact Lie group of dimension n. Let σ ∈ Σ( G) be a symbol satisfying σ L ∞ ( G) + σ · 1 {[ξ]∈ G: 2 j−1 ≤ ξ <2 j } L1 s 0 ( G) 2 j(n−s 0 ) , j ≥ 0, (3.32) uniformly in j, for some 0 ≤ s 0 ≤ n, s 0 ∈ N. Then A ≡ T σ extends to a bounded operator from F r p,q (G) into F r p,q (G) for all 1 < p, q < ∞, and all r ∈ R. For p = 1, A admits a bounded extension from F r 1,q (G) into weak-F r 1,q (G). Proof. That a symbol satisfying (3.32) for some integer s 0 with 0 ≤ s 0 ≤ n, satisfies also the Hörmander condition (3.3) for all s > n 2 was proved in Theorem 6.16 of [23]. In view of Theorem 3.1 we conclude the proof.