Palestine Journal of Mathematics
Vol. 5(2) (2016)
, 35–44
© Palestine Polytechnic University-PPU 2016
Besov continuity for Multipliers dened on compact Lie groups
Duván Cardona
Communicated by A. Aldroubi
MSC 2010 Classications: Primary 43A22, 43A77; Secondary 43A15, 22A30.
Keywords and phrases: Besov spaces, Multipliers, Compact Lie groups, Pseudo-differential operators.
I would like to thank the anonymous referee for his/her remarks which helped to improve the manuscript.
Abstract In this note we give estimates for Fourier multipliers on Besov spaces. We present
a class of bounded non-invariant pseudo-differential operators on Compact Lie groups.
1 Introduction
Given a compact Lie group G, in this paper we establish continuity properties of Fourier multipliers dened on compact Lie groups when acting on Besov spaces. In our analysis on a Lie group
b denotes the unitary dual of G, i.e. the set of equivalence classes of all strongly
G, the set G
continuous irreducible unitary representations of G. So, if ξ : G → U (Hξ ) is an irreducible
unitary representation and σ (ξ ) ∈ Cdξ ×dξ , the corresponding Fourier multiplier Op(σ ) is the
pseudo-differential operator formally dened by the formula
Op(σ )f (x) =
∑
dξ Tr[ξ (x)σ (ξ )(F f )(ξ )]
(1.1)
b
[ξ ]∈G
where, the summations is understood so that from each class [ξ ] we pick just one representative
ξ ∈ [ξ ], dξ = dim(Hξ ) and (F f )(ξ ) is the Fourier transform at ξ :
(F f )(ξ ) := fb(ξ ) =
∫
f (x)ξ (x)∗ dx ∈ Cdξ ×dξ .
(1.2)
G
Besov spaces on compact Lie groups where introduced and analyzed in [23] and they form scales
r
Bp,q
(G) carrying three indices r ∈ R, 0 < p, q ≤ ∞. There are several possibilities concerning
the conditions to impose on a symbol σ in the attempt to establish a Fourier multiplier theorem
of boundedness on Besov spaces and Lebesgue spaces (see [5, 6, 9, 10, 11, 30] ). However, our
work is closely related with a recent result by Michael Ruzhansky and Jens Wirth [30]:
Theorem 1.1. Let G be a compact Lie group. Denote by κ the smallest even integer larger than
1
2 dim(G). Let Op(σ ) be a Fourier multiplier and assume that its symbol σ (ξ ) satises
b
∥Dα σ (ξ )∥op ≤ Cα ⟨ξ⟩−|α| , for all |α| ≤ κ , and [ξ ] ∈ G.
(1.3)
Then, Op(σ ) is of weak type (1, 1) and bounded on Lp (G) for all 1 < p < ∞.
In this paper we prove the following theorems
Theorem 1.2. Let G be a compact Lie group and Op(σ ) be a Fourier multiplier. If Op(σ ) is
bounded from Lp1 (G) into Lp2 (G), then Op(σ ) extends to a bounded operator from Bpr1 ,q (G)
into Bpr2 ,q (G), for all r ∈ R and 0 < q ≤ ∞.
Theorem 1.3. Let G be a compact Lie group and Op(σ ) be a Fourier multiplier. If Op(σ ) is of
r
weak type (1, 1) then Op(σ ) extends to a bounded operator from B1r,q (G), into Bp,q
(G) for all
r ∈ R, 0 < q ≤ ∞ and 0 < p < 1.
As a consequence of our theorems and the estimate by Ruzhansky and Wirth we obtain:
36
Duván Cardona
Theorem 1.4. Let G be a compact Lie group. Denote by κ the smallest even integer larger than
1
2 dim(G). Let Op(σ ) be a Fourier multiplier and assume that its symbol σ (ξ ) satises
b
∥Dα σ (ξ )∥op ≤ Cα ⟨ξ⟩−|α| , for all |α| ≤ κ , and [ξ ] ∈ G.
(1.4)
r
Then, Op(σ ) is a bounded operator on Bp,q
(G) for all 1 < p < ∞, r ∈ R and 0 < q ≤ ∞.
Theorem 1.5. Let G be a compact Lie group. Denote by κ the smallest even integer larger than
1
2 dim(G). Let Op(σ ) be a Fourier multiplier and assume that its symbol σ (ξ ) satises
b
∥Dα σ (ξ )∥op ≤ Cα ⟨ξ⟩−|α| , for all |α| ≤ κ , and [ξ ] ∈ G.
(1.5)
r
Then, Op(σ ) is a bounded operator from Bp,q
(G) into B1r,q (G) for all 0 < p < 1, r ∈ R and
0 < q ≤ ∞.
Theorem 1.4 can be extended to the case of non-invariant pseudo-differential operators. For this
we use a version of the Sobolev embedding theorem. This approach was used by Ruzhansky and
Wirth [30], (see also Ruzhansky-Turunen [28] and [29]) in order to get Lp multiplier theorems
for non-invariant pseudo-differential operators. In this work we consider the global quantization
of operators on compact Lie groups proposed in [24, 25]. If
b → ∪ b Cdξ ×dξ
σ (x, ξ ) : G × G
[ξ ]∈G
is a measurable function, the corresponding non-invariant operator is the pseudo-differential
operator dened formally by
Op(σ )f (x) =
∑
dξ Tr[ξ (x)σ (x, ξ )(F f )(ξ )].
(1.6)
b
[ξ ]∈G
With this in mind, we present the following result.
Theorem 1.6. Let G be a compact Lie group and n = dim(G) its dimension. Denote by κ
the smallest even integer larger than n2 , and l = [n/p] + 1. Let Op(σ ) be a pseudo-differential
operator on G and assume that its symbol σ (x, ξ ) satises
b
∥∂xβ Dα σ (x, ξ )∥op ≤ Cα ⟨ξ⟩−|α| , for all |α| ≤ κ , |β| ≤ l and [ξ ] ∈ G.
(1.7)
r
Then, Op(σ ) is a bounded operator on Bp,q
(G) for all 1 < p < ∞, r ∈ R and 0 < q ≤ ∞.
We note that in the commutative case of the n-dimensional torus G = Tn conditions on the
number of derivatives on the symbol in the ξ -variable, and assumptions on the size of these
derivatives can be relaxed. This is possible by the following result which was proved by Julio
Delgado in [18].
Theorem 1.7. Let 0 < ε < 1 and k ∈ N with k > n2 , let σ be a symbol such that |Dα σ (x, ξ )| ≤
n
n
Cα ⟨ξ⟩− 2 ε−(1−ε)|α| , |∂xβ σ (x, ξ )| ≤ Cβ ⟨ξ⟩− 2 , for |α|, |β| ≤ k. Then, Op(σ ) is a bounded operator
from Lp (Tn ) into Lp (Tn ) for 2 ≤ p < ∞.
Now, by using Theorem 1.2 and the estimate by Delgado we deduce the following Besov
estimate.
Theorem 1.8. Let 0 < ε < 1 and k, l ∈ N with k >
and l > np , let σ (x, ξ ) be a symbol
such that |∂xβ Dα σ (x, ξ )| ≤ Cα,β ⟨ξ⟩
for |α| ≤ k, |β| ≤ l. Then, Op(σ ) is a bounded
r
r
operator from Bp,q
(Tn ) into Bp,q
(Tn ) for all 1 < p < ∞, r ∈ R and 0 < q ≤ ∞.
−n
2 ε−(1−ε)|α|
n
2
This paper is organized as follows. In Section 2, we summarizes basic properties on the
harmonic analysis on compact Lie groups including the Ruzhansky-Turunen quantization of
global pseudo-differential operators on compact-Lie groups and the denition of Besov spaces
on such groups. Finally, in Section 3 we proof our results on the boundedness of invariant and
non-invariant pseudo-differential operators on Besov spaces.
Besov continuity for Multipliers dened on compact Lie groups
37
2 Preliminaries
In this section we will introduce some preliminaries on pseudo-differential operators on compact
Lie groups. There are two notions of pseudo-differential operators on compact Lie groups. The
rst notion in the case of general manifolds (based on the idea of local symbols, as in Hörmander [21]) and, in a much more recent context, the one of global pseudo-differential operators on
compact Lie groups as dened by [28] (from full symbols, for which the notations and terminologies are taken from [28]). We∫will always∫ equip a compact Lie group with the Haar measure
µG . For simplicity, we will write G f dx for G f dµG , Lp (G) for Lp (G, µG ), etc. The following
assumptions are based on the group Fourier transform
∫
φ
b(ξ ) =
φ(x)ξ (x)∗ dx,
φ(x) =
G
∑
dξ Tr(ξ (x)φ
b(ξ )).
b
[ξ ]∈G
The Peter-Weyl Theorem on G implies the Plancherel identity on L2 (G),

∥φ∥L2 (G) = 
∑
 12
dξ Tr(φ
b(ξ )φ
b(ξ )∗ ) = ∥φ∥
b L2 (Gb) .
b
[ξ ]∈G
Here
∥A∥HS = Tr(AA∗ ),
denotes the Hilbert-Schmidt norm of matrices. Any linear operator A on G mapping C ∞ (G)
into D′ (G) gives rise to a matrix-valued global (or full) symbol σA (x, ξ ) ∈ Cdξ ×dξ given by
σA (x, ξ ) = ξ (x)∗ (Aξ )(x),
(2.1)
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 [28]
Af (x) =
∑
dξ Tr[ξ (x)σA (x, ξ )fb(ξ )].
(2.2)
b
[ξ ]∈G
b ) spaces on the unitary dual can be well
In this paper we use the notation Op(σA ) = A. Lp (G
b ) is dened by the norm
dened. If p = 2, L2 (G
∥G∥2L2 (Gb) =
∑
dξ ∥G(ξ )∥2HS .
b
[ξ ]∈G
Now, we want to introduce Sobolev spaces and, for this, we give some basic tools. Let ξ ∈
b if x ∈ G is xed, ξ (x) : Hξ → Hξ is an unitary operator and dξ := dim Hξ < ∞.
Rep(G) := ∪G,
ˆ
There exists a non-negative real number λ[ξ] depending only on the equivalence class [ξ ] ∈ G,
but not on the representation ξ, such that −LG ξ (x) = λ[ξ] ξ (x); here LG is the Laplacian on
the group G (in this case, dened as the Casimir element on G). Let ⟨ξ⟩ denote the function
1
⟨ξ⟩ = (1 + λ[ξ] ) 2 .
Denition 2.1. For every s ∈ R, the Sobolev space H s (G) on the Lie group G is dened by the
b ).
condition: f ∈ H s (G) if only if ⟨ξ⟩s fb ∈ L2 (G
The Sobolev space H s (G) is a Hilbert space endowed with the inner product ⟨f, g⟩s =
⟨Ls f, Ls g⟩L2 (G) , where, for every r ∈ R, Ls : H r → H r−s is the bounded pseudo-differential
operator with symbol ⟨ξ⟩s Iξ .
(G)
Denition 2.2. Let (Yj )dim
be a basis for the Lie algebra g of G, and let ∂j be the left-invariant
j =1
vector elds corresponding to Yj . We dene the differential operator associated to such a basis
by DYj = ∂j and, for every α ∈ Nn , the differential operator ∂xα is the one given by ∂xα =
∂1α1 · · · ∂nαn . Now, if ξ0 is a xed irreducible representation, the matrix-valued difference operator
d
is the given by Dξ0 = (Dξ0 ,i,j )i,jξ0=1 = ξ0 (·) −Idξ0 . If the representation is xed we omit the index
ξ0 so that, from a sequence D1 = Dξ0 ,j1 ,i1 , · · · , Dn = Dξ0 ,jn ,in of operators of this type we dene
Dα = Dα1 1 · · · Dαnn , where α ∈ Nn .
38
Duván Cardona
Denition 2.3. We introduce the Besov spaces on compact Lie groups using the Fourier transform on the group G as follow. Let r ∈ R, 0 ≤ q < ∞ and 0 < p ≤ ∞. If f is a measurable
r
function on G, we say that f ∈ Bp,q
(G) if f satises

r
∥f ∥Bp,q
:= 
∞
∑
∑
2mrq ∥
m=0
 q1
dξ Tr[ξ (x)fb(ξ )]∥qLp (G)  < ∞.
(2.3)
2m ≤⟨ξ⟩<2m+1
r
If q = ∞, Bp,∞
(G) consists of those functions f satisfying
∑
r
∥f ∥Bp,∞
:= sup 2mr ∥
m∈N
dξ Tr[ξ (x)fb(ξ )]∥Lp (G) < ∞.
(2.4)
2m ≤⟨ξ⟩<2m+1
r
In the case of p = q = ∞ and 0 < r < 1 and G = T we obtain B∞,∞
(T) = Lr (T), that is
the familiar space of Hölder continuous functions of order r. According to the usual denition,
a function f belongs to Lr if there exists a constant A such that: |f (x)| ≤ A and
|f |Lr := sup
x,y
|f (x − y ) − f (x)|
≤ A.
|y|r
These are Banach spaces together with the norm
∥f ∥Lr = |f |Lr + sup |f (x)|.
x∈T
3 Proofs
In this section we prove our results which were presented above. We treat the theory with
basic methods. In our tools we consider the Sobolev embedding theorem and basics on Fourier
analysis. We star with the proof of the Theorem 1.2.
Theorem 3.1. Let G be a compact Lie group and Op(σ ) be a Fourier multiplier. If Op(σ ) is
bounded from Lp1 (G) into Lp2 (G), then Op(σ ) extends to a bounded operator from Bpr1 ,q (G)
into Bpr2 ,q (G), for all r ∈ R and 0 < q ≤ ∞.
Proof. First, let us consider a multiplier operator Op(σ ) bounded from Lp1 (G) into Lp2 (G), and
f ∈ C ∞ (G). Then, we have
∥T f ∥Lp2 (G) ≤ C∥f ∥Lp1 (G) ,
where C = ∥T ∥B(Lp1 ,Lp2 ) is the usual operator norm. We denote by χm (ξ ) the characteristic
function of
Dm := {ξ : 2m ≤ ⟨ξ⟩ < 2m+1 }
and Op(χm ) the corresponding Fourier multiplier of the symbol χm (ξ )Iξ . Here, Iξ is the identity
operator in Cdξ ×dξ . By the denition of Besov norm, if 0 < q < ∞ we have
∥Op(σ )f ∥qB r p
2 ,q
=
∞
∑
m=0
=
∞
∑
∞
∑
2mrq ∥
∞
∑
m=0
=
∞
∑
m=0
∑
dξ · χm (ξ )Tr[ξ (x)F (Op(σ )f )(ξ )]∥qLp2 (G)
b
[ξ ]∈G
2mrq ∥
m=0
=
dξ Tr[ξ (x)F (Op(σ )f )(ξ )]∥qLp2 (G)
2m ≤⟨ξ⟩<2m+1
m=0
=
∑
2mrq ∥
∑
dξ · Tr[ξ (x)χm (ξ )σ (ξ )(F f )(ξ )]∥qLp2 (G)
b
[ξ ]∈G
2mrq ∥
∑
dξ · Tr[ξ (x)σ (ξ )F (Op(χm )f )(ξ )]∥qLp2 (G)
b
[ξ ]∈G
2mrq ∥Op(σ )[(Op(χm )f )]∥qLp2 (G) .
Besov continuity for Multipliers dened on compact Lie groups
39
By the boundedness of Op(σ ) from Lp1 (G) into Lp2 (G) we get,
∥Op(σ )f ∥qB r p
2 ,q
≤
∞
∑
2mrq C q ∥Op(χm )f ∥qLp1 (G)
m=0
=
∞
∑
m=0
=
∞
∑
=
dξ · Tr[ξ (x)χm (ξ )Iξ F (f )(ξ )]∥qLp1 (G)
b
[ξ ]∈G
∑
2mrq C q ∥
m=0
∞
∑
∑
2mrq C q ∥
dξ · Tr[ξ (x)Iξ F (f )(ξ )]∥qLp1 (G)
2m ≤⟨ξ⟩<2m+1
∑
2mrq C q ∥
m=0
dξ · Tr[ξ (x)Iξ F (f )(ξ )]∥qLp1 (G)
2m ≤⟨ξ⟩<2m+1
= C q ∥f ∥qB r p ,q
1
Hence,
∥Op(σ )f ∥B r p2 ,q ≤ C∥f ∥B r p1 ,q .
If q = ∞ we have
∥Op(σ )f ∥B r p2 ,∞ = sup 2mr ∥
m∈N
= sup 2mr ∥
m∈N
= sup 2mr ∥
m∈N
= sup 2mr ∥
m∈N
∑
dξ Tr[ξ (x)F (Op(σ )f )(ξ )]∥Lp2 (G)
2m ≤⟨ξ⟩<2m+1
∑
dξ · χm (ξ )Tr[ξ (x)F (Op(σ )f )(ξ )]∥Lp2 (G)
b
[ξ ]∈G
∑
dξ · Tr[ξ (x)χm (ξ )σ (ξ )(F f )(ξ )]∥Lp2 (G)
b
[ξ ]∈G
∑
dξ · Tr[ξ (x)σ (ξ )F (Op(χm )f )(ξ )]∥Lp2 (G)
b
[ξ ]∈G
= sup 2mr ∥Op(σ )[(Op(χm )f )]∥Lp2 (G) .
m∈N
Newly, by using the fact that Op(σ ) is a bounded operator from Lp1 (G) into Lp2 (G) we have,
∥Op(σ )f ∥B r p2 ,∞ ≤ sup 2mr C∥Op(χm )f ∥Lp1 (G)
m∈N
= sup 2mr C∥
m∈N
∑
∑
= sup 2mr C∥
m∈N
dξ · Tr[ξ (x)Iξ F (f )(ξ )]∥Lp1 (G)
2m ≤⟨ξ⟩<2m+1
∑
= sup 2mr C∥
m∈N
dξ · Tr[ξ (x)χm (ξ )Iξ F (f )(ξ )]∥Lp1 (G)
b
[ξ ]∈G
dξ · Tr[ξ (x)Iξ F (f )(ξ )]∥Lp1 (G)
2m ≤⟨ξ⟩<2m+1
= C∥f ∥B r p1 ,∞ .
This implies that,
∥Op(σ )f ∥B r p2 ,∞ ≤ C∥f ∥B r p1 ,∞ .
With the last inequality we end the proof.
The proof of Theorem 1.3 depends on the following result due to Kolmogorov (see Lemma
5.16 in [19]).
40
Duván Cardona
Lemma 3.2. Given an operator S : (X, µ) → (Y, ν ) which is of weak type (1,1), 0 < v < 1, and
a set E ⊂ Y of nite measure, there exists a C > 0 such that
∫
E
|Sf (x)|v dx ≤ Cµ(E )1−v ∥f ∥vL1 (X ) .
(3.1)
Theorem 3.3. Let G be a compact Lie group and Op(σ ) be a Fourier multiplier. If Op(σ ) is of
r
weak type (1, 1) then Op(σ ) extends to a bounded operator from from B1r,q (G), into Bp,q
(G) for
all r ∈ R, 0 < q ≤ ∞ and 0 < p < 1.
Proof. Let us consider a Fourier multiplier Op(σ ) on G of weak type (1,1). By Lemma 3.2
we deduce the boundedness of Op(σ ) from L1 (G) into Lp (G) for all 0 < p < 1. Finally, by
r
(G) for all
Theorem 1.2 we obtain that Op(σ ) is a continuous multiplier from B1r,q (G), into Bp,q
0 < p < 1, r ∈ R and 0 < q ≤ ∞.
Remark 3.4. We observe that Theorems 1.4 and 1.5 are an immediate consequence of the Theorems 1.1, 1.2 and 1.3.
Now we extend the boundedness of Fourier multipliers on Besov spaces, to the case of noninvariant pseudo-differential operators.
Theorem 3.5. Let G be a compact Lie group and n = dim(G) its dimension. Denote by κ
the smallest even integer larger than n2 , and l = [n/p] + 1. Let Op(σ ) be a pseudo-differential
operator and assume that its symbol σ (x, ξ ) satises
b
∥∂xβ Dα σ (x, ξ )∥op ≤ Cα ⟨ξ⟩−|α| , for all |α| ≤ κ , |β| ≤ l and [ξ ] ∈ G.
(3.2)
r
Then, Op(σ ) is a bounded operator on Bp,q
(G) for all 1 < p < ∞, r ∈ R and 0 < q ≤ ∞.
Proof. Let f ∈ C ∞ (G). To prove this theorem we write
Op(σ )f (x) =
∑
dξ Tr[ξ (x)σ (x, ξ )fb(ξ )]
b
[ξ ]∈G

∫

=
G

∫

=
G
∑

dξ Tr[ξ (y −1 x)σ (x, ξ )] f (y )dy
b
[ξ ]∈G
∑

dξ Tr[ξ (y )σ (x, ξ )] f (xy −1 )dy.
b
[ξ ]∈G
Hence Op(σ )f (x) = (κ(x, ·) ∗ f )(x), where
κ(z, y ) =
∑
dξ Tr[ξ (y )σ (z, ξ )]
(3.3)
b
[ξ ]∈G
and ∗ is the right convolution operator. Moreover, if we dene Az f (x) = (κ(z, ·) ∗ f )(x) for
every element z ∈ G, we have
Ax f (x) = Op(σ )f (x), x ∈ G.
For all 0 ≤ |β| ≤ [n/p] + 1 we have ∂zβ Az f (x) = Op(∂zβ σ (z, ·))f (x). So, by the condition 3.2
r
and Theorem 1.2, for every z ∈ G, ∂zβ Az f = Op(∂zβ σ (z, ·))f is a bounded operator on Bp,q
(G)
for all 1 < p < ∞, r ∈ R and 0 < q ≤ ∞. Now, we want to estimate the Besov norm of
Besov continuity for Multipliers dened on compact Lie groups
Op(σ (·, ·)). First, we observe that
p
∑
dξ Tr[ξ (x)F (Op(σ )f )(ξ )]
2m ≤⟨ξ⟩<2m+1
p
L
p
∫ ∫
∑
∗
:= dξ Tr[ξ (x)
Op(σ )f (y )ξ (y ) dy ] dx
G m
G
2 ≤⟨ξ⟩<2m+1
p
∫ ∫
∑
∗
dx
=
d
Tr
[
ξ
(
x
)
A
f
(
y
)
ξ
(
y
)
dy
]
ξ
y
G m
G
2 ≤⟨ξ⟩<2m+1
p
∫
∑
dξ Tr[ξ (x)F (Az f )(ξ )] dx
≤
sup G z∈G m
2 ≤⟨ξ⟩<2m+1
By the Sobolev embedding theorem (Theorem 3.1.3 of [1]), we have
p
∑
sup dξ Tr[ξ (x)F (Az f )(ξ )]
z∈G m
2 ≤⟨ξ⟩<2m+1
p
∑ ∫ ∑
β
dz
.
σ
(
z,
·
))
f
)(
ξ
)]
d
Tr
[
ξ
(
x
)
F
(
Op
(
∂
ξ
z
G
m
m
+
1
|β|≤l
2 ≤⟨ξ⟩<2
p
∫ ∑
β
. sup
dξ Tr[ξ (x)F (Op(∂z σ (z, ·))f )(ξ )] dz
|β|≤l G m
2 ≤⟨ξ⟩<2m+1
From this, and the Sobolev embedding theorem we have
p
∑
sup dξ Tr[ξ (x)Az f (y )ξ (y )∗ ] dy
G z∈G m
2 ≤⟨ξ⟩<2m+1
p
∑ ∫ ∫ ∑
β
.
dξ Tr[ξ (x)F (Op(∂z σ (z, ·))f )(ξ )] dzdx
|β|≤l G G 2m ≤⟨ξ⟩<2m+1
p
∫ ∫ ∑
β
. sup
dξ Tr[ξ (x)F (Op(∂z σ (z, ·))f )(ξ )] dxdz
|β|≤l G G m
2 ≤⟨ξ⟩<2m+1
p
∫ ∑
β
dx
≤ sup
d
Tr
[
ξ
(
x
)
F
(
Op
(
∂
σ
(
z,
·
))
f
)(
ξ
)]
ξ
z
|β|≤l,z∈G G m
2 ≤⟨ξ⟩<2m+1
∑
= sup ∥
dξ Tr[ξ (x)F (Op(∂zβ σ (z, ·))f )(ξ )]∥pLp
∫
|β|≤l,z∈G
2m ≤⟨ξ⟩<2m+1
Hence,
∑
∥
dξ Tr[ξ (x)F (Op(σ )f )(ξ )]∥Lp
2m ≤⟨ξ⟩<2m+1
.
sup
|β|≤l,z∈G
∥
∑
2m ≤⟨ξ⟩<2m+1
dξ Tr[ξ (x)F (Op(∂zβ σ (z, ·))f )(ξ )]∥Lp
41
42
Duván Cardona
Thus, considering 0 < q < ∞ we obtain
q  q1
∑

:= 
2mrq dξ Tr[ξ (x)F (Op(σ )f )(ξ )]
2m ≤⟨ξ⟩<2m+1
p
m=0

∥Op(σ )f ∥Bp,q
r (T)
∞
∑
L
q  q1
∞
∑
∑
mrq
β


.
2
sup dξ Tr[ξ (x)F (Op(∂z σ (z, ·))f )(ξ )]
|β|≤l,z∈G
m=0
2m ≤⟨ξ⟩<2m+1
p

L
We dene for every z ∈ T the non-negative function z 7→ g (z ) by
q
∑
β
dξ Tr[ξ (x)F (Op(∂z σ (z, ·))f )(ξ )]
g (z ) = sup .
|β|≤l m
p
2 ≤⟨ξ⟩<2m+1
L
We can choose a sequence (zn )n∈N ⊂ T such that (g (zn ))n is increasing and
lim g (zn ) = sup g (z ).
n→∞
z∈T
So, by an application of the monotone convergence theorem we have
(
∞
∑
) q1
2
mrq
sup g (z )
=
z∈T
m=0
(
(
=
∞
∑
2
n→∞
m=0
lim
n→∞
(
= lim
n→∞
(
≤ sup
z∈T
) q1
lim g (zn )
mrq
) q1
∞
∑
2
mrq
g (zn )
m=0
) q1
∞
∑
2
mrq
m=0
∞
∑
g (zn )
) q1
2
mrq
g (z )
m=0
Hence, we can write
q  q1
∑

.
2mrq sup dξ Tr[ξ (x)F (Op(∂zβ σ (z, ·))f )(ξ )]
|β|≤l,z∈G m
p
m=0
2 ≤⟨ξ⟩<2m+1

∥Op(σ )f ∥Bp,q
r (T)
∞
∑
L
≤
sup
|β|≤l,z∈T
∥Op
(∂zβ σ (z, ·))f ∥Bp,q
r (T)
[
]
sup
≤
|β|≤l,z∈G
∥Op
(∂zβ σ (z, ·))∥B (Bp,q
r )
r .
∥f ∥Bp,q
So, we deduce the boundedness of Op(σ ). Now, we treat of a similar way the boundedness of
Op(σ ) if q = ∞. In fact, from the inequality
∑
∥
dξ Tr[ξ (x)F (Op(σ )f )(ξ )]∥Lp
2m ≤⟨ξ⟩<2m+1
.
sup
|β|≤l,z∈G
we have
∑
∥
2m ≤⟨ξ⟩<2m+1
∑
2mr ∥
dξ Tr[ξ (x)F (Op(σ )f )(ξ )]∥Lp
2m ≤⟨ξ⟩<2m+1
. 2mr
sup
|β|≤l,z∈G
dξ Tr[ξ (x)F (Op(∂zβ σ (z, ·))f )(ξ )]∥Lp
∥
∑
2m ≤⟨ξ⟩<2m+1
dξ Tr[ξ (x)F (Op(∂zβ σ (z, ·))f )(ξ )]∥Lp .
Besov continuity for Multipliers dened on compact Lie groups
So we get
[
∥Op(σ )f ∥Bp,∞
r
(T) .
43
]
sup
|β|≤l,z∈G
r
∥Op(∂zβ σ (z, ·))∥B (Bp,∞
.
r
) ∥f ∥Bp,∞
With the last inequality we end the proof.
We note that in the n−dimensional case of the torus some conditions on the derivatives of the
symbol (number of derivatives and size of them) can be relaxed:
Theorem 3.6. Let 0 < ε < 1 and k, l ∈ N with k >
and l > np , let σ (x, ξ ) be a symbol
such that |∂xβ Dα σ (x, ξ )| ≤ Cα,β ⟨ξ⟩
for |α| ≤ k, |β| ≤ l. Then, Op(σ ) is a bounded
r
r
operator from Bp,q
(Tn ) into Bp,q
(Tn ) for all 1 < p < ∞, r ∈ R and 0 < q ≤ ∞.
−n
2 ε−(1−ε)|α|
n
2
Proof. First, we prove this theorem in the case of Fourier multipliers. If we consider a toroidal
symbol σ satisfying
n
|Dα σ (ξ )| ≤ Cα,β ⟨ξ⟩− 2 ε−(1−ε)|α| ,
by the Theorem 1.2 we obtain the boundedness of Op(σ ) on Lp (Tn ) for 2 ≤ p < ∞. If 1 < p ≤ 2
we apply the boundedness of Op(σ ) for p′ ≥ 2, where 1/p + 1/p′ = 1 :
∥Op(σ )f ∥Lp (Tn ) = sup{|⟨Op(σ )f, g⟩| : ∥g∥Lp′ (Tn ) ≤ 1}
= sup{|⟨f, Op(σ )∗ g⟩| : ∥g∥Lp′ (Tn ) ≤ 1}
= sup{|⟨f, Op(σ (·))g⟩| : ∥g∥Lp′ (Tn ) ≤ 1}
≤ ∥Op(σ (·))∥B (Lp′ ) ∥f ∥Lp (Tn ) .
Now we have the global continuity of Op(σ ) on Lp (Tn ) for all 1 < p < ∞. By Theorem 1.2
r
we deduce the boundedness of the multiplier Op(σ ) on Bp,q
(Tn ) for all 1 < p < ∞, r ∈ R and
0 < q ≤ ∞. Finally, if we reproduce the proof of theorem 1.4 in the particular case of the torus
Tn , we extend this estimate to the case of pseudo-differential operators Op(σ (·, ·)) with symbol
σ (x, ξ ) satisfying the following toroidal inequalities
|∂xβ Dα σ (x, ξ )| ≤ Cα,β ⟨ξ⟩− 2 ε−(1−ε)|α| , |α| ≤ k, |β| ≤ l.
n
Remark 3.7. We end this section with some references on the topic. The boundedness of Fourier
multipliers in Lp -spaces, Hölder spaces and Besov spaces has been considered by many authors
for a long time. In the general case of Compact Lie groups we refer the reader to the works of
Alexopoulos, Anker, Coifman, Ruzhansky and Wirth [2, 3, 17, 26, 27, 28, 29, 30]. The particular
case of Fourier multipliers on the torus has been investigated in [4, 5, 6, 9, 10, 11]. Lp and Hölder
estimates of periodic pseudo-differential operators can be found in [12, 13, 14, 18] and [22].
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Author information
Duván Cardona, Department of Mathematics, Universidad de los Andes, Bogotá , Colombia.
E-mail: [email protected] [email protected]
Received:
June 29, 2015.
Accepted:
December 4, 2015.