Endpoint boundedness of toroidal pseudo-differential operators

Benhamoud Ramla

doi:10.1515/math-2024-0023

Article Open Access

Endpoint boundedness of toroidal pseudo-differential operators

Benhamoud Ramla

Published/Copyright: August 2, 2024

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$Open Mathematics$

From the journal Open Mathematics Volume 22 Issue 1

Abstract

In this note, we prove that the toroidal pseudo-differential operator is bounded from L ∞ ( T n ) to BMO ( T n ) if the symbol belongs to the toroidal Hörmander class S ρ , δ n ( ρ − 1 ) ∕ 2 ( T n × Z n ) with 0 < ρ ≤ 1 and 0 ≤ δ < 1 . As a corollary, we obtain a result of toroidal pseudo-differential operators on L p when 2 < p < ∞ for symbols in the class S ρ , δ m ( T n × Z n ) with m ≤ n ( ρ − 1 ) 1 2 − 1 p + n p min { 0 , ρ − δ } .

Keywords: Torus; toroidal pseudo-differential operator; toroidal Hörmander symbol; BMO(𝕋 n )

MSC 2010: 43A75; 35S05

1 Introduction and main results

The theory of pseudo-differential operators was initiated by Kohn and Nirenberg [1] and Hörmander [2]. These operators have a powerful role in the study of partial differential equations. A pseudo-differential operator on R n is defined by

T a f ( x ) = ∫ R n e i 2 π x ξ a ( x , ξ ) f ˆ ( ξ ) d ξ ,

where the function a is called the symbol of the operator T a . Symbols are classified according to their asymptotic behavior at infinity. The Hörmander class introduced in [3] is an important one.

Let N be the set { 0 , 1 , 2 , … } and

⟨ ξ ⟩ ≔ ( 1 + ∣ ξ ∣ 2 ) 1 2 .

For m ∈ R , 0 ≤ ρ , δ ≤ 1 , the Hörmander class S ρ , δ m ( R n × R n ) consists of all functions a that are smooth in ( x , ξ ) and satisfy

A N , M = sup x , ξ ∈ R n ⟨ ξ ⟩ − m + ρ N − δ M ∣ ∇ ξ N ∇ x M a ( x , ξ ) ∣ < + ∞ ,

for any N , M ∈ N .

The boundedness of pseudo-differential operators on R n has been investigated by many researchers. In particular, in [4–7], it was shown that if a ∈ S ρ , δ m with δ < 1 and m ≤ min 0 , n ( ρ − δ ) 2 , then the operator T a is bounded on L 2 and the bound on m is sharp. Additionally, Rodino [8] proved the L 2 boundedness of T a for a ∈ S ρ , 1 m with m < n ( ρ − 1 ) 2 and constructed an example a ∈ S ρ , 1 n ( ρ − 1 ) ∕ 2 such that T a is unbounded on L 2 . For S 1 , 1 0 , one can also see Ching [9] and Stein [10]. Stein showed in an unpublished lecture note that if a ∈ S ρ , δ n ( ρ − 1 ) ∕ 2 and either 0 ≤ δ < ρ ≤ 1 or 0 < δ = ρ < 1 , then T a is of weak type ( 1 , 1 ) and bounded from H 1 to L 1 . Alvarez and Hounie [11] extended this result for 0 < ρ ≤ 1 , 0 ≤ δ < 1 . They proved the weak type ( 1 , 1 ) and H 1 − L 1 boundedness for a ∈ S ρ , δ m with m = n 2 ( ρ − 1 + min { 0 , ρ − δ } ) . Recently, Guo and Zhu [12] studied the various endpoint estimates when a ∈ S ρ , 1 n ( ρ − 1 ) .

Agranovich [13] developed the notion of toroidal pseudo-differential operators. He proposed a global quantization on the unit circle S 1 , which was generalizable for any torus T n . Toroidal pseudo-differential operators are associated with symbols on T n × R n or T n × Z n . The equivalence of them has been shown in [14].

Let T n be the torus ( R ∕ Z ) n . It can be considered as the metric space ( [ 0 , 1 ) n , ρ ) . For x , y ∈ [ 0 , 1 ) n , the distance of x and y in T n is given by

∣ x − y ∣ = ρ ( x , y ) = ∑ k = 1 n min { ∣ x k − y k ∣ 2 , ( 1 − ∣ x k − y k ∣ ) 2 } .

The toroidal Fourier transform F T n is defined by

( F T n f ) ( ξ ) = f ˆ ( ξ ) = ∫ T n e − i 2 π x ξ f ( x ) d x .

Let S ( Z n ) be the restriction of the Schwartz space S ( R n ) on Z n . Specifically, a function φ : Z n → C belongs to S ( Z n ) if and only if φ can be extended to a function φ ˜ ∈ S ( R n ) such that φ ˜ ( ξ ) = φ ( ξ ) for any ξ ∈ Z n . It is well known that F T n is a bijection from C ∞ ( T n ) to S ( Z n ) and its inverse is given by

f ( x ) = ∑ ξ ∈ Z n e i 2 π x ξ ( F T n f ) ( ξ ) = ∑ ξ ∈ Z n e i 2 π x ξ f ˆ ( ξ ) .

A toroidal pseudo-differential operator a ( x , D ) is defined by

a ( x , D ) f ( x ) = ∑ ξ ∈ Z n e i 2 π x ξ a ( x , ξ ) f ˆ ( ξ ) .

As in [15], we recall the definition of the toroidal Hörmander class S ρ , δ m ( T n × Z n ) . Let { e j : j = 1 , … , n } be the standard orthogonal basis of R n and Δ j ( j = 1 , … , n ) be the partial difference operator on a function f on Z n , i.e.,

Δ j f ( ξ ) = f ( ξ + e j ) − f ( ξ ) .

For any multi-index α ∈ N n , Δ α = Δ 1 α 1 … Δ n α n .

Definition 1.1

The toroidal Hörmander class S ρ , δ m ( T n × Z n ) ( m ∈ R , 0 ≤ ρ , δ ≤ 1 ) is the set of functions a ( x , ξ ) , which are smooth in x for all ξ ∈ Z n and satisfy

∀ α , β ∈ N n , A α , β = sup ξ ∈ Z n ⟨ ξ ⟩ − m + ρ ∣ α ∣ − δ ∣ β ∣ ∣ Δ ξ α ∂ x β a ( x , ξ ) ∣ < ∞ .

We define the quasi-norms for a symbol a ∈ S ρ , δ m ( T n × Z n ) as follows:

(1.1) ‖ a ‖ S ρ , δ m , N = sup ∣ α ∣ , ∣ β ∣ ≤ N sup x ∈ T n sup ξ ∈ Z n ⟨ ξ ⟩ − m + ρ ∣ α ∣ − δ ∣ β ∣ ∣ Δ ξ α ∂ x β a ( x , ξ ) ∣ ,

where N ∈ N .

Because of the equivalence of symbols on T n × R n and T n × Z n , which was shown in [14], the L p ( 1 < p < ∞ ) boundedness of toroidal pseudo-differential operators with symbols in S ρ , δ m ( T n × Z n ) can be derived from the corresponding result of pseudo-differential operators. But as T n is compact, there are many better results for the toroidal pseudo-differential operators, especially on L 2 . For example, Ruzhansky and Turunen [16] proved the L 2 boundedness only if ∣ ∂ x β a ( x , ξ ) ∣ ≤ C for all ( x , ξ ) ∈ T n × Z n and ∣ β ∣ ≤ n + 1 . Along this direction, there are a lot of interesting results on T n or a compact Lie group. For example, one can see [17–24].

On the other hand, it was proved in [25] that T a is bounded on L ∞ when a ∈ S ρ , 1 m with m < n ( ρ − 1 ) 2 and T a is not bounded on L ∞ in general when a ∈ S ρ , 0 n ( ρ − 1 ) ∕ 2 . It is natural to think that this result is also true for toroidal pseudo-differential operators. But, to the best of our knowledge, there are no results for the toroidal pseudo-differential operator a ( x , D ) on L ∞ if a ∈ S ρ , δ n ( ρ − 1 ) ∕ 2 ( T n × Z n ) with ρ < δ ≤ 1 .

Motivated by previous results in the field, particularly the absence of results regarding the behavior of toroidal pseudo-differential operators on the torus T n , our aim is to explore the L ∞ ( T n ) − BMO ( T n ) boundedness. This investigation aims to deepen our understanding of the behavior of toroidal pseudo-differential operators on the torus T n .

In this note, our main result is the L ∞ ( T n ) − BMO ( T n ) boundedness of a ( x , D ) if a ∈ S ρ , δ n ( ρ − 1 ) ∕ 2 ( T n × Z n ) with ρ < δ < 1 , which is stated as the following theorem.

Theorem 1.1

If a ∈ S ρ , δ n ( ρ − 1 ) ∕ 2 ( T n × Z n ) with 0 ≤ δ < 1 , 0 < ρ ≤ 1 , then a ( x , D ) is bounded from L ∞ ( T n ) to BMO ( T n ) , i.e.,

‖ a ( x , D ) f ‖ BMO ≤ C ‖ f ‖ L ∞ ,

where C depends only on n , ρ , and δ and some quasi norms of a in S ρ , δ n ( ρ − 1 ) ∕ 2 .

Remark

When 0 ≤ δ < ρ ≤ 1 or 0 < δ = ρ < 1 , this theorem is well known.

As a corollary of Theorem 1.1 and Fefferman-Stein interpolation, we obtain that

Theorem 1.2

If a ∈ S ρ , δ m ( T n × Z n ) with 0 ≤ δ < 1 , 0 < ρ ≤ 1 and

m ≤ n ( ρ − 1 ) 1 2 − 1 p + n p min { 0 , ρ − δ } ,

then a ( x , D ) is bounded on L p ( T n ) when 2 < p < ∞ , i.e.,

‖ a ( x , D ) f ‖ p ≤ C ‖ f ‖ p ,

where C depends only on n , ρ , δ , and p and some quasi norms of a in S ρ , δ m .

Our results are generalizations of the related results for a ( x , D ) if a belongs to some toroidal Hörmander class S ρ , δ m .

This note is organized as follows. In Section 2, we recall some basic lemmas. In Section 3, we prove Theorem 1.1, and we conclude L p ( T n ) boundedness using interpolation.

Throughout this note, we use C to denote a positive constant, which may vary from line to line and depends only on n , ρ , δ , and p and some quasi-norms.

2 Some fundamental lemmas

Before giving the main proof, in this section, we recall some fundamental lemmas.

Lemma 2.1

([20, Theorem 3.8]) If a ∈ S ρ , 0 n ( ρ − 1 ) ∕ 2 ( T n × Z n ) with 0 < ρ < 1 , then a ( x , D ) is bounded from L ∞ ( T n ) to BMO ( T n ) .

Lemma 2.1 is a special case of [20, Theorem 3.8].

Lemma 2.2

If a ∈ S ρ , δ m ( T n × Z n ) with 0 ≤ δ < 1 , 0 < ρ ≤ 1 and m ≤ min 0 , n ( ρ − δ ) 2 , then a ( x , D ) is bounded on L 2 ( T n ) .

Proof

It is a direct conclusion from Theorem 1 in [7] and Theorem 6 in [17] or [14].□

For two multi-indices α , β ∈ N n , β ≤ α means that β k ≤ α k for all 1 ≤ k ≤ n , and we denote

( α β ) = ∏ k = 1 n ( α k β k ) = ∏ k = 1 n α k ! ( α k − β k ) ! β k ! .

For the partial difference operator, we introduce two fundamental properties.

Lemma 2.3

([20, Proposition 2.3], [23, Proposition 3.3.4, p. 311]) For a function f : Z n → C , we have

Δ ξ α f ( ξ ) = ∑ β ≤ α ( − 1 ) ∣ α − β ∣ ( α β ) f ( ξ + β ) .

Lemma 2.4

([20, Lemma 2.4], discrete Leibniz formula) For f , g : Z n → C and for all multi-index α , there holds

Δ ξ α ( f g ) ( ξ ) = ∑ β ≤ α ( α β ) Δ ξ β f Δ ξ α − β g ( ξ + β ) .

Now, we introduce the discrete dyadic decomposition { φ j : j ∈ N } ⊂ S ( Z n ) as in [26,27]. This sequence satisfies:

supp φ 0 ⊂ { ξ ∈ Z n : ∣ ξ ∣ ≤ 2 } and φ 0 ( ξ ) = 1 when ∣ ξ ∣ ≤ 1 .
supp φ j ⊂ { ξ ∈ Z n : 2 j − 1 < ∣ ξ ∣ ≤ 2 j + 1 } for j ≥ 1 .
For any ξ ∈ Z n , one have 0 ≤ φ j ( ξ ) ≤ 1 ( j ∈ N ) and ∑ j ∈ N φ j ( ξ ) = 1 .
For any α ∈ N n , there exists a constant C α > 0 such that
∣ Δ ξ α φ j ( ξ ) ∣ ≤ C α ⟨ ξ ⟩ − ∣ α ∣ ,
for any j ∈ N , ξ ∈ Z n .

We can divide the operator a ( x , D ) as

a ( x , D ) f ( x ) = ∑ ξ ∈ Z n ∑ j ∈ N e i 2 π x ξ a ( x , ξ ) φ j ( ξ ) f ^ ( ξ ) = ∑ ξ ∈ Z n ∑ j ∈ N ∫ T n e i 2 π ( x − y ) ξ a ( x , ξ ) φ j ( ξ ) f ( y ) d y = ∑ j ∈ N a j ( x , D ) f ( x ) .

It is easy to see that a j ( x , D ) f is well defined for any f ∈ L 1 ( T n ) .

In the next lemma, we estimate ‖ a j ( x , D ) f ‖ ∞ for j ∈ N .

Lemma 2.5

If 0 ≤ ρ ≤ 1 and a : T n × Z n → C satisfies that

∣ Δ ξ N a ( x , ξ ) ∣ = ∑ ∣ α ∣ = N ∣ Δ ξ α a ( x , ξ ) ∣ ≤ C N ⟨ ξ ⟩ m − ρ N ,

for any N ∈ N , then

‖ a j ( x , D ) f ‖ ∞ ≤ C 2 j ( m − n ( ρ − 1 ) ∕ 2 ) ‖ f ‖ ∞ ,

where C depends only on n, m, and ρ and some quasi-norms of a.

Proof

When 2 j ≤ 9 n , the proof is more simple. We assume that 2 j > 9 n . At first, for any N ≤ n 2 + 1 , using Lemma 2.4, we obtain that

(2.1) ∣ Δ ξ N [ a ( x , ξ ) φ j ( ξ ) ] ∣ ≤ C ∑ ∣ α ∣ + ∣ β ∣ = N ∣ Δ ξ α a ( x , ξ ) Δ ξ β φ j ( ξ + β ) ∣ ≤ C sup 2 j − 1 < ∣ ξ ∣ < 2 j + 1 ∑ ∣ α ∣ + ∣ β ∣ = N ≤ n 2 + 1 ⟨ ξ ⟩ m − ρ ∣ α ∣ ( 1 + ∣ ξ + β ∣ ) − ∣ β ∣ ≤ C 2 j ( m − ρ N ) .

For x = ( x 1 , … , x n ) , y = ( y 1 , … , y n ) ∈ T n , we denote

∣ ( x − y ) α ∣ = ∏ k = 1 n ( min { ∣ x k − y k ∣ , 1 − ∣ x k − y k ∣ } ) α k .

It is easy to see that

∣ x − y ∣ N = ∑ k = 1 n min { ∣ x k − y k ∣ 2 , ( 1 − ∣ x k − y k ∣ ) 2 } N 2 ≤ ∑ k = 1 n min { ∣ x k − y k ∣ , ( 1 − ∣ x k − y k ∣ ) } N ≤ C N ∑ ∣ α ∣ = N ∏ k = 1 n ( min { ∣ x k − y k ∣ , 1 − ∣ x k − y k ∣ } ) α k = C ∑ ∣ α ∣ = N ∣ ( x − y ) α ∣ .

One can also check that

(2.2) ∣ ( x − y ) α ∣ = ∏ k = 1 n ( min { ∣ x k − y k ∣ , 1 − ∣ x k − y k ∣ } ) α k ≤ π 2 ∣ α ∣ ∏ k = 1 n ∣ e − i 2 π ( x k − y k ) − 1 ∣ α k .

Using Lemma 2.3, we can obtain the following summation by parts formula (one can also see [26,27]):

(2.3) ∏ k = 1 n ( e − i 2 π ( x k − y k ) − 1 ) α k ∑ ξ ∈ Z n e i 2 π ( x − y ) ξ a ( x , ξ ) φ j ( ξ ) = ∑ ξ ∈ Z n e i 2 π ( x − y ) ξ a ( x , ξ ) φ j ( ξ ) ∏ k = 1 n ∑ β k = 0 α k ( − 1 ) α k − β k α k β k e − i 2 π ( x k − y k ) β k = ∑ ξ ∈ Z n e i 2 π ( x − y ) ξ a ( x , ξ ) φ j ( ξ ) ∑ β ≤ α ( − 1 ) ∣ α − β ∣ α β e − i 2 π ( x − y ) β = ∑ η ∈ Z n e i 2 π ( x − y ) η ∑ β ≤ α ( − 1 ) ∣ α − β ∣ α β a ( x , η + β ) φ j ( η + β ) = ∑ η ∈ Z n e i 2 π ( x − y ) η Δ η α [ a ( x , η ) φ j ( η ) ] .

Let N be the smallest integer that is bigger than n 2 . From (2.1)–(2.3) and the Plancherel theorem, for any x ∈ T n , one can obtain that

∣ a j ( x , D ) f ( x ) ∣ = ∑ ξ ∈ Z n ∫ T n e i 2 π ( x − y ) ξ a ( x , ξ ) φ j ( ξ ) f ( y ) d y ≤ C ∫ T n ( 1 + 2 j ρ ∣ x − y ∣ ) − N ( 1 + 2 j ρ ∣ x − y ∣ ) N ∑ ξ ∈ Z n e i 2 π ( x − y ) ξ a ( x , ξ ) φ j ( ξ ) d y ‖ f ‖ ∞ ≤ C ∫ T n ( 1 + 2 j ρ ∣ x − y ∣ ) − 2 N d y 1 2 ∑ ∣ α ∣ ≤ N ∫ T n 2 j ρ ∣ α ∣ ( x − y ) α ∑ ξ ∈ Z n e i 2 π ( x − y ) ξ a ( x , ξ ) φ j ( ξ ) 2 d y 1 2 ‖ f ‖ ∞ ≤ C 2 − j n ρ ∕ 2 ∑ ∣ α ∣ ≤ N ∫ T n 2 j ρ ∣ α ∣ ∏ k = 1 n ( e − i 2 π ( x k − y k ) − 1 ) α k ∑ ξ ∈ Z n e i 2 π ( x − y ) ξ a ( x , ξ ) φ j ( ξ ) 2 d y 1 2 ‖ f ‖ ∞ ≤ C 2 − j n ρ ∕ 2 ∑ ∣ α ∣ ≤ N ∫ T n ∑ η ∈ Z n e i 2 π ( x − y ) η 2 j ρ ∣ α ∣ Δ η α ( a ( x , η ) φ j ( η ) ) 2 d y 1 2 ‖ f ‖ ∞ ≤ C 2 − j n ρ ∕ 2 ∑ ∣ α ∣ ≤ N ∑ 2 j − 1 < ∣ η ∣ < 2 j + 1 ∣ 2 j ρ ∣ α ∣ Δ η α [ a ( x , η ) φ j ( η ) ] ∣ 2 1 2 ‖ f ‖ ∞ ≤ C 2 − j n ρ ∕ 2 ∑ ∣ α ∣ ≤ N ∑ 2 j − 1 < ∣ η ∣ < 2 j + 1 ∣ 2 j ρ ∣ α ∣ 2 j m − j ρ ∣ α ∣ ∣ 2 1 2 ‖ f ‖ ∞ ≤ C 2 − j n ρ ∕ 2 ∑ ∣ α ∣ ≤ N 2 j ( m + n ∕ 2 ) ‖ f ‖ ∞ ≤ C 2 j ( m − n ( ρ − 1 ) ∕ 2 ) ‖ f ‖ ∞ .

This completes the proof.□

3 Proof of main results

In this section, we present the proof of our main results. First, we establish the L ∞ -BMO boundedness. Second, we demonstrate the L p ( 2 < p < ∞ ) boundedness.

3.1 L ∞ -BMO boundedness

It is enough for us to prove Theorem 1.1 only when ρ < δ < 1 as Theorem 1.1 is already known for either 0 ≤ δ < ρ ≤ 1 or 0 < δ = ρ < 1 .

Proof

At first, the Bounded Mean Oscillation (BMO) norm on the torus is defined as

‖ f ‖ BMO ( T n ) = sup Q ⊂ T n 1 ∣ Q ∣ ∫ Q ∣ f ( x ) − f Q ∣ d x ,

where f Q = 1 ∣ Q ∣ ∫ Q f ( y ) d y is the mean value of f over Q .

We introduce an equivalent norm of BMO,

‖ f ‖ BMO ( T n ) ≈ sup Q ⊂ T n inf c ∈ R 1 ∣ Q ∣ ∫ Q ∣ f ( x ) − c ∣ d x ,

where the supremum is taken over all cubes Q ⊂ T n . So, for any cube Q , it is enough for us to choose a λ Q such that

1 ∣ Q ∣ ∫ Q ∣ a ( x , D ) f ( x ) − λ Q ∣ d x ≤ C ‖ f ‖ ∞ ,

where C is independent of Q .

Let l ( Q ) be the length of the side of Q . As Q ⊂ T n , there must be l ( Q ) ≤ 1 . So we can always take j Q > 0 such that

2 − j Q ( 1 + δ ) ∕ 2 < l ( Q ) ≤ 2 1 − j Q ( 1 + δ ) ∕ 2 .

For j ≥ 0 , define the operator S j by S j f ^ ( ξ ) = ∑ k = 0 j φ k ( ξ ) f ^ ( ξ ) . Now, we can decompose the operator a ( x , D ) into two parts,

(3.1) a ( x , D ) f ( x ) = ∑ j ∈ N ∑ ξ ∈ Z n e i 2 π x ξ a ( x , ξ ) φ j ( ξ ) f ^ ( ξ ) = ∑ j = 0 j Q ∑ ξ ∈ Z n e i 2 π x ξ a ( x , ξ ) φ j ( ξ ) f ^ ( ξ ) + ∑ j = j Q + 1 ∞ ∑ ξ ∈ Z n e i 2 π x ξ a ( x , ξ ) φ j ( ξ ) f ^ ( ξ ) = a ( x , D ) ( S j Q f ) ( x ) + ∑ j = j Q + 1 ∞ a j ( x , D ) f ( x ) .

Step 1. We start with the first term in (3.1). Set

a ( x Q , D ) f ( x ) = ∑ ξ ∈ Z n e i 2 π x ξ a ( x Q , ξ ) f ˆ ( ξ ) ,

where x Q is the center of Q .

It is obvious that a ( x Q , ξ ) ∈ S ρ , 0 n ( ρ − 1 ) ∕ 2 ( T n × Z n ) and ‖ S j Q f ‖ ∞ ≤ C ‖ f ‖ ∞ . Using Lemma 2.1, we obtain that

‖ a ( x Q , D ) ( S j Q f ) ‖ BMO ≤ C ‖ S j Q f ‖ ∞ ≤ C ‖ f ‖ ∞ .

Therefore, we can choose a λ Q such that

(3.2) 1 ∣ Q ∣ ∫ Q ∣ a ( x Q , D ) ( S j Q f ) ( x ) − λ Q ∣ d x ≤ C ‖ f ‖ ∞ .

Let B r be the ball in R n centered at the origin with radius r . Take a nonnegative function η ∈ C c ∞ ( B 2 ) with 0 ≤ η ≤ 1 and η ≡ 1 on B 1 . Set b Q ( x , ξ ) = a ( x , ξ ) − a ( x Q , ξ ) l ( Q ) η x − x Q n l ( Q ) . It is easy to see that b Q ( x , ξ ) = 0 when ∣ x − x Q ∣ ≥ 2 n l ( Q ) and b Q ( x , ξ ) = a ( x , ξ ) − a ( x Q , ξ ) l ( Q ) when x ∈ Q . For any N ∈ N , one can easily obtain that

∣ Δ ξ N b Q ( x , ξ ) ∣ = ∣ Δ ξ N [ a ( x , ξ ) − a ( x Q , ξ ) ] η ( x − x Q n l ( Q ) ) ∣ l ( Q ) ≤ sup ∣ y − x Q ∣ < 2 n l ( Q ) ∣ Δ ξ N ∇ y a ( y , ξ ) ∣ ≤ C N ⟨ ξ ⟩ n ( ρ − 1 ) ∕ 2 + δ − ρ N ,

where C N is independent of Q . Using Lemma 2.5, we obtain that

(3.3) ‖ b Q , j ( x , D ) f ‖ ∞ ≤ C 2 j ( n ( ρ − 1 ) ∕ 2 + δ − n ( ρ − 1 ) ∕ 2 ) ‖ f ‖ ∞ ≤ C 2 j δ ‖ f ‖ ∞ .

Now, from (3.2), (3.3), and the choice of j Q , we obtain that

(3.4) 1 ∣ Q ∣ ∫ Q ∣ a ( x , D ) ( S j Q f ) ( x ) − λ Q ∣ d x ≤ 1 ∣ Q ∣ ∫ Q ∣ a ( x , D ) ( S j Q f ) ( x ) − a ( x Q , D ) ( S j Q f ) ( x ) ∣ d x + 1 ∣ Q ∣ ∫ Q ∣ a ( x Q , D ) ( S j Q f ) ( x ) − λ Q ∣ d x ≤ 1 ∣ Q ∣ ∫ Q ∑ ξ ∈ Z n ∑ j = 0 j Q e i 2 π x ξ ( a ( x , ξ ) − a ( x Q , ξ ) ) φ j ( ξ ) f ^ ( ξ ) d x + C ‖ f ‖ ∞ = l ( Q ) ∣ Q ∣ ∫ Q ∑ j = 0 j Q b Q , j ( x , D ) f ( x ) d x + C ‖ f ‖ ∞ ≤ C l ( Q ) ∑ j = 0 j Q ‖ b Q , j ( x , D ) f ‖ ∞ + C ‖ f ‖ ∞ ≤ C ( 2 − j Q ( 1 + δ ) ∕ 2 ∑ j = 0 j Q 2 j δ + 1 ) ‖ f ‖ ∞ ≤ C ( 2 j Q ( δ − 1 ) ∕ 2 + 1 ) ‖ f ‖ ∞ ≤ C ‖ f ‖ ∞ .

Step 2. For j > j Q , as 2 − j ( 1 + δ ) ∕ 2 < l ( Q ) , the cube Q can be divided into K pairwise disjoint cubes Q j , μ such that

K ≤ C 2 j n ( 1 + δ ) ∕ 2 ∣ Q ∣ , 2 − j ( 1 + δ ) ∕ 2 ≤ l ( Q j , μ ) ≤ 2 1 − j ( 1 + δ ) ∕ 2 .

Let x j , μ be the center of Q j , μ . We use the same definition as in Step 1 to define b Q j , μ . Then, from Lemma 2.5,

‖ b Q j , μ , j ( x , D ) f ‖ ∞ ≤ C 2 j ( n ( ρ − 1 ) ∕ 2 + δ − n ( ρ − 1 ) ∕ 2 ) ‖ f ‖ ∞ ≤ C 2 j δ ‖ f ‖ ∞ .

Using the same arguments in Step 1, we obtain that

(3.5) 1 ∣ Q ∣ ∑ j = j Q + 1 ∞ ∑ μ = 1 K ∫ Q j , μ ∣ a j ( x , D ) f ( x ) − a j ( x j , μ , D ) f ( x ) ∣ d x = 1 ∣ Q ∣ ∑ j = j Q + 1 ∞ ∑ μ = 1 K l ( Q j , μ ) ∫ Q j , μ ∣ b Q j , μ , j ( x , D ) f ( x ) ∣ d x ≤ 1 ∣ Q ∣ ∑ j = j Q + 1 ∞ ∑ μ = 1 K l ( Q j , μ ) ∣ Q j , μ ∣ ‖ b Q j , μ , j ( x , D ) f ‖ ∞ ≤ C ∣ Q ∣ ∑ j = j Q + 1 ∞ ∑ μ = 1 K 2 − j ( 1 + δ ) ∕ 2 ∣ Q j , μ ∣ 2 j δ ‖ f ‖ ∞ ≤ C ∣ Q ∣ ∑ j = j Q + 1 ∞ 2 j n ( 1 + δ ) ∕ 2 ∣ Q ∣ 2 − j ( 1 + δ ) ∕ 2 2 − j n ( 1 + δ ) ∕ 2 2 j δ ‖ f ‖ ∞ ≤ C ∑ j = j Q + 1 ∞ 2 − j ( 1 + δ ) ∕ 2 2 j δ ‖ f ‖ ∞ ≤ C ‖ f ‖ ∞ ,

where C is independent of Q . Note that it is necessary that δ < 1 here.

Step 3. Set

Q ˜ j , μ = 2 1 + j 3 + δ 4 − ρ Q j , μ = Q ( x j , μ , 2 1 + j 3 + δ 4 − ρ l ( Q j , μ ) ) , f j , μ = f χ Q ˜ j , μ ,

where χ Q ˜ j , μ is the characteristic function of Q ˜ j , μ .

It is easy to see that

(3.6) ∣ Q ˜ j , μ ∣ ≤ C 2 j n 3 + δ 4 − ρ ∣ Q j , μ ∣ .

One can easily check from (2.1) that 2 j n ( 1 − ρ ) 2 a ( x j , μ , ξ ) φ j ( ξ ) ∈ S ρ , 0 0 and quasi-norms (1.1) are independent of j and Q . Using Lemma 2.2 and (3.6), we obtain that

‖ a j ( x j , μ , D ) f j , μ ‖ 2 = 2 j n ( ρ − 1 ) 2 ∑ ξ ∈ Z n e i 2 π x ξ 2 j n ( 1 − ρ ) 2 a ( x j , μ , ξ ) φ j ( ξ ) f j , μ ^ ( ξ ) 2

≤ C 2 j n ( ρ − 1 ) 2 ‖ f j , μ ‖ 2 ≤ C 2 j n ( ρ − 1 ) 2 ∣ Q ˜ j , μ ∣ 1 2 ‖ f ‖ ∞ ≤ C 2 j n ( ρ − 1 ) 2 2 j n 2 3 + δ 4 − ρ ∣ Q j , μ ∣ 1 2 ‖ f ‖ ∞ = C 2 − j n 8 ( 1 − δ ) ∣ Q j , μ ∣ 1 2 ‖ f ‖ ∞ ,

where C is independent of Q , j , μ . Therefore, one has

(3.7) 1 ∣ Q ∣ ∑ j = j Q + 1 ∞ ∑ μ = 1 K ∫ Q j , μ ∣ a j ( x j , μ , D ) f j , μ ( x ) ∣ d x ≤ 1 ∣ Q ∣ ∑ j = j Q + 1 ∞ ∑ μ = 1 K ∣ Q j , μ ∣ 1 2 ‖ a j ( x j , μ , D ) f j , μ ‖ 2 ≤ C ∣ Q ∣ ∑ j = j Q + 1 ∞ ∑ μ = 1 K ∣ Q j , μ ∣ 1 2 2 − j n 8 ( 1 − δ ) ∣ Q j , μ ∣ 1 2 ‖ f ‖ ∞ ≤ C ∣ Q ∣ ∑ j = j Q + 1 ∞ ∑ μ = 1 K 2 − j n 8 ( 1 − δ ) ∣ Q j , μ ∣ ‖ f ‖ ∞ ≤ C ∣ Q ∣ ∑ j = j Q + 1 ∞ 2 j n ( 1 + δ ) ∕ 2 ∣ Q ∣ 2 − j n 8 ( 1 − δ ) 2 − j n ( 1 + δ ) ∕ 2 ‖ f ‖ ∞ ≤ C ∑ j = j Q + 1 ∞ 2 − j n 8 ( 1 − δ ) ‖ f ‖ ∞ ≤ C ‖ f ‖ ∞ .

Here, δ < 1 is also necessary.

Step 4. From the selection of Q ˜ j , μ and the assumption ρ < δ < 1 , when y ∉ Q ˜ j , μ and x ∈ Q j , μ , we have

∣ y − x ∣ ≥ ( 2 1 + j 3 + δ 4 − ρ − 1 ) l ( Q j , μ ) ≥ 2 j 3 + δ 4 − ρ 2 − j ( 1 + δ ) ∕ 2 = 2 j 1 − δ 4 − ρ .

Let N be the smallest integer that is bigger than n 2 . Using the same arguments in the proof of Lemma 2.5, from (2.1)–(2.3) and the Plancherel theorem, when x ∈ Q j , μ , we obtain that

(3.8) ∣ a j ( x j , μ , D ) ( f − f j , μ ) ( x ) ∣ = ∫ Q ˜ j , μ c ∑ ξ ∈ Z n e i 2 π ( x − y ) ξ a ( x Q j , μ , ξ ) φ j ( ξ ) f ( y ) d y ≤ C ∫ ∣ y − x ∣ ≥ 2 j 1 − δ 4 − ρ ∣ x − y ∣ − N ∣ x − y ∣ N ∑ ξ ∈ Z n e i 2 π ( x − y ) ξ a ( x j , μ , ξ ) φ j ( ξ ) d y ‖ f ‖ ∞ ≤ C ∫ ∣ y − x ∣ ≥ 2 j 1 − δ 4 − ρ ∣ x − y ∣ − 2 N d y 1 ∕ 2 ∑ ∣ α ∣ = N ∫ T n ∑ ξ ∈ Z n ( x − y ) α e i 2 π ( x − y ) ξ a ( x j , μ , ξ ) φ j ( ξ ) 2 d y 1 ∕ 2 ‖ f ‖ ∞ ≤ C 2 j 1 − δ 4 − ρ ( n ∕ 2 − N ) ∑ ∣ α ∣ = N ∫ T n ∑ η ∈ Z n e i 2 π ( x − y ) η Δ η α ( a ( x , η ) φ j ( η ) ) 2 d y 1 2 ‖ f ‖ ∞ = C 2 j 1 − δ 4 − ρ ( n ∕ 2 − N ) ∑ ∣ α ∣ = N ∑ 2 j − 1 < ∣ η ∣ < 2 j + 1 ∣ Δ η α [ a ( x , η ) φ j ( η ) ] ∣ 2 1 2 ‖ f ‖ ∞ ≤ C ∑ ∣ α ∣ = N 2 j 1 − δ 4 − ρ ( n ∕ 2 − N ) 2 j ( n ∕ 2 + n ( ρ − 1 ) ∕ 2 − ρ ∣ α ∣ ) ‖ f ‖ ∞ ≤ C 2 − j ( 1 − δ ) ( 2 N − n ) ∕ 8 ‖ f ‖ ∞ .

Finally, from (3.1), (3.4), (3.5), (3.7), and (3.8), we can obtain

1 ∣ Q ∣ ∫ Q ∣ a ( x , D ) f ( x ) − λ Q ∣ d x ≤ 1 ∣ Q ∣ ∫ Q ∣ a ( x , D ) ( S j Q f ) ( x ) − λ Q ∣ + ∑ j = j Q + 1 ∞ ∣ a j ( x , D ) f ( x ) ∣ d x ≤ C ‖ f ‖ ∞ + 1 ∣ Q ∣ ∑ j = j Q + 1 ∞ ∑ μ = 1 K ∫ Q j , μ ( ∣ a j ( x , D ) f ( x ) − a j ( x j , μ , D ) f ( x ) ∣ + ∣ a j ( x j , μ , D ) f ( x ) ∣ ) d x ≤ C ‖ f ‖ ∞ + 1 ∣ Q ∣ ∑ j = j Q + 1 ∞ ∑ μ = 1 K ∫ Q j , μ ( ∣ a j ( x j , μ , D ) f j , μ ( x ) ∣ + ∣ a j ( x j , μ , D ) ( f − f j , μ ) ( x ) ∣ ) d x ≤ C ‖ f ‖ ∞ + C ∣ Q ∣ ∑ j = j Q + 1 ∞ ∑ μ = 1 K ∫ Q j , μ 2 − j ( 1 − δ ) ( 2 N − n ) 8 ‖ f ‖ ∞ d x ≤ C 1 + ∑ j = j Q + 1 ∞ 2 − j ( 1 − δ ) ( 2 N − n ) 8 ‖ f ‖ ∞ ≤ C ‖ f ‖ ∞ .

So we complete the proof.□

3.2 L p ( 2 < p < ∞ ) boundedness

We adopt the arguments in [10, p. 411] to prove Theorem 1.2.

Proof

As Theorem 1.2 is already known when either 0 ≤ δ < ρ ≤ 1 or 0 < δ = ρ < 1 , we can assume that 0 < ρ < δ < 1 . Denote

m p = n ( ρ − 1 ) 1 2 − 1 p + n ( ρ − δ ) p .

For a ∈ S ρ , δ m p , we set

T z f ( x ) = e 1 − 2 p − z 2 ∑ ξ ∈ Z n e i 2 π x ξ a ( x , ξ ) ⟨ ξ ⟩ n ( 1 − δ ) 1 2 − 1 p − z 2 f ^ ( ξ ) .

One can check that T z f is holomorphic in z (see [10, p. 175]) and

T 1 − 2 p f ( x ) = ∑ ξ ∈ Z n e i 2 π x ξ a ( x , ξ ) f ^ ( ξ ) = a ( x , D ) f ( x ) .

For any t ∈ R , we have

∣ T i t f ( x ) ∣ = e ( 1 − 2 p ) 2 ∑ ξ ∈ Z n e i 2 π x ξ e − t 2 ⟨ ξ ⟩ i t n ( δ − 1 ) 2 ⟨ ξ ⟩ n ( 1 − δ ) 1 2 − 1 p a ( x , ξ ) f ^ ( ξ ) , ∣ T 1 + i t f ( x ) ∣ = e 4 p 2 ∑ ξ ∈ Z n e i 2 π x ξ e − t 2 ⟨ ξ ⟩ i t n ( δ − 1 ) 2 ⟨ ξ ⟩ n ( δ − 1 ) p a ( x , ξ ) f ^ ( ξ ) .

Set

b 1 ( x , ξ ) = e − t 2 ⟨ ξ ⟩ i t n ( δ − 1 ) 2 ⟨ ξ ⟩ n ( 1 − δ ) 1 2 − 1 p a ( x , ξ ) .

As a ∈ S ρ , δ m p , using simple computations b 1 ∈ S ρ , δ n ( ρ − δ ) 2 and quasi-norms do not depend on t (see [10, p. 411]). Using Lemma 2.2, we have

‖ T i t f ‖ 2 ≤ C ‖ f ‖ 2 .

Similarly, set

b 2 ( x , ξ ) = e − t 2 ⟨ ξ ⟩ i t n ( δ − 1 ) 2 ⟨ ξ ⟩ n ( δ − 1 ) p a ( x , ξ ) .

One can check that b 2 ∈ S ρ , δ n ( ρ − 1 ) 2 and quasi-norms do not depend on t (see [10, p. 411]). Using Theorem 1.1, we obtain that

‖ T 1 + i t f ‖ BMO ≤ C ‖ f ‖ ∞ .

Using the Fefferman-Stein interpolation [10, p. 175], we show that

‖ a ( x , D ) f ‖ p = T 1 − 2 p f p ≤ C ‖ f ‖ p .

This completes the proof of Theorem 1.2.□

Acknowledgements

The author is grateful for the reviewer’s valuable comments that improved the manuscript.

Funding information: Benhamoud Ramla was supported by the National Natural Science Foundation of China (No. 12071437).
Author contributions: The author confirms the sole responsibility for the conception of the study and presented results and manuscript preparation.
Conflict of interest: The author declares no competing interests.

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Received: 2023-11-08

Revised: 2024-05-16

Accepted: 2024-06-03

Published Online: 2024-08-02

This work is licensed under the Creative Commons Attribution 4.0 International License.

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https://doi.org/10.1515/math-2024-0023

Keywords for this article

Torus; toroidal pseudo-differential operator; toroidal Hörmander symbol; BMO(𝕋 n )

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