Iterated paraproducts and iterated commutator estimates in Besov spaces
aa r X i v : . [ m a t h . A P ] M a y ITERATED PARAPRODUCTS AND ITERATED COMMUTATOR ESTIMATES INBESOV SPACES
MASATO HOSHINO
Abstract.
In the previous study [6], the author provided an algebraic proof of the multicomponentcommutator estimate in Besov spaces C α = B α ∞ , ∞ with 0 < α <
1. In this paper, we extend that resultto general Besov spaces B αp,q with p, q ∈ [1 , ∞ ] and 0 < α < Introduction
It is well known that, the definition of Besov space in Euclidean space R d by Littlewood-Paley theoryis equivalent to the one based on the estimate of Taylor remainder, when the regularity parameter ispositive. See [1, Theorem 2.36] for instance. Especially, Besov space B α ∞ , ∞ ( R d ) is the same as H¨olderspace C α ( R d ) if α is a positive noninteger.In [6], the author showed a similar equivalence result for Bony’s paraproduct and its iterated versions.For any distributions f, g ∈ S ′ ( R d ), the paraproduct f ≺ g is defined via Littlewood-Paley theory, sothis is not a local operator. Nevertheless, in the case f ∈ C α ( R d ) and g ∈ C β ( R d ) with 0 < α, β and α + β <
1, the previous result [6, Theorem 3.1] implies( f ≺ g )( y ) = ( f ≺ g )( x ) + f ( x )( g ( y ) − g ( x )) + O ( | y − x | α + β ) . (1.1)Conversely, we can show that the function h of such a local behavior is essentially the same as f ≺ g. In[6], the author studied a generalized version of (1.1) for the iterated paraproducts, and as a consequence,provided an algebraic proof of the commutator estimate [4, Lemma 2.4], which has an important role inthe theory of paracontrolled calculus [4].In this paper, we consider the Besov type extension of the results in [6]. First we show the estimatelike (1.1), see Theorem 3.1 below. The result is no longer a uniform bound on R d , but an L p L q typeestimate of Taylor remainder. As a consequence, we also show the commutator estimate in Besov spaces,stated as below. Commutators discussed in this paper is defined as follows. Definition 1.1.
For any functions ξ, f , f , . . . in S ( R d ) , define C ( f , ξ ) := f (cid:23) ξ (:= f ξ − f ≺ ξ ) , C ( f , f , ξ ) := C ( f ≺ f , ξ ) − f C ( f , ξ ) , C ( f , . . . , f n , ξ ) := C ( f ≺ f , f , . . . , f n , ξ ) − f C ( f , f , . . . , f n , ξ ) . We denote by B α, p,q the closure of S ( R d ) in the space B αp,q ( R d ). The following theorem is a generalizationof [6, Theorem 4.2] onto Besov norms. Theorem 1.1.
Let α , . . . , α n ∈ (0 , and α ◦ < be such that α + · · · + α n < ,α + · · · + α n + α ◦ < < α + · · · + α n + α ◦ , and let α := α + · · · + α n + α ◦ . Let p , . . . , p n , p ◦ , q , . . . , q n , q ◦ ∈ [1 , ∞ ] be such that p := 1 p + · · · + 1 p n + 1 p ◦ ≤ , q := 1 q + · · · + 1 q n + 1 q ◦ ≤ . Then there exists a unique multilinear continuous operator ˜ C : B α , p ,q × · · · × B α n , p n ,q n × B α ◦ , p ◦ ,q ◦ → B α, p,q such that, ˜ C ( f , . . . , f n , ξ ) = C ( f , . . . , f n , ξ ) for any smooth inputs ( f , . . . , f n , ξ ) . To show the main theorem, we introduce a regularity structure suitable for our context, and impose B p,q type bounds on models and modelled distributions. Thus in our case, each basis vector τ of themodel space has three homogeneity parameters ( α τ , p τ , q τ ), which is slightly different from the originalsetting [5]. See [7, 8, 9] for relevant studies. In [7, 8], the authors defined B p,q type modelled distributionsand proved a generalized reconstruction theorem, while B ∞ , ∞ type bounds are imposed on models. In[9], the authors defined B p,p (Sobolev) type models and modelled distributions to consider the Sobolevtype rough paths.This paper is organized as follows. In Section 2, we define some important notions used in thispaper; Besov type norms, paraproducts, and the word Hopf algebra. In Section 3, we show the Besovtype estimates of Taylor remainders of iterated paraproducts. In Section 4, we show the Besov typecommutator estimates. 2. Preliminaries
We introduce some important notions used throughout this paper.2.1.
Besov type norms.
In this paper, we often use a sequence { a j } ∞ j = − of numbers, functions, oroperators. We use simplifying notations for partial sums as follows. a Let q ∈ [1 , ∞ ] and let { c j } ∞ j = − be a sequence of nonnegative numbers. If α > , then wehave k{ jα c ≥ j } j k ℓ q . k{ jα c j } j k ℓ q , (2.1) k{ − jα c ≤ j } j k ℓ q . k{ − jα c j } j k ℓ q . (2.2) proof. We extend { c j } ∞ j = − into a sequence { c j } j ∈ Z by setting c j = 0 if j ≤ − 2. For (2.1), by usingYoung’s inequality on the group Z , (cid:13)(cid:13) { Jα c ≥ J } J (cid:13)(cid:13) ℓ q = (cid:13)(cid:13)(cid:13)n X j ≥ J ( J − j ) α jα c j o J (cid:13)(cid:13)(cid:13) ℓ q ≤ X −∞
Denote by S = S ( R d ) the space of Schwartz functions, and by S ′ its dual space. Fix smooth radialfunctions χ and ρ such that, • supp( χ ) ⊂ { x ; | x | < } and supp( ρ ) ⊂ { x ; < | x | < } , • χ ( x ) + P ∞ j =0 ρ (2 − j x ) = 1 for any x ∈ R d .Set ρ − := χ and ρ j := ρ (2 − j · ) for j ≥ 0. We define the Littlewood-Paley blocks∆ j f := F − ( ρ j F f )for f ∈ S ′ , where F is the Fourier transform and F − is its inverse. It is useful to write∆ j f ( x ) = Z R d Q j ( x, y ) f ( y ) dy, where Q j ( x, y ) = F − ( ρ j )( x − y ). We also write Q j ( h ) = F − ( ρ j )( h ). Definition 2.1. For any α ∈ R and p, q ∈ [1 , ∞ ] , we define the (nonhomogeneous) Besov space B αp,q bythe space of all f ∈ S ′ such that k f k B αp,q := (cid:13)(cid:13)(cid:13)(cid:8) jα k ∆ j f k L p (cid:9) j ≥− (cid:13)(cid:13)(cid:13) ℓ q < ∞ . As stated in [1, Theorem 2.36], it is possible to define Besov norms without Littlewood-Paley theory.The aim of this paper is to study the following norm for two parameter functions. Definition 2.2. Let α ∈ R and p, q ∈ [1 , ∞ ] . For any two parameter measurable function ω ( x, y ) on R d × R d , define k ω k D αp,q := (cid:13)(cid:13) | h | − α k ω ( x, x + h ) k L p ( dx ) (cid:13)(cid:13) L q ( dh/ | h | d ) . TERATED PARAPRODUCTS AND ITERATED COMMUTATOR ESTIMATES IN BESOV SPACES 3 The following well-known result provides an alternative definition of Besov norm. A self-containedproof appears in the next subsection. Proposition 2.2 ([1, Theorem 2.36]) . If α ∈ (0 , , then k f k B αp,q ≃ k f k L p + k ω f k D αp,q , where ω f ( x, y ) = f ( y ) − f ( x ) . Technical lemmas. We prove some technical lemmas used throughout this paper. Definition 2.3. Let α ∈ R and p, q ∈ [1 , ∞ ] . For any sequence { f j ( x ) } ∞ j = − of measurable functions on R d , define k{ f j } j k B αp,q := (cid:13)(cid:13)(cid:13)(cid:8) jα k f j k L p (cid:9) j (cid:13)(cid:13)(cid:13) ℓ q . By definition, k f k B αp,q = k{ ∆ j f } j k B αp,q . We often emphasize the variables j and x and write k f j ( x ) k B αp,q = (cid:13)(cid:13)(cid:13)(cid:8) jα k f j ( x ) k L p ( dx ) (cid:9) j (cid:13)(cid:13)(cid:13) ℓ q . by an abuse of notation. Lemma 2.3. Let α > and q ∈ [1 , ∞ ] . Let F be a nonnegative function on R d such that (cid:13)(cid:13) | h | − α F ( h ) (cid:13)(cid:13) L q ( dh/ | h | d ) ≤ C. Then for any nonnegative function ϕ ∈ S , one has (cid:13)(cid:13)(cid:13)(cid:13) jα Z R d jd ϕ (2 j h ) F ( h ) dh (cid:13)(cid:13)(cid:13)(cid:13) ℓ q . C. proof. The proof is essentially contained in the latter half part of the proof of [1, Theorem 2.36]. There α ∈ (0 , 1) is assumed, but we can see that Lemma 2.3 holds for any α > 0. The only point to be modifiedis the integration over 2 j | h | > q < ∞ . Indeed, for any ε > jα (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z j | h | > jd ϕ (2 j h ) F ( h ) dh (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ − jε Z j | h | > | j h | d + α + ε | ϕ (2 j h ) | F ( h ) | h | α + ε dh | h | d . − jε Z j | h | > F ( h ) q | h | ( α + ε ) q dh | h | d ! /q by H¨older’s inequality for the measure dh/ | h | d , and we have (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) − jε Z j | h | > F ( h ) q | h | ( α + ε ) q dh | h | d ! /q (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ℓ q . (cid:13)(cid:13) | h | − α F ( h ) (cid:13)(cid:13) L q ( dh/ | h | d ) , since the sum of 2 − jεq over j such that 2 j | h | > | h | εq . (cid:3) Lemma 2.4. Let α > . For any ω ∈ D αp,q , one has the bound (cid:13)(cid:13) ∆ Lemma 2.5. Let { ω j ( x, y ) } ∞ j = − be a sequence of two parameter functions. Assume that for some C > and α > , the bound k ω j ( x + h, x ) k B α − θp,q := (cid:13)(cid:13)(cid:8) j ( α − θ ) k ω j ( x + h, x ) k L p (cid:9) j ≥− (cid:13)(cid:13) ℓ q ≤ C | h | θ (2.3) holds for any h ∈ R d and any θ in a neighborhood of α . Then ω = P j ≥− ω j converges in D αp,q and onehas the bound k ω k D αp,q . C. proof. We follow the proof of [1, Theorem 2.36]. Since the case q = ∞ is the same as [6, Lemma 3.7],we consider q < ∞ . Assume C ≤ A N = (cid:8) h ∈ R d ; 2 − N − ≤ | h | < − N (cid:9) ( N ≥ ,A − = { h ∈ R d ; 1 ≤ | h |} . Fix a small ε > θ = α ± ε . If h ∈ A N with N ≥ | h | − α k ω ( x + h, x ) k L p ( dx ) . X j | h | − α k ω j ( x + h, x ) k L p ( dx ) . X − ≤ j 0, by Young’s inequality we have X N ≥ X j ≥− −| j − N | ε j < ∞ . If h ∈ A − , similarly to above, Z A − (cid:16) | h | − α k ω ( x + h, x ) k L p ( dx ) (cid:17) q dh | h | d . X j ≥− j − jε Z | h |≥ | h | − εq − d dh . , which completes the proof. (cid:3) Using above lemmas, we can prove Proposition 2.2. Proof of Proposition 2.2. Note that k ∆ − f k L p . k f k L p . For j ≥ 0, since R Q j = 0 we have∆ j f ( x ) = ∆ j ( ω f ( x, · ))( x ) . Lemma 2.4 yields k f k B αp,q . k f k L p + k ω f k D αp,q . To show the converse, let ω j ( x, y ) = ∆ j f ( y ) − ∆ j f ( x )and apply Lemma 2.5. Obviously, k ω j ( x, x + h ) k L p ( dx ) . k ∆ j f k L p . − jα qj k f k B αp,q . TERATED PARAPRODUCTS AND ITERATED COMMUTATOR ESTIMATES IN BESOV SPACES 5 By Minkowski’s inequality and the continuity of the differentiation B αp,q ∋ f 7→ ∇ f ∈ ( B α − p,q ) d (see [1,Proposition 2.78]), we have k ω j ( x, x + h ) k L p ( dx ) ≤ (cid:13)(cid:13)(cid:13)(cid:13) h · Z ∇ ∆ j f ( x + θh ) dθ (cid:13)(cid:13)(cid:13)(cid:13) L p ( dx ) ≤ | h |k ∆ j ( ∇ f ) k L p . | h | j (1 − α ) qj k f k B αp,q . By an interpolation, for any θ ∈ [0 , 1] we have k ω j ( x + h, x ) k L p ( dx ) . | h | θ j ( θ − α ) qj k f k B αp,q , so k ω f k D αp,q . k f k B αp,q by Lemma 2.5. (cid:3) Paraproduct. For any smooth functions f, g , we can decompose the product f g by f g = X j,k ≥− ∆ j f ∆ k g = (cid:16) X j Let α > . If ω ∈ D αp,q , then P ( ω ) ∈ B αp,q . The mapping ω P ( ω ) is continuous. proof. In view of [2, Proposition 8], k P ( ω ) k B αp,q . k P j ( ω ) k B αp,q . For the right hand side, exchanging variables y = z + h and x = z + h + k and using Minkowski’s inequality,we have k P j ( ω ) k L p . Z Z | Q We introduce a specific regularity structure. Fix an integer n . For anyintegers 1 ≤ k ≤ ℓ ≤ n , denote by ( k . . . ℓ ) the sequence from k to ℓ , which is called a word throughoutthis paper. We discuss the algebras made from the set W of all such words. Let Alg( W ) be thecommutative algebra freely generated by W with unit . We regard as an empty word and considerthe extended set W = W ∪ { } of words. For any nonempty words σ = ( k . . . ℓ ) and η = (( ℓ + 1) . . . m )in W , we define σ ⊔ η = ( k . . . m ). We also define ⊔ τ = τ ⊔ = τ . Definition 2.4. Define the linear map ∆ : Alg( W ) → Alg( W ) ⊗ Alg( W ) by ∆ τ = X σ,η ∈ W , σ ⊔ η = τ σ ⊗ η for any τ ∈ W . The map ∆ is coassociative; (∆ ⊗ id)∆ = (id ⊗ ∆)∆. It is easy to show the existence of the algebramap A : Alg( W ) → Alg( W ) such that A = ,M ( A ⊗ id)∆ τ = M (id ⊗ A )∆ τ = 0 ( τ ∈ W ) , where M : Alg( W ) ⊗ Alg( W ) → Alg( W ) is the product map. Such A is called an antipode. In other words,Alg( W ) is a Hopf algebra . The existence of A yields that, the set G of all algebra maps γ : Alg( W ) → R forms a group by the product ( γ ∗ γ )( τ ) = ( γ ⊗ γ )∆ τ. TERATED PARAPRODUCTS AND ITERATED COMMUTATOR ESTIMATES IN BESOV SPACES 6 The inverse of γ ∈ G is given by γ − = γ ◦ A .In Section 3, we study the family { f τ = f τ ( x ) } τ ∈ W of functions on R d , indexed by words. We regard f ( x ) ∈ G by extending the map τ f τ ( x ) algebraically. Then we define the G -valued two parameterfunction by ω ( x, y ) = f ( x ) − ∗ f ( y ) . In other words, we have a family { ω τ ( x, y ) := ω ( x, y )( τ ) } τ ∈ W of two parameter functions, indexed bywords. The following relationships between f and ω are useful in Section 3. Lemma 2.7. For any ≤ k ≤ ℓ ≤ n , one has ω k...ℓ ( x, y ) = f k...ℓ ( y ) − f k...ℓ ( x ) − ℓ − X m = k f k...m ( x ) ω ( m +1) ...ℓ ( x, y ) , (2.4) ω k...ℓ ( x, z ) = ω k...ℓ ( x, y ) + ω k...ℓ ( y, z )+ ℓ − X m = k ω k...m ( x, y ) ω ( m +1) ...ℓ ( y, z ) . (2.5) proof. Immediate consequences of the simple formulas. (2.4): f ( y ) = f ( x ) ∗ ω ( x, y ), (2.5): ω ( x, z ) = ω ( x, y ) ∗ ω ( y, z ). (cid:3) Taylor remainders of iterated paraproducts For a given sequence f , f , . . . of functions, we define the iterated paraproducts ( f ) ≺ := f , ( f , . . . , f n ) ≺ := ( f , . . . , f n − ) ≺ ≺ f n . The aim of this section is to show the following Besov type estimate, which is an extension of [6, Theorem3.1]. We write f ≺ k...ℓ := ( f k , . . . , f ℓ ) ≺ . Theorem 3.1. For any measurable functions f , . . . , f n , we define the family { ω ≺ k...ℓ ( x, y ) } ≤ k ≤ ℓ ≤ n of two parameter functions by the recursive formula (2.4) with f k...ℓ replaced by f ≺ k...ℓ . Let α , . . . , α n ∈ (0 , , p , . . . , p n , q , . . . , q n ∈ [1 , ∞ ] , and f i ∈ B α i p i ,q i for each i . If α := α + · · · + α n < , p = p + · · · + p n ≤ , and q = q + · · · + q n ≤ , then we have ω ≺ ...n ∈ D αp,q and k ω ≺ ...n k D αp,q . k f k B α p ,q · · · k f n k B αnpn,qn . (3.1)3.1. Simplified iterated paraproducts. Fix the parameters and the functions as in Theorem 3.1. Forany 1 ≤ k ≤ ℓ ≤ n , we use the following simplifying notations. α k...ℓ := α k + · · · + α ℓ , p k...ℓ := 1 p k + · · · + 1 p ℓ , q k...ℓ := 1 q k + · · · + 1 q ℓ . First we show the existence of the family { ˜ f k...ℓ } ≤ k ≤ ℓ ≤ n such that the corresponding { ˜ ω k...ℓ } ≤ k ≤ ℓ ≤ n satisfies the bound (3.1). Definition 3.1. For any j ≥ − , we define ( ˜ f k ) j := ∆ j f k , ( ˜ f k...ℓ ) j := ( ˜ f k... ( ℓ − ) Define (˜ ω k...ℓ ) j recursively by (˜ ω k...ℓ ) j ( x, y ) = ( ˜ f k...ℓ ) j ( y ) − ( ˜ f k...ℓ ) j ( x ) − ℓ − X m = k ˜ f k...m ( x )(˜ ω ( m +1) ...ℓ ) j ( x, y ) . Then one has the following formulas. TERATED PARAPRODUCTS AND ITERATED COMMUTATOR ESTIMATES IN BESOV SPACES 7 (1) (˜ ω k ) j ( x, y ) = ∆ j f k ( y ) − ∆ j f k ( x ) . (2) If k < ℓ , (˜ ω k...ℓ ) j ( x, y ) = (˜ ω k... ( ℓ − ) 1. The case n = 1 is already proved in the proof ofProposition 2.2, in Section 2.2. Let n ≥ 2. By Lemma 3.2-(3), we inductively have (cid:13)(cid:13) ( C ...n ) j (cid:13)(cid:13) B α ...np ...n,q ...n ≤ (cid:13)(cid:13) ( C ... ( n − ) ≥ j − (cid:13)(cid:13) B α ... ( n − p ... ( n − ,q ... ( n − (cid:13)(cid:13) ( ˜ f n ) j (cid:13)(cid:13) B αnpn,qn . , where we use Lemma 2.1-(2.1) for the bound of ( C ... ( n − ) ≥ j − . Assume (3.2) holds for the word(1 . . . ( n − θ ∈ ( α ... ( n − , (cid:13)(cid:13) (˜ ω ... ( n − ) We show the bound (3.1) for ω ≺ , which is really required. For any word τ = ( k . . . ℓ ), denote by Π( τ ) the set of all partitions of τ , that is, we write { τ , . . . , τ m } ∈ Π( τ )if τ , . . . , τ m are nonempty words of the form τ j = ( k j . . . ℓ j ) for each j , where k = k , ℓ m = ℓ , and ℓ j + 1 = k j +1 for any j . Recall the definitions of α τ = α k...ℓ , p τ , and q τ as before. Lemma 3.3. For any word τ = ( k . . . ℓ ) , there exists a function [ ˜ f ] τ ∈ B α τ p τ ,q τ continuously depending on f k , . . . , f ℓ , such that, one has the formula ˜ f τ = X { σ,η }∈ Π( τ ) ˜ f σ ≺ [ ˜ f ] η + [ ˜ f ] τ . (3.3) Moreover, one has the atomic decomposition ˜ f τ = ∞ X m =1 X { τ ,...,τ m }∈ Π( τ ) ([ ˜ f ] τ , . . . , [ ˜ f ] τ m ) ≺ . (3.4) proof. Second formula (3.4) is an immediate consequence of the first one (3.3). The proof of (3.3) isessentially the same as [2, Proposition 12]. The point is that we use Besov norms B αp,q , while in [2] theparticular case p = q = ∞ is considered. TERATED PARAPRODUCTS AND ITERATED COMMUTATOR ESTIMATES IN BESOV SPACES 8 Here we give a proof of (3.3). Write ω f ( x, y ) = f ( y ) − f ( x ) for simplicity. Expanding ˜ ω τ by repeating(2.4), we have ˜ ω τ ( x, y )= ω ˜ f τ ( x, y ) − X { τ ,τ }∈ Π( τ ) ˜ f τ ( x )˜ ω τ ( x, y )= · · · = ω ˜ f τ ( x, y ) − ∞ X m =2 ( − m X { τ ,...,τ m }∈ Π( τ ) ( ˜ f τ . . . ˜ f τ m − )( x ) ω ˜ f τm ( x, y ) . (3.5)Applying the two parameter operator P to both sides, we have P (˜ ω τ ) = 1 ≺ ˜ f τ − ∞ X m =2 ( − m X { τ ,...,τ m }∈ Π( τ ) ( ˜ f τ . . . ˜ f τ m − ) ≺ ˜ f τ m . By Lemma 2.6, P (˜ ω τ ) belongs to B α τ p τ ,q τ and continuously depends on f k , . . . , f ℓ . If ˜ f τ m has a decompo-sition (3.3), ∞ X m =2 ( − m X { τ ,...,τ m }∈ Π( τ ) ( ˜ f τ . . . ˜ f τ m − ) ≺ ˜ f τ m = ∞ X m =2 ( − m X { τ ,...,τ m }∈ Π( τ ) ( ˜ f τ . . . ˜ f τ m − ) ≺ [ ˜ f ] τ m + ∞ X m =2 ( − m X { τ ,...,τ m ,τ m +1 }∈ Π( τ ) ( ˜ f τ . . . ˜ f τ m − ) ≺ ( ˜ f τ m ≺ [ ˜ f ] τ m +1 )= X { τ ,τ }∈ Π( τ ) ˜ f τ ≺ [ ˜ f ] τ + ∞ X m =2 ( − m X { τ ,...,τ m ,τ m +1 }∈ Π( τ ) R ( ˜ f τ . . . ˜ f τ m − , ˜ f τ m , [ ˜ f ] τ m +1 ) , where R is the correcting operator defined by R ( a, b, c ) := a ≺ ( b ≺ c ) − ( ab ) ≺ c. The sum of all R terms belongs to B α τ p τ ,q τ and continuously depends on f k , . . . , f ℓ . Its proof is left toLemma 3.4 below. Then we obtain the formula (3.3) since k ˜ f τ − ≺ ˜ f τ k B rpτ ,qτ = k ∆ ≤ ˜ f τ k B rpτ ,qτ . k ˜ f τ k B αℓpτ .qτ for any r > (cid:3) Lemma 3.4. Let σ = ( k . . . ℓ ) , α ′ > , p ′ , q ′ ∈ [1 , ∞ ] , and g ∈ B α ′ p ′ ,q ′ . Assume that α = α σ + α ′ < , /p = 1 /p σ + 1 /p ′ ≤ , and /q = 1 /q σ + 1 /q ′ ≤ . Then one has the bound (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ X m =2 ( − m X { τ ,...,τ m }∈ Π( σ ) R ( ˜ f τ . . . ˜ f τ m − , ˜ f τ m , g ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) B αp,q . k f k k B αkpk,qk . . . k f ℓ k B αℓpℓ,qℓ k g k B α ′ p ′ ,q ′ . proof. Just an analogue of [2, Proposition 10], so see it for details. In view of the formula (3.5), it issufficient to show that (cid:13)(cid:13) P j (cid:0) (˜ ω σ ( x, · ) ≺ g )( y ) (cid:1)(cid:13)(cid:13) B αp,q < ∞ , where we write P j (Ω) = P j (Ω( x, y )) as an abuse of notation. Since the integral R Q j ( z, y ) Q
We emphasize the dependence of ω ≺ k...ℓ on f k . . . , f ℓ by writing ω ≺ k...ℓ = ω ≺ ( f k , . . . , k ℓ ) . We prove the result by an induction on the number of the components of ω ≺ . By the formula (3.3),[ ˜ f ] ( k ) = f k for a word with only one letter. Hence if Ξ ∈ Π( τ ) has the same cardinality as the length of τ (denoted by | Ξ | = | τ | ), we have [ ˜ f ] ≺ Ξ = f ≺ τ and [˜ ω ] ≺ Ξ = ω ≺ τ . Hence by (3.9), ω ≺ τ = ˜ ω τ − X Ξ ∈ Π( τ ) , | Ξ | < | τ | [˜ ω ] ≺ Ξ . The bound for ˜ ω τ was already obtained. By an assumption of the induction, ω ≺ is continuous as a lessthan | τ | -component operator. Hence (cid:13)(cid:13) [˜ ω ] ≺ τ ...τ m (cid:13)(cid:13) D ατpτ ,qτ = (cid:13)(cid:13) ω ≺ ([ ˜ f ] τ , . . . , [ ˜ f ] τ m ) (cid:13)(cid:13) D ατpτ ,qτ . k [ ˜ f ] τ k B ατ pτ ,qτ . . . k [ ˜ f ] τ m k B ατmpτm,qτm . k f k k B αkpk,qk . . . k f ℓ k B αℓpℓ,qℓ , where we use the continuity of ( f k j , . . . , f ℓ j ) [ ˜ f ] τ j (Lemma 3.3). As a result, we obtain the continuityof ω ≺ τ with respect to f k , . . . , f ℓ . (cid:3) TERATED PARAPRODUCTS AND ITERATED COMMUTATOR ESTIMATES IN BESOV SPACES 10 Besov type regularity structure and commutator estimates We prove Theorem 1.1 in the rest of this paper. We show only the existence of the continuous map ˜ C .The uniqueness of ˜ C and its multilinearity follows from the denseness argument.4.1. Besov type regularity structure. We return to the Hopf algebra Alg( W ) with the charactergroup G . We consider a subset V = { } ∪ { ( k . . . n ) ; k = 1 , . . . , n } . of W and a linear subspace T = h V i . Since ∆ T ⊂ Alg( W ) ⊗ T , for any γ ∈ G we can define the linearmap Γ γ : T → T by Γ γ = ( γ ⊗ id)∆ . The pair ( T, G ) is an example of the regularity structure . Remark 4.1. Note that the position of γ is opposite to the original definition [5] . Because of it, in themapping γ Γ γ , the order of multiplication is turned over as follows. Γ γ Γ γ = Γ γ ∗ γ (4.1)We define a model (Π , Γ) on the regularity structure ( T, G ). Fix the parameters and the functionssatisfying the assumptions in Theorem 1.1. For any 1 ≤ k ≤ n , we define α ′ k...n = α k + · · · + α n + α ◦ , p ′ k...n = 1 p k + · · · + 1 p n + 1 p ◦ , q ′ k...n = 1 q k + · · · + 1 q n + 1 q ◦ . Moreover, set α ′ = α ◦ , p ′ = p ◦ , and q ′ = q ◦ . Definition 4.1. Let f ≺ k...ℓ = ( f k , . . . , f ℓ ) ≺ be the iterated paraproduct. We regard f ≺ ( x ) ∈ G by extendingthe map τ f ≺ τ ( x ) algebraically, and define ω ≺ ( x, y ) = f ≺ ( x ) − ∗ f ≺ ( y ) , Γ xy = Γ ω ≺ ( x,y ) . Definition 4.2. For any linear map Π : T → S ′ , define Π x τ = ( f ≺ ( x ) − ⊗ Π)∆ τ. Denote by M the set of all maps Π such that k Π k M := sup τ ∈ V (cid:13)(cid:13) ∆ T, G ). Note that these formulas are slightlydifferent from the original ones [5], like the formula (4.1).As an analogue of [6, 3], we can show that the space M has a simple topological structure. Let V − = { } ∪ { ( k . . . n ) ; k = 2 , . . . , n } . Note that α ′ τ < τ ∈ V − by assumption. Theorem 4.2. For any Π ∈ M , define the linear map [Π] : T → S ′ by Π τ = X σ ⊔ η = τ, σ = f ≺ σ ≺ [Π] η + [Π] τ. (4.3) Then [Π] τ ∈ B α ′ τ p ′ τ ,q ′ τ and the mapping ( f , . . . , f n , Π) [Π] τ ∈ B α ′ τ p ′ τ ,q ′ τ is continuous. Conversely, for any given family { [Π] τ } τ ∈ V − ∈ Y τ ∈ V − B α ′ τ p ′ τ ,q ′ τ , there exists a unique element Π ∈ M satisfying (4.3) . Moreover, the map { [Π] τ } τ ∈ V − Π is continuous. The above theorem is just an analogue of [2, Theorem 14 and Corollary 15], so we leave the detailsto the reader. The only modification is that we have to use the Besov type reconstruction theorem inAppendix. TERATED PARAPRODUCTS AND ITERATED COMMUTATOR ESTIMATES IN BESOV SPACES 11 Proof of Theorem 1.1. Now we show the iterated commutator estimates. This part is strictly ananalogue of [6, Section 4]. Proof of Theorem 1.1. For any given ξ ∈ B α ◦ p ◦ ,q ◦ , we can define Π ξ ∈ M by[Π ξ ] = ξ, [Π ξ ]( k . . . n ) = 0 (2 ≤ k ≤ n ) . Note that Π ξ = ξ, Π ξ ( k . . . n ) = f ≺ k...n ≺ ξ (2 ≤ k ≤ n ) , Π ξ (1 . . . n ) = f ≺ ...n ≺ ξ + [Π ξ ](1 . . . n ) , (4.4)by the formula (4.3). Then the map ( f , . . . , f n , ξ ) [Π ξ ](1 . . . n )is continuous, which turns out to be the required map ˜ C . It remains to show that[Π ξ ](1 . . . n ) = C ( f , . . . , f n , ξ )(4.5)if all inputs ( f , . . . , f n , ξ ) are in S ( R d ). Since Π ξ = ( f ≺ ( x ) ⊗ Π ξx )∆,Π ξ ( k . . . n )( x ) = Π ξx ( k . . . n )( x ) + f ≺ k...n ( x ) ξ ( x )+ n − X ℓ = k f ≺ k...ℓ ( x ) (cid:0) Π ξx (( ℓ + 1) . . . n ) (cid:1) ( x ) , (4.6)for any 1 ≤ k ≤ n . By using it and (4.4), we can inductively show thatΠ ξx ( k . . . n )( x ) = − C ( f k , . . . , f n , ξ )( x )for 2 ≤ k ≤ n . Then letting k = 1 in (4.6) and usingΠ ξx (1 . . . n )( x ) = lim j →∞ ∆ 0, we have (4.5) by the definition of C . (cid:3) Appendix A. Besov type reconstruction theorem We define Besov type modelled distributions. Recall that V = { } ∪ { ( k . . . n ) ; k = 1 , . . . , n } . and T = h V i . Definition A.1. For any function g : R d → T , define ω g ( x, y ) = g ( y ) − Γ xy g ( x ) and denote by ω g τ ( y, x ) its τ -component. Let k be the smallest integer such that ω g k...n ( y, x ) does notvanish, and let α > α ′ k...n , p ∈ [1 , p ′ k...n ] , and q ∈ [1 , q ′ k...n ] . For such parameters, we define k g k D αp,q := sup τ (cid:13)(cid:13) ω g τ (cid:13)(cid:13) D α − α ′ τp \ p ′ τ ,q \ q ′ τ , where p \ p ′ τ = p − p ′ τ , q \ q ′ τ = q − q ′ τ . Let D αp,q be the set of functions g : R d → T such that k g k D αp,q < ∞ . Such g is called a modelled distribution controlled by Γ. We show the Besov type reconstructiontheorem. Proposition A.1. For any g ∈ D αp,q and Π ∈ M , we define P g ( z ) = X j Z Z R d × R d P j ( z, x ) Q j ( z, y )Π x (cid:0) g ( x ) (cid:1) ( y ) dxdy. (1) If α > , there exists a unique continuous bilinear map Q : D αp,q × M → B αp,q such that (cid:13)(cid:13) ∆ The proof is almost the same as [2, Proposition 9]. In view of it, here it is sufficient to show thebound (cid:13)(cid:13) ∆ j (cid:0) P g − Π x g ( x ) (cid:1) ( x ) (cid:13)(cid:13) B αp,q < ∞ . We have only to consider j ≥ 1. For such j ,∆ j (cid:0) P g − Π x g ( x ) (cid:1) ( x )= X i ; i ∼ j Z Z Z Q j ( x, y ) Q
The author is supported by JSPS KAKENHI Early-Career Scientists 19K14556.The author thanks the anonymous referee for reading the paper carefully and providing helpful comments. References [1] H. Bahouri, J.-Y. Chemin, and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, Springer,2011.[2] I. Bailleul and M. Hoshino, Paracontrolled calculus and regularity structures, arXiv:1812.07919.[3] I. Bailleul and M. Hoshino, Regularity structures and paracontrolled calculus, arXiv:1912.08438.[4] M. Gubinelli, P. Imkeller, and N. Perkowski, Paracontrolled distributions and singular PDEs, Forum Math. Pi, (2015), e6, 75pp.[5] M. Hairer, A theory of regularity structures, Invent. Math., (2014), no. 2, 269-504.[6] M. Hoshino, Commutator estimates from a viewpoint of regularity structures, arXiv:1903.00623.[7] M. Hairer and C, Labb´e, The reconstruction theorem in Besov spaces, J. Funct. Anal., (2017), no. 8, 2578-2618.[8] C. Liu, D. J. Pr¨omel, and J. Teichmann, Stochastic Analysis with Modelled Distributions, arXiv:1609.03834.[9] C. Liu, D. J. Pr¨omel, and J. Teichmann, Optimal extension to Sobolev rough paths, arXiv:1811.05173. Faculty of Mathematics, Kyushu University E-mail address ::