Littlewood--Paley--Rubio de Francia Inequality for the Two-parameter Walsh System
aa r X i v : . [ m a t h . F A ] F e b Littlewood–Paley–Rubio de Francia Inequality forthe Two-parameter Walsh System
Viacheslav Borovitskiy ∗ St. Petersburg Department of Steklov Mathematical InstituteSt. Petersburg State University
Abstract
A version of Littlewood–Paley–Rubio de Francia inequality for the two-parameter Walsh system is proved: for any family of disjoint rectangles I k = I k × I k in Z + × Z + and a family of functions f k with Walsh spectruminside I k the following is true (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)X k f k (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p ≤ C p (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X k | f k | ! / (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p , < p ≤ , where C p does not depend on the choice of rectangles { I k } or func-tions { f k } . The arguments are based on the atomic theory of two-parametermartingale Hardy spaces. In the course of the proof, a two-parameterversion of Gundy’s theorem on the boundedness of operators taking mar-tingales to measurable functions is formulated, which might be of inde-pendent interest. Consider a countable index set Z and an orthonormal basis { φ n } n ∈ Z in the space L . Define operators M I for I ⊆ Z by the expression M I f = P n ∈ I h f, φ n i φ n .Whenever M I f = f , we say that the spectrum of f lies in I and write spec f ⊆ I .Consider also a partition { I k } k ∈ N of the index set Z and a family of func-tions f k ∈ L such that spec f k ⊆ I k . Then the following equality holds (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ X k =1 f k (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ X k =1 | f k | ! / (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L . (1) ∗ This research was supported by the Ministry of Science and Higher Education of theRussian Federation (agreement No. 075-15-2019-1620), and by the Foundation for the Ad-vancement of Theoretical Physics and Mathematics “BASIS”.
Keywords : Littlewood-Paley inequality, Rubio de Francia inequality, Walsh system, Gundy’stheorem, martingale, Hardy space, two-parameter, multi-parameter singular integral operator. I k are singletons, we recover precisely Parseval’s identity.Of course, if we replace both L norms in equation (1) by L p norms withsome p = 2, the identity will not be valid. In this case it is interesting to study aweaker kind of relationships between the left hand side and the right hand sideof (1). For instance, for some bases { φ n } and specific partitions Z = ∪ k ∈ N I k ,this or that one-sided inequality with a multiplicative constant might be true.The most famous assertion of this kind is the Littlewood–Paley inequality c p (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ X k =1 | f k | ! / (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p ≤ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ X k =1 f k (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p ≤ C p (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ X k =1 | f k | ! / (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p , < p < ∞ , (2)where φ n ( t ) = e πint , n ∈ Z is a standard trigonometric system over the in-terval [0 , L = L ([0 , I k is a partition of the set Z of integers in aHadamard lacunary sequence of intervals. The corresponding statement for trigonometric system and partitions of Z into arbitrary intervals was established by Rubio de Francia [21] in 1985. Heshowed that in this case the following pair of inequalities holds (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ X k =1 f k (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p ≤ C p (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ X k =1 | f k | ! / (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p , < p ≤ , (3) c p (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ X k =1 | f k | ! / (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p ≤ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ X k =1 f k (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p , p ≥ . (4)These are called Littlewood–Paley–Rubio de Francia inequalities or simply Ru-bio de Francia inequalities. Establishing these sparked a whole new line ofresearch, yielding a number of extensions of this result published to date.The majority of the extensions study the case of the trigonometric system.Specifically, in the papers [2, 11] inequality (3) was generalized to arbitrary expo-nents 0 < p ≤
2. In the papers [8, 22, 17, 19] generalization for the D -parametertrigonometric system φ n , n ∈ Z D , and partitions of Z D into arbitrary productsof intervals was formulated, with inequality (3) similarly extended to arbitraryexponents 0 < p ≤
2. Rubio de Francia himself in the original paper [21] aswell as other authors in [9, 1] considered some weighted generalizations. In thepapers [18, 14] some versions of these inequalities for the Morrey–Companatoand Tribel–Lizorkin spaces were proved. There is a review by Lacey [12] thatconsiders some of the mentioned, as well as some other extensions of Rubio deFrancia inequalities.Recently, Osipov [16] proved a version of inequality (3) where { φ n } , n ∈ Z + ,is the Walsh system and the I k partition the positive integers Z + into arbitrary The very same year when Littlewood and Paley introduced the pair of inequalities (2) forthe trigonometric system (see [13]), the paper [20] of Paley appeared, proving the same pairof inequalities for the Walsh system that we will study in this paper. φ n , n ∈ Z + × Z + and partitions of Z + × Z + into arbitrary pairwise nonintersecting rectangles.Formally, we prove the following statement. Theorem 1.
Consider a family of pairwise nonintersecting rectangles I k = I k × I k inside Z + × Z + and a family of functions f k with Walsh spectrum inside I k , meaning that f k ( x , x ) = X ( n ,n ) ∈ I k ( f k , w n w n ) w n ( x ) w n ( x ) , (5) where w n i are the standard Walsh functions in the Paley ordering.If < p ≤ , then (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)X k f k (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p ≤ C p (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X k | f k | ! / (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p , (6) where C p does not depend on the choice of rectangles { I k } or functions { f k } . The proof is based upon a martingale version of the two-parameters singularintegral theory of R. Fefferman and Journe [5, 8], formulated by Weisz [23].We use the theory of Weisz to prove Theorem 4 — a two-parameter analogof Gundy’s Theorem [7] on the boundedness of operators taking martingales tomeasurable functions. Theorem 4 helps proving the boundedness of operatorsthat map two-parameter martingales to measurable functions in a rather generalsetting, and thus stands out as interesting on its own. By applying the combinatorial argument from Osipov’s work on the one-parameter Walsh system [16] independently for each variable, we essentiallyreduce the Rubio de Francia inequality for the two-parameter Walsh system tothe question of boundedness for a certain operator, which in its turn we resolveby means of Theorem 4.
Here we present some preliminaries that will help us prove the main Theo-rem 1. First, we define two-parameter dyadic martingales and introduce thecorresponding Hardy spaces. Then we present some notions from the atomictheory of Hardy spaces that are useful for establishing boundedness of operatorsmapping martingales to measurable functions. Finally, we recall the definitionof the classical Walsh basis and define the two-parameter Walsh system.Though we will need the theory of l -valued functions and martingales toprove the main theorem, to avoid cumbersome notation we study in this sectiononly the scalar-valued case. We do so because every definition, notion andassertion introduced here will be trivially transferable to the l -valued case. To the best knowledge of the author, this assertion has not been explicitly formulated inthe contemporary literature. .1 Two-parameter dyadic martingales We define two-parameter dyadic filtration to be the family {F n ,n } n ∈ Z + ,n ∈ Z + of σ -algebras generated by the dyadic rectangles of size 2 − n × − n , that is F n ,n = σ (cid:18)(cid:26)(cid:20) k n , k + 12 n (cid:21) × (cid:20) k n , k + 12 n (cid:21) : 0 ≤ k i < n i (cid:27)(cid:19) , (7)where σ ( H ) denotes the σ -algebra generated by the elements of the set H .Define operator E n ,n to be the conditional expectation with respect to the σ -algebra F n ,n .Hereinafter we will often denote elements ( n , n ) ∈ Z by a single symbol n .For n, m ∈ Z we write n ≤ m if and only if n ≤ m and n ≤ m . With this,we introduce the following definition. Definition.
A family of integrable functions u = { u n } n ∈ Z is a two-parameterdyadic martingale (from now on referred to as a martingale ) if the followingconditions are fulfilled:1) for all n ∈ Z the function u n is F n -measurable,2) we have E n u m = u n for all n, m such that n ≤ m .We say that a martingale u is in L p and write u ∈ L p for some 0 < p ≤ ∞ if u n ∈ L p for all n ∈ Z and k u k L p := sup n ∈ Z k u n k L p < ∞ . For two-parametermartingales, as in the classical one-parameter case, the following is true [24]:if u ∈ L p for 1 < p < ∞ , then there exists a function g ∈ L p such that u n = E n g and lim min( n ,n ) →∞ k u n − g k L p = 0 , k u k L p = k g k L p . (8)Following the common practice, we will henceforth identify a martingale u withthe function g and denote g by the same symbol u .Another important objects that we define are the martingale differences ∆ n :∆ n ,n u := u n ,n − u n − ,n − u n ,n − + u n − ,n − , (9)where the formal symbols u n , − and u − ,n are assumed to be equal to zero. We start with introducing a version of the Littlewood–Paley square function fortwo-parameter dyadic martingales.
Definition.
Littlewood–Paley square function is denoted by S and is given by S ( u ) := (cid:18) X n ∈ Z | ∆ n u | (cid:19) / . (10)4he expression k S ( u ) k L p constitutes a norm that defines Hardy spaces. Definition.
For 0 < p < ∞ the martingale Hardy space H p (from now onreferred to as the Hardy space ) consists of martingales u such that k u k H p := k S ( u ) k L p < ∞ . (11)It is known (cf. [3, 4, 15]) that k S ( u ) k L p ∼ k u k L p for 1 < p < ∞ , meaning thatfor such exponents p the spaces L p and H p coincide. However for p ≤
1, theHardy spaces constitute an independent and very useful entity.We finish with formulating the following interpolation result for Hardy spaces.
Theorem 2.
Consider a sublinear operator V that is bounded between H p and L p and between H p and L p . Then V is bounded between H p and L p for p < p < p .Proof. Cf. [23, Theorem A]. H p R. Fefferman’s theorem [5] is an extremely important tool for establishing theboundedness of operators on two-parameter Hardy classes in trigonometric har-monic analysis. It allows one to check rather simple quasi locality conditions,similar to those often used in one-parameter case. As it turns out, the situa-tion in the two-parameter martingale case is similar. The corresponding claimis based, as in the trigonometric case, on the atomic decomposition of Hardyspace. Here we will formulate this claim. We start with two definitions.First, we define the martingale counterpart of R. Fefferman’s rectangle atoms.
Definition.
We call a function a ∈ L a martingale H p rectangle atom (fromnow on referred to as a rectangle atom ) if the following conditions are satisfied1) supp a ⊆ F , where F ⊆ [0 , is some dyadic rectangle,2) k a k L ≤ | F | / − /p ,3) for all x, y ∈ [0 ,
1) we have R a ( u, y ) du = R a ( x, u ) du = 0.In accordance with the convention mentioned in Subsection 2.1, we view rect-angle atom as a function or a martingale depending on the context.Second, we introduce a class of operators for which the aforementioned quasilocality condition is satisfied. Definition.
An operator V mapping martingales to measurable functions issaid to be H p quasi local , if there exists δ > r ∈ N , for alldyadic rectangles R ⊆ [0 , , and for all H p rectangle atoms supported on R we have Z [0 , \ R r | V a | p ≤ C p − δr , (12)5here R r is a dyadic rectangle such that R ⊆ R r and | R r | = 2 r | R | , and C p isa constant depending only on p .Finally, we formulate the claim. Theorem 3.
Consider a sublinear operator V that is H p quasi local for someexponent < p ≤ . If V is bounded between L and L , then k V u k L p ≤ C p k u k H p for all u ∈ H p . (13) Proof.
Cf. [23, Theorem 2].
We conclude the preliminaries with defining the two-parameter Walsh system.We start with recalling the definition of the classical one-parameter Walsh sys-tem.
Definition.
The Walsh system { w n } n ∈ Z + is a family of piecewise constant func-tions of one real variable defined as follows. First, put w = 0. Then, if n > n = 2 k + · · · + 2 k s , k > k > · · · > k s ≥
0, put w n ( x ) := s Y i =1 r k i +1 ( x ) , where r k ( x ) = sgn sin 2 k πx. (14)Here { r k } k ∈ Z + is the Rademacher system. Different orderings of Walsh functionsare considered throughout the literature. The ordering that corresponds to thedefinition above is called the Paley ordering . Hereinafter we consider preciselythis ordering.The two-parameter Walsh system is defined by the expression w n ,n ( x , x ) = w n ( x ) w n ( x ) , n ∈ Z . (15)It is an orthonormal basis in L (cid:0) [0 , (cid:1) . Moreover, for any function f we have( E k ,k f )( x , x ) = k − X n =0 2 k − X n =0 h f, w n i w n ( x , x ) , (16)(∆ k ,k f )( x , x ) = X n ∈ δ k ,k h f, w n i w n ( x , x ) , (17)where h· , ·i is the inner product in L (cid:0) [0 , (cid:1) and δ k ,k = [2 k − , k − × [2 k − , k − , k , k > ,δ ,k = { } × [2 k − , k − , k > ,δ k , = [2 k − , k − × { } , k > ,δ , = { (0 , } . (18)6or a pair w n ( · ) , w m ( · ), n, m ∈ Z + of two one-parameter Walsh functionswe have w n ( x ) w m ( x ) = w n ˙+ m ( x ), where ˙+ is the bitwise exclusive disjunction(xor) operation acting upon the binary representations of numbers n and m : (cid:16)X ∞ k =0 α k k (cid:17) ˙+ (cid:16)X ∞ k =0 β k k (cid:17) := X ∞ k =0 ( α k + β k mod 2)2 k . (19)If we define the corresponding operation ˙+ acting on a pair of n = ( n , n ) and m = ( m , m ) by putting n ˙+ m = ( n ˙+ m , n ˙+ m ) , (20)then obviously w n ( x , x ) w m ( x , x ) = w n ˙+ m ( x , x ) . (21) Theorem 3 from the previous section enables us to formulate a new version ofGundy’s theorem that he introduced in the papers [7, 6]. Note that our for-mulation will be closer to the version formulated much later in the Kislyakov’spaper [10]. This theorem, due to the simplicity of its conditions, can be notablyuseful in proving the boundedness of operators taking martingales into measur-abe functions. It is thus an interesting result on its own and the key to provingTheorem 1.We with a definition. A martingale u is a simple martingale if there existssome m ∈ Z such that u n = E m u n for all indices n ∈ Z . With this, we mayformulate the following version of Gundy’s theorem. Theorem 4.
Consider a sublinear operator V mapping martingales to measur-able functions. Assume the following two conditions. The operator V is bounded between L and L . If u is a simple martingale for which u , = 0 and ∆ n u = e n ∆ n u, where e n ∈ F m for some m ≤ n, m = n, (22) then {| V u | > } ⊆ S n ∈ Z \{ } e n .Then V is bounded between H p and L p for any < p ≤ .Proof. Fix 0 < p ≤
1. We will show that V is H p quasi local, then the claimwill follow from Theorem 3.Take some H p rectangle atom a supported on a dyadic rectangle R ⊆ [0 , .We need to check that for all r ∈ N and for some δ not depending on a and r we have: Z [0 , \ R r | V a | p ( x , x ) dx dx ≤ C p − δr . (23)7e claim that condition (23) can be checked only for atoms that are sim-ple martingales. Indeed, assume that it is true indeed for all rectangle atomsthat are simple martingales, and let us prove that it is true for arbitrary rect-angle atom a . It is easy to check that the simple martingale a n = E n a re-mains a rectangle atom. From lim min( n ,n ) →∞ k a n − a k L = 0 it follows thatlim min( n ,n ) →∞ k V a n − V a k L = 0, because V is bounded between L and L .Hence lim min( n ,n ) →∞ k V a n − V a k L p = 0. This justifies the passage to thelimit in inequality (23), which proves inequality for the initial rectangle atom a . Thus hereinafter in this proof we assume all rectangle atoms to be simplemartingales.Now we find an element N ∈ Z such that R ∈ F N and for any n suchthat R ∈ F n , we have N ≤ n . Since supp a ⊆ R and thanks to item 3 in thedefinition of a rectangle atom, we have∆ n a = R ∆ n a for n ≥ N, n = N, (24)∆ n a = ∅ ∆ n a otherwise. (25)Moreover, a , = 0 due to item 3 in the definition of rectangle atom. Usingcondition 2 of this theorem, we have { ( V a ) > } ⊆ R , hence Z [0 , \ R r | V a | p ( x , x ) dx dx ≤ Z [0 , \ R | V a | p ( x , x ) dx dx = 0 . (26)The right-hand side is trivially bounded by C p − δr for any δ >
0, which provesthe claim.
Corollary 1.
If the conditions of Theorem are fulfilled, then V is boundedbetween L s and L s for < s ≤ .Proof. Interpolation between the boundedness of V : H p → L p for some p ≤ V : L → L by means of Theorem 2 gives the result. G In this section we introduce the auxiliary operator G , the two-parameter coun-terpart of the auxiliary operator G introduced by Osipov in [16], and proveits boundedness using the results of the previous section. It is this particularoperator that will appear in the course of the proof of the main theorem.Consider a family of multi-indices A ⊆ N × Z . Its elements are pairs( j, k ), where j ∈ N , k ∈ Z . Let δ k be as in equation (18) and consider a family { a j,k } ( j,k ) ∈A ⊆ Z such that (cid:8) a j,k ˙+ δ k (cid:9) ( j,k ) ∈A consists of pairwise nonintersect-ing subsets of Z . We define the operator G induced by the family { a j,k } ( j,k ) ∈A and prove its boundedness in the following lemma.8 emma 1. The operator G maps any vector-valued function h = { h j,k } ( j,k ) ∈ N × Z from L p ( l N × Z ) , < p ≤ , to a measurable function by the following law ( Gh )( x , x ) := X ( j,k ) ∈A w a j,k ( x , x )(∆ k h j,k )( x , x ) . (27) This operator is bounded between L p ( l N × Z ) and L p , that is k Gh k L p ≤ C p k h k L p ( l N × Z ) , (28) where the constant C p depends only on p .Proof. Since for 1 < p ≤ L p and the martingales from L p , the operator G may be viewed asan operator mapping l ( N × Z )-valued martingales (rather than l ( N × Z )-valued functions) to measurable functions. We will prove that G fulfills theconditions of Theorem 4, or rather, strictly speaking, generalization to the caseof l ( N × Z )-valued martingales. Here we rely on the fact that everything canbe transferred to the l -valued case, as mentioned at the beginning of Section 2. G is linear, and thus of course it is sublinear. Further, Plancherel’s identityand the fact that (cid:8) a j,k ˙+ δ k (cid:9) ( j,k ) ∈A is a family of pairwise nonintersecting setsgive the boundedness of G between L and L .Finally, we claim that for a simple l ( N × Z )-valued martingale for which u , = 0 and ∆ n u = e n ∆ n u , where e n ∈ F m , m ≤ n, m = n , it we have {| Gu | > } ⊆ S n ∈ Z \{ } e n . Indeed, write {| Gu | > } ⊆ [ ( j,k ) ∈A (cid:8)(cid:12)(cid:12) w a j,k ∆ k u j,k (cid:12)(cid:12) > (cid:9) = [ ( j,k ) ∈A {| ∆ k u j,k | > } (29)= [ ( j,k ) ∈A {| e k ∆ k u j,k | > } ⊆ [ ( j,k ) ∈A {| e k | > } ⊆ [ k ∈ Z \{ } e k . (30)This proves the claim. Finally, here we use the theory established in the preceding sections to provethe main theorem. We begin with recalling its formulation.
Theorem 1.
Consider a family of pairwise nonintersecting rectangles I k = I k × I k inside Z + × Z + and a family of functions f k with Walsh spectrum inside I k , meaning that f k ( x , x ) = X ( n ,n ) ∈ I k ( f k , w n w n ) w n ( x ) w n ( x ) , (5)9 here w n i are the standard Walsh functions in the Paley ordering.If < p ≤ , then (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)X k f k (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p ≤ C p (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X k | f k | ! / (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p , (6) where C p does not depend on the choice of rectangles { I k } or functions { f k } .Proof. As in [16], we partition the rectangles I k into fragments that behave wellunder shifts induced by the operation ˙+. This, together with Lemma 1 andthe classical assertion about the boundedness of the Littlewood–Paley squarefunction, will allow us to prove the claim. Let I k = I k × I k = [ a (1) k , b (1) k − × [ a (2) k , b (2) k − . (31)We build the partition of I k by forming the direct product of partitions ofintervals I k and I k , while partitioning these individual intervals in the same wayas it was done by Osipov in [16].Let us recall that in [16] an interval I = [ a, b − ⊆ Z + was partitioned into I = { a } ∪ (cid:18) r [ j =1 J j (cid:19) ∪ (cid:18) s [ j =1 ˜ J j (cid:19) = (cid:18) r [ j =0 J j (cid:19) ∪ (cid:18) s [ j =1 ˜ J j (cid:19) , (32)where r, s ∈ Z + are some numbers, J = { a } , and J j , ˜ J j are pairwise noninter-secting sets. Moreover, for j > | J j | = 2 κ j , | ˜ J j | = 2 γ j , where κ j is astrictly increasing and γ j is a strictly decreasing Z + -valued sequence.The most important property of the intervals J j and ˜ J j is that they can beshifted to become the dyadic intervals a ˙+ J = { } , a ˙+ J j = [2 κ j , κ j +1 − , b ˙+ ˜ J j = [2 γ j , γ j +1 − , (33)hence the following holds∆ κ j +1 w a f = w a f, if spec f ⊆ J j , (34)∆ γ j +1 w b f = w b f, if spec f ⊆ ˜ J j . (35)To partition a rectangle I = I × I = [ a (1) , b (1) − × [ a (2) , b (2) − ⊆ Z ,we partition each interval I i as above and consider all direct products, yielding I = (cid:18)[ j A j (cid:19) ∪ (cid:18)[ j B j (cid:19) ∪ (cid:18)[ j C j (cid:19) ∪ (cid:18)[ j D j (cid:19) , (36)where A j = J (1) j (1) × J (2) j (2) , B j = ˜ J (1) j (1) × ˜ J (2) j (2) , (37) C j = ˜ J (1) j (1) × J (2) j (2) , D j = J (1) j (1) × ˜ J (2) j (2) , (38)10here a superscript indicates whether the object belongs to the partition of I or I .This partition of I possesses properties similar to those in (34), (35). Define a, b, c, d to be the vertices of the rectangle I , that is a := ( a (1) , a (2) ) , b := ( b (1) , b (2) ) , c := ( b (1) , a (2) ) , d := ( a (1) , b (2) ) , (39)then ∆ κ (1) j +1 ,κ (2) j +1 w a f = w a f, if spec f ⊆ A j , (40)∆ γ (1) j +1 ,γ (2) j +1 w b f = w b f, if spec f ⊆ B j , (41)∆ γ (1) j +1 ,κ (2) j +1 w c f = w c f, if spec f ⊆ C j , (42)∆ κ (1) j +1 ,γ (2) j +1 w d f = w d f, if spec f ⊆ D j . (43)This behavior under shifts will be of utter importance in what follows.Let us similarly partition each I k , adding yet another index k to all objectsthat arise from this construction. Then f k can be represented as the sum f k = X j f Ak,j + X j f Bk,j + X j f Ck,j + X j f Dk,j , (44)where spec f Ak,j ⊆ A k,j , spec f Bk,j ⊆ B k,j , spec f Ck,j ⊆ C k,j , spec f Dk,j ⊆ D k,j .Define g Ak,j = w a k f Ak,j , g
Bk,j = w b k f Bk,j , g
Ck,j = w c k f Ck,j , g
Dk,j = w d k f Dk,j . (45)Then P k f k can be represented as follows: X k w a k X j g Ak,j + w b k X j g Bk,j + w c k X j g Ck,j + w d k X j g Dk,j ! . (46)Application of Lemma 1 to this expression (justified by none other than prop-erties (40)–(43)), followed by applying the triangle inequality, gives us (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)X k f k (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p . (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X k X j (cid:12)(cid:12) g Ak,j (cid:12)(cid:12) + X j (cid:12)(cid:12) g Bk,j (cid:12)(cid:12) + X j (cid:12)(cid:12) g Ck,j (cid:12)(cid:12) (47)+ X j (cid:12)(cid:12) g Dk,j (cid:12)(cid:12) !! / (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p (48) ≤ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X k X j (cid:12)(cid:12) g Ak,j (cid:12)(cid:12) ! / (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p + (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X k X j (cid:12)(cid:12) g Bk,j (cid:12)(cid:12) ! / (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p (49)+ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X k X j (cid:12)(cid:12) g Ck,j (cid:12)(cid:12) ! / (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p + (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X k X j (cid:12)(cid:12) g Dk,j (cid:12)(cid:12) ! / (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p . (50) Note that k and j here correspond, respectively, to j and k in the formulation of Lemma 1. . denotes the inequality up to some implicit multiplica-tive constant. Consider now separately, e.g., the third term. Write w c k f k = w c k ˙+ a k X j g Ak,j + w c k ˙+ b k X j g Bk,j + X j g Ck,j + w c k ˙+ d k X j g Dk,j . (51)We note that ∆ γ (1) k,j +1 ,κ (2) k,j +1 w c k f k = g Ck,j , hence in the decomposition w c k f k = P n ∈ Z ∆ n w c k f k , the functions g Ck,j are among the right-hand side terms. Itfollows that X j (cid:12)(cid:12) g Ck,j (cid:12)(cid:12) ≤ X n ∈ Z | ∆ n w c k f k | = (cid:16) S ( w c k f k ) (cid:17) , (52)where S is the Littlewood–Paley square function. By leveraging its boundedness(cf. the papers [3, 4, 15], also the book [24], where the scalar-valued version ofthis statement was proved, from which the vector-valued version follows easily),we have (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X k X j (cid:12)(cid:12) g Ck,j (cid:12)(cid:12) ! / (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p ≤ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X k X n ∈ Z | ∆ n w c k f k | ! / (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p (53) . (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X k | w c k f k | ! / (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X k | f k | ! / (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p . (54)Similarly, we can bound each of the four terms in (49) and (50). Collectingthese inequalities, we finally obtain (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)X k f k (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p . (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X k | f k | ! / (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p , (55)which proves the claim. Remark.
In the light of the papers [17] and [19], it is natural to ask whethera similar statement holds for a general multi-parameter Walsh system and apartition of Z D + into arbitrary products of intervals. The author is going toaddress this question in the near future. For now we only mention that there isno direct analog of Theorem 3 in the general multi parameter case and a finerstatement would be required. References [1] Viacheslav Borovitskiy. “Weighted Littlewood–Paley Inequality for Ar-bitrary Rectangles in R ”. In: St. Petersburg Mathematical Journal ??(2021), ? (Cit. on p. 2). 122] J. Bourgain. “On square functions on the trigonometric system”. In:
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