Fourier multipliers for Hardy spaces on graded Lie groups
aa r X i v : . [ m a t h . C A ] J a n FOURIER MULTIPLIERS FOR HARDY SPACES ON GRADEDLIE GROUPS
QING HONG, GUORONG HU, AND MICHAEL RUZHANSKY
Abstract.
In the recent work [23], one studied Fourier multiplies on graded Liegroups defined via group Fourier transform, proving that Fourier multiplier opera-tors satisfying H¨ormander type conditions on graded Lie groups are L p bounded for1 < p < ∞ and are of weak-type (1 , H p → L p and H p → H p boundedness of such Fourier multiplier operators for 0 < p ≤ H p is the Hardy space on graded Lie groups. Contents
1. Introduction and main results 22. Preliminaries 62.1. Graded Lie groups and their homogeneous structure 62.2. Fourier analysis on graded Lie groups 82.3. Rockland operators 102.4. Difference operators 112.5. Sobolev spaces on b G G H p ( G ) 143.3. Littlewood-Paley characterization of H p ( G ) in terms of Rocklandoperators 154. Proofs of main results 244.1. Proof of Theorem 1.5 244.2. Proof of Theorem 1.6 295. An application to Riesz operators on graded Lie groups 32References 33 Date : January 20, 2021.2010
Mathematics Subject Classification.
Key words and phrases.
Graded nilpotent Lie groups, representations of Lie groups, Fouriermultipliers, Hardy spaces.Guorong Hu was supported by NSF of China (Grant No. 11901256) and by NSF of JiangxiProvince (Grant No. 20192BAB211001). Qing Hong was supported by NSF of China (Grant No.12001251) and by NSF of Jiangxi Province (Grant No. 20202BAB211001). Michael Ruzhansky wassupported by the FWO Odysseus 1 grant G.0H94.18N: Analysis and Partial Differential Equations,by the EPSRC grant EP/R003025/2 and by the FWO grant G022821N. Introduction and main results
Many problems in harmonic analysis and partial differential equations are relatedto the study of Fourier multipliers for certain function spaces. We start by recallingthe classical Mihlin multiplier theorem. It says that if a function σ ( ξ ) defined on R n \{ } has continuous derivatives up to ( ⌊ n/ ⌋ + 1)-th order, and satisfies | ∂ αξ σ ( ξ ) | ≤ C α | ξ | −| α | (1.1)for all ξ ∈ R n \{ } and all multi-indices α ∈ N n with length | α | ≤ ⌊ n/ ⌋ + 1, then theFourier multiplier operator T σ associated with σ , initially defined for f ∈ S ( R n ) via T σ f = F − ( σ b f ) , extends to a bounded operator on L p ( R n ) for all 1 < p < ∞ . H¨ormander [33]improved this result by showing that the regularity condition obeyed by σ ( ξ ) couldbe allowed to be of fractional order. More precisely he proved that if σ ∈ S ′ ( R n )satisfies sup t> k η ( · ) σ ( t · ) k W s ( R n ) < ∞ (1.2)for some s > n/
2, where η is a function in C ∞ ( R n \{ } ) such that | η ( ξ ) | ≥ c > { r < | ξ | < r } , then T σ extends to a bounded operator on L p ( R n ),for all 1 < p < ∞ . Here W s ( R n ) denote the Bessel potential spaces (or fractionalSobolev spaces) on R n . It is well known that the condition (1.2) is weaker thanthan the condition (1.1). Typical examples of Mihlin and H¨ormander-type Fouriermultiplier operators are the Riesz transforms on R n defined by R j f ( x ) = F − (cid:18) − iξ j | ξ | b f ( ξ ) (cid:19) ( x ) , j = 1 , , · · · , n. (1.3)Indeed these operators satisfy the conditions (1.1) and (1.2) for arbitrary order.Mihlin and H¨ormander’s multiplier theorems have been generalized in several direc-tions, one of which is extending these results to the case 0 < p ≤
1. In this direction,Calder´on and Torchinsky [5] proved that if σ satisfies (1.2) for some s > n (1 /p − / < p ≤
1, then T σ is bounded on the Hardy space H p ( R n ). Product (or bi-parameter) version of Fourier multipliers were studied in [27, 20, 10, 11, 35, 42, 8, 9]and the references therein. In particular, Chen and Lu [9] obtained sharp H¨ormandertype result for boundedness of bi-parameter Fourier multipliers on product Hardyspaces.Another direction is generalizing Mihlin and H¨ormander’s multiplier theoremsto the setting of Lie groups. For the context of Heisenberg-type and more gen-eral nilpotent groups, the L p -boundedness of multiplier operators was studied in[39, 12, 40, 41, 29, 43], while for the context of Lie groups of polynomial growth,this was investigated in [1]. We note that most of these mentioned works were pri-marily concerned with spectral multipliers of one (or several) operator such as asub-Laplacian, and the optimality of a Mihlin-H¨ormander condition in terms of thetopological or homogeneous dimensions is very difficult and still open [29, 43, 40]. Ageneral result concerning L p -boundedness of spectral multipliers, in the context of ametric measure space satisfying the doubling condition (spaces of homogeneous type OURIER MULTIPLIERS FOR HARDY SPACES ON GRADED LIE GROUPS 3 in the sense of Coifman and Weiss [13]) and endowed with a nonnegative self-adjointoperator whose heat kernel satisfies a Gaussian estimate, was proved in [18].Hulanicki, Folland and Stein [34, 26] studied multipliers for Hardy spaces on strat-ified Lie groups. They proved that if L is a sub-Laplacian on a stratified Lie group G and F is a function on (0 , ∞ ) satisfying the Mihlin type condition | F ( j ) ( λ ) | ≤ Cλ − j , j = 1 , , · · · , N, (1.4)for some positive integer N > ( Q/p ) + (3 Q/
2) + 1 (where Q is the homogeneousdimension of G ), then the operator F ( L ) = Z ∞ F ( λ ) dE L ( λ )is bounded on the Hardy space H p ( G ), where E L is the resolution of the identityassociated to L . De Michele and Mauceri [16] improved Hulanicki and Stein’s H p multiplier theorem by replacing the regularity condition (1.4) with a weaker one in-volving Besov spaces on the real line, but still they only considered spectral mul-tipliers. We point out that in the Euclidean setting spectral multipliers can bethought of as the radial case of Fourier multipliers. Indeed, when G = ( R n , +)and L = − ∆ = − P nj =1 ∂ ∂x j , the spectral multiplier operator F ( − ∆) coincides withthe Fourier multiplier operator defined by T f = F − (cid:2) F (cid:0) | · | (cid:1) b f (cid:3) . We also note that Riesz transforms defined by (1.3) are Fourier multiplier operators,but not included in spectral multiplier operators associated with the Laplacian − ∆.Much fewer works have been devoted to the study of Fourier multipliers on Liegroups. To our best knowledge, the first study of Fourier multipliers on Lie groupswas done by Coifman and Weiss in [13], where they developed Calder´on-Zygmundtheory on spaces of homogeneous type and as an application they studied the Fouriermultipliers of SU (2), see also [14]. After that, investigations of Fourier multipliers oncompact Lie groups has been focused on the central multipliers [48, 50, 51, 52], untillthe appearance of the recent works of Ruzhansky and Wirth [46, 47] and Fischer [21].The rest of the literature concerning Fourier multipliers on Lie groups is restrictedto the motion group [45] and to the Heisenberg group [15].Recently, Fischer and Ruzhansky [23] initiated the study of Fourier multipliers ongraded Lie groups. They proved that H¨ormander type conditions (defined via Sobolevspaces on the dual of the group) on the Fourier multipliers imply L p -boundedness.They also expressed these conditions using difference operators (which could bethought of as a analogue of the Euclidean derivatives with respect to the Fouriervariable) and positive Rockland operators. We shall recall the definition of gradedLie groups and related notions in Section 2. The main result of [23] is the followingH¨ormander type Fourier multiplier theorem for L p spaces on graded Lie groups. Theorem 1.1. ([23, Theorem 1.2])
Let G be a graded Lie group with homogeneousdimension Q , and let R be a Rockland operator on G . Let η ∈ C ∞ c (0 , ∞ ) be a non-zero QING HONG, GUORONG HU, AND MICHAEL RUZHANSKY function, and σ = { σ ( π ) , π ∈ b G } be a measurable field of operators such that sup t> (cid:13)(cid:13) { σ ( t · π ) η ( π ( R )) , π ∈ b G } (cid:13)(cid:13) W s ( b G ) < ∞ , (1 . a sup t> (cid:13)(cid:13) { η ( π ( R )) σ ( t · π ) , π ∈ b G } (cid:13)(cid:13) W s ( b G ) < ∞ , (1 . b (1.5) for some s > Q/ . Then the Fourier multiplier operator T σ corresponding to σ isbounded on L p ( G ) for all < p < ∞ , and is of weak type (1 , . Furthermore, k T σ k L ( L p ( G )) ≤ C (cid:18) sup t> (cid:13)(cid:13) { σ ( t · π ) η ( π ( R )) π ∈ b G } (cid:13)(cid:13) W s ( b G ) + sup t> (cid:13)(cid:13) { η ( π ( R )) σ ( t · π ) , π ∈ b G } (cid:13)(cid:13) W s ( b G ) (cid:19) , where C > is a constant independent of σ but may depend on p, s, G, R and a choiceof the cut-off function η . Remark 1.2.
Analogously to the Euclidean case, if σ satisfies the Mihlin type con-ditions X [ α ] ≤ N sup π ∈ b G (cid:13)(cid:13) π ( R ) [ α ] ν ∆ α σ (cid:13)(cid:13) L ( H π ) < ∞ , (1 . a X [ α ] ≤ N sup π ∈ b G (cid:13)(cid:13) ∆ α σπ ( R ) [ α ] ν (cid:13)(cid:13) L ( H π ) < ∞ , (1 . b (1.6)for some positive integer N > s divisible by the dilation weights v , v , · · · , v n (seeSection 2 for the definition of dilation weights), then σ satisfies the H¨ormander typeconditions (1.5). See [23, Proposition 4.9] for the proof of this fact. Here, ν isthe homogeneous degree of R , and ∆ α are difference operators introduced in [23],which play a role analogous to derivatives with respect to the Fourier variable in theEuclidean case. All unfamiliar notions and notation appeared here will be recalled inSection 2. Remark 1.3.
Theorem 1.1 yields the classical H¨ormander multiplier theorem if itis applied to the abelian Euclidean setting, that is, G = ( R n , +) with the isotropicdilations and R = − ∆ being the Laplace operator. Remark 1.4.
The following more precise version of Theorem 1.1 holds (see [23,Corollary 4.12]):(i) If σ = { σ ( π ) , π ∈ b G } satisfies (1 . a , then T σ is of weak-type (1 , T σ is bounded on L p ( G ) for 1 < p ≤ σ = { σ ( π ) , π ∈ b G } satisfies (1 . b , then T ∗ σ is of weak-type (1 , T σ is bounded on L p ( G ) for 2 ≤ p < ∞ .The purpose of this paper is to extend Theorem 1.1 by considering the case 0
1. More precisely we investigate the H p → L p and H p → H p boundedness ofFourier multiplier operators on graded Lie groups for 0 < p ≤
1. Our main resultsare stated as follows.
Theorem 1.5.
Let G be a graded Lie group with homogeneous dimension Q , andlet R be a positive Rockland operator on G . Let < p ≤ and η ∈ C ∞ c (0 , ∞ ) be acut-off function not identically zero. If σ = { σ ( π ) , π ∈ b G } is a measurable field of OURIER MULTIPLIERS FOR HARDY SPACES ON GRADED LIE GROUPS 5 operators such that sup t> (cid:13)(cid:13) { σ ( t · π ) η ( π ( R )) , π ∈ b G } (cid:13)(cid:13) W s ( b G ) < ∞ (1.7) for some s > Q (1 /p − / , then the Fourier multiplier operator T σ corresponding to σ is bounded on the Hardy space H p ( G ) . Furthermore, k T σ k L ( H p ( G ) ,L p ( G )) ≤ C sup t> (cid:13)(cid:13) { σ ( t · π ) η ( π ( R )) , π ∈ b G } (cid:13)(cid:13) W s ( b G ) , (1.8) where C > is a constant independent of σ but may depend on p, s, G, R and a choiceof the cut-off function η . Theorem 1.6.
Let G be a graded Lie group with homogeneous dimension Q , andlet R be a positive Rockland operator on G . Let < p ≤ and η ∈ C ∞ c (0 , ∞ ) be acut-off function not identically zero. If σ = { σ ( π ) , π ∈ b G } is a measurable field ofoperators such that sup t> (cid:13)(cid:13) { η ( π ( R )) σ ( t · π ) , π ∈ b G } (cid:13)(cid:13) W s ( b G ) < ∞ (1.9) for some s > Q (1 /p − /
2) + ( v n − v ) (cid:0)(cid:4) Lv (cid:5) + 1 (cid:1) where L = max (cid:8) [ α ] : α ∈ N n , [ α ] ≡ α v + α v + · · · + α n v n ≤ Q (1 /p − (cid:9) , and v , v , · · · , v n are dilation weights (see Section 2 for their definition), then T σ isbounded on H p ( G ) . Furthermore, k T σ k L ( H p ( G )) ≤ C sup t> (cid:13)(cid:13) { η ( π ( R )) σ ( t · π ) , π ∈ b G } (cid:13)(cid:13) W s ( b G ) . We briefly discuss the relation and comparison between Theorems 1.5 and 1.6.(i) Theorem 1.5 is about H p → L p boundedness, while Theorem 1.6 is about H p → H p boundedness. Note that H p → H p boundedness implies H p → L p boundedness (see e.g. [28]), that is, under the assumption of Theorem 1.6,the operator T σ is also H p → L p bounded.(ii) The H¨ormander-type condition (1.7) in Theorem 1.5 is a “right-side” condi-tion (i.e., η ( π ( R )) is on the right side of σ ( t · π )), while the H¨ormander-typecondition (1.9) in Theorem 1.6 is a “left-side” one. Note that, if the “left-side”condition is replaced by the “right-side” one, the argument in Theorem 1.6does not work, and thus we can not obtain the H p → H p boundedness of T σ .(iii) Compared with Theorem 1.5, the parameter s in Theorem 1.6 satisfies astronger condition. This means that, when one considers H p → L p bounded-ness of T σ , the “left-side” H¨ormander-type condition requires a bigger param-eter s .(iv) When G is the abelian Euclidean setting ( R n , +) with isotropic dilations thenone has ( v n − v ) (cid:0)(cid:4) Lv (cid:5) + 1 (cid:1) = 0 (since v = v = · · · = v n = 1). ThusTheorem 1.6 recovers the classical H¨ormander-type Fourier multiplier theoremfor Hardy spaces obtained by Calder´on and Torchinsky [5].We now discuss the main ingredients of our proofs. To prove the H p ( G ) → L p ( G )boundedness of T σ , we will utilize atomic decomposition of H p ( G ), which was estab-lished by Folland and Stein [26]. It suffices to show that for any atom a , we have QING HONG, GUORONG HU, AND MICHAEL RUZHANSKY k T σ a k L p ( G ) ≤ C sup t> (cid:13)(cid:13) { σ ( t · π ) η ( π ( R )) , π ∈ b G } (cid:13)(cid:13) W s ( b G ) with the constant C indepen-dent of a . To do this we decompose G into B ∪ B c where B is a suitable dilation ofthe ball to which the atom a is associated, and we estimate the integral of | T σ a | p over B and B c separately. The spectral theory, Taylor’s inequality on G , some propertiesof W s ( b G ) and Hulanicki’s theorem will be crucial in our argument. To derive the H p ( G ) → H p ( G ) boundedness of T σ , we will use atomic decomposition as well asa Littlewood-Paley characterization of H p ( G ) in terms of Rockland operators. TheLittlewood-Paley characterization of H p ( G ) that we need is not a direct consequenceof known results in the literature and will be established in Section 3 (see Theorem3.7 below), by using some ideas from [44]. It is worth noting that the methods usedin [3, 16, 19] to prove H p → H p boundedness of spectral multiplier operators cannot be adapted to our Fourier multiplier problem. Indeed, the arguments in thesementioned papers rely on the fact that Φ( L )Ψ( L ) = Ψ( L )Φ( L ) whenever L is anonnegative self-adjoint operator and Φ , Ψ are bounded Borel measure functions on[0 , ∞ ); however, two Fourier multiplier operators T σ and T σ on G need not to becommutative. The non-commutative nature of our setting makes it harder to ob-tain H p → H p boundedness of Fourier multipliers. This is why we have to assume s > Q (1 /p − /
2) + ( v n − v ) (cid:0)(cid:4) Lv (cid:5) + 1 (cid:1) in Theorem 1.2 instead of s > Q (1 /p − / Notation.
Throughout this article we shall use N (resp. N ) to denote the set ofall nonnegative integers (resp. the set of all strictly positive integers). For any non-negative number s , we denote by ⌊ s ⌋ the largest integer less than or equal to s . For abounded linear operator T : X → Y where X and Y are quasi-Banach spaces, we de-note k T k L ( X , Y ) := sup k x k X =1 k T x k Y . If Y = X then we write k T k L ( X ) = k T k L ( X , X ) .The letter C will be used to denote positive constants, which are independent of themain variables involved and whose values may vary at every occurrence. By writing f . g we mean that f ≤ Cg . If f . g . f , we write f ∼ g .2. Preliminaries
In this section, we first recall the definition of graded Lie groups and their homo-geneous structure. Then we recall some basic representation theory, definitions ofgroup Fourier transform and Rockland operators, and Sobolev spaces on the unitarydual of a graded Lie group.2.1.
Graded Lie groups and their homogeneous structure.
A Lie group G is said to be graded if it is connected and simply connected, and its Lie algebra g is endowed with a vector space decomposition g = ⊕ ∞ k =1 g k (where all but finitelymany of the g k ’s are { } ) such that [ g k , g k ′ ] ⊂ g k + k ′ for all k, k ′ ∈ N . Such a groupis necessarily nilpotent, and the exponential map exp : g → G is a diffeomorphism. OURIER MULTIPLIERS FOR HARDY SPACES ON GRADED LIE GROUPS 7
Examples of graded Lie groups include the Euclidean space R n , the Heisenberg group H n and, more generally, all stratified Lie groups.We choose and fix a basis { X , · · · , X n } of g , so that it is adapted to the gradation,i.e., { X , · · · , X n } (possibly ∅ ) is a basis of g , { X n +1 , · · · , X n + n } (possibly ∅ ) isa basis of g , and so on. Via the map R n ∋ ( x , · · · , x n ) exp( x X + · · · + x n X n ) ≡ x ∈ G, (2.1)each point ( x , · · · , x n ) ∈ R n is identified with the point x ∈ G . This map takes theLebesgue measure on R n to a bi-invariant Haar measure dµ on G . We denote thegroup identity of G by e .The Lie algebra g is equipped with a natural family of dilations { δ t } t> which arelinear mappings from g to g determined by δ t X = t k X if X ∈ g k , k = 1 , , · · · . For each j ∈ { , · · · , n } , let v j be the unique positive integer such that X j ∈ g v j .Then we have δ t X j = t v j X j , j = 1 , · · · , n . The associated group dilation is given by δ t x = ( t v x , · · · , t v n x n ) , for x = ( x , · · · , x n ) ∈ G and t >
0. In what follows we generally write tx insteadof δ t x . The integers v , · · · , v n are referred to as weights of the dilations { δ t } t> , andthe positive integer Q := ∞ X k =1 k (dim g k ) = n X j =1 v j is called the homogeneous dimension of G .A homogeneous norm on G is a continuous function x → | x | from G to [0 , ∞ )which vanishes only at e and satisfies that | x − | = | x | and | tx | = t | x | for all x ∈ G and t >
0. It is known that there exists at least one homogeneous norm on G andany two homogeneous norms on G are equivalent (see e.g. [26]). Henceforth we fixa homogeneous norm on G. It satisfies a triangle inequality: there exists a constant γ ≥ | xy | ≤ γ ( | x | + | y | ) for all x, y ∈ G. (2.2)There is an analogue of polar coordinates on homogeneous groups with the homo-geneous dimension Q replacing the topological dimension n , see [26]: ∀ f ∈ L ( G ) Z G f ( x ) dµ ( x ) = Z ∞ Z S f ( ry ) r Q − dσ ( y ) dr, (2.3)where dσ is a (unique) positive Borel measure on the unit sphere S := { x ∈ G : | x | =1 } . This implies that for 0 < r < R < ∞ and θ ∈ R , Z r ≤| x |≤ R | x | θ − Q dµ ( x ) = ( Cθ − ( R θ − r θ ) if θ = 0 ,C log( R/r ) if θ = 0 . (2.4)Consequently, if θ > | · | θ − Q is integrable near the group identity e , and if θ < | · | θ − Q is integrable near ∞ .We next recall the Taylor’s inequality on graded Lie groups. See [26, Theorem1.37] or [22, Proposition 2.1]. QING HONG, GUORONG HU, AND MICHAEL RUZHANSKY
Proposition 2.1.
For each M ∈ N there exists a positive group constant C M suchthat for any function f ∈ C M +1 ( G ) , we have | f ( yx ) − P x ( y ) | ≤ C M X | α |≤⌈ M ⌋ +1[ α ] >M | y | [ α ] sup | z |≤ β M +1 | y | | e X α f ( zx ) | , (2.5) where P x is the right Taylor polynomial of f at x of homogeneous degree M , and ⌈ M ⌋ := max {| α | : α ∈ N n with [ α ] ≤ M } . Since G has been identified with R n via the map given in (2.1), functions on G can be viewed as functions on R n , and vise versa. This leads naturally to thenotions of test function classes D ( G ), S ( G ) and the distribution spaces D ′ ( G ), S ′ ( G ).For example, a function f is said to be in the Schwartz class S ( G ) if f ◦ exp is aSchwartz function on R n . The coordinate function G ∋ x = ( x , · · · , x n ) x ∈ R is denoted by x . For a multi-index α = ( α , · · · , α n ) ∈ N n , set x α = x α · · · x α n n and X α = X α · · · X α n n . With respect to the dilations D t on G and on g , the homogeneousdegree of x α and X α is [ α ] := n X j =1 v j α j . A function P : G → C is called a polynomial, if it is of the form P ( x ) = X α ∈ N n c α x α where all but finitely many of the complex coefficients c α vanish. The homogeneousdegree of the polynomial P is defined as max { [ α ] : c α = 0 } . For M ∈ N , we set P M := { all polynomials on G with homogeneous degree ≤ M } . ˆWe denote by e X , · · · , e X n the corresponding basis for right-invariant vector fields,that is, e X j f ( x ) = ddt f (cid:0) exp( tX j ) x (cid:1)(cid:12)(cid:12) t =0 , j = 1 , · · · , n. Also, for α ∈ N n , we set e X α = e X α · · · e X α n n .If f and g are measurable functions on G , then their convulution is defined by f ∗ g ( x ) = Z G f ( y ) g ( y − x ) dµ ( y ) = Z G f ( xy − ) g ( y ) dµ ( y ) , provided that the integrals converge. For any multi-index α ∈ N n and sufficientlygood functions f and g , we have (see [26, Chapter 1]) X α ( f ∗ g ) = f ∗ ( X α g ) , e X α ( f ∗ g ) = ( e X α f ) ∗ g, ( X α f ) ∗ g = f ∗ ( e X α g ) . Fourier analysis on graded Lie groups.
The general theory of representationof Lie groups may be found in [17]. Here we also refer to [24] for a description whichis more adapted to our particular context.A representation π of a Lie group G on a Hilbert space H π = { } is a homo-morphism from G into the group of bounded linear operators on H π with boundedinverse. More precisely, OURIER MULTIPLIERS FOR HARDY SPACES ON GRADED LIE GROUPS 9 • for every x ∈ G , the linear mapping π ( x ) : H π → H π is bounded and hasbounded inverse; • for every x, y ∈ G , we have π ( xy ) = π ( x ) π ( y ).A representation π of G is called irreducible if it has no closed invariant subspaces. π is called unitary if π ( x ) is unitary for every x ∈ G , and is called strongly continuousif the mapping π : G → L ( H π ) is continuous with respect to the strong operatortopology in L ( H π ). Two representations π and π are said to be equivalent if thereexists a bounded linear mapping A : H π → H π between their representation spaceswith a bounded inverse such that the relation Aπ ( x ) = π ( x ) A holds for all x ∈ G .In this case we write π ∼ π , and denote their equivalence class by [ π ] = [ π ]. Theset of all equivalence classes of strongly continuous irreducible unitary representationsof G is called the unitary dual of G and is denoted by b G . In what follows, we willidentify one representation π with its equivalent class [ π ].For a unitary representation of G , the corresponding infinitesimal representationwhich acts on the universal enveloping algebra U ( g ) of the Lie algebra g is still denotedby π . This is characterized by its action on g : π ( X ) = ∂ t =0 π ( e tX ) , X ∈ g . The infinitesimal action acts on the space H ∞ π of smooth vectors, that is, the spaceof vectors v ∈ H π such that the function G ∋ x π ( x ) v ∈ H π is of class C ∞ .The Fourier coefficients or group Fourier transform of a function f ∈ L ( G ) at π ∈ b G is defined by F G f ( π ) ≡ b f ( π ) ≡ π ( f ) := Z G f ( x ) π ( x ) ∗ dµ ( x ) . It is readily seen that k b f ( π ) k L ( H π ) ≤ k f k L ( G ) . For f, g ∈ L ( G ), we also have [ f ∗ g ( π ) = b g ( π ) b f ( π ) . There exists a unique positive Borel measure b µ on b G , called the Plancherel measure,such that for any continuous function f on G with compact support, one has Z G | f ( x ) | dµ ( x ) = Z b G kF G f ( π ) k HS ( H π ) d b µ ( π ) , where k · k HS ( H π ) denotes the Hilbert-Schmidt norm on the space HS ( H π ) ∼ H π ⊗ H ∗ π of Hilbert-Schmidt operators on the Hilbert space H π . Since L ( G ) ∩ L ( G ) is densein L ( G ), the Fourier transform F G extends to a unitary operator from L ( G ) onto L ( b G ).By the general theory on locally compact unimodular groups of type I (see e.g.[17]), if T is an L -bounded operator on G which commutes with left-translations,then there exists a field of bounded operators b T ( π ) such that for all f ∈ L ( G ), F G ( T f )( π ) = b T ( π ) b f ( π ) a.e. π ∈ b G. Moreover, we have k T k L ( L ( G )) = sup π ∈ b G k b T ( π ) k L ( H π ) , where the supremum here is understood as the essential supremum with respect tothe Plancherel measure µ . Conversely, given any σ = { σ ( π ) , π ∈ b G } ∈ L ∞ ( b G ), thereis a corresponding operator T σ given by F G ( T σ f )( π ) = σ ( π ) b f ( π ) , f ∈ L ( G ) . By the Plancherel theorem, T σ is bounded on L ( G ) with k T σ k L ( L ( G )) = k σ k L ∞ ( b G ) .If π is a unitary irreducible representation of G and t >
0, we define t · π to be theunitary irreducible representation such that t · π ( x ) = π ( tx ) , x ∈ G. Rockland operators.
Let G be a graded Lie group. A left-invariant differentialoperator R on G is called a Rockland operator if it is homogeneous of positive degreeand for each unitary irreducible non-trivial representation π of G , the operator π ( R )is injective on H ∞ π . Rockland operators may be defined on any homogeneous group,however it turns out that the existence of a Rockland operator on a homogeneousgroup implies that (the Lie algebra of) the group admits a gradation. This is thereason why we and the authors in [23] consider the setting of graded Lie groups. Onany graded Lie group G , the operator n X j =1 ( − ν vj c j X ν vj j with c j > ν if ν is any commonmultiple of v , · · · , v n .We will mainly consider positive Rockland operators. A Rockland operator R issaid to be positive, if Z G R f ( x ) f ( x ) dµ ( x ) ≥ f ∈ S ( G ). If a Rockland operator R is positive then R and π ( R ) admitself-adjoint extensions on L ( G ) and H π , respectively. We use the same notation fortheir self-adjoint extensions. By the spectral theory, we have R = Z ∞ λdE R ( λ ) and π ( R ) = Z ∞ λdE π ( R ) ( λ ) , where E R ( λ ) (resp. E π ( R ) ( λ )) is the resolution of the identity associated to R (resp. π ( R )).We now recall Hulanicki’s theorem, which will play a crucial role in the proofs ofour main results. Proposition 2.2. (Hulanicki [32])
Let G be a graded Lie group and let R be a positiveRockland operator on G . Given any s ∈ (0 , ∞ ) , p ∈ [1 , ∞ ) and α ∈ N n , there exist C > and d ∈ N such that for any Φ ∈ C d (0 , ∞ ) , Z G (1 + | x | ) s | X α φ ( x ) | p dµ ( x ) ≤ C sup λ> ,ℓ =0 , , ··· ,d | Φ ( ℓ ) ( λ ) | , OURIER MULTIPLIERS FOR HARDY SPACES ON GRADED LIE GROUPS 11 where φ denotes the convolution kernel of the operator Φ( R ) .The same results with the right-invariant vector fields e X j ’s instead of the left-invariant vector fields X j ’s hold. Remark 2.3.
Hulanicki’s theorem implies that if Φ ∈ S ( R ) then the convolutionkernel of the operator Φ( R ) belongs to S ( G ). This allows us to extend the domainof the operator Φ( R ) from L ( G ) to S ′ ( G ): if Φ ∈ S ( R ) and f ∈ S ′ ( G ), we defineΦ( R ) f := f ∗ φ, where φ ∈ S ( G ) is the convolution kernel of Φ( R ). Remark 2.4.
Hulanicki’s theorem also implies that the map associating a multiplierfunction with its convolution kernel S ( R ) ∋ Φ φ ∈ S ( G )is continuous between the Schwartz spaces. In addition, Z G φ ( x ) dµ ( x ) = Φ(0) . (2.6)For the proof of (2.6), see, for example, [36] and [37], using the fact that the heatkernel of R satisfies Gaussian upper bound estimate (cf. [2]).2.4. Difference operators.
The Mihlin condition (1.1) is formulated in terms of thederivatives with respect to the Fourier variable ξ . However, for a field σ = { σ ( π ) , π ∈ b G } of operators, there is no direct way to define an analogue of derivatives withrespect to the Fourier variable π . To generalize the symbolic conditions to the settingof graded Lie groups, Fischer and Ruzhansky [24] introduced the so-called differenceoperators, whose definition we now recall.For a, b ∈ R , we denote by L L ( L a ( G ) , L b ( G )) the subspace of all T ∈ L ( L a ( G ) , L b ( G ))which are left-invariant. Here L a ( G ) is the Bessel potential space (fractional Sobolevspace) defined in [25]. Define K a,b ( G ) := (cid:8) K ∈ S ′ ( G ) : the operator S ( G ) ∋ f f ∗ K extends toa bounded operator from L a ( G ) to L b ( G )) (cid:9) and define L ∞ a,b ( b G ) to be the space of all fields σ = { σ ( π ) , π ∈ b G } such that k σ k L ∞ a,b ( b G ) := sup π ∈ b G k π ( I + R ) bν σ ( π ) π ( I + R ) − aν k L ( H π ) < ∞ . From [24, Proposition 5.1.24] we see that, if σ ∈ L ∞ a,b ( b G ) then the Fourier multiplieroperator T σ corresponding to σ belongs to L L ( L a ( G ) , L b ( G )) with k T σ k L ( L a ( G ) ,L b ( G )) = k σ k L ∞ a,b ( b G ) . Conversely, if T ∈ L L ( L a ( G ) , L b ( G )), then there exists a unique σ ∈ L ∞ a,b ( b G ) suchthat F G ( T f )( π ) = σ ( π ) b f ( π ) , f ∈ L ( G ) . In this case, denoting by K ∈ K a,b ( G ) the convolution kernel of T , we define F G K = σ and F − G σ = K. This extends the definition of Fourier transform to the space K a,b ( G ). See [24, Defi-nition 5.1.25].For α ∈ N n and σ = { σ ( π ) , π ∈ b G } ∈ L ∞ a,b ( b G ), the difference operator ∆ α actingon σ is defined according to the formula (see [24, Definition 5.2.1])∆ α σ ( π ) = F G ( q α F − G σ )( π ) for a.e. π ∈ b G, where q α ( x ) = x α . Analogously to the derivatives in the Euclidean setting, theoperator ∆ α satisfies the Lebnitz rule [24, Section 5.2.2]:∆ α ( στ ) = X α + α = α C α ,α ∆ α ( σ )∆ α ( τ ) , σ, τ ∈ L ∞ a,b ( b G ) . Sobolev spaces on b G . To generalize the H¨ormander type conditions to gradedLie groups, Fischer and Ruzhansky [25] introduced Sobolev spaces on b G , which aredefined as follows. Definition 2.5.
For each s ≥
0, we define W s ( b G ) as the space of measurablefields σ = { σ ( π ) , π ∈ b G } such that ∆ (1+ |·| ) s σ ∈ L ( b G ). In other words, W s ( b G ) = F (cid:0) L ( G, (1 + | · | ) s ) (cid:1) . For σ ∈ W s ( b G ), we define k σ k W s ( b G ) = k ∆ (1+ |·| ) s σ k L ( b G ) . Remark 2.6.
By the Plancherel theorem, if σ = { σ ( π ) , π ∈ b G } ∈ W s ( b G ), then k σ k W s ( b G ) = (cid:18)Z G (1 + | x | ) s | K ( x ) | dµ ( x ) (cid:19) / , where K = F − G σ .We will use the properties of W s ( b G ) stated in the following two lemmas. Both ofthem were proved in [23]. Lemma 2.7. ([23, Lemma 3.8])
For any σ and τ in W s ( b G ) , the product στ = { σ ( π ) τ ( π ) , π ∈ b G } satisfies (with possibly unbounded norms) k στ k W s ( b G ) ≤ C (cid:0) k σ k W s ( b G ) kF − G τ k L ( G ) + kF − G σ k L ( G ) k τ k W s ( b G ) (cid:1) with a constant C > independent of σ and τ .Hence for s > Q/ , if σ, τ ∈ W s ( b G ) then στ ∈ W s ( b G ) , and W s ( b G ) is a (noncom-mutative) algebra. Lemma 2.8. ([23, Corollary 4.7]) η ∈ C ∞ c (0 , ∞ ) be a cut-off function not identicallyzero. Suppose σ = { σ ( π ) , π ∈ b G } is a measurable field of operators such that sup t> (cid:13)(cid:13) { σ ( t · π ) η ( π ( R )) , π ∈ b G } (cid:13)(cid:13) W s ( b G ) < ∞ or sup t> (cid:13)(cid:13) { η ( π ( R )) σ ( t · π ) , π ∈ b G } (cid:13)(cid:13) W s ( b G ) < ∞ for some s > Q (1 /p − / , where η ∈ C ∞ c (0 , ∞ ) is a cut-off function not identicallyzero. Then σ ∈ L ∞ ( b G ) . Moreover, there is a constant C such that k σ k L ∞ ( b G ) ≤ C min (cid:26) sup t> (cid:13)(cid:13) { η ( π ( R )) σ ( t · π ) , π ∈ b G } (cid:13)(cid:13) W s ( b G ) , sup t> (cid:13)(cid:13) { η ( π ( R )) σ ( t · π ) , π ∈ b G } (cid:13)(cid:13) W s ( b G ) (cid:27) . OURIER MULTIPLIERS FOR HARDY SPACES ON GRADED LIE GROUPS 13 Hardy spaces on graded Lie groups
A comprehensive theory of Hardy spaces on general homogeneous groups was de-veloped by Folland and Stein [26]. Since all graded Lie groups are homogeneous, thetheory in [26] applies to our setting. In this section, we will recall definition of Hardyspaces on graded Lie groups and their atomic decomposition. Then we will establisha Littlewood-Paley characterization of Hardy spaces on graded Lie groups in termsof Rockland operators.Throughout this section, G is a graded Lie group with homogeneous dimension Q .For any function φ on G and t >
0, we define the normalized dilation D t φ of φ by D t φ ( x ) := t Q φ ( tx ) , x ∈ G. Maximal definition of Hardy spaces on G . Following [26], for any φ ∈ S ( G )and N ∈ N , we set k φ k ( N ) := sup | α |≤ N,x ∈ G (1 + | x | ) ( N +1)( Q +1) | e X α φ ( x ) | . Moreover, given N ∈ N , we define the grand maximal function M ( N ) f of f ∈ S ′ ( G )by M ( N ) f ( x ) := sup φ ∈S ( G ) , k φ k ( N ) ≤ M φ f ( x ) , x ∈ G, where M φ f is the nontangential maximal function, defined by M φ f ( x ) := sup | x − y | For 0 < p < ∞ , the Hardy space H p ( G ) is defined as H p ( G ) := { f ∈ S ′ ( G ) : M ( N p ) f < ∞} , where N p := min { [ α ] : α ∈ N n with [ α ] > Q (1 /p − } . The quasi-norm on H p ( G ) is defined by k f k H p ( G ) := k M ( N p ) f k L p ( G ) . The Hardy spaces H p ( G ), initially defined via grand maximal function, can becharacterized by radial maximal function and nontangential maximal function equiv-alently. To recall these maximal characterizations, we need the notion of commutativeapproximate identities introduced in [26]. A commutative approximate identity on G is a function φ ∈ S ( G ) such that R G φ ( x ) dx = 1 and ( D s φ ) ∗ ( D t φ ) = ( D t φ ) ∗ ( D s φ )for all s, t > 0. Given f ∈ S ′ ( G ) and φ ∈ S ( G ), the radial maximal function M φ f isdefined by M φ f ( x ) = sup t> | f ∗ φ t ( x ) | , x ∈ G. Proposition 3.2. ([26, Corollary 4.17]) Suppose φ is a commutative approximateidentity. If f ∈ S ′ ( G ) and < p < ∞ , then the following are equivalent: (i) M φ f ∈ L p ( G ) ; (ii) M φ f ∈ L p ( G ) ; (iii) M ( N p ) f ∈ L p ( G ) . Moreover, we have k M φ f k L p ( G ) ∼ k M φ f k L p ( G ) ∼ k M ( N p ) f k L p ( G ) with the implicit constants depending only on φ and p . The space H p ( G ) can be identified with L p ( G ) when 1 < p < ∞ (see [26, p. 75]).However this is not true when 0 < p ≤ 1. In the latter case, it is known that f ∈ L q ( G ) ∩ H p ( G ) for some 1 < q < ∞ = ⇒ f ∈ L p ( G ) and k f k L p ( G ) ≤ k f k H p ( G ) ;see the proof of [28, Theorem 1.1]. Using this result and the fact that L ( G ) ∩ H p ( G )is dense in H p ( G ), one can deduce that, if T is a linear operator bounded on L q ( G )for some 1 < q < ∞ , and bounded on H p ( G ), then T is bounded from H p ( G ) to L p ( G ).We recall from [26] the notion of distributions vanishing weakly at infinity. Definition 3.3. A distribution f ∈ S ′ ( G ) is said to vanish weakly at infinity, if forall φ ∈ S ( G ), f ∗ ( D t φ ) → S ′ ( G ) as t → f belongingto H p ( G ) is that f vanishes weakly at infinity. Proposition 3.4. ([26, Proposition 7.9]) If f ∈ H p ( G ) (0 < p < ∞ ) , then f vanishesweakly at infinity. Atomic decomposition of H p ( G ) . Atomic decomposition is a very useful toolfor the study of boundedness of operators on Hardy spaces. Analogously to the Eu-clidean case, Hardy spaces on graded Lie groups also admit an atomic decomposition,which we now recall. See [26] for a more comprehensive description.A triplet ( p, q, M ) is said to be admissible, if 0 < p ≤ ≤ q ≤ ∞ , p = q and M ∈ N with M ≥ max (cid:8) [ α ] : α ∈ N n with [ α ] ≤ Q (1 /p − (cid:9) . Definition 3.5. Given an admissible triplet ( p, q, M ), we say that a function a on G is a ( p, q, M )-atom, if it is a compactly supported L q function such that(i) there is a ball B such that supp a ⊂ B and k a k L q ( G ) ≤ | B | /q − /p , where | B | is the Haar measure of B ;(ii) for all polynomials P on G with homogeneous degree ≤ M , Z G a ( x ) P ( x ) dµ ( x ) = 0 . The atomic decomposition of H p ( G ) can be stated as follows. Proposition 3.6. ([26]) Let ( p, q, M ) be an admissible triplet. There is a constant c > such that for all any ( p, q, M ) -atom a , one has k a k H p ( G ) ≤ c. Conversely, given any f ∈ H p ( G ) , there is a sequence { a j } ∞ j =1 of ( p, q, L ) -atomsand a sequence { λ j } ∞ j =1 of complex numbers such that f = ∞ X j =1 λ j a j in S ′ ( G ) OURIER MULTIPLIERS FOR HARDY SPACES ON GRADED LIE GROUPS 15 with (cid:0) P ∞ j =1 | λ j | p (cid:1) /p ≤ C k f k H p ( G ) , where C is a constant independent of f . Further-more, k f k H p ( G ) ∼ inf ∞ X j =1 | λ j | p ! /p : f = ∞ X j =1 λ j a j in S ′ ( G ) , and each a j is an ( p, q, M ) -atom Littlewood-Paley characterization of H p ( G ) in terms of Rockland op-erators. Our main aim in this subsection is to characterize H p ( G ) by means ofLittlewood-Paley decomposition associated with Rockland operators. More preciselywe will establish the following result. Theorem 3.7. Let R be a Rockland operator on G , homogeneous of degree ν . Let Φ ∈ S ( R ) such that supp Φ ⊂ [2 − ν , ν ] and ∞ X j = −∞ | Φ(2 − νj λ ) | > for all λ ∈ (0 , ∞ ) . (3.1) Then for any p ∈ (0 , ∞ ) and any f ∈ S ′ ( G ) that vanishes weakly at infinity, we have k f k H p ( G ) ∼ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ X j = −∞ | Φ(2 − νj R ) f | ! / (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p ( G ) , f ∈ S ′ ( G ) . (3.2)To obtain Littlewood-Paley type characterizations, a useful idea is applying vector-valued estimates of singular integral operators, as was done in e.g. [44] and [7]. Wenote that in [7] a Littlewood-Paley characterization of L p ( G ) in terms of Rocklandoperators was established for 1 < p < ∞ .The proof of Theorem 3.7 needs some preparations. First we introduce the ℓ ( Z )-valued Hardy space H p ( G ; ℓ ( Z )). Definition 3.8. Let φ be a commutative approximate identity on G . For 0 < p < ∞ ,we define H p ( G ; ℓ ( Z )) to be the space of all sequences { f j } ∞ j = −∞ ⊂ S ′ ( G ) such that (cid:13)(cid:13) { f j } ∞ j = −∞ (cid:13)(cid:13) H p ( G ; ℓ ( Z )) := (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) sup k ∈ Z ∞ X j = −∞ | f j ∗ ( D k φ ) | ! / (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p ( G ) < ∞ . Next we state an ℓ ( Z )-valued extension of the H p -boundedness of singular integraloperators of convolution type on graded Lie groups. This is a special case of [26, The-orem 6.20] which concerns general vector-valued singular integrals on homogeneousgroups. Lemma 3.9. (cf. [26, Proposition 6.20]) Suppose < p < ∞ , M is a positive integer, r > Q (1 /p − , and T = { T k } k ∈ Z is a collection of L ( G ; ℓ ( Z )) - L ( G ) boundedoperators such that there exists a collection { K ij } i,j ∈ Z ⊂ S ( G ) with the followingproperties: (i) For any α ∈ N n with | α | ≤ M , there is a constant C α > such that (cid:13)(cid:13)(cid:13) { e X α K ij ( x ) } i,j ∈ Z (cid:13)(cid:13)(cid:13) L ( ℓ ( Z ) ,ℓ ( Z )) ≤ C α | x | − Q − [ α ] for all x ∈ G \{ o } . (ii) For any { f j } ∞ j = −∞ ∈ L ( G ; ℓ ( Z )) such that each f j is compactly supported,we have T i (cid:2) { f j } ∞ j = −∞ (cid:3) ( x ) = ∞ X j = −∞ f j ∗ K ij ( x ) for all x ∈ G and i ∈ Z . (iii) There is a positive integer N such that K ij ≡ if | i | + | j | > N .Then we have (cid:13)(cid:13)(cid:13)(cid:8) T i (cid:2) { f j } ∞ j = −∞ (cid:3)(cid:9) ∞ i = −∞ (cid:13)(cid:13)(cid:13) H p ( G ; ℓ ( Z )) . (cid:13)(cid:13) { f j } ∞ j = −∞ (cid:13)(cid:13) H p ( G ; ℓ ( Z )) . The result stated in the following lemma is proved in [26]. Lemma 3.10. ([26, Proposition 1.49]) Suppose φ ∈ S ( G ) and R G φ ( x ) dx = a . Thenfor any f ∈ S ′ ( G ) , f ∗ ( D t φ ) → af in S ′ ( G ) as t → ∞ . We will also need the following fundamental estimate. Lemma 3.11. Suppose φ, ψ ∈ S ( G ) , M ∈ N and < ε < . (i) If R G ψ ( x ) P ( x ) dx = 0 for all P ∈ P M − , then there is a constant C ′ = C ′ φ,ψ,ε such that for all j, k ∈ Z with j ≤ k , | ( D j φ ) ∗ ( D k ψ )( x ) | ≤ C ( j − k )( M − ε ) − jM (2 − j + | x | ) Q + M . (ii) If R G φ ( x ) P ( x ) dx = 0 for all P ∈ P M − , then there is a constant C = C φ,ψ,ε such that for all j, k ∈ Z with j ≥ k , | ( D j φ ) ∗ ( D k ψ )( x ) | ≤ C ( k − j )( M − ε ) − kM (2 − k + | x | ) Q + M . Proof. For a function f on G , we define the function e f by e f ( x ) := f ( x − ). Then wehave ( D j φ ) ∗ ( D k ψ )( x ) = ( D k e ψ ) ∗ ( D j e φ )( x − ). Hence, once (i) is proved, (ii) willfollow immediately.To prove (i), let j ≤ k and let P j,φx,M − ( · ) ∈ P M − be the (left) Taylor polynomialof D j φ at x of homogeneous degree M − 1. By the vanishing moment condition on D k ψ , we have | ( D j φ ) ∗ ( D k ψ )( x ) | = (cid:12)(cid:12)(cid:12)(cid:12)Z G h ( D j φ )( xy − ) − P j,φx,M − ( y − ) i ( D k ψ )( y ) dµ ( y ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ Z | y |≤ − j + | x | γβM (cid:12)(cid:12)(cid:12) ( D j φ )( xy − ) − P j,φx,M − ( y − ) (cid:12)(cid:12)(cid:12) | ( D k ψ )( y ) | dµ ( y )+ Z | y |≥ − j + | x | γβM | ( D j φ )( xy − ) || ( D k ψ )( y ) | dµ ( y )+ Z | y |≥ − j + | x | γβM (cid:12)(cid:12)(cid:12) P j,φx,M ( y − ) (cid:12)(cid:12)(cid:12) | ( D k ψ )( y ) | dµ ( y ) ≡ I + I + I . OURIER MULTIPLIERS FOR HARDY SPACES ON GRADED LIE GROUPS 17 By Proposition 2.1 and (2.4), we have I . Z | y |≤ − j + | x | γβM | ( D k ψ )( y ) | X | α |≤⌈ M − ⌋ +1[ α ] >M − j [ α ] | y | [ α ] sup | z |≤ β M | y | | D j ( e X α φ )( xz ) | dµ ( y ) . Z | y |≤ − j + | x | γβM − kM (2 − k + | y | ) Q + M X | α |≤⌈ M − ⌋ +1[ α ] >M − j [ α ] | y | [ α ] sup | z |≤ β M | y | − j ( M +[ α ]) (2 − j + | xz | ) Q + M +[ α ] dµ ( y ) ∼ X | α |≤⌈ M − ⌋ +1[ α ] >M − − kM − jM (2 − j + | x | ) Q + M +[ α ] Z | y |≤ − j + | x | γβM | y | [ α ] (2 − k + | y | ) Q + M dµ ( y ) ≤ X | α |≤⌈ M − ⌋ +1[ α ] >M − − kM − jM kε (2 − j + | x | ) Q + M +[ α ] Z | y |≤ − j + | x | γβM | y | [ α ] (2 − k + | y | ) Q + M − ε dµ ( y ) ≤ X | α |≤⌈ M − ⌋ +1[ α ] >M − − kM − jM kε (2 − j + | x | ) Q + M +[ α ] Z | y |≤ − j + | x | γβM | y | Q + M − [ α ] − ε dµ ( y ) ∼ − kM − jM kε (2 − j + | x | ) Q +2 M − ε ≤ ( j − k ) ε − jM (2 − j + | x | ) Q + M where, for the third line we used that if | y | ≤ − j + | x | γβ M and | z | ≤ β M | y | then 2 − j + | xz | ∼ − j + | x | , which follows by (2.2).To estimate I , we note that if | y | > − j + | x | γβ M then (2 − k + | y | ) − . (2 − j + | x | ) − .Hence I . Z | y | > − j + | x | γβM − jM (2 − j + | xy − | ) Q + M − kM (2 − k + | y | ) Q + M dµ ( y ) . − kM (2 − j + | x | ) Q + M Z G − jM (2 − j + | xy − | ) Q + M dµ ( y ) . − kM (2 − j + | x | ) Q + M = 2 − ( k − j ) M − jM (2 − j + | x | ) Q + M ≤ − ( k − j )( M − ε ) − jM (2 − j + | x | ) Q + M . Finally we estimate I . Note that P j,φx,M − ( y ) can be written in the form (see [26]): P j,φx,M − ( y ) = X α :[ α ] ≤ M − X β :[ β ]=[ α ] c β X β ( D j φ )( x ) y α . Since φ is Schwartz, for every multi-index β with [ β ] = [ α ] we have | X β ( D j φ )( x ) | . j [ α ] jQ (1 + 2 j | x | ) Q +[ α ] . It follows that (cid:12)(cid:12)(cid:12) P j,φx,M − ( y − ) (cid:12)(cid:12)(cid:12) . X [ α ] ≤ M − j [ α ] jQ | y | [ α ] (1 + 2 j | x | ) Q +[ α ] ∼ X [ α ] ≤ M − | y | [ α ] (2 − j + | x | ) Q +[ α ] . Hence I . Z | y | > − j + | x | γβM − kM (2 − k + | y | ) Q + M X [ α ] ≤ M − | y | [ α ] (2 − j + | x | ) Q +[ α ] dµ ( y ) . X [ α ] ≤ M − − kM (2 − j + | x | ) Q +[ α ] Z | y | > − j + | x | γβM | y | Q + M − [ α ] dy = 2 − kM (2 − j + | x | ) Q + M ∼ − ( k − j ) M − jM (2 − j + | x | ) Q + M < − ( k − j )( M − ε ) − jM (2 − j + | x | ) Q + M . Therefore, the assertion (i) is proved. (cid:3) We will also need an analogue of Peetre’s inequality. To describe it we introducethe following notation. For every t > 0, letΣ t := { f ∈ S ′ ( G ) : Θ( R ) f = f for all Θ ∈ S ( R ) such that Θ ≡ , t ν ] } . For example, if Φ ∈ S ( R ) such that supp Φ ⊂ ( −∞ , t ν ], then Φ( R ) f ∈ Σ t . Note thatany distribution f in Σ t coincides with a smooth function f ( x ) on G . Lemma 3.12. Let t, r > . Then there exists a constant C > such that for all f ∈ Σ t , sup y ∈ G | f ( y ) | (1 + t | x − y | ) Q/r ≤ C M ( | f | r )( x ) r , x ∈ G, where M is the Hardy-Littlewood maximal operator on G . The proof of this lemma was given in [30] in the setting of stratified Lie groups.Note that, with slight modification, the argument given there can be adapted to thesetting of graded Lie groups. For reader’s convenience we include the proof. Proof. Let y be any fixed point in G . By the mean value theorem (see [26, Theorem1.33] or [24, Proposition 3.1.46]), for every δ > y ∈ B ( y , δ ), | f ( y ) − f ( y ) | ≤ C n X k =1 | y − y | v k sup | z |≤ β | y − y | | ( X k f )( y z ) | ≤ C n X k =1 δ v k sup | z |≤ βδ | ( X k f )( y z ) | . Hence | f ( y ) | ≤ C n X k =1 δ v k sup | y − w |≤ βδ | ( X k f )( w ) | + δ − Q/r (cid:18)Z B ( y ,δ ) | f ( y ) | r dµ ( y ) (cid:19) /r . OURIER MULTIPLIERS FOR HARDY SPACES ON GRADED LIE GROUPS 19 Dividing both sides by (1 + t | y − x | ) Q/r , we have | f ( y ) | (1 + t | y − x | ) Q/r ≤ C n X k =1 δ v k sup | y − w |≤ βδ | X k f ( w ) | (1 + t | w − x | ) Q/r (1 + t | w − x | ) Q/r (1 + t | y − x | ) Q/r + δ − Q/r (1 + t | y − x | ) Q/r (cid:18)Z B ( y ,δ ) | f ( y ) | r dµ ( y ) (cid:19) /r . (3.3)Choose a function Φ ∈ S ( R ) such that Φ = 1 on [0 , f ∈ Σ t , we have f = Φ( t − ν R ) f = f ∗ ( D t φ ) , where φ ∈ S ( G ) is the convolution kernel of Φ( R ). Hence | X k f ( w ) | (1 + t | w − x | ) Q/r ≤ t v k (1 + t | w − x | ) Q/r Z G | f ( z ) D t ( X k f )( z − w ) | dµ ( z ) ≤ C t v k (1 + t | w − x | ) Q/r Z G t Q | f ( z ) | (1 + t | z − w | ) ( Q/r )+ Q +1 dµ ( z ) ≤ C t v k sup z ∈ G | f ( z ) | (1 + t | z − x | ) Q/r , (3.4)where we used the inequality (1 + t | w − x | ) Q/r (1 + t | z − w | ) Q/r ≥ (1 + t | z − x | ) Q/r .Also, note that (cid:18)Z B ( y ,δ ) | f ( y ) | r dµ ( y ) (cid:19) /r ≤ Z B ( x,γ ( δ + | y − x | ) ) | f ( y ) | r dµ ( y ) ! /r ≤ C γ Q/r ( δ + | y − x | ) Q/r M ( | f | r )( x ) r (3.5)Inserting (3.4) and (3.5) into (3.3), and using the inequality(1 + t | w − x | ) Q/r (1 + t | y − x | ) − Q/r ≤ (1 + t | y − w | ) Q/r , we obtain | f ( y ) | (1 + t | y − x | ) Q/r ≤ C C n X k =1 δ v k t v k (1 + tβδ ) Q/r sup z ∈ G | f ( z ) | (1 + t | z − x | ) Q/r + C δ − Q/r γ Q/r ( δ + | y − x | ) Q/r (1 + t | y − x | ) Q/r M ( | f | r )( x ) r . (3.6)Let ε be a sufficiently small positive number such that C C nε v (1 + βε ) Q/r < / δ = t − ε . Then from (3.6) it follows thatsup y ∈ G | f ( y ) | (1 + t | y − x | ) Q/r ≤ 12 sup z ∈ G | f ( z ) | (1 + t | z − x | ) Q/r + C δ − Q/r γ − Q/r (1 + ε − ) M ( | f | r )( x ) r . This yields the desired estimate immediately. (cid:3) Finally, we have a Calder´on type reproducing formula on G . Lemma 3.13. Let R be a Rockland operator homogeneous of degree ν , and Φ be afunction in S ( R ) , vanishing near the origin and satisfying ∞ X j = −∞ Φ(2 − νj λ ) = 1 for all λ ∈ (0 , ∞ ) . (3.7) Then for any f ∈ S ′ ( G ) which vanishes weakly at infinity, we have f = ∞ X j = −∞ Φ(2 − νj R ) f with the sum converging in S ′ ( G ) .Proof. Let φ denote the convolution kernel of Φ( R ). Then φ ∈ S ( G ) (see Remark2.3), and the convolution kernel of Φ(2 − νj R ) is D j φ .We first show that the sum P ∞ j = −∞ D j φ converges in S ′ ( G ) to the Dirac delta-function δ , that is, for any ψ ∈ S ( G ),lim N ,N →∞ * N X j = − N D j φ, ψ + = ψ (0) , where h· , ·i denote the pairing between elements in S ′ ( G ) and S ( G ).To this end, we fix an arbitrary ψ ∈ S ( G ). First note that since Φ vanishes near theorigin, we have R G φ ( x ) P ( x ) dµ ( x ) = 0 for all polynomials P on G . Hence it followsfrom Lemma 3.11 that for all j ≥ x ∈ G , | ( D j φ ) ∗ ψ ( x ) | . − j ( M − ) | x | ) M + Q ≤ − j ( M − ) , where M is an arbitrary positive integer. For j < 0, using the fact that φ is boundedit is easy to see that | ( D j φ ) ∗ ψ ( x ) | . jQ . Therefore, the sum P ∞ j = −∞ | ( D j φ ) ∗ ψ ( x ) | converges uniformly with respect to x ∈ G .Since R is a positive self-adjoint operator and 0 can be neglected in the spectralresolution (cf. [24, Remark 4.2.8]), it follows from (3.7) and the spectral theoremthat lim N ,N →∞ N X j = − N ( D j φ ) ∗ ψ = lim N ,N →∞ N X j = − N Φ(2 − νj R ) ψ = ψ in L ( G ) . Hence, there exist two increasing monotonically sequences { m k } ∞ k =1 and { n k } ∞ k =1 ofpositive integers, such thatlim k →∞ n k X j = − m k ( D j φ ) ∗ ψ ( x ) = ψ ( x ) for a.e. x ∈ G. (Here we used the following fact: if a sequence { g j } j of functions converges in L to g , then there is a subsequence { g j k } k converging to g almost everywhere.) But on OURIER MULTIPLIERS FOR HARDY SPACES ON GRADED LIE GROUPS 21 the other hand, P ∞ j = −∞ | ( D j φ ) ∗ ψ ( x ) | converges uniformly with respect to x ∈ G .Therefore, lim N ,N →∞ N X j = − N ( D j φ ) ∗ ψ ( x ) = ψ ( x ) for all x ∈ G. In particular,lim N ,N →∞ * N X j = − N D j φ, ψ + = lim N ,N →∞ N X j = − N Z G ( D j φ )( x ) ψ ( x ) dµ ( x )= lim N ,N →∞ N X j = − N ( D j φ ) ∗ ψ ( o )= ψ ( o ) . This shows that lim N ,N →∞ N X j = − N D j φ = δ in S ′ ( G ) . (3.8)Following an idea from the proof of [26, Theorem 1.64], we define the distributions g and g by g := ∞ X j =1 D j φ and g := X j = −∞ D j φ, (3.9)respectively. We have seen that both sums in (3.9) converge in S ′ ( G ) and thus g , g are well-defined and g , g ∈ S ′ ( G ); moreover, g + g = δ . An argument in the proofof [26, Theorem 1.64] (using (3.8) and the fact that R G φ ( x ) dx = 0) shows that thedistribution g coincides with a Schwartz function on G , and R G g ( x ) dx = 1.Observe that for any k ∈ Z , D k g = X j = −∞ D j + k φ = k X j = −∞ D j φ, so that N X j = − N D j φ = N X j = −∞ D j φ − − ( N +1) X j = −∞ D j φ = D N g − D − ( N g . Since g ∈ S ( G ) with R G g ( x ) dx = 1, for any f ∈ S ′ ( G ), we have lim N →∞ f ∗ ( D N g ) = f by Lemma 3.10, and if f vanishes weakly at infinity, lim N →∞ f ∗ ( D − ( N g ) = 0. Therefore, for any f ∈ S ′ ( G ) which vanishes weakly at infinity, wehave lim N ,N →∞ N X j = − N f ∗ ( D j φ ) = f. This completes the proof of Lemma 3.13. (cid:3) Now we are ready to give the proof of Theorem 3.7. Proof of Theorem 3.7. Define the operator T = { T i } i ∈ Z mapping ℓ -valued functionsto ℓ -valued functions as follows: T i (cid:2) { f j } ∞ j = −∞ (cid:3) = Φ(2 − νi R ) f = f ∗ ϕ i , i ∈ Z , where ϕ i ( x ) := 2 Qi ϕ (2 i x ) and ϕ ∈ S ( G ) is the convolution kernel of the oper-ator Φ( R ). Then the L ( ℓ ( Z ) , ℓ ( Z ))-valued kernel { K ij } i,j ∈ Z corresponding to T = { T i } i ∈ Z is given by K ij = ( ϕ i , i ∈ Z , j = 0 , , i ∈ Z , j ∈ Z \{ } . For this kernel we have (cid:13)(cid:13)(cid:13) { e X α K ij ( x ) } i,j ∈ Z (cid:13)(cid:13)(cid:13) L ( ℓ ( Z ) ,ℓ ( Z )) = (cid:13)(cid:13)(cid:13) { e X α ϕ i ( x ) } i ∈ Z (cid:13)(cid:13)(cid:13) ℓ ( Z ) = ∞ X i = −∞ Q +[ α ]) i (cid:12)(cid:12) ( e X α ϕ )(2 i x ) (cid:12)(cid:12) . (3.10)Let i x be the unique integer such that 2 − i ≤ | x | < − i +1 . We split the last sum in(3.10) and estimate it as follows. ∞ X i = −∞ n +[ α ]) i (cid:12)(cid:12) ( e X α ϕ )(2 i x ) (cid:12)(cid:12) = i x X i = −∞ Q +[ α ]) i (cid:12)(cid:12) ( e X α ϕ )(2 i x ) (cid:12)(cid:12) + ∞ X i = i x +1 Q +[ α ]) i (cid:12)(cid:12) ( e X α ϕ )(2 i x ) (cid:12)(cid:12) . i x X i = −∞ Q +[ α ]) i + ∞ X i = i x +1 Q +[ α ]) i | i x | − Q +[ α ]+1) . | x | − Q − [ α ] , This shows that the vector-valued kernel { K ij } i,j ∈ Z satisfies the condition (ii) inLemma 3.9. Hence, for all f ∈ S ′ ( G ), (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ X i = −∞ | Φ(2 − νi R ) f | ! / (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p ( G ) = (cid:13)(cid:13) { T i f } ∞ i = −∞ (cid:13)(cid:13) L p ( G ; ℓ ( Z )) ≤ (cid:13)(cid:13) { T i f } ∞ i = −∞ (cid:13)(cid:13) H p ( G ; ℓ ( Z )) . k f k H p ( G ) . (3.11)Strictly speaking, Lemma 3.9 only yields an estimate for (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) N X i = − N | Φ(2 − νi R ) f | ! / (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p ( G ) uniformly over N (because the corresponding truncated vector-valued kernel satis-fies the condition (ii) in Lemma 3.9 uniformly), and we have to use the monotoneconvergence theorem to obtain (3.11). Thus the direction “ & ” of (3.2) holds for all f ∈ S ′ ( G ). OURIER MULTIPLIERS FOR HARDY SPACES ON GRADED LIE GROUPS 23 Next we show the direction “ . ” of (3.2). Since Φ satisfies (3.1), it is possible tofind Ψ ∈ S ( R ) such that supp Ψ ⊂ [2 − ν , ν ] and ∞ X j = −∞ Ψ(2 − νj λ )Φ(2 − νj λ ) = 1 for all λ ∈ (0 , ∞ ) . (3.12)An example of such a Ψ isΨ( λ ) = Φ( λ ) P ∞ j = −∞ | Φ(2 − νj λ ) | , λ ∈ (0 , ∞ ) , , λ ∈ ( −∞ , . Define the operator T = { T i } i ∈ Z acting on ℓ -valued functions by T i (cid:2) { f j } ∞ j = −∞ (cid:3) = (P ∞ j = −∞ Ψ(2 − νj R ) f j , i = 0 , j ∈ Z , , i ∈ Z \{ } , j ∈ Z . Let ψ be the convolution kernel of Ψ( R ) and set ψ j ( x ) := 2 Qj ψ (2 j x ). The L ( ℓ ( Z ) , ℓ ( Z ))-valued kernel { e K ij } i,j ∈ Z corresponding to T = { T i } i ∈ Z is given by e K ij = ( ψ j , i = 0 , j ∈ Z , , i ∈ Z \{ } , j ∈ Z . This vector-valued kernel { e K ij } i,j ∈ Z also satisfies the condition (ii) in Lemma 3.9.Indeed, by the Cauchy-Schwarz inequality we have (cid:13)(cid:13)(cid:13) { e X α e K ij ( x ) } i,j ∈ Z (cid:13)(cid:13)(cid:13) L ( ℓ ( Z ) ,ℓ ( Z )) = (cid:13)(cid:13)(cid:13) { e X α ψ j ( x ) } j ∈ Z (cid:13)(cid:13)(cid:13) ℓ ( Z ) and, similarly to the estimate of (cid:13)(cid:13) { e X α ϕ i ( x ) } i ∈ Z (cid:13)(cid:13) ℓ ( Z ) we have done before, (cid:13)(cid:13)(cid:13) { e X α ψ j ( x ) } j ∈ Z (cid:13)(cid:13)(cid:13) ℓ ( Z ) . | x | − Q − [ α ] . Hence, we can apply Lemma 3.9 to obtain (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ X j = −∞ Ψ(2 − νj R ) f j (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) H p ( G ) . (cid:13)(cid:13) { f j } ∞ j = −∞ (cid:13)(cid:13) H p ( G ; ℓ ) (3.13)Let f ∈ S ′ ( G ) such that f vanishes weakly at infinity. Then, taking f j = Φ(2 − νj R ) f in (3.13) gives (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ X j = −∞ Ψ(2 − νj R )Φ(2 − νj R ) f (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) H p ( G ) . (cid:13)(cid:13) { f j } ∞ j = −∞ (cid:13)(cid:13) H p ( G ; ℓ ) . (3.14)On the other hand, by (3.12) and Lemma 3.13, we have f = ∞ X j = −∞ Ψ(2 − νj R )Φ(2 − νj R ) f in S ′ ( G ) . Combining this with (3.14) yields k f k H p ( G ) . (cid:13)(cid:13) { Φ(2 − νj R ) f } ∞ j = −∞ (cid:13)(cid:13) H p ( G ; ℓ ) . (3.15) Now we choose a function Ω ∈ S ( R ) such that supp Ω ⊂ ( −∞ , ν ] and Ω ≡ , − νk R )Φ(2 − νj R ) = ( Φ(2 − νj R ) , j ≤ k − , j ≥ k + 2 . (3.16)Consider now the case k − < j < k + 2. Denoting by ω the convolution kernel ofΩ( R ), setting ω j ( x ) = 2 Qj ω (2 j x ) for j ∈ Z , and letting L ≥ Q/r + Q + 1, we have | Ω(2 − νk R )Φ(2 − νj R ) f ( x ) | ≤ Z G (cid:12)(cid:12) Φ(2 − νj R ) f ( y )2 Qk ω (cid:0) k ( y − x ) (cid:1)(cid:12)(cid:12) dµ ( y ) . sup y ∈ G | Φ(2 − νj R ) f ( y ) | (1 + 2 j | x − y | ) Q/r Z G Qk (1 + 2 j | x − y | ) Q/r (1 + 2 k | y − x | ) L dµ ( y ) ∼ sup y ∈ G | Φ(2 − νj R ) f ( y ) | (1 + 2 j | x − y | ) Q/r Z G Qk (1 + 2 k | y − x | ) L − Q/r dµ ( y ) . M (cid:0) | Φ(2 − νj R ) f | r (cid:1) ( x ) r , (3.17)where for the last line we used Lemma 3.12 and the fact that Φ(2 − νj R ) f ⊂ Σ j +1 .Combining (3.16) with (3.17) yieldssup k ∈ Z ∞ X j = −∞ | Ω(2 − νk R )Φ(2 − νj R ) f ( x ) | ! / . ∞ X j = −∞ (cid:2) M (cid:0) | Φ(2 − νj R ) f | r (cid:1) ( x ) (cid:3) /r ! / . We then apply the vector-valued Hardy-Littlewood maximal inequality to obtain (cid:13)(cid:13) { Φ(2 − νj R ) f } ∞ j = −∞ (cid:13)(cid:13) H p ( G ; ℓ ) . (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ X j = −∞ | Φ(2 − νj R ) f | ! / (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p ( G ) . This along with (3.15) yields that the direction “ . ” in (3.2) holds for all f ∈ S ′ ( G )that vanishes weakly at infinity. The proof of Theorem 3.7 is thus complete. (cid:3) Proofs of main results Proof of Theorem 1.5. Let 0 < p ≤ σ = { σ ( π ) , π ∈ b G } isa measurable field of operators satisfying (1.7), i.e.,sup t> (cid:13)(cid:13) { σ ( t · π ) η ( π ( R )) , π ∈ b G } (cid:13)(cid:13) W s ( b G ) < ∞ (4.1)for some s > Q (1 /p − / η ∈ C ∞ (0 , ∞ ), not identically zero. Since thefunction s 7→ k · k W s ( b G ) is monotonically increasing, we may assume without loss ofgenerality that Q (1 /p − / < s < ⌊ Q (1 /p − ⌋ + Q/ . (4.2)Fix a function Φ ∈ S ( R ) such that supp Φ ⊂ [2 − ν , ν ] and Φ satisfies (3.1). Thenas in the proof of Theorem 3.7 we can find Ψ ∈ S ( R ) such that supp Ψ ⊂ [2 − ν , ν ] OURIER MULTIPLIERS FOR HARDY SPACES ON GRADED LIE GROUPS 25 and X j ∈ Z Φ(2 − νj λ )Ψ(2 − νj λ ) = 1 for all λ ∈ (0 , ∞ ) . (4.3)By [23, Proposition 4.6], we also havesup t> (cid:13)(cid:13) { σ ( t · π )Φ( π ( R )) , π ∈ b G } (cid:13)(cid:13) W s ( b G ) ≤ C sup t> (cid:13)(cid:13) { σ ( t · π ) η ( π ( R )) , π ∈ b G } (cid:13)(cid:13) W s ( b G ) , (4.4)where C is a constant depending on s, η and Φ, but independent of σ .Let L = max { [ α ] : α ∈ N n with [ α ] ≤ Q (1 /p − } . By Proposition 3.6, to provethat T σ is bounded from H p ( G ) to L p ( G ), it suffices to show that for any ( p, , L )-atom a , k T σ a k L p ( G ) ≤ C sup t> (cid:13)(cid:13) { σ ( t · π ) η ( π ( R )) , π ∈ b G } (cid:13)(cid:13) W s ( b G ) . Suppose a is any ( p, , L )-atom associated to a ball B = B ( x , r ). Since T σ commuteswith left translations, we may assume without loss of generality that x = e , i.e.,the ball B is centered at the group identity e . Let c be a fixed constant such that c ≥ γ (1 + β L ), where γ and β are constants from (2.2) and (2.5), respectively. Wethen write k T σ a k pL p ( G ) = Z B ( e,cr ) | T σ a ( x ) | p dµ ( x ) + Z B ( e,cr ) c | T σ a ( x ) | p dµ ( x )=: I + I . First we estimate I . From Lemma 2.8 we see that the condition (4.14) implies σ ∈ L ∞ ( b G ) with k σ k L ∞ ( b G ) ≤ C sup t> (cid:13)(cid:13) { σ ( t · π ) η ( π ( R )) , π ∈ b G } (cid:13)(cid:13) W s ( b G ) . Hence T σ isbounded on L ( G ). Using this and H¨older’s inequality, we have I . k T σ a k pL ( G ) | B ( e, cr ) | − p . k σ k L ∞ ( b G ) k a k pL ( G ) | B ( e, cr ) | − p . sup t> (cid:13)(cid:13) { σ ( t · π ) η ( π ( R )) , π ∈ b G } (cid:13)(cid:13) W s ( b G ) | B ( e, r ) | ( − p ) p | B ( e, cr ) | − p . sup t> (cid:13)(cid:13) { σ ( t · π ) η ( π ( R )) , π ∈ b G } (cid:13)(cid:13) W s ( b G ) . (4.5)To estimate I , we note that by (4.3) and the spectral theorem (recalling that 0can be neglected in the spectral resolution), a = X j ∈ Z Φ(2 − νj R )Ψ(2 − νj R ) a in L ( G ) . It follows that T σ a = T σ X j ∈ Z Φ(2 − νj R )Ψ(2 − νj R ) a ! = X j ∈ Z T σ Φ(2 − νj R )Ψ(2 − νj R ) a. (4.6)We set σ j ( π ) = σ (2 νj · π )Φ( π ( R ))Ψ( π ( R )), so that T σ Φ(2 − νj R )Ψ(2 − νj R ) = T σ j (2 − νj · π ) . Substituting this into (4.6) gives T σ a ( x ) = X j ∈ Z T σ j (2 − νj · π ) a ( x ) = X j ∈ Z Z B ( e,r ) K j ( y − x ) a ( y ) dµ ( x ) = X j ∈ Z F j ( x ) , where K j := F − G [ σ j (2 − νj · π )] and F j ( x ) := R B ( e,r ) K j ( y − x ) a ( y ) dµ ( y ). Then byH¨older’s inequality and (2.4), Z B ( e,cr ) c | T a ( x ) | p dµ ( x ) = Z B ( e,cr ) c (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X j ∈ Z F j ( x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p dµ ( x ) ≤ X j ∈ Z Z B ( e,cr ) c | F j ( x ) | p dµ ( x )= X j ∈ Z Z B ( e,cr ) c | x | − ps | x | ps | F j ( x ) | p dµ ( x )(4.7) ≤ X j ∈ Z (cid:18)Z B ( e,cr ) c | x | − ps − p dµ ( x ) (cid:19) − p (cid:18)Z B ( e,cr ) c | x | s | F j ( x ) | dµ ( x ) (cid:19) p ∼ X j ∈ Z r (2 − p ) Q − ps (cid:18)Z B ( e,cr ) c | x | s | F j ( x ) | dµ ( x ) (cid:19) p . Let j be the unique integer such that 2 − j ≤ r < − j +1 (i.e., r ∼ − j ). To estimatethe last integral in (4.7), we consider two cases. Case 1: j ≥ j . Observe that | x | ∼ | y − x | whenever x ∈ B ( e, cr ) c and y ∈ B ( e, r ).Hence for all x ∈ B ( e, cr ) c , | x | s | F j ( x ) | = | x | s (cid:12)(cid:12)(cid:12)(cid:12)Z B ( e,r ) K j ( y − x ) a ( y ) dµ ( y ) (cid:12)(cid:12)(cid:12)(cid:12) ∼ Z B ( e,r ) | y − x | s | K j ( y − x ) || a ( y ) | dµ ( y ) ≤ (cid:18)Z B ( e,r ) | y − x | s | K j ( y − x ) | dµ ( y ) (cid:19) / k a k L ( G ) . | B ( e, r ) | − p (cid:18)Z B ( e,r ) | y − x | s | K j ( y − x ) | dµ ( y ) (cid:19) / . It follows by Fubini’s theorem that Z B ( e,cr ) c | x | s | F j ( x ) | dµ ( x ) . | B ( e, r ) | − p Z B ( e,cr ) c (cid:18)Z B ( e,r ) | y − x | s | K j ( y − x ) | dy (cid:19) dµ ( x )= | B ( e, r ) | − p Z B ( e,r ) (cid:18)Z B ( e,cr ) c | y − x | s | K j ( y − x ) | dµ ( x ) (cid:19) dµ ( y ) ≤ | B ( e, r ) | − p Z B ( e,r ) (cid:18)Z G | y − x | s | K j ( y − x ) | dµ ( x ) (cid:19) dµ ( y )(4.8) OURIER MULTIPLIERS FOR HARDY SPACES ON GRADED LIE GROUPS 27 = | B ( e, r ) | − p Z B ( e,r ) (cid:18)Z G | x | s | K j ( x ) | dµ ( x ) (cid:19) dµ ( y )= | B ( e, r ) | − p Z G | x | s | K j ( x ) | dµ ( x ) . Let H j := F − G [ σ j ( π )]. Then K j ( x ) = 2 jQ H j (2 j x ), and (4.8) implies Z B ( e,cr ) c | x | s | F j ( x ) | dµ ( x ) . jQ − js − jQ | B ( e, r ) | − p Z G | j w | s | H j (2 j w ) | dµ (2 j w ) ∼ j ( Q − s ) r Q (2 − p ) Z G | w | s | H j ( w ) | dµ ( w ) . (4.9) Case 2: j < j . Let P K j ,x,L ( y ) be the Taylor polynomial of the function K j at x of homogeneous degree L . Then by the vanishing moments of a , Taylor’s inequality(Proposition 2.1), and the fact that | x | ∼ | zx | whenever x ∈ B ( e, cr ) c , y ∈ B ( e, r )and | z | ≤ β L +1 | y | , we have, for all x ∈ B ( e, cr ) c , | x | s | F j ( x ) | = | x | s (cid:12)(cid:12)(cid:12)(cid:12)Z B ( e,r ) [ K j ( y − x ) − P K j ,x,L ( y − )] a ( y ) dµ ( y ) (cid:12)(cid:12)(cid:12)(cid:12) . Z B ( e,r ) | x | s X | α |≤⌈ L ⌋ +1[ α ] >L sup | z |≤ β L +1 | y | | y | [ α ] | ( e X α K j )( zx ) || a ( y ) | dµ ( y ) ∼ Z B ( e,r ) X | α |≤⌈ L ⌋ +1[ α ] >L sup | z |≤ β L +1 | y | | y | [ α ] | zx | s | ( e X α K j )( zx ) || a ( y ) | dµ ( y ) ≤ X | α |≤⌈ L ⌋ +1[ α ] >L r [ α ] Z B ( e,r ) sup | z |≤ β L +1 | y | | zx | s | ( e X α K j )( zx ) | dµ ( y ) ! / k a k L ( G ) . | B ( e, r ) | − p X | α |≤⌈ L ⌋ +1[ α ] >L r [ α ] Z B ( e,r ) sup | z |≤ β L +1 | y | | zx | s | ( e X α K j )( zx ) | dµ ( y ) ! / . It follows by Fubini’s theorem that Z B ( e,cr ) c | x | s | F j ( x ) | dµ ( x ) . | B ( e, r ) | − p X | α |≤⌈ L ⌋ +1[ α ] >L r α ] Z B ( e,cr ) c Z B ( e,r ) sup | z |≤ β L +1 | y | | zx | s | ( e X α K j )( zx ) | dµ ( y ) ! dµ ( x ) . | B ( e, r ) | − p X | α |≤⌈ L ⌋ +1[ α ] >L r α ] Z B ( e,r ) Z B ( e,cr ) c sup | z |≤ β L +1 | y | | zx | s | ( e X α K j )( zx ) | dµ ( x ) ! dµ ( y )= | B ( e, r ) | − p X | α |≤⌈ L ⌋ +1[ α ] >L r α ] Z B ( e,r ) sup | z |≤ β L +1 | y | Z B ( e,cr ) c | zx | s | ( e X α K j )( zx ) | dµ ( x ) ! dµ ( y )(4.10) ≤ | B ( e, r ) | − p X | α |≤⌈ L ⌋ +1[ α ] >L r α ] Z B ( e,r ) sup | z |≤ β L +1 | y | (cid:18)Z G | zx | s | ( e X α K j )( zx ) | dµ ( x ) (cid:19) dµ ( y )= | B ( e, r ) | − p X | α |≤⌈ L ⌋ +1[ α ] >L r α ] Z B (cid:18)Z G | w | s | ( e X α K j )( w ) | dµ ( w ) (cid:19) dµ ( y )= | B ( e, r ) | − p X | α |≤⌈ L ⌋ +1[ α ] >L r α ] Z G | w | s | ( e X α K j )( w ) | dµ ( w ) . As in Case 1 we still set H j := F − G [ σ j ( π )]. Then K j ( x ) = 2 jQ H j (2 j x ), and (4.10)implies Z B ( e,cr ) c | x | s | F j ( x ) | dµ ( x ) . X | α |≤⌈ L ⌋ +1[ α ] >L jQ j [ α ] r α ] | B ( e, r ) | − p Z G | w | s | e X α H j (2 j w ) | dµ ( w )= X | α |≤⌈ L ⌋ +1[ α ] >L jQ j [ α ] − js − jQ r α ] | B ( e, r ) | − p Z G | j w | s | ( e X α H j )(2 j w ) | dµ (2 j w )(4.11) ∼ X | α |≤⌈ L ⌋ +1[ α ] >L j ( Q +2[ α ] − s ) r α ]+2 Q − Qp Z G | w | s | ( e X α H j )( w ) | dµ ( w ) . By Hulanicki’s theorem, F − G (cid:0) { Ψ( π ( R )) π ( X α ) : π ∈ b G } (cid:1) ∈ S ( G ). Hence, for all j ∈ Z and all α ∈ N n , by Remark 2.6, Lemma 2.7 and (4.4), there is a constant C α independent of j such that Z G | w | s | ( e X α H j )( w ) | dµ ( w ) ≤ (cid:13)(cid:13) { σ (2 νj π )Φ( π ( R ))Ψ( π ( R )) π ( X α ) , π ∈ b G } (cid:13)(cid:13) W s ( b G ) ≤ C α (cid:13)(cid:13) { σ (2 νj π )Φ( π ( R )) , π ∈ b G } (cid:13)(cid:13) W s ( b G ) ≤ C α sup t> (cid:13)(cid:13) { σ ( t · π )Φ( π ( R )) , π ∈ b G } (cid:13)(cid:13) W s ( b G ) ≤ C α sup t> (cid:13)(cid:13) { σ ( t · π ) η ( π ( R )) , π ∈ b G } (cid:13)(cid:13) W s ( b G ) . (4.12)From (4.7), (4.9), (4.11) and (4.12), it follows that I = Z B ( e,cr ) c | T a ( x ) | p dµ ( x ) . X j ≥ j ( j − j ) p ( Q − s ) ! sup t> (cid:13)(cid:13) { σ ( t · π ) η ( π ( R )) , π ∈ b G } (cid:13)(cid:13) pW s ( b G ) OURIER MULTIPLIERS FOR HARDY SPACES ON GRADED LIE GROUPS 29 + X j Suppose 0 < p ≤ σ = { σ ( π ) , π ∈ b G } is ameasurable field of operators such thatsup t> (cid:13)(cid:13) { η ( π ( R )) σ ( t · π ) , π ∈ b G } (cid:13)(cid:13) W s ( b G ) < ∞ (4.14)for some s > Q (1 /p − / 2) + ( v n − v ) (cid:0)(cid:4) Lv (cid:5) + 1 (cid:1) , not identically zero. Here L =max { [ α ] : α ∈ N n with [ α ] ≤ Q (1 /p − } . Without loss of generality we may assume s = s ′ + ( L + v )( v n v − 1) with Q (1 /p − / < s ′ < ⌊ Q (1 /p − ⌋ + Q/ p, , L )-atom a , T σ a vanishes weakly at infinity and there is a constant C suchthat k g R Φ ( T σ a ) k L p ( G ) ≤ C sup t> (cid:13)(cid:13) { η ( π ( R )) σ ( t · π ) , π ∈ b G } (cid:13)(cid:13) W s ( b G ) , (4.15)where g R Φ ( T σ a ) = ∞ X j = −∞ | Φ(2 − νj R ) T σ f | ! / . Note that the operator g R Φ is bounded on L ( G ) (see [26, Theorem 7.7]), and that by[23, Proposition 4.6], there is a constant C depending on s, η and Φ, but independentof σ , such thatsup t> (cid:13)(cid:13) { Φ( π ( R )) σ ( t · π ) , π ∈ b G } (cid:13)(cid:13) W s ( b G ) ≤ C sup t> (cid:13)(cid:13) { η ( π ( R )) σ ( t · π ) , π ∈ b G } (cid:13)(cid:13) W s ( b G ) . (4.16)Suppose a is an ( p, , L )-atom atom associated to a ball B = B ( x , r ). Since both T σ and g R Φ commute with left translations, we may assume without loss of generalitythat x = e , i.e., the ball B is centered at the group identity e . As we have seen in the proof of Theorem 1.5, T σ is bounded on L ( G ). Hence T σ a ∈ L ( G ), whichimplies that T σ a vanishes weakly at infinity. To prove (4.15), we fix a constant c suchthat c ≥ γ (1 + β L ), where γ and β are constants from (2.2) and (2.5), respectively,and write k g R Φ ( T σ a ) k pL p ( G ) = Z B ( e,cr ) | g R Φ ( T σ a )( x ) | p dµ ( x ) + Z B ( e,cr ) c | g R Φ ( T σ a )( x ) | p dµ ( x )=: I + I . Using L boundedness of T σ and g R Φ , and similarly to the proof of (4.5), we have I . sup t> (cid:13)(cid:13) { η ( π ( R )) σ ( t · π ) , π ∈ b G } (cid:13)(cid:13) W s ( b G ) . (4.17)To estimate I , choose a function Θ ∈ S ( R ) such that Θ ≡ I = Z B ( e,cr ) c | g R Φ ( T σ a )( x ) | p dµ ( x )= Z B ( e,cr ) c | x | − ps ′ | x | s ′ X j ∈ Z | Θ(2 − νj R )Φ(2 − νj R ) T σ a ( x ) | ! p dµ ( x ) ≤ (cid:18)Z B ( e,cr ) c | x | − ps ′ − p dµ ( x ) (cid:19) − p (4.18) × Z B ( e,cr ) c | x | s ′ X j ∈ Z | Θ(2 − νj R )Φ(2 − νj R ) T σ a ( x ) | dµ ( x ) ! p ∼ r (2 − p ) Q − ps ′ X j ∈ Z Z B ( e,cr ) c | x | s ′ | Θ(2 − νj R )Φ(2 − νj R ) T σ a ( x ) | dµ ( x ) ! p . Set σ j ( π ) := Θ( π ( R ))Φ( π ( R )) σ (2 νj · π ), K j := F − G [ σ j (2 − νj · π )] and H j := F − G [ σ j ( π )].Then K j ( x ) = 2 jQ H j (2 j x ), andΘ(2 − νj R )Φ(2 − νj R ) T σ a ( x ) = Z G K j ( y − x ) a ( y ) dµ ( y ) . Let j be the unique integer such that 2 − j ≤ r < − j +1 (i.e., r ∼ − j ). For j ≥ j , similarly to (4.9) we have Z B ( e,cr ) c | x | s ′ | Θ(2 − νj R )Φ(2 − νj R ) T σ a ( x ) | dµ ( x ) . j ( Q − s ′ ) r Q (2 − p ) Z G | w | s ′ | H j ( w ) | dµ ( w ) , (4.19) OURIER MULTIPLIERS FOR HARDY SPACES ON GRADED LIE GROUPS 31 while for j < j , similarly to (4.11) we have Z B ( e,cr ) c | x | s ′ | Θ(2 − νj R )Φ(2 − νj R ) T σ a ( x ) | dµ ( x ) . X | α |≤⌈ L ⌋ +1[ α ] >L j ( Q +2[ α ] − s ′ ) r α ]+2 Q − Qp Z G | w | s ′ | ( e X α H j )( w ) | dµ ( w ) . (4.20)Hulanicki’s theorem along with Lemma 2.7 and (4.16) yields that (similarly to(4.12) with α = (0 , · · · , Z G | w | s ′ | H j ( w ) | dµ ( w ) . sup t> (cid:13)(cid:13) { Φ( π ( R )) σ ( t · π ) , π ∈ b G } (cid:13)(cid:13) W s ′ ( b G ) . sup t> (cid:13)(cid:13) { η ( π ( R )) σ ( t · π ) : π ∈ b G } (cid:13)(cid:13) W s ′ ( b G ) . (4.21)For each α ∈ N n with | α | ≤ ⌈ L ⌋ + 1 and [ α ] > L , we can write e X α = X | α ′ |≤| α | , [ α ′ ] ≥ [ α ] P α,α ′ ( x ) X α ′ , where each P α,α ′ is a homogeneous polynomial of degree [ α ′ ] − [ α ] (see [26, Proposition1.29]), and we observe that | α | ≤ ⌈ L ⌋ + 1 , [ α ] > L, | α ′ | ≤ | α | , [ α ′ ] ≥ [ α ] = ⇒ ≤ [ α ′ ] − [ α ] ≤ ( v n − v ) (cid:18)(cid:22) Lv (cid:23) + 1 (cid:19) . By this observation, Hulanicki’s theorem (which implies that F − G (cid:2) π ( X α ′ )Θ( π ( R )) (cid:3) ∈S ( G )), Lemma 2.7 and (4.16), we can deduce that Z G | w | s ′ | ( e X α H j ( w ) | dµ ( w ) . X | α ′ |≤| α | , [ α ′ ] ≥ [ α ] Z G (1 + | w | ) s ′ +2( v n − v )( ⌊ Lv ⌋ +1) | ( X α ′ H j )( w ) | dµ ( w ) . X | α ′ |≤| α | , [ α ′ ] ≥ [ α ] (cid:13)(cid:13) { π ( X α ′ )Θ( π ( R ))Φ( π ( R )) σ (2 νj · π ) , π ∈ b G } (cid:13)(cid:13) W s ′ +( vn − v ⌊ Lv ⌋ +1) ( b G ) (4.22) . sup t> (cid:13)(cid:13) { Φ( π ( R )) σ ( t · π ) , π ∈ b G } (cid:13)(cid:13) W s ′ +( vn − v ⌊ Lv ⌋ +1) ( b G ) . sup t> (cid:13)(cid:13) { η ( π ( R )) σ ( t · π ) , π ∈ b G } (cid:13)(cid:13) W s ( b G ) . Inserting (4.19), (4.20), (4.21) and (4.22) into (4.18), we obtain I = Z B ( e,cr ) c | g R Φ ( T σ a )( x ) | p dµ ( x ) . X j ≥ j ( j − j ) p ( Q − s ′ ) ! sup t> (cid:13)(cid:13) { η ( π ( R )) σ ( t · π ) , π ∈ b G } (cid:13)(cid:13) pW s ′ ( b G ) + X j An application to Riesz operators on graded Lie groups Our results can be applied to Riesz operators on graded Lie groups. Let G bea graded Lie group and let R be a positive Rockland operator on G , homogeneousof degree ν . For each α ∈ N n with α = (0 , , · · · , X α R − [ α ] /ν a Rieszoperator on G . As we mentioned before, this type of operators are not included inthe spectral multiplier operators studied in the literature, but may be viewed as aFourier multiplier operator T σ with multiplier symbol given via σ ( π ) = π ( X ) α π ( R ) − [ α ] /ν . It is easy to check that σ is 0-homogeneous, i.e., σ ( t · π ) = σ ( π ).Let η ∈ C ∞ c (0 , ∞ ) be a cut-off function not identically zero. Define η ∈ C ∞ (0 , ∞ )via η ( λ ) = λ − [ α ] /ν η ( λ ), so that σ ( π ) η ( π ( R ) = π ( X ) α η ( π ( R )). By Hulanicki’s The-orem, (cid:13)(cid:13) { π ( X ) α η ( π ( R )) , π ∈ b G } (cid:13)(cid:13) W s ( b G ) < ∞ for all s > 0. It follows thatsup t> (cid:13)(cid:13) { σ ( t · π ) η ( π ( R )) , π ∈ b G } (cid:13)(cid:13) W s ( b G ) = (cid:13)(cid:13) { σ ( π ) η ( π ( R ) , π ∈ b G } (cid:13)(cid:13) W s ( b G ) < ∞ (5.1)for all s > σ ∗ = { σ ( π ) ∗ , π ∈ b G } . Then σ ∗ ( π ) = ( − [ α ] π ( R ) − [ α ] /ν π ( X ) α . By a similarargument as above one can show thatsup t> (cid:13)(cid:13) { η ( π ( R )) σ ∗ ( t · π ) , π ∈ b G } (cid:13)(cid:13) W s ( b G ) < ∞ . (5.2)Theorem 1.5 coupled with (5.1) implies that X α R − [ α ] /ν is bounded from H p ( G ) to L p ( G ) for 0 < p ≤ 1, while Theorem 1.6 coupled with (5.2) implies that ( − [ α ] R − [ α ] /ν X α (the adjoint of X α R − [ α ] /ν ) is bounded on H p ( G ) for 0 < p ≤ 1. Note that X α R − [ α ] /ν and its adjoint are also bounded on L ( G ). Hence, by interpolation, X α R − [ α ] /ν andits adjoint are bounded on L p ( G ) for 1 < p ≤ 2. Therefore we have the followingresult. Corollary 5.1. For any α ∈ N n , the Riesz operator X α R − [ α ] /ν is bounded from H p ( G ) to L p ( G ) if < p ≤ , and is bounded on L p ( G ) if < p < ∞ . Remark 5.2. It is worth mentioning that the L p ( G )-boundedness of X α R − [ α ] /ν for1 < p < ∞ has been proved in [25]. OURIER MULTIPLIERS FOR HARDY SPACES ON GRADED LIE GROUPS 33 References [1] G. Alexopoulos, Spectral multipliers on Lie groups of polynomial growth , Proc. Amer. Math.Soc. (3) (1994) 973–979.[2] P. Auscher, A. F. M. ter Elst and D. W. Robinson, On positive Rockland operators , Colloq.Math. (1994), 197–216.[3] T. A. Bui and X. T. Duong, Spectral multipliers of self-adjoint operators on Besov and Triebel-Lizorkin spaces associated to operators , to appear in Int. Math. Res. Not. IMRN.[4] A. P. Calder´on and A. Torchinsky, Parabolic maximal functions associated with a distribution ,Advances in Math. (1975), 1–64.[5] A. P. Calder´on and A. Torchinsky, Parabolic maximal functions associated with a distribution.II , Advances in Math. (1977), 101–171.[6] D. Cardona, J. Delgado and M. Ruzhansky, L p -bounds for pseudo-differential operators ongraded Lie groups, arXiv:1911.03397.[7] D. Cardona and M. Ruzhansky, Littlewood-Paley theorem, Nikolskii inequality, Besov spaces,Fourier and spectral multipliers on graded Lie groups , arXiv:1610.04701.[8] J. Chen and G. Lu, H¨ormander type theorems for multi-linear and multi-parameter Fouriermultiplier operators with limited smoothness , Nonlinear Anal. (2014), 98–112.[9] J. Chen and G. Lu, H¨ormander type theorem on bi-parameter Hardy spaces for bi-parameterFourier multipliers with optimal smoothness , Rev. Mat. Iberoam. (2018), 1541–1561.[10] L. K. Chen, The multiplier operators on the product spaces , Illinois. J. Math. (1994), 420–433.[11] L. K. Chen and D. Fan, The multiplier operators on the weighted product spaces , Proc. Amer.Math. Soc. (1996), 3755–3765.[12] M. Christ, L p bounds for spectral multipliers on nilpotent groups , Trans. Amer. Math. Soc. (1991), 73–81.[13] R. Coifman and G. Weiss, Analyse harmonique non-commutative sur certains espaces ho-mog`enes , Lecture Notes in Mathematics, Vol. 242 (1971), Springer-Verlag, Berlin-New York.[14] R. Coifman and G. Weiss, Multiplier transformations of functions on SU (2) and Σ , Collectionof articles dedicated to Alberto Gonz´alez Dom´ınguez on his sixty-fifth birthday, Rev. Un. Mat.Argentina (1970/71), 145–166.[15] L. De Michele and G. Mauceri, L p multipliers on the Heisenberg group , Michigan Math. J. (1979), 361–371.[16] L. De Michele and G. Mauceri, H p multipliers on stratified groups , Ann. Mat. Pura Appl. (4)148 (1987), 353–366.[17] J. Dixmier, C ∗ -algebras , Translated from the French by Francis Jellett, North-Holland Mathe-matical Library, Vol. 15 (1977), North-Holland Publishing Co., Amsterdam-New York-Oxford.[18] X. T. Duong, E. M. Ouhabaz and A. Sikora, Plancherel-type estimates and sharp spectralmultipliers , J. Funct. Anal. (2002), 443–485.[19] X. T. Duong and L. Yan, Spectral multipliers for Hardy spaces associated to non-negative self-adjoint operators satisfying Davies-Gaffney estimates , J. Math. Soc. Japan (2011), 295–319.[20] R. Fefferman and K. C. Lin, A sharp result on multiplier operators . Unpublished.[21] V. Fischer, Differential structure on the dual of a compact Lie group , J. Funct. Anal. (2020), art. ID: 108555.[22] V. Fischer and M. Ruzhansky, A pseudo-differential calculus on graded nilpotent Lie groups ,arXiv:1209.2621.[23] V. Fischer and M. Ruzhansky, Fourier multipliers on graded Lie groups , Colloq. Math., toappear.[24] V. Fischer and M. Ruzhansky, Quantization on nilpotent Lie groups , Progress in Mathematics,Vol. 314 (2016), Birkh¨auser Basel.[25] V. Fischer and M. Ruzhansky, Sobolev spaces on graded Lie groups , Ann. Inst. Fourier (Greno-ble), (2017), 1671–1723.[26] G. B. Folland and E. M. Stein, Hardy Spaces on Homogeneous Groups. Mathematical Notes28, Princeton University Press, Princeton, 1982. [27] R. F. Gundy and E. M. Stein, H p theory for the poly-disc , Proc. Natl. Acad. Sci. USA (1979), 1026–1029.[28] Y. Han, J. Li, G. Lu and P. Wang, H p → H p boundedness implies H p → L p boundedness ,Forum Math. (2011), 729–756.[29] W. Hebisch and J. Zienkiewicz, Multiplier theorem on generalized Heisenberg groups. II , Colloq.Math. (1995), 29–36.[30] G. Hu, Homogeneous Triebel-Lizorkin spaces on stratified Lie groups , J. Funct. Spaces Appl.,Vol. 2013, Art. ID 475103, 16pp.[31] G. Hu, Littlewood-Paley characterization of H¨older-Zygmund spaces on stratified Lie groups ,Czechoslovak Math. J. (2019), 131–159.[32] A. Hulanicki, A functional calculus for Rockland operators on nilpotent Lie groups , Stud. Math. (1984), 253–266.[33] L. H¨ormander, Estimates for translation invariant operators in L p spaces , Acta Math. (1960), 93–140.[34] A. Hulanicki and E. M. Stein, Marcinkiewicz multiplier theorem for stratified groups , unpub-lished manuscript.[35] V. L. Hung, Multiplier operators on product spaces , Studia. Math. (2002), 265–275.[36] G. Kerkyacharian and P. Petrushev, Heat kernel based decomposition of spaces of distributionsin the framework of Dirichlet spaces , Trans. Amer. Math. Soc. (2014), 121–189.[37] L. Liu, D. Yang and W. Yuan, Besov-type and Triebel-Lizorkin-type spaces associated with heatkernels , Collect. Math. (2016), 247–310.[38] A. Martini and D. M¨uller, Spectral multipliers on 2-step groups: topological versus homogeneousdimension , Geom. Funct. Anal. (2016), 680–702.[39] G. Mauceri and S. Meda, Vector-valued multipliers on stratified groups , Revista Mat. Iberoamer-icana (1990), 141–154.[40] D. M¨uller and E. M. Stein, On spectral multipliers for Heisenberg and related groups , J. Math.Pures Appl. (1994), 413–440.[41] D. M¨uller, F. Ricci and E. M. Stein, Marcinkiewicz multipliers and multi-parameter structureon Heisenberg (-type) groups. II , Math. Z. (1996), 267–291.[42] C. Muscalu, J. Pipher, T. Tao and C. Thiele, Bi-parameter paraproducts , Acta Math. (2004), 269–296.[43] G. Mauceri and S. Meda, Vector-valued multipliers on stratified groups , Rev. Mat. Iberoameri-cana (1990), 141–154.[44] E. Nakai and Y. Sawano, Hardy spaces with variable exponents and generalized Campanatospaces , J. Funct. Anal. (2012), 3665–3748.[45] R. Rubin, Multipliers on the rigid motions of the plane and their relations to multipliers ondirect products , Proc. Amer. Math. Soc. (1976), 89–98.[46] M. Ruzhansky and J. Wirth, On multipliers on compact Lie groups , Funct. Anal. Appl. (2013), 72–75.[47] M. Ruzhansky and J. Wirth, L p Fourier multipliers on compact Lie groups , Math. Z. (2015), 621–642.[48] R. Strichartz, Multiplier transformations on compact Lie groups and algebras , Trans. Amer.Math. Soc. (1974), 99–110.[49] M. Taibleson and G. Weiss, The molecular characterization of certain Hardy spaces . Repre-sentation theorems for Hardy spaces, pp. 67–149, Asterisque, 77, Soc. Math. France, Paris,1980.[50] L. Vretare, On L p Fourier multipliers on a compact Lie-group , Math. Scand. (1974), 49–55.[51] N. Weiss, L p estimates for bi-invariant operators on compact Lie groups , Amer. J. Math. (1972), 103–118.[52] N. Weiss, A multiplier theorem for SU ( n ), Proc. Amer. Math. Soc. (1976), 366–370. OURIER MULTIPLIERS FOR HARDY SPACES ON GRADED LIE GROUPS 35 Department of Mathematics, Jiangxi Normal University, Nanchang, Jiangxi 330022,P.R. China Email address : [email protected] Department of Mathematics, Jiangxi Normal University, Nanchang, Jiangxi 330022,P.R. China Email address : [email protected] Ghent University, Department of Mathematics, Krijgslaan 281, Building S8, B9000 Ghent, Belgium AND Queen Mary University of London, School of Mathemat-ical Sciences, Mile End Road, London E1 4NS, United Kingdom Email address ::