A note on K -functional, Modulus of smoothness, Jackson theorem and Nikolskii-Stechkin inequality on Damek-Ricci spaces
aa r X i v : . [ m a t h . C A ] M a y A NOTE ON K -FUNCTIONAL, MODULUS OF SMOOTHNESS,JACKSON THEOREM AND NIKOLSKII-STECHKIN INEQUALITY ONDAMEK-RICCI SPACES VISHVESH KUMAR AND MICHAEL RUZHANSKY
Abstract.
In this paper we study approximation theorems for L -space on Damek-Ricci spaces. We prove direct Jackson theorem of approximations for the modulus ofsmoothness defined using spherical mean operator on Damek-Ricci spaces. We also proveNikolskii-Stechkin inequality. To prove these inequalities we use functions of boundedspectrum as a tool of approximation. Finally, as an application we prove equivalence ofthe K -functional and modulus of smoothness for Damek-Ricci spaces. Contents
1. Introduction 12. Essentials about harmonic
N A groups 33. Main results 73.1. Direct Jackson theorem 83.2. Nikolskii-Stechkin inequality 113.3. Equivalence of the K -functional and modulus of smoothness 13Acknowledgment 14References 141. Introduction
The main purpose of this paper is to study the equivalence of the K -functional and themodulus of smoothness generated by the spherical mean operator on Damek-Ricci spaces.Damek-Ricci spaces, also known as Harmonic NA groups, are solvable (non-unimodular) Mathematics Subject Classification.
Primary 22E30, 41A17 Secondary 41A10.
Key words and phrases.
K-functional, Damek-Ricci spaces, Harmonic
N A groups, Fourier transform,Spherical mean operator, Modulus of smoothness, Direct Jackson theorem, Jacobi transform.
Lie groups. It is worth mentioning that Damek-Ricci spaces contain non-compact sym-metric spaces of rank one as a very small subclass and, in general, Damek-Ricci spacesare not symmetric. Damek-Ricci spaces were introduced by Eva Damek and Fulvo Ricciin [14] and the geometry of these spaces was studied by Damek [13] and Cowling-Dooley-Koranyi [6]. Fourier analysis on these spaces has been developed and studied by manyauthors including Anker-Damek-Yacoub [1], Astengo-Comporesi-Di Blasio [2] , Damek-Ricci [15], Di Blasio [16], Ray-Sarkar [40], Kumar-Ray-Sarkar [22]. One of the interestingfeatures of these spaces is that the radial analysis on these spaces behaves similar to thehyperbolic spaces as observed in [1] and therefore it fits into the perfect setting of Jacobianalysis developed by Flensted-Jensen and Koornwinder [23, 19, 20].The study of the K -functional is a classical and important topic in interpolation the-ory and approximation theory. Peetre the K -functional is useful for describing the in-terpolation spaces between two Banach spaces. First, let us recall the definition of the K -functional. For two Banach spaces A and A , the Peetre the K -functional is given by K ( f, δ, A , A ) := inf {k f k A + δ k f k A : f = f + f , f ∈ A , f ∈ A } , where δ is a positive parameter. Now, the Peetre interpolation space ( A , A ) θ,r for 0 <θ < , < r ≤ ∞ , is defined by the norm | f | ( A ,A ) θ,r := (cid:0)R ∞ [ δ − θ K ( f, δ, A , A )] r dδδ (cid:1) r if 0 < r < ∞ , sup δ> δ − θ K ( f, δ, A , A ) if r = ∞ . The characterizations of the K -functional has several applications in approximationtheory [12]. In [27], Peetre started characterization of the K -functional by proving anequivalence of it with the modulus of smoothness for L p -spaces on R n which proved tobe very helpful to study apporximation theory. Later, in [10] the authors showed itsequivalence in terms of the rearrangement of derivatives for a pair of Sobolev spaces W mp and for the pair ( L p , W mp ) . In particular, a characterization of the K -functional for( L ( R ) , W m ( R )) can be found in the classical book of Berens and Buter [4]. The char-acterizations of the K -functional for the pair ( L ( X ) , W m ( X )) were explored by severalauthors for different choices of X. Classically, this equivalence was proved for X = R n byPeetre [27] and after that it was proved for X = [ a, b ] by De Vore-Scherer [10], for weightedsetting by Ditzian [11], for X = R n with Dunkl translation by Belkina and Platonov [3], -FUNCTIONAL AND MODULUS OF SMOOTHNESS ON DAMEK-RICCI SPACES 3 for rank one symmetric spaces [18], for Jacobi analysis in [17] and for compact symmetricspaces on [38]. In this paper our aim is to extend this characterization to more generalsetting of solvable (non unimodular) Lie groups. We consider the pair ( L ( X ) , W m ( X ))for X being the Damek-Ricci spaces. We will prove the equivalence of the K -functionaland modulus of smoothness generated by spherical mean operator on Damek-Ricci space.Modulus of smoothness for Damek-Ricci space has been introduced in [22]. We prove ourmain result by establishing two classical results, namely, Direct Jackson theorem [26] andNikolskii-Stechkin inequality [25] for Damek-Ricci spaces. Platonov studied Direct Jack-son theorem and Nikolskii-Stechkin inequality for compact homogeneous manifolds andfor noncompact symmetric spaces of rank one ([33, 35, 34, 38, 32, 39]).2. Essentials about harmonic
N A groups
For basics of harmonic
N A groups and Fourier analysis on them, one can refer toseminal research papers [13, 14, 15, 16, 1, 2, 6, 40, 22, 21]. However, we give necessarydefinitions, notation and terminology that we shall use in this paper.Let n be a two-step nilpotent Lie algebra, equipped with an inner product h , i . Denoteby z the center of n and by v the orthogonal complement of z in n with respect to theinner product of n . We assume that dimensions of v and z are m and l respectively as realvector spaces. The Lie algebra n is H -type algebra if for every Z ∈ z , the map J Z : v → v defined by h J Z X, Y i = h Z, [ X, Y ] i , X, Y ∈ v , Z ∈ z , satisfies the condition J Z = −k Z k I v , where I v is the identity operator on v . It followsthat for Z ∈ z with k Z k = 1 one has J Z = − I v ; that is, J Z induced a complex structure on v and hence m = dim( v ) is always even. A connected and simply connected Lie group N is called H -type if its Lie algebra is of H -type. The exponential map is a diffeomorphismas N is nilpotent, we can parametrize the element of N = exp n by ( X, Z ), for X ∈ v and Z ∈ z . The multiplication on N follows from the Campbell-Baker-Hausdorff formulagiven by ( X, Z )( Z ′ , Z ′ ) = ( X + X ′ , Z + Z ′ + 12 [ X, X ′ ]) . The group A = R ∗ + acts on N by nonisotropic dilations as follows: ( X, Y ) ( a X, aZ ) . Let S = N ⋉ A be the semidirect product of N with A under the aforementioned action. VISHVESH KUMAR AND MICHAEL RUZHANSKY
The group multiplication on S is defined by( X, Z, a )( X ′ , Z ′ , a ′ ) = ( X + a X ′ , Z + aZ ′ + 12 a [ X, X ′ ] , aa ′ ) . Then S is a solvable (connected and simply connected) Lie group with Lie algebra s = z ⊕ v ⊕ R and Lie bracket[( X, Z, ℓ ) , ( X ′ , Z ′ , ℓ ′ )] = ( 12 ℓX ′ − ℓ ′ X, ℓZ ′ − ℓ ′ Z + [ X, X ] ′ , . The group S is equipped with the left-invariant Riemannian metric induced by h ( X, Z, ℓ ) , ( X ′ , Z ′ , ℓ ′ ) i = h X, X ′ i + h Z, Z ′ i + ℓℓ ′ on s . The homogneous dimension of N is equal to m + l and will be denoted by Q. Attimes, we also use symbol ρ for Q . Hence dim( s ) = m + l + 1 , denoted by d. The associatedleft Haar measure on S is given by a − Q − dXdZda, where dX, dZ and da are the Lebesguemeasures on v , z and R ∗ + respectively. The element of A will be identified with a t = e t ,t ∈ R . The group S can be realized as the unit ball B ( s ) in s using the Cayley transform C : S → B ( s ) (see [1]).To define (Helgason) Fourier transform on S we need to introduce the notion of Poissonkernel [2]. The Poisson Kernel P : S × N → R is defined by P ( na t , n ′ ) = P a t ( n ′− n ) , where P a t ( n ) = P a t ( X, Z ) = Ca Qt (cid:18) a t + | X | (cid:19) + | Z | ! − Q , n = ( X, Z ) ∈ N. The value of C is suitably adjusted so that R N P a ( n ) dn = 1 and P ( n ) ≤ . The Poissonkernel satisfies several useful properties (see [22, 40, 2]), we list here a few of them. For λ ∈ C , the complex power of the Poisson kernel is defined as P λ ( x, n ) = P ( x, n ) − iλQ . It is known ([40, 2]) that for each fixed x ∈ S, P λ ( x, · ) ∈ L p ( N ) for 1 ≤ p ≤ ∞ if λ = iγ p ρ, where γ p = p − . A very special feature of P λ ( x, n ) is that it is constant on thehypersurfaces H n,a t = { nσ ( a t n ′ ) : n ′ ∈ N } . Here σ is the geodesic inversion on S, that isan involutive, measure-preserving, diffeomorphism which can be explicitly given by [6]: σ ( X, Z, a t ) = (cid:18) e t + | V | (cid:19) + | Z | ! − (cid:18)(cid:18) − (cid:18) e t + | X | (cid:19) + J Z (cid:19) X, − Z, a t (cid:19) . -FUNCTIONAL AND MODULUS OF SMOOTHNESS ON DAMEK-RICCI SPACES 5 Let ∆ S be the Laplace-Beltrami operator on S. Then for every fixed n ∈ N, P λ ( x, n )is an eigenfunction of ∆ S with eigenvalue − ( λ + Q ) (see [2]). For a measurable function f on S, the (Helgason) Fourier transform is defined as e f ( λ, n ) = Z S f ( x ) P λ ( x, n ) dx whenever the integral converge. For f ∈ C ∞ c ( S ) , the following inversion formula holds ([2,Theorem 4.4]): f ( x ) = C Z R Z N e f ( λ, n ) P − λ ( λ, n ) | c ( λ ) | − dλdn, where C = c m,l π . The authors also proved that the (Helgason) Fourier transform extendsto an isometry from L ( S ) onto the space L ( R + × N, C | c ( λ ) | − dλdn ) . In fact they havethe precise value of constants, we refer the reader to [2]. The following estimates for thefunction | c ( λ ) | holds: c | λ | d − ≤ | c ( λ ) | − ≤ (1 + | λ | ) d − for all λ ∈ R (e. g. see [40]). In [40,Theorem 4.6], the authors proved the following version of the Hausdorff-Young inequality:For 1 ≤ p ≤ (cid:18)Z R Z N | e f ( λ + iγ p ′ ρ, n ) | p ′ dn | c ( λ ) | − dλ (cid:19) p ′ ≤ C p k f k p . (1)A function f on S is called radial if for all x, y ∈ S, f ( x ) = f ( y ) if µ ( x, e ) = µ ( y, e ) , where µ is the metric induced by the canonical left invariant Riemannian structure on S and e is the identity element of S. Note that radial functions on S can be identified withthe functions f = f ( r ) of the geodesic distance r = µ ( x, e ) ∈ [0 , ∞ ) to the identity. It isclear that µ ( a t , e ) = | t | for t ∈ R . At times, for any radial function f we use the notation f ( a t ) = f ( t ) . For any function space F ( S ) on S , the subspace of radial functions will bedenoted by F ( S ) . The elementary spherical function φ λ ( x ) is defined by φ λ ( x ) := Z N P λ ( x, n ) P − λ ( x, n ) dn. It follows ([1, 2]) that φ λ is a radial eigenfunction of the Laplace-Beltrami operator ∆ S of S with eigenvalue − ( λ + Q ) such that φ λ ( x ) = φ − λ ( x ) , φ λ ( x ) = φ λ ( x − ) and φ λ ( e ) = 1 . It is also evident from the fact that, for every fixed n ∈ N, P λ ( x, n ) is an eigenfunctionof ∆ S with eigenvalue − ( λ + Q ), that, for suitable function f on S, we have g ∆ lS f ( λ, n ) = − ( λ + Q l e f ( λ, n ) VISHVESH KUMAR AND MICHAEL RUZHANSKY for every natural number l (see [2, p. 416]). In [1], the authors showed that the radial part(in geodesic polar coordinates) of the Laplace-Beltrami operator ∆ S given byrad ∆ S = ∂ ∂t + { m + l t k t } ∂∂t , is (by substituting r = t ) equal to L α,β with indices α = m + l +12 and β = l − , where L α,β is the Jacobi operator studied by Koornwinder [23] in detail. It is worth noting that we arein the ideal situation of Jacobi analysis with α > β > − . In fact, the Jacobi functions φ α,βλ and elementary spherical functions φ λ are related as ([1]): φ λ ( t ) = φ α,β λ ( t ) . As consequenceof this relation, the following estimates for the elementary spherical functions hold true(see [36]).
Lemma 2.1.
The following inequalities are valid for spherical functions φ λ ( t ) ( t, λ ∈ R + ) : • | φ λ ( t ) | ≤ . • | − φ λ ( t ) | ≤ t ( λ + Q ) . • There exists a constant c > , depending only on λ, such that | − φ λ ( t ) | ≥ c for λt ≥ . Let σ t be the normalized surface measure of the geodesic sphere of radius t . Then σ t is a nonnegative radial measure. The spherical mean operator M t on a suitable functionspace on S is defined by M t f := f ∗ σ t . It can be noted that M t f ( x ) = R ( f x )( t ), where f x denotes the right translation of function f by x and R is the radialization operatordefined, for suitable function f, by R f ( x ) = Z S ν f ( y ) dσ ν ( y ) , where ν = r ( x ) = µ ( C ( x ) , , here C is the Cayley transform, and dσ ν is the normalizedsurface measure induced by the left invariant Riemannian metric on the geodesic sphere S ν = { y ∈ S : µ ( y, e ) = ν } . It is easy to see that R f is a radial function and forany radial function f, R f = f. Consequently, for a radial function f, M t f is the usualtranslation of f by t. In [22], the authors proved that, for a suitable function f on S, g M t f ( λ, n ) = e f ( λ, n ) φ λ ( t ) whenever both make sense. Also, M t f converges to f as t → , i.e., µ ( a t , e ) → . It is also known that M t is a bounded operator on L ( S ) with operator -FUNCTIONAL AND MODULUS OF SMOOTHNESS ON DAMEK-RICCI SPACES 7 norm equal to φ ( a t ) . In particular, for f ∈ L ( S ) , we have k M t f k ≤ φ ( a t ) k f k . Thefollowing Lemmata are taken from [5].
Lemma 2.2.
Let α > − . Then there are positive constant c ,α and c ,α such that c ,α min { , ( λt ) } ≤ − j α ( λt ) ≤ c ,α min { , ( λt ) } , where j α is the usual Bessel function of first kind normalized by j α (0) = 1 . Lemma 2.3.
Let α > − and t > . Then, for all λ ∈ R , there exist a constant c > such that for all ≤ t ≤ t , the function φ λ satisfies | − φ λ ( t ) | ≥ c | − j α ( λt ) | , where j α is the usual Bessel function of first kind normalized by j α (0) = 1 . Main results
In this section we present our main results. Throughout this section, we denote a Damek-Ricci space by S . We denote by L ( S ) the Hilbert space of all square integrable functionon S with respect to Haar measure λ on S. We begin this section by recalling the definitionof Sobolev spaces on Damek-Ricci spaces.The Sobolev space W m ( S ) on Damek-Ricci space S is defined by W m ( S ) := { f ∈ L ( S ) : ∆ lS f ∈ L ( S ) , l = 1 , , . . . , m } . The space W m ( S ) can be equipped with seminorm | f | W m ( S ) := k ∆ mS f k and with thenorm k f k W m ( S ) = k f k + k ∆ mS f k . The modulus of smoothness (continuity) Ω k is defined by using the spherical meanoperator M t as follows: Ω k ( f, δ ) := sup This subsection is devoted for proving the Direct Jack-son theorem of approximations theory for Damek-Ricci spaces. For the approximation wewill use the functions of bounded spectrum. The functions of bounded spectrum were usedby Platonov [39, 32, 33, 34] to prove Jackson type direct theorem for Jacobi transformand for symmetric spaces. Such kind of functions also appear in the work of Pesenson[31] under the name of Paley-Wiener functions for studying approximation theory onhomogeneous manifolds.A function f ∈ L ( S ) is called a function with bounded spectrum (or a Paley-Wienerfunction ) of order ν > F f ( λ, n ) = 0 for | λ | > ν. Denote the space of all function on S with bounded spectrum of order ν by BS ν ( S ) . Thebest approximation of a function f ∈ L ( S ) by the functions in BS ν ( S ) is defined by E ν ( f ) := inf g ∈ BV ν ( S ) k f − g k L ( S ) . Lemma 3.1. Let ν > . For any function f ∈ L ( S ) , the function P ν ( f ) defined by P ν ( f )( x ) := F − ( F f ( λ, n ) χ ν ( λ )) , where χ ν is a function defined by χ ν ( λ ) = 1 for | λ | ≤ ν and otherwise, satisfies thefollowing properties: -FUNCTIONAL AND MODULUS OF SMOOTHNESS ON DAMEK-RICCI SPACES 9 (i) For every f ∈ L ( S ) , P ν ( f ) ∈ BS ν ( S ) . (ii) For every function f ∈ BS ν ( S ) , P ν ( f ) = f. (iii) If f ∈ L ( S ) then k P ν ( f ) k L ( S ) ≤ k f k L ( S ) and k f − P ν ( f ) k L ( S ) ≤ E ν ( f ) . Proof. (i) This is trivial to see. Indeed, by definition we have F P ν ( f )( x ) = F f ( λ, n ) χ ν ( λ ) = 0for | λ | > ν. Therefore, P ν ( f ) ∈ BS ν ( S ) . (ii) Let f ∈ BS ν ( S ) . Then F f ( λ, n ) = 0 for | λ | > ν and F P ν ( f )( λ, n ) = F f ( λ, n ) for | λ | ≤ ν. So, by using the inversion formula we have P ν ( f )( x ) = C Z R Z N F P ν ( f )( λ, n ) | c ( λ ) | − dλ dn = C Z | λ |≤ ν Z N F f ( λ, n ) | c ( λ ) | − dλ dn = C Z R Z N F f ( λ, n ) | c ( λ ) | − dλ dn = f ( x ) . (iii) Take f ∈ L ( S ) . By Plancherel formula, we get k P ν ( f ) k L ( S ) = Z ∞ Z N |F P ν ( f )( λ, n ) | | c ( λ ) | − dλ dn = Z ν Z N |F f ( λ, n ) | | c ( λ ) | − dλ dn ≤ Z ∞ Z N |F f ( λ, n ) | | c ( λ ) | − dλ dn = k f k L ( S ) . Also, for proving second inequality take any g ∈ BS ν ( S ) such that k f − g k ≤ E ν ( f ) . Now, by using the fact that P ν ( g ) = g we get k f − P ν ( f ) k L ( S ) = k f − g − P ν ( g − f ) k L ( S ) ≤ k f − g k L ( S ) + k f − g k L ( S ) ≤ E ν ( f ) . (cid:3) The following two theorems are analogues of Jackson’s direct theorem in classical ap-proximation theorem for Damek-Ricci spaces. Theorem 3.2. If f ∈ L ( S ) then for every ν > we have E ν ( f ) ≤ c k Ω k (cid:18) f, ν (cid:19) , k ∈ N , (2) where c k is a constant.Proof. The Plancherel formula gives that k f − P ν ( f ) k L ( S ) = Z ∞ Z N |F ( f − P ν ( f ))( λ, n ) | | c ( λ ) | − dλ dn = Z ∞ Z N | − χ ν ( λ ) | |F ( f ( λ, n ) | | c ( λ ) | − dλ dn = Z λ ≥ ν Z N |F ( f ( λ, n ) | | c ( λ ) | − dλ dn. By Lemma 2.1 we have | − φ λ (cid:0) ν (cid:1) | ≥ c for λ ≥ ν. Therefore, by Plancherel formula, weget k f − P ν ( f ) k L ( S ) ≤ c − k Z λ ≥ ν Z N | − φ λ (1 /ν ) | k |F f ( λ, n ) | | c ( λ ) | − dλ dn = c − k Z λ ≥ ν Z N |F (( I − M /ν ) k f )( λ, n ) | | c ( λ ) | − dλ dn ≤ c − k Z ∞ Z N |F (( I − M /ν ) k f )( λ, n ) | | c ( λ ) | − dλ dn = c − k k ( I − M /ν ) k f k L ( S ) . Therefore, as P ν ( f ) ∈ BS ν ( S ) , we get E ν ( f ) = inf g ∈ BV ν ( S ) k f − g k L ( S ) ≤ k f − P ν ( f ) k L ( S ) ≤ c − k k ( I − M /ν ) k f k L ( S ) = c − k k ∆ k /ν f k L ( S ) ≤ c k Ω k (cid:18) f, ν (cid:19) , proving (2) and hence the theorem is proved. (cid:3) Theorem 3.3. Let r ∈ N and ν > . Assume that f, ∆ S f, ∆ f, . . . , ∆ r f are in L ( S ) . Then E ν ( f ) ≤ c ′ k ν − r Ω k (cid:18) ∆ rS f, ν (cid:19) , k ∈ N , (3) where c ′ k is a constant. -FUNCTIONAL AND MODULUS OF SMOOTHNESS ON DAMEK-RICCI SPACES 11 Proof. Let r ∈ N and t > . Suppose that f, ∆ S f, ∆ S f, . . . , ∆ rS f are in L ( S ) . ThenLemma 2.1 and Plancherel formula give that k ( I − M t ) f k L ( S ) = Z ∞ Z N |F (( I − M t ) f )( λ, n ) | | c ( λ ) | − dλ dn = Z ∞ Z N | − φ λ ( a t ) | |F f ( λ, n ) | | c ( λ ) | − dλ dn ≤ t Z ∞ Z N ( λ + Q |F f ( λ, n ) | | c ( λ ) | − dλ dn = t Z ∞ Z N |F (∆ S f )( λ, n ) | | c ( λ ) | − dλ dn = t k ∆ S f k L ( S ) . Therefore, k ( I − M t ) f k L ( S ) ≤ t k ∆ S f k L ( S ) . (4)By proceeding similar to the proof of Theorem 3.2 we get k f − P ν ( f ) k L ( S ) ≤ c − ( k + r ) k ( I − M /ν ) k + r f k L ( S ) . (5)By applying inequality (4) on the right hand side of (5) r -times we obtain that k f − P ν ( f ) k L ( S ) ≤ c − ( k + r ) − r ν − r k ( I − M /ν ) k ∆ rS f k L ( S ) = c ′ k ν − r Ω k (cid:18) ∆ rS f, ν (cid:19) , where c ′ k = c − ( k + r ) − r . Now, the theorem follows from the definition of E ν ( f ) by notingthat E ν ( f ) = inf g ∈ BV ν ( S ) k f − g k L ( S ) ≤ k f − P ν ( f ) k L ( S ) ≤ c ′ k ν − r Ω k (cid:18) ∆ rS f, ν (cid:19) , completing the proof. (cid:3) Nikolskii-Stechkin inequality. In this subsection, we will prove Nikolskii-Stechkininequality [25] for Damek-Ricci spaces. Theorem 3.4. For any f ∈ L ( S ) and ν > we have k ∆ kS ( P ν ( f )) k L ( S ) ≤ c ν k k ∆ k /ν f k L ( S ) , k ∈ N . (6) Proof. First note that F (∆ kS P ν ( f ))( λ, n ) = ( − k (cid:18) λ + Q (cid:19) k F ( P ν ( f ))( λ, n ) . Using Plancherel formula we have k ∆ kS ( P ν ( f )) k L ( S ) = Z ∞ Z N |F (∆ kS P ν ( f ))( λ, n ) | | c ( λ ) | − dλ dn = Z | λ |≤ ν Z N (cid:0) λ + Q / (cid:1) k |F f ( λ, n ) | | c ( λ ) | − dλ dn = Z ∞ Z N ( λ + Q / k χ ν ( λ ) | − φ λ (1 /ν ) | k | − φ λ (1 /ν ) | k |F f ( λ, n ) | | c ( λ ) | − dλ dn. Now note that by Lemma 2.3 we havesup λ ∈ R ( λ + Q / k χ ν ( λ ) | − φ λ (1 /ν ) | k = ν k sup | λ |≤ ν (( λ + Q / /ν ) k | − φ λ (1 /ν ) | k ≤ ν k c sup | λ |≤ ν (( λ + Q / /ν ) k | − j α ( λ/ν ) | k = ν k c sup | t |≤ ( t + Q / ν ) k | − j α ( t ) | k = C ′ c ν k , where C ′ = sup | t |≤ ( t + Q / ν ) k | − j α ( t ) | k . Therefore, we get k ∆ kS ( P ν ( f )) k L ( S ) ≤ C ′ c ν k Z ∞ Z N | − φ λ (1 /ν ) | k |F f ( λ, n ) | | c ( λ ) | − dλ dn = C ′ c ν k Z ∞ Z N |F (∆ k /ν f )( λ, n ) | | c ( λ ) | − dλ dn = C ′ c ν k k ∆ k /ν f k L ( S ) . Hence, k ∆ kS ( P ν ( f )) k L ( S ) ≤ c ν k k ∆ k /ν f k L ( S ) . (cid:3) As noted in Lemma 3.1 that P ν ( f ) = f for any f ∈ BS ν ( S ) , the following corollary isimmediate. Corollary 3.5. For ν > , k ∈ N and f ∈ BV ν ( S ) we have the following inequality: k ∆ kS f k L ( S ) ≤ c ν k k ∆ k /ν f k L ( S ) . The following corollary follows from the definition of modulus of smoothness. -FUNCTIONAL AND MODULUS OF SMOOTHNESS ON DAMEK-RICCI SPACES 13 Corollary 3.6. For ν > , k ∈ N and f ∈ L ( S ) we have the following inequality: k ∆ kS f k L ( S ) ≤ c ν k Ω k (cid:18) f, ν (cid:19) . Equivalence of the K -functional and modulus of smoothness. Our mainobjective will be proved here. We will prove in the following theorem that the K -functionalfor the pair ( L ( S ) , W m ( S )) and modulus of smoothness generated by spherical meanoperators are equivalent. The Peetre the K -functional K ( f, δ, L ( S ) , W m ( S )) for the pair( L ( S ) , W m ( S )) is defined by K m ( f, δ ) := inf {k f − g k L ( S ) + δ k ∆ mS g k L ( S ) : f ∈ L ( S ) g ∈ W m ( S ) } . The next theorem presents the equivalence of the K -functional K m ( f, δ m ) and themodulus of smoothness Ω m ( f, δ ) for f ∈ L ( S ) and δ > . Theorem 3.7. For f ∈ L ( S ) and δ > we have Ω m ( f, δ ) ≍ K m ( f, δ m ) . (7) In other words, there exist c > , c > such that for all f ∈ L ( S ) and δ > we have c Ω m ( f, δ ) ≤ K m ( f, δ m ) ≤ c Ω m ( f, δ ) . Proof. Take g ∈ W m ( S ) . Now by using the properties of modulus of continuity Ω m ( f, δ ) we get Ω m ( f, δ ) ≤ Ω m ( f − g, δ ) + Ω m ( g, δ ) ≤ ( φ ( a t ) + 1) m k f − g k L ( S ) + δ m k ∆ mS g k L ( S ) ≤ ˜ c ( k f − g k L ( S ) + δ m k ∆ mS g k L ( S ) ) , where ˜ c = ( φ ( a t ) + 1) m . By taking the infimum over all g ∈ W m ( S ) , we obtainΩ m ( f, δ ) . K m ( f, δ m ) . Now, to prove the other side we take g = P ν ( f ) for ν > , then, from the definition of K m ( f, δ m ) , it follows that K m ( f, δ m ) ≤ k f − P ν ( f ) k L ( S ) + δ m k ∆ mS ( P ν ( f )) k L ( S ) . (8) Now, from Lemma 3.1 (iii), (2) and Corollary 3.6 we get that K m ( f, δ m ) ≤ E v ( f ) + c δ m ν m Ω m (cid:18) f, ν (cid:19) ≤ c Ω m (cid:18) f, ν (cid:19) + c ( δν ) m Ω m (cid:18) f, ν (cid:19) ≤ c (1 + ( δν ) m )Ω m (cid:18) f, ν (cid:19) . By taking ν = δ we get K m ( f, δ m ) . Ω m ( f, δ ) proving (7). (cid:3) Acknowledgment VK and MR are supported by FWO Odysseus 1 grant G.0H94.18N: Analysis and PartialDifferential Equations. MR is also supported by the Leverhulme Grant RPG-2017-151 andby EPSRC Grant EP/R003025/1. References 1. Anker, J-P., Damek, E., Yacoub, C. Spherical analysis on harmonic AN groups. Ann. Scuola Norm.Sup. Pisa Cl. Sci. (4) 23 (1996), no. 4, 643-679.2. Astengo, F., Camporesi, R., Di Blasio, B. 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J. 42 (2001), no. 1, 119-13036. Platonov, S. S. The Fourier transform of functions satisfying a Lipschitz condition on symmetricspaces of rank 1. Sibirsk. Mat. Zh. Sibirsk. Mat. Zh. 46 (2005), no. 6, 1374-1387; translation in SiberianMath. J. 46 (2005), no. 6, 1108-1118.38. Platonov, S. S. On some problems in the theory of the approximation of functions on compacthomogeneous manifolds. (Russian) Mat. Sb. Sb. Math. L metric: Jacksons type direct theorems. Integral Transforms Spec. Funct. Trans. Amer. Math.Soc. 361 (2009), no. 8, 4269-4297. Vishvesh KumarDepartment of Mathematics: Analysis, Logic and Discrete MathematicsGhent University, Belgium E-mail address : [email protected] Michael RuzhanskyDepartment of Mathematics: Analysis, Logic and Discrete MathematicsGhent University, BelgiumandSchool of Mathematics -FUNCTIONAL AND MODULUS OF SMOOTHNESS ON DAMEK-RICCI SPACES 17 Queen Mary University of LondonUnited Kingdom