Riesz Means on Graphs and Discrete Groups

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8 Feb 2011 - Keywords Riesz means·Graphs·Discrete Laplacian·Discrete groups· ... the vertices x and y neighbors and we write x ∼ y when σxy = 0. A path ...
Potential Anal DOI 10.1007/s11118-011-9261-x

Riesz Means on Graphs and Discrete Groups Anestis Fotiadis · Athanasios G. Georgiadis

Received: 8 February 2011 / Accepted: 11 October 2011 © Springer Science+Business Media B.V. 2011

 Suppose that  is a weighted graph or a discrete group. Let mα,R (λ) = 1 − Abstract    λ  α be the Riesz means and let  be the discrete Laplacian on . We prove that if R + D is the homogeneous dimension of  then the operator mα,R () is bounded on L p , provided that α > D| 1p − 12 |. Keywords Riesz means · Graphs · Discrete Laplacian · Discrete groups · Markov kernels Mathematics Subject Classifications (2010) 42B15 · 47B99 · 43A15

1 Introduction and Statement of the Results The Riesz means have already been extensively studied on Rn , on Lie groups, on symmetric spaces and other contexts (see [2–4, 6–9, 12, 14, 15, 17, 19–21, 24–27] and the references therein). In [3] Alexopoulos and Lohoué study the Riesz means on Lie groups of polynomial volume growth and Riemannian manifolds of non negative curvature. For

A. Fotiadis (B) Department of Mathematics and Statistics, University Of Cyprus, P.O. Box 20537, 1678 Nicosia, Cyprus e-mail: [email protected] A. G. Georgiadis Department of Mathematics, Aristotle University of Thessaloniki, Thessaloniki 54.124, Greece e-mail: [email protected]

A. Fotiadis, A.G. Georgiadis

example, in the case of D–dimensional manifolds with non negative Ricci curvature, they prove in [3, Theorem 1–4] the following result. (i) If α >

D , 2

then mα,R () is bounded on L p , for p ∈ [1, ∞].

(ii) If 0 < α < D2 , then mα,R () is bounded on L p , for p ∈ (1, ∞), provided that α > D| 1p − 12 |. In this article we prove the above result in the setting of graphs and discrete groups of polynomial volume growth. In order to state our result we have to introduce some notation and present the geometric setting. 1. Graphs Let  be a countable infinite set, and let σxy = σ yx ≥ 0 be a symmetric weight on  satisfying σxx > 0, x ∈ . This weight induces a graph structure on . We call the vertices x and y neighbors and we write x ∼ y when σxy  = 0. A path of length n joining x, y is a sequence x = x1 , x2 , ..., xn = y such that xi ∼ xi+1 . The infimum of those lengths we set to be the distance d(x, y). We assume that  is connected, i.e. any two vertices are joined by a path. We consider the discrete measure  μ ({x}) = σxy , x ∈ . y∼x

For simplicity, we set μx = μ ({x}) , |A| = μ(A) and L p = L p (, μ). We shall assume (cf. [11, 13]) that there is a constant c > 0 such that σxy ≥ cμx , x ∈ . Let B(x, r) = {y ∈  : d(x, y) ≤ r}. We shall assume as in [23] that  satisfies the doubling volume property, i.e. there is c > 0 such that |B (x, 2r) | ≤ c|B (x, r) |, x ∈ , r > 0. This implies that there are c, D > 0 such that  r D |B (x, s) |, r > s > 0. |B (x, r) | ≤ c s We call D the homogeneous dimension of . If f is a function on , then we set  1 f (x)μx , x0 ∈ , r > 0. f B(x0 ,r) = |B(x0 , r)|

(1.1)

(1.2)

x∈B(x0 ,r)

We shall assume that  satisfies a Poincaré inequality, i.e. that there is c > 0 such that for every function f      f (x) − f B(x ,r) 2 μx ≤ cr2 | f (y) − f (x)|2 σ (x, y), x0 ∈ , r > 0. 0 x∈B(x0 ,r)

x,y∈B(x0 ,2r)

(1.3) We consider the symmetric nonnegative Markov kernel σxy , x, y ∈ , p (x, y) = μx μ y

Riesz Means on Graphs and Discrete Groups

 and the operator Pf (x) = y∈ p (x, y) f (y) μ y , x ∈ . Then, the operator  = I − P is called the discrete Laplacian on . It is symmetric and we have  f 2 ≤  f 2 + Pf 2 ≤ 2  f 2 . 2. Discrete groups Discrete groups can be considered as graphs. Because of their algebraic structure they present a special interest and for that we present them briefly below. Let  be a finitely generated discrete group. If e is the identity element of , then let U be a generating subset of  with e ∈ U. Assume that when x ∈ U then x−1 ∈ U. Let U 0 = {e}, U n = {x1 · · · xn : x1 , . . . , xn ∈ U} and |x| = min{n ∈ N : x ∈ U n }. We denote by |A| the number of elements of A ⊂ . We set d(x, y) = |xy−1 | and let B(x, r) = {y ∈  : d(x, y) ≤ r}. Note that B(x, r) = U [r] x, where [r] denotes the integer part of r. We shall assume that  has polynomial volume growth, i.e. there are b , c > 0 such that  n U  ≤ cnb , n ∈ N.

(1.4)

Then [1, p.417], there are c, D > 0 such that D |U n1 | n1 , n1 > n2 , n1 , n2 ∈ N. ≤c |U n2 | n2

(1.5)

We call D the homogeneous dimension of . Assume that p is a discrete probability measure on  that is symmetric, i.e. p(x) = p(x−1 ), x ∈  and that its support is finite and generates . We set p(x, y) = p(xy−1 ).  We consider the operator Pf (x) = y p(x, y) f (y). Then, the operator  = I − P is called the discrete Laplacian on . It is symmetric, positive, such that  f 2 ≤ 2 f 2 . Therefore, in both of the above cases, the discrete Laplacian admits a spectral 2 resolution  = 0 λdEλ , where dEλ is the spectral measure on . If m (λ) is a bounded Borel function then we can define, by using the spectral theorem, the operator

2

m () =

m (λ) dEλ .

(1.6)

0

This operator is bounded on L2 , with m () 2 ≤ m∞ . The function m is called a multiplier and the operator m() is called a spectral multiplier. For α, R > 0, the Riesz means of order α are the operators mα,R () defined by the multiplier   α λ mα,R (λ) = 1 −   , λ ∈ R. R + In this article we shall prove the following result.

(1.7)

A. Fotiadis, A.G. Georgiadis

Theorem 1 (i) If α > D2 , then mα,R () are bounded on L p , for p ∈ [1, ∞]. (ii) If 0 < α < D2 , then mα,R () are bounded on L p , for p ∈ (1, ∞), provided that α > D| 1p − 12 |. For the Laplace operator on R D the critical power in the above result is (D − 1)/2 rather than D/2. According to the Hörmander multiplier theorem [1, 18], the spectral multiplier m() is bounded on L p , p ∈ (1, ∞), provided that m belongs in the Lipschitz space Cα , for some α > D/2. Note that the Riesz means mα,R (λ) belong in Cα , but our result is not covered by the Hörmander multiplier theorem  since we also  treat the case p = 1, p = ∞ when α > D/2, as well as the case α ∈ D| 1p − 12 |, D2 . Theorem 1, as it is already mentioned, has been proved for Riemannian manifolds M with Ric(M) ≥ 0 and Lie groups of polynomial volume growth [3]. In Section 2 we decompose mα,R (λ) into compactly supported components and then we estimate the corresponding kernels. In Section 3 we apply these estimates in order to prove Theorem 1. Throughout this article the different constants will always be denoted by the same letter c. When their dependence or independence is significant, it will be clearly stated.

2 Compactly Supported Multipliers In this section we shall decompose the Riesz means into compactly supported components and we shall estimate the corresponding kernels. 2.1 An Approximation Lemma Consider A > 0 such that A = n + ε, with n ∈ N and ε ∈ (0, 1]. If f ∈ C0n then we set 

   f (n) (x + t) − f (n) (x) (2.1) ; t > 0, x ∈ R . M A ( f ) = sup tε As in [1, 18] we shall need the following lemma [22]. Lemma 1 Let n ∈ N, ε ∈ (0, 1], A = n + ε and assume that f ∈ C0n (R) is even. Then, there is c > 0 such that for all k ∈ N there is a polynomial Q with terms of only even order such that degQ ≤ 2k , || f − Q||∞ ≤

cM A ( f ) . kA

√ √ Then, √ Q( λ) is a polynomial in λkof degree k, consequently Q( ) can be written as Q( ) = a0 I + a1 P + · · · + ak P . It follows that if supp(g) ⊆ B(x, m) then  √   supp Q  g ⊆ B(x, m + k). (2.2)

Riesz Means on Graphs and Discrete Groups

The relation above has been first exploited by Carne [5], and it is the discrete analogue to the finite propagation speed property of the wave operator [28]. 2.2 The Riesz Means Decomposition Consider the multiplier   α λ mα,R (λ) = 1 −   , λ ∈ R, R +

√ and set r = R. Let i ∈ Z be such that 2(i−1)/2 < r ≤ 2i/2 , and set I = |i| ∈ N. Using a cut-off function, we can consider a decomposition mα,R () = m0α,r () + m∞ α,r (), as follows: If r ≤ 1 then take m0α,r to be identically 0. If r > 1 then consider that m0α,r is 0 when λ2 ∈ [ r+1 , r], m0α,r = mα,R when λ2 ∈ [0, r+3 ] and m0α,r ∈ C∞ . 2 4 ∞ 0 0 Therefore, in both cases it is mα,r ∈ C0 . Let Kα,r be the kernel of m0α,r . It follows 0 (·, y)1 < c, thus m0α,r () is bounded on L∞ . But by [18, Corollary 1] that sup y∈ Kα,r 2 it is also bounded on L , so by interpolation and duality we conclude that m0α,r () is bounded on L p , for all p ∈ [1, +∞]. So we only need to treat m∞ α,r (). We consider a smooth function ϕ such that      1 5 ϕ 2 j x = 1. and supp(ϕ) ⊆ − , − 4 4 j≥0   We set χ j,r (λ) = ϕ 2 j(| λr |2 − 1) , j ≥ 0 (see [3, p.213] for more details). We divide m∞ α,r into compactly supported terms m j,r (λ) = m∞ α,r (λ)χ j,r (λ), j ≥ 0. ∞ Let Kα,r (x, y) and K j,r (x, y) be the Schwartz kernels corresponding to m∞ α,r () and m j,r () respectively. Then   ∞ m∞ m j,r (λ) and Kα,r (x, y) = K j,r (x, y). (2.3) α,r (λ) = j≥0

j≥0

Case 1 r  = 1. In this case 1 is not in the support of m∞ α,r . We set −2 I  h j,r (λ) = m j,r (λ2 ) 1 − λ2 , j ≥ 0.

(2.4)

We observe that |supp(h j,r )| ≤ cr2− j. In the support of h j,r we have by induction that    2 (k)    λ  1−  ≤ cr−k 2−(α−k) j, k, j ∈ N.   r   Thus, by the product rule (cf. [3, p.214]) we find that    (k)  h j,r  ≤ ck r−k 2−(α−k) j , k, j ∈ N. ∞

(2.5)

A. Fotiadis, A.G. Georgiadis

By the definition of h j,r follows that m j,r () = h j,r

√  I √     P2 and K j,r (x, y) = h j,r  p2 I (., y) (x), j ≥ 0.

(2.6)

Case 2 r = 1. In this case we set h j,1 (λ) = m j,1 (λ2 ), j ≥ 0. Then, m j,1 () = h j,1

  √  √   and K j,1 (x, y) = h j,1  p0 (., y) (x), j ≥ 0.

(2.7)

We observe that Eq. 2.5 is still valid for r = 1. 2.3 Kernel Estimates For the proof of our result, as in [1, 18], the basic tool is the discrete analogue of the heat kernel we shall now present. Let p0 (x, y) = δx (y), where δx is the Dirac mass at x. We denote by pn (x, y) the n−th iterate of the kernel p (x, y) , defined by pn (x, y) =



p (x, z) pn−1 (z, y) μz , x, y ∈ , n ≥ 1.

z∈

Then, pn (x, y) is the discrete analogue of heat kernel. The doubling volume property Eq. 1.1 and the Poincaré inequality (1.3), in the case of graphs, and the polynomial volume growth Eq. 1.4, in the case of discrete groups, imply [1, 10, 13, 16] that there is c > 0 such that   c d (x, y)2 pn (x, y) ≤   √  exp − , x, y ∈ . (2.8)  B x, n  cn Integrating estimate Eq. 2.8 we obtain as in [18, p.1057] the following result. Lemma 2 If q ∈ N , q ≥ I, then there is c > 0 such that c (i)  p2 I (., y)2 ≤    , y ∈   B y, 2 I/2 

ce−c2  , y ∈ . ) ) ≤    B y, 2 I/2  q−I

(ii)  p2 I (., y) L2 ( B(

c y,2q/2

Let us set     Aq (y) = B y, 2(q+1)/2 \ B y, 2q/2 , y ∈ , q ∈ N.

Riesz Means on Graphs and Discrete Groups

Lemma 3 If j, q ∈ N, q ≥ I, A > 0, then there is a c > 0 such that   c2−α j (i)  K j,r (., y)2 ≤    , y ∈   B y, 2 I/2 

  c2−A(q−I)/2 2−(α−A) j   (ii)  K j,r (., y) L2 ( Aq (y)) ≤  , y ∈ .  B y, 2 I/2  Proof For simplicity, consider that r  = 1. If r = 1 then the proof is similar, but simpler, and we will omit it. (i) By Lemma 2, Eqs. 2.5 and 2.6, it follows that   √      K j,r (., y) ≤  I (., y)  p h j,r 2 2 L2   ≤ h j,r ∞  p2 I (., y)2   −1/2 ≤ c2−α j  B y, 2 I/2  . (ii) By Lemma 1, it follows that there is a polynomial Q with terms of only even order such that     degQ ≤ 2q/2 and h j,r − Q∞ ≤ cM A h j,r 2−A(q−2)/2 . Using Eq. 2.5 we find that     (1− )     M A (h j,r ) ≤ hn+1 ≤ cr−A 2−(α−A) j, j,r  · supp h j,r ∞

where A = n + , n ∈ N, ∈ (0, 1]. Hence   h j,r − Q



≤ c2−A(q−I)/2 2−(α−A) j.

(2.9)

√ We have from Eq. 2.2 that supp(Q( )1 B(y,2(q−2)/2 ) p2 I (., y)) ⊆ B(y, 2q/2 ). Taking also into account Lemma 2, Eqs. 2.5, 2.6 and 2.9, we find that    K j,r (., y)

L2 (Aq (y))

 √      ≤ h j,r  1 B(y,2(q−2)/2 ) p2 I (., y) 

L2 ( Aq (y))   √      1 B( y,2(q−2)/2 )c p2 I (., y)  2 + h j,r L ( Aq (y))  √      ≤ (h j,r − Q)  1 B( y,2(q−2)/2 ) p2 I (., y)  2 L ( Aq (y))   √      1 B( y,2(q−2)/2 )c p2 I (., y)  2 + h j,r L ( Aq (y))

A. Fotiadis, A.G. Georgiadis

        ≤ h j,r − Q∞ · p2 I (.,y)2 + h j,r ∞ · 1 B( y,2(q−2)/2 )c p2 I (.,y)

2

c2−α je−c2 c2−A(q−I)/2 2−(α−A) j    +     B y, 2 I/2   B y, 2 I/2  q−I



≤c

2−A(q−I)/2 2−(α−A) j    .  B y, 2 I/2 



3 Proof of Theorem 1 For simplicity, we shall again consider that r  = 1. If r = 1 then the proof is similar, but 0 simpler, and we will omit it. Recall that mα,R () = m0α,r () + m∞ α,r (), where mα,r () p ∞ is bounded on L for all p ∈ [1, ∞]. Therefore, it is enough to show that mα,r () is bounded on L p . (i) By the Cauchy–Schwarz inequality, Lemma 3 and Eq. 1.2, we find that    K j,r (., y)

L1 ( B( y,2(2 j+I)/2 ))

   1/2  ≤  B y, 2(2 j+I)/2   K j,r (., y) L2 ( B( y,2(2 j+I)/2 )) ≤ c2−(α−D/2) j,

and for q ≥ I we find that    K j,r (., y)

L1 ( Aq (y))

   1/2  ≤  B y, 2(q+1)/2   K j,r (., y) L2 ( Aq (y)) ≤ c2−(A−D/2)(q−I)/2 2−(α−A) j.

As a result, choosing A > D/2, we have    K j,r (., y)

≤ L1 ( B( y,2(2 j+I)/2 )) c

∞     K j,r (., y) q=2 j+I

L1 ( Aq (y))

≤ c2−(A−D/2) j2−(α−A) j = c2−(α−D/2) j. From the above we conclude that K j,r (., y)1 ≤ c2−(α−D/2) j, and from this estimate and the inequality α > D/2 follows that  ∞ Kα,r (., y)1 ≤ K j,r (., y)1 < c. j≥0 ∞ 2 Thus, m∞ α,r () is bounded on L . But it is also bounded on L , so by interpolap tion and duality we conclude that m∞ () is bounded on L , for all p ∈ [1, +∞]. α,r

Riesz Means on Graphs and Discrete Groups p (ii) It is enough to prove that m∞ α,r () is bounded on L for 1 < p < 2. Then, by p duality, it follows that it is also bounded on L for p > 2. Hence, consider 1 < p < 2 and proceed as in [3, p.217]. In particular, if 0 < t < 1 is such that

t 1−t 2 1 = + , i.e. t = − 1, p 1 2 p then, by interpolation, we have that using the estimate on K j,r (., y)1 and h j,r ∞   m j,r ()



p, p

  ≤ m j,r () ≤ c2



2 p −1

  − 2p +2

  m j,r ()

1 2     −(α−D/2) j 2p −1 −α j − 2p +2

2

= c2

   −α+D 1p − 12 j

.

From this estimate and the inequality α > D| 1p − 12 | follows that     ∞ m j,r () < c. m () ≤ α,r p, p p, p j≥0 p Hence, m∞ α,r () is bounded on L , p ∈ (1, ∞).

Acknowledgements The first author would like to thank Michel Marias for his constant help. The second author wants to express his gratitude to Onaseio Foundation of Greece for its support. Both authors would like to thank the referee for his remarks and suggestions.

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