Atomic Decomposition of H p Spaces Associated with Some Schrödinger Operators

Transcription

1 age 75) Atomic Decomposition of H p Spaces Associated with Some Schrödinger Operators Jacek Dziubański Abstract. Let {T t } t>0 be the semigroup generated by a Schrödinger operator A = V,whereV is a nonnegative polynomial on R d.wesaythatfis in H p A associated with the operator A if the maximal function Mfx) =sup t>0 T t fx) belongs to L p R d ). We characterize elements of the space H p A for 0 <p 1 by a special atomic decomposition. 1. Introduction. Let A be a Schrödinger operator on R d which has the form 1.1) A = +V, where V x) = β α a βx β is a nonnegative nonzero polynomial on R d, α = α 1,α 2,...,α d ), N d, N = {0,1,2,...}. These operators were studied by a number of authors, cf. [Fe], [HN], [Z]. We say that f is in the space H p A if the maximal function 1.2) Mfx) =sup T t fx) t>0 is in L p R d ), where {T t } t>0 is the semigroup generated by A. The quasi-norm in H p A is defined by 1.3) f p H p A = Mf p L p R d ). Let 0 λde Aλ) be the spectral resolution of A. For a function ϕ from the 75 Indiana University Mathematics Journal c, Vol. 47, No )

2 76 J. Dziubański Schwartz class S[0, )), ϕ0) 0, we define the maximal operator M ϕ setting 1.4) M ϕ fx) =sup ϕta)fx), t>0 where ϕta)f = 0 ϕtλ)de Aλ)f. We shall show that the quasi-norms Mf p L and M ϕf p p Lp are equivalent. Our aim is to present an atomic characterization of the spaces H p A for p 0,1]. Operators of the form 1.1) appear as images of certain homogeneous operators on homogeneous nilpotent Lie groups which, in fact, are differential operators if V is a sum of squares of polynomials, cf. e.g. [DHJ], [HJ], and references there. These methods are also used here. For p = 1 similar results have been obtained in [DZ1]. We define an auxiliary function mx,v ), see [Z], by 1.5) mx,v )= β α D β Vx) 1/ β +2), where β = β 1,β 2,...,β d ) = β 1 +β 2 + +β d. Since V is a nonzero polynomial, there is a constant c > 0 such that c mx,v ) for every x R d. We set 1.6) B 0 = {x R d : c mx,v ) < 1}, B n = {x R d :2 n 1)/2 mx,v ) < 2 n/2 } for n =1,2,3.... We have R d = n=0 B n. We will denote by Bx,r) the ball in R d with the center at x and radius r. We say that a function a is an atom for the space H p A associated to a ball Bx 0,r)if 1.7) supp a Bx 0,r), 1.8) a L volbx 0,r)) 1/p, 1.9) if x 0 B n, then r 2 1 n/2, 1.10) if x 0 B n and r 2 1 n/2, then x β ax)dx =0 for all ) 1 β d p 1.

3 H p Spaces Associated with Some Schrödinger Operators 77 The atomic quasi-norm in the space H p A is defined by { 1.11) f p H p atom = inf c j p}, A where the infimum is taken over all decompositions f = j c ja j, with a j being H p A atoms and c j being scalars. Our main result in this paper is as follows: Theorem Let ϕ S[0, )), ϕ0) 0. Then for every p 0,1], there is a constant C>0such that 1.13) M ϕ f L p C Mf L p, 1.14) Mf L p C f H p, A 1.15) f H p C M A ϕ f L p. Our spaces H p A are of a different nature than the classical Hardy spaces H p R d ), which may be thought of as the spaces H p Rd ), being the ordinary Laplacian on R d. It follows from Theorem 1.12 and properties of the sets B n, see Section 2, that H p R d ) is a proper subspace of the space H p A. Every element of H p A can be decomposed into atoms that are supported on small balls, but for certain atoms no moment condition is required. Actually our H p A atoms are scaled local atoms in the sense of Goldberg, cf. Section 8. The idea of the proof of Theorem 1.12 is based on the fact that the kernels of the operators T t look like the classical heat kernels multiplied by 1 + mx,v )) b, which have fast enough decay when t is large cf )). The kernels of the operators ϕta) have a similar feature. If A is the Hermite operator, this type of decay of the kernels T t can be read from Mehler s formula. In order to get appropriate estimates for the kernels ϕta), where A is a Schrödinger operator with a positive polynomial potential, we use the fact that A can be obtained as Π P,wherePis a regular kernel on a special nilpotent Lie group G and Π is a unitary representation of G, cf. Section 3. The theory of regular kernels on nilpotent Lie groups, which was developed by P. G lowacki, turns out to be crucial here, cf. Section 3. Acknowledgments. The author is greatly indebted to Andrzej Hulanicki for his remarks. He also wishes to express his gratitude to the referee for his several helpful comments. 2. Resolution of identity associated with B n. The following two lemmas below present important properties of the sets B n. j

4 78 J. Dziubański Lemma 2.1. n, ifx B n, then There is a constant C such that for every R>2and every {n : Bx,2 n/2 R) B n Ø} [n Clog 2 R, n+c log 2 R]. Proof. Assume that y Bx,2 n/2 R) B n, where x B n. By the definition of B n there is β α such that 1/C)2 n 1) β +2)/2 D β Vy). Applying the Taylor formula and the fact that x B n, we obtain 1 1) β +2)/2 C 2n γ α 1 γ! Dγ+β V x)y x) γ C2 β +2)n/2 R α, which implies n n + C log 2 R. In the same manner we can see that n n + C log 2 R. Lemma 2.2. There is a constant C and a collection of balls B n,k) = Bx n,k),2 1 n/2 ), n = 0,1,2,..., k = 1,2,..., such that x n,k) B n, B n k Bx n,k),2 n/2 ), and #{n,k ):Bx n,k),r2 n/2 ) Bx n,k ),R2 n /2 ) Ø} R C for every n,k) and R 2. Proof. For fixed n let Bx n,k),2 2 n/2 ) be a countable sequence of balls such that x n,k) B n,bx n,k),2 2 n/2 ) Bx n,k ),2 2 n/2 )=Øfor k k, and B n k Bx n,k),2 n/2 ). It is now easy to check, using Lemma 2.1, that the family of balls Bx n,k),2 1 n/2 ), n =0,1,..., k =1,2,... satisfies the conclusion of Lemma 2.2. As a consequence of Lemma 2.2, we obtain the following result: Lemma 2.3. There are nonnegative functions ψ n,k) such that 2.4) ψ n,k) Cc Bx n,k),2 1 n/2 )), 2.5) ψ n,k) x) =1, 2.6) n,k) β x β ψ n,k) C β 2 β n/2. L 3. Functional calculus of operators on homogeneous groups and Schrödinger operators. Let G be a homogeneous nilpotent group equipped with a family of dilations δ t cf. [FS]). A distribution P on G is called a regular

5 H p Spaces Associated with Some Schrödinger Operators 79 kernel of order r>0, if P coincides with a smooth function away from the origin and 3.1) P,f δ t = t r P,f. It was proved in[d2]that for every Schrödinger operator of the form 1.1) there exist a homogeneous nilpotent Lie group G, a unitary representation Π of G, and a regular symmetric kernel P of order 2 such that 3.2) Π P = A. We shall denote by the same letter P the convolution operator f f P. The kernel P satisfies the following maximal subelliptic estimates proved by P. G lowacki cf. [G], [D2]); for every left-invariant homogeneous differential operator on G and every N /2 there is a constant C>0 such that 3.3) f L2 G) C 1 + P ) N f L2 G), for f C c G), where is the degree of homogeneity of. Topologically G can be written as G = X Y = R d R D, where D = α 1 +1)α 2 +1)...α d + 1). The construction of the group G and the representation Π is such that for every multi-index β N d there is a left-invariant differential operator on G such that 3.4) β x β =Π. Let {S t } t>0 be the semigroup of linear operators on L 2 G) generated by the essentially self-adjoint operator P. The estimates 3.3) and the homogeneity of P imply that the semigroup {S t } t>0 is of the form S t f = f q t, q t g)=t Q/2 q 1 δ t 1/2g), where q t C G) L 2 G), Q denotes the homogeneous dimension of G. Moreover, the following estimates for q t hold cf. [G], [D2]): for every left-invariant homogeneous differential operator on G there is a constant C>0 such that 3.5) q t g) Ct + g ) Q 2, where g is a fixed homogeneous norm on G and is the degree of homogeneity of.

6 80 J. Dziubański For a positive integer k let W k G) denote the set of all functions F C k G) such that sup Fg)1 + g ) k < g G for every left-invariant differential operator on G of degree k. Similarly, we define W k R d R D ) to be the set of all functions F = F x,ξ) C k R d R D ) such that sup x,ξ) R d R D β β x β ξ β Fx,ξ)1 + x + ξ ) k < for all β N d, β N D such that β + β k. Obviously, cf. [FS], for every k > 0 there exists k such that W k G) W k R d R D )andw k R d R D ) W k G). For an integer m > 0 let us denote by S m 0 [0, )) the subspace of all functions ϕ from S[0, )) such that d j dλ j ϕ0+ )=0 forj=1,2,3,...m. We set S0 [0, )) = m=1 Sm 0 [0, )). Let 0 λde Pλ) be the spectral resolution of the positive definite operator P. The theorem below was actually proved in [D1]. Theorem 3.6 cf. [D1]). For every k>0there is an integer m>0such that for ϕ S0 m [0, )) there is a function F [ϕ] W k G) such that ϕp )f = f F [ϕ], where ϕp )f = 0 ϕλ)de P λ)f. Moreover, for every left-invariant differential operator on G of degree k there is a constant C such that for ϕ S m 0 [0, )) we have F [ϕ] g) 1 + g ) k C m sup {1 + λ) m d j } dλ j ϕλ). j=0 λ 0 Corollary 3.7. If ϕ S0 [0, )), then there is a Schwartz class function F [ϕ] on G such that ϕp )f = f F [ϕ]. Moreover, ϕtp )f = f F [ϕ] t, where F [ϕ] t g) =t Q/2 F [ϕ] δ t 1/2g).

7 H p Spaces Associated with Some Schrödinger Operators 81 We shall need the following result: Proposition 3.8. Assume that ψ S[0, )), 0 / suppψ. LetF [ψ] be the convolution kernel of ψp ). Then 3.9) G x β F [ψ] x)dx =0 for every multi-index β N d+d. Proof. For a multi-index β let N be a positive integer such that N > β +Q,where β is the homogeneous degree of the monomial x β on G. Then by the spectral theorem 3.10) F [ψ] x) =P N F [ω] x), where ωλ) =λ N ψλ). Observe that the operator P N is a convolution with a regular kernel of order 2N. Therefore, x β P N is well defined, and F [ψ] x)x β dx = F [ω] P N )x)x β dx = F [ω] x)x β P N )dx. G G Note that x β P N is a C function on G homogeneous of degree β 2N<0. Thus x β P N 0, which implies 3.9). The properties of the kernels F [ϕ] t of the operators ϕtp ) on the group G allow us to obtain suitable estimates for the integral kernels Φ t x,w) ofthe operators ϕta) onr d. One can deduce from the equality Π P = A that 3.11) Π ϕtp ) = ϕta) for ϕ S[0, )). The construction of G, Π,andPis such that if F L 1 G), then the integral kernel F x,w) of the of the operator Π F on R d is given by G 3.12) F x,w) = Fw x,v x),...,d β Vx),...), F x,ξ) =F Y F)x,ξ), where F Y F is the partial Fourier transform of F with respect to Y variables cf. [D2], [DHJ]). Consequently, if F t g) =t Q/2 Fδ t 1/2g), then the integral kernel F t x,w) of the operator Π Ft is of the form ) w x 3.13) F t x,w) =t d/2 F,tVx),...,t β +2)/2 D β V x),....

8 82 J. Dziubański For a function ϕ S0 m [0, )) we denote by F [ϕ] t the convolution kernel of the operator ϕtp )ong, and by Φ t x,w) the integral kernel of the operator ϕta) = Π [ϕ] F. By 3.12), 3.13), Theorem 3.6, Corollary 3.7, and the t homogeneity of P, we have the following: Corollary For every integer k>0there exists m>0such that if ϕ S m 0 [0, )), then the kernels Φ t and F [ϕ] = F [ϕ] 1 are related by ) w x 3.15) Φ t x,w) =t d/2 Φ,tVx),...,t β +2)/2 D β V x),..., where 3.16) Φx,ξ) =F Y F [ϕ] )x,ξ) W k R d R D ). If, moreover, ϕ S 0 [0, )), then Φx,ξ) SR d R D ). Here and subsequently, SR d R D ) denotes the Schwartz class of functions on R d R D. LetusdenotebyT t x,w) the integral kernels of the semigroup T t =Π qt. Proposition For every b>0there is a constant C b such that 3.18) 0 T t x,w) C b t d/2 1 + t 1/2 x w ) b β α1 + t β +2)/2 D β V x) ) b. For every multiindex β N d there is a constant C>0such that 3.19) β x β T tx,w) Ct d+ β )/2 1 + t 1/2 x w ) d 2 β. Proof. The estimate 3.18) was proved in [D2, Proposition 3.17]. To prove 3.19) we use 3.5), 3.12), and 3.4). It was pointed out by the referee that the estimate 3.18) can be written as 3.18 ) 0 T t x,w) C b t d/2 1 + t 1/2 x w ) b 1 + mx,v )) b.

9 H p Spaces Associated with Some Schrödinger Operators 83 Assume that ψ S[0, )) be such that 0 / suppψ. Similarly, we denote by F [ψ] t the convolution kernel of the operator ψtp ) and by Ψ t x,w) the integral kernel of the operator ψta). Obviously, by Corollary 3.14, we have ) w x 3.15 ) Ψ t x,w) =t d/2 Ψ,tVx),...,t β +2)/2 D β V x),..., where 3.16 ) Ψx,ξ) =FY F [ψ] )x,ξ) SR d R D ). The corollary below follows from Proposition ) Corollary For every multi-index γ N D and β N d we have ξ Ψu γ w,ξ) w β dw =0. ξ=0 Rd γ 4. Tangential maximal functions associated to A. Lemma 4.1. Assume that m {1,2,3,... } and θ, ϕ S0[0, )), m ϕ0) 0. Then there are ψ 0) S0 m [0, )) and ψ Cc 0,1) such that 4.2) 1 = ψ 0) λ)ϕλ)+ ψ2 j λ)ϕ2 j λ) for λ 0, j=1 and, consequently θλ) =ψ 0) λ)θλ)ϕλ)+ ψ2 j λ)θλ)ϕ2 j λ) for λ 0. j=1 Proof. Since ϕ0) 0 there exists a positive integer k and a constant c > 0 such that ϕλ) > c for λ [0,2 k ]. Therefore there is a function ψ Cc 2 k 7,2 k 5 ) such that j= ψ2 j λ)ϕ2 j λ)=1forλ>0. Set ηλ) = ψ2 j λ)ϕ2 j λ). j=1

10 84 J. Dziubański Obviously ηλ) =1forλ>2 k 1. Put 1 ηλ) for λ [0,2 k ] ψ 0) λ) = ϕλ) 0 for λ>2 k. It is easy to verify that ψ 0) S m 0 [0, )) and 4.2) holds. For functions ψ 0), ψ, andθfrom Lemma 4.1 let us denote by K 0) t x,y) and K 2 j tx,y) the integral kernels of the operators ψ 0) ta)θta)andψ2 j ta)θta) respectively. The following lemma can be deduced from 3.4), 3.13), Theorem 3.6 and Corollary Lemma 4.3. For every M, M,k>0there there exists m>0such that if θ, ϕ, ψ 0),ψ S0 m [0, )) are from Lemma 4.1. Then there exists a constant C>0such that β t x, y) Ct d+ β )/2 1+ x y ) M 4.4), 4.5) x β K0) β x β K 2 j tx,y) C2 j t) d+ β )/2 2 jm for every β, β k. 1+ x y ) M, 2 j t) 1/2 Remark. One should not confuse the functions ψ 0), ψ that are defined on [0, ) with the functions ψ n,k) from Lemma 2.3) that are defined on R d. For a positive real number N and ϕ S[0, )) we define an analogue of Peetre tangential maximal operator M ϕ,n by 4.6) M ϕ,n fx) = sup t>0, y R d { ϕta)fy) 1 + t 1/2 y x ) N }. The operators Mϕ,N have many of the properties of classical maximal operators. We present some of them which we shall need later. For the convenience of the reader we provide the proofs. Lemma 4.7. For every N > 0 there exists m>0such that if θ, ϕ S0 m [0, )) such that ϕ0) 0, then there exists a constant C such that 4.8) M θ fx) CM ϕ,nfx).

11 H p Spaces Associated with Some Schrödinger Operators 85 Proof. Lemmas 4.1 and 4.3 imply M θ fx) =supψ 0) ta)θta)ϕta)fx)+ ψ2 j ta)θta)ϕ2 j ta)fx) t>0 sup t>0 supc t>0 K 0) t x, y)ϕta)fy) dy t>0 j=1 j=1 +sup K 2 j tx,y)ϕ2 j ta)fy) dy 2 jm j=0 2 j t) d/2 1+ x y ) M 2 j t) 1/2 1+ x y ) N M 2 j t) 1/2 ϕ,n fx)dy, which gives 4.8). Lemma 4.9. For every N > 0 there exists m > 0 such that if ϕ S0 m [0, )), ϕ0) 0, then there is a constant C>0such that 4.10) ϕta)fx) x i Ct 1/2 1+ x w ) N Mϕ,N fw). Proof. Applying Lemmas 4.1 and 4.3 to θ = ϕ, we obtain ϕta)fx) x i K 0) t x, y)ϕta)fy) x i dy C j=0 + K x 2 j tx,y)ϕ2 j ta)fy) i dy j=1 2 j t) d+1)/2 2 jm 1+ x y ) M 2 j t) 1/2 1+ w y ) N 1+ w y ) N ϕ2 j ta)fy) dy 2 j t) 1/2 2 j t) 1/2

12 86 J. Dziubański C j=0 t d+1)/2 2 jm 2 jd+n+1)/2 1+ x y ) M 1+ x y ) N 1+ x w ) N M ϕ,n fw)dy Ct 1/2 1+ x w ) N M ϕ,n fw). Proposition For every N>0there is m>0such that for every function ϕ S0 m [0, )), ϕ0) 0, there is a constant C>0such that 4.12) M ϕ,n fx) C[M [H L] M ϕ f) r x)] 1/r, where r = d/n and M [H L] is the classical Hardy-Littlewood maximal operator. get Proof. Fix 0 <δ 1 <1 and set δ = δ 1. By the Mean Value Theorem we ϕta)fx u) 1/r Cδ d/r ϕta)fy) dy) r + Cδ sup ϕta)fy) x u y δ x y u δ Cδ d/r δ + u ) d/r [M [H L] ϕta)f r x)] 1/r + Cδ sup ϕta)fy) 1+ x y ) N 1+ x y ) N. x u y δ Lemma 4.9 leads to ϕta)fx u) Cδ d/r δ + u ) d/r [M [H L] ϕta)f r x)] 1/r + Ct 1/2 δmϕ,nfx) 1+ δ u + t1/2 Cδ d/r 1 1+ u ) d/r [M [H L] ϕta)f r x)] 1/r + Cδ 1 Mϕ,N fx) 1+δ 1 + u ) N. ) N

13 H p Spaces Associated with Some Schrödinger Operators 87 Therefore, ϕta)fx u) 1+ u ) N Cδ d/r 1 [M [H L] ϕta)f r x)] 1/r + Cδ 1 M ϕ,n fx). Taking δ 1 sufficiently small, we get 4.12). Proposition For every p 0,1] there exists m>0such that if θ, ϕ S0 m [0, )), ϕ0) 0, then there exists a constant C>0such that 4.14) M θ f L p C M ϕ f L p. Proof. Fix N>0 such that r = d/n < p. Letm>0 be such that Lemma 4.7 and Proposition 4.11 hold. By Lemma 4.7 it suffices to show that 4.15) M ϕ,nf L p C M ϕ f L p. By virtue of Proposition 4.11, we obtain M ϕ,n f L p C [M [H L] M ϕ f) r ] 1/r L p = C M [H L] M ϕ f) r 1/r L p/r C M ϕ f) r 1/r L p/r = C M ϕ f L p. 5. Proof of 1.13). For p 0,1] let m>0 be large enough such that Proposition 4.13 holds. If ϕ S[0, )), then there are constants c 1,c 2,...,c m+1 such that θλ) =ϕλ) m+1 n=1 c ne nλ S m 0 [0, )). Therefore M ϕ f L p C M θ f L p + Mf L p). By Proposition 4.13 it suffices to show that there exists a function ϕ S m 0 [0, )), ϕ0) 0, such that 5.1) M ϕ f L p C Mf L p. Such a function can be constructed as a linear combination of functions e nλ,

14 88 J. Dziubański that is, there are constants d 1,d 2,...,d m+1 such that m+1 5.2) ϕλ) = d n e nλ S0 m [0, )) and ϕ0) = 1. n=1 Obviously for ϕ of the form 5.2) the inequality 5.1) is satisfied. 6. Proof of 1.14). It suffices to show that there is a constant C>0 such that 6.1) Ma p L p C for every H p A atom a. Letabe an Hp A atom associated to a ball Bx 0,r). Assume that x 0 B n. By the definition of the atoms r 2 1 n/2. It follows from 1.8) and the estimates 3.18) that 6.2) Max) Cr d/p for x B x 0,r)=Bx 0,2r), and, consequently, 6.3) Bx 0,2r) Max)) p dx C. In order to show that 6.4) Bx 0,2r) c Max)) p dx C, we consider two cases: o Case 1: r 2 1 n/2 Then, by definition, a satisfies the moment conditions 1.10). Let τ be the smallest integer >d1/p 1). For fixed x Bx 0,2r) c let us consider the Taylor expansion of the function w T t x,w) atx 0. According to 3.19) and the equality T t x,w) =T t w,x), we get

15 H p Spaces Associated with Some Schrödinger Operators 89 T t ax) = aw)t t x,w)dw = aw)[t t x,w) Therefore 6.5) Ct τ/2 t d/2 γ <τ C x x 0 d τ r τ+d r d/p. ) 1 γ γ! w γ T t Max)) p dx Cr pτ+pd r d Bx 0,2r) c x,w) w x 0 ) γ ]dw w=x0 aw) 1+ x x ) d τ 0 w x 0 τ dw Bx 0,2r) c x x 0 d+τ)p dx C. o Case 2: 2 1 n/2 <r 2 n/2 Note that in this case no moment condition on a is required. For 0 <t 2 n and x Bx 0,2r) c, we have T t ax) C Ct d/2 aw) T t x,w)dw Bx 0,r) Ct d/2 r d/p r d/p 1+ x w ) M dw Bx 0,r) C2 nd/2p) 1+ x x ) M 0, 2 n/2 with M>0 being large, cf. 3.18). This gives 1+ x x ) M 0 dw 6.6) Bx 0,2r) c ) p sup T t ax) dx C 0<t 2 n 2 nd/2 1+ x x ) Mp 0 dx C. 2 n/2 It remains to estimate Bx 0,2r) c sup t>2 n T t ax) ) p dx.

16 90 J. Dziubański Let Pm t be the operator defined by Pmfx) t = fw) t d/2 χ B0,m) t 1/2 w x)) χ [ m,m] t β +2)/2 D β V x) )) dw. As a consequence of 3.18), we obtain β α 6.7) T t ax) m 2b m P t max), where b m C q 1 + m) q for every q>0. We shall use the following lemma. Lemma 6.8. There is a constant C 1 1 independent of n such that for every bounded function f such that suppf Bx n,2 1 n/2 ), x n B n, and every m 2, we have 6.9) P t mf 0 for t>m C 1 2 n. Proof. Let f be a bounded function such that suppf Bx n,2 1 n/2 ), x n B n. Then, by Lemma 2.1, suppf n+c 2 k=n C 2 B k. Assume that Pmf t 0. Then there is x R d and w suppf such that t β +2)/2 D β V x) m for all β, β α, and t 1/2 w x) m. Since D γ V w) = β α 1/β!)Dβ+γ V x)w x) β for every γ α, On the other hand, there is γ such that D γ V w) Cm α +1 t γ +2)/2. D γ V w) 1/ γ +2) 1 2D 2n C 2)/2. Thus 2 n C2) γ +2))/2 Cm α +1) t γ +2)/2. This implies t m C 1 2 n, which completes the proof of Lemma 6.8.

17 H p Spaces Associated with Some Schrödinger Operators 91 By Lemma 6.8 Bx 0,2r) c ) p sup T t ax) dx sup t>2 n t>2 n b m Pmax) t m=2 b m sup m=2 2 n <t 2 n m C 1 p L p P t max) p L p dx). Note that sup 2 n <t 2 n m C 1 P t max) C2 nd/2p) χ Bx0,2 n/2 m C 3 )x). Therefore Bx 0,2r) c ) p sup T t ax) dx C t>2 n m=2 b p m2 nd/2 χ Bx0,2 n/2 m C 3 )x)dx C. This ends the proof of 6.4). 7. Local maximal functions. For a function ϕ S[0, )) a real number N>0, and for a nonnegative integer n we define the local maximal functions M n) ϕ,n and M n) ϕ by 7.1) 7.2) M n) ϕ,n fx) = M n) ϕ fx) = sup sup { ϕta)fw) 1 + t 1/2 w x ) N }, 0<t 2 n,w R d Φt w x,0)fw)dw, 0<t 2 n R d and where Φ t x,0) = t d/2 Φx/, 0), and Φx,0) is defined by 3.15) and 3.16), here x,0) R d R D. Let us observe, cf. [D2], that the function x Φ x,0) is the convolution kernel of the operator ϕ ), where is the Laplacian on R d. Therefore Φ x,0) as a function of x) belongs to the Schwartz class SR d ) and Φ x,0)dx = ϕ0). Our aim in this section is to prove the following result: Theorem 7.3. For every l, N > 0 and every p 0,1] there exists m>0 such that if ϕ S m 0 [0, )), ϕ0) = 1, then there is a constant C>0such that M n) ϕ ψ n,k) f) L p C 1+2 n/2 x x n,k) ) l M n) ϕ,n fx)) L p dx), see Section 2 for the definitions of ψ n,k) and x n,k).

18 92 J. Dziubański For ϕ S0 m [0, )), ϕ0) = 1 let ψ 0) S0 m [0, )), and ψ Cc be such that ψ 0) λ) =0forλ>1and 0, )) 7.4) 1 = ϕλ)ψ 0) λ)+ ϕ2 j λ)ψ2 j λ) for λ 0. j=1 Proposition 7.5. For every M, l>0there is a constant C>0such that for every 0 <t 2 n we have ψ n,k) w) Φ t w x, 0)Ψ 2 j tw,u)dw C2 jm t d/2 1+2 j/2 t 1/2 x u ) l. Proof. For γ N D we shall denote by D γ ξ the differential operator γ / ξ γ. From 3.15 ) it follows that ψ n,k) w) Φ t w x,0)ψ 2 j tw,u)dw ψ n,k) w) Φ t w x,0) [Ψ 2 j tw,u) ) 1 γ! 2jd/2 t d/2 D γ Ψ u w ξ 2 j t), 0 ξ γ] dw 1/2 γ <M + ψ n,k) w) Φ t w x,0) ) 1 γ! 2jd/2 t d/2 D γ Ψ u w ξ 2 j t), 0 ξ γ dw 1/2 γ <M =I 1 +I 2, where ξ =2 j tv w),...,2 j t) β +2)/2 D β V w),...). In order to estimate I 1 we observe that ξ C2 j, whenever w B n,k) and 0 <t 2 n. By virtue of the Taylor formula and Corollary 3.14, we obtain I 1 C 2 jm t d/2 1+ x w ) b 2 jd/2 t d/2 1+ w u ) l dw B n,k) 2 j t) 1/2 C 2 jm t d/2 B n,k) 1+ x w ) b 2 jd/2 t d/2 1+ w u ) l 2 j t) 1/2 1+ x u ) l 1+ x u ) l dw 2 j t) 1/2 2 j t) 1/2

19 H p Spaces Associated with Some Schrödinger Operators 93 C2 jm t d/2 1+ x u ) l 2 jd/2 t d/2 1+ x w ) b 2 j t) 1/2 B n,k) 1+ x w ) l dw 2 j t) 1/2 C2 jm t d/2 1+ x u ) l 2 jd/2 t d/2 1+ x w ) b 2 j t) 1/2 B n,k) C2 jm t d/2 2 jl+d)/2 1+ x u ) l. 2 j t) 1/2 2 jl/2 1+ x w ) l dw To estimate I 2 we study each summand separately. Let Hx,w) = 1 γ! ψ n,k)w) Φ t w x,0)ξ γ, where ξ =2 j tv w),...,2 j t) β +2)/2 D β V w)...). We denote by D κ whx,w) the differentiation of the function Hx, w) with respect to the second variable. For fixed x and u we consider Taylor expansion of the function w Hx,w) at the point u. By Corollary 3.20, ) I 2,γ = 1 γ! ψ n,k)w) Φ t w x,0)2 jd/2 t d/2 D γ Ψ u w ξ 2 j t), 0 ξ γ dw 1/2 ) = Hx,w)2 jd/2 t d/2 D γ Ψ u w ξ 2 j t), 0 dw 1/2 [ = Hx,w) 1 κ! Dκ whx,u)w u) κ] κ <M ) 2 jd/2 t d/2 D γ Ψ u w ξ 2 j t), 0 dw 1/2 w u x u /2 = I 2,γ + I 2,γ. + w u x u /2

20 94 J. Dziubański Since 0 <t 2 n and M is large, using Taylor s formula, 2.6), and Corollary 3.14, we have I 2,γ C w u M t d/2 t M /2 1+ x u ) l w u x u /2 2 j t) d/2 1+ w u ) b dw 2 j t) 1/2 Likewise, C2 jm /2 t d/2 1+ x u ) l C2 jm t d/2 1+ x u ) l. 2 j t) 1/2 I 2,γ C w u M t d/2 t M /2 2 j t) d/2 1+ w u ) b dw w u x u /2 2 j t) 1/2 C2 jm /2 t d/2 1+ x u ) l, 2 j t) 1/2 which completes the proof of Proposition 7.5. Corollary 7.6. For every M,l > 0 there is a constant C such that for every 0 <t 2 n we have 7.7) ψ n,k) w) Φ t w x,0)ψ 2 j tw,u)dw C2 jm t d/2 1+ x u ) l 1+2 n/2 x x n,k) ) l. 2 j t) 1/2 Proof. Corollary 3.14 implies 7.8) ψ n,k) w) Φ t w x,0)ψ 2 j tw,u)dw C ψ n,k) w)t d/2 1+ x w ) l 2 dj/2 t d/2 1+ w u ) l dw 2 j t) 1/2 Ct d/2 1+2 n/2 x x n,k) ) l 2 dj/2 t d/2 1+ w u ) l dw 2 j t) 1/2

21 H p Spaces Associated with Some Schrödinger Operators 95 Ct d/2 1+2 n/2 x x n,k) ) l. Now 7.7) follows from 7.8) and Proposition 7.5. Proof of Theorem 7.3. Let 0 <t 2 n. From 7.4) we conclude that ψ n,k) w)fw) Φ t w x,0)dw [ = ψ n,k) w) Φ t w x,0) ψ 0) ta)ϕta)fw) ] + ψ2 j ta)ϕ2 j ta)fw) dw j=1 ψ n,k) w) Φ t w x,0)ψ 0) t w,u) 1+ x u ) N 1+ x u ) N ϕta)fu)dudw + j=1 ψ n,k) w) Φ t w x,0)ψ 2 j tw,u) 1+ x u ) N 2 j/2 1+ x u ) N ϕ2 j ta)fu)dudw 2 j/2. Applying Corollaries 7.6 and 3.14 with m sufficiently large, we get ψ n,k) w)fw) Φ t w x,0)dw C ψ n,k) w)t d/2 1+ x w ) l N t d/2 1+ w u ) l N + C 1+ x u ) N M n) ϕ,n fx)dw du 2 jm t d/2 1+2 n/2 x x n,k) ) l j=1 1+ x u ) l N 1+ x u ) N M n) 2 j t) 1/2 2 j t) 1/2 ϕ,n fx)du

22 96 J. Dziubański C1+2 n/2 x x n,k) ) l M n) ϕ,n fx). 8. Proof of 1.15). Atomic decomposition. Before we turn to our proof of atomic decomposition we state some results from the theory of local Hardy spaces, cf. [Go]. Let Φ be a function from the Schwartz class on R d such that Φ =1. For every nonnegative integer n we define the local maximal function M n) Φ by 8.1) M n) fx) = sup f Φ Φ t x), 0<t 2 n where Φ t x) =t d/2 Φx/ ). We say that f is in the local Hardy space h p n if 8.2) f p h p n = M n) Φ f p L p <. A function ã is an atom for the local Hardy space h p n associated to a ball Bx 0,r), r 2 n/2 if 8.3) suppã Bx 0,r), r 2 n/2, 8.4) ã L volbx 0,r)) 1/p, 8.5) if r 2 1 n/2, ) 1 then ãx)x β dx =0for β d p 1. The atomic quasi-norm in h p n is defined by { 8.6) f p = inf c h p j p}, ã,n j where the infimum is taken over all decompositions f = c j ã j, where ã j are h p n atoms and c j are scalars. Theorem 8.7 [Go]. The quasi-norms h p n and h p are equivalent with ã,n constants independent of n Z, that is, there exists a constant C p > 0 such that for every n we have C 1 p f h p n f h p ã,n C p f h p n.

23 H p Spaces Associated with Some Schrödinger Operators 97 Moreover, if f h p n, suppf Bx,2 1 n/2 ), then there are h p n atoms ã j such that suppã j Bx,2 2 n/2 ) and 8.8) f = j c j ã j, c j p C f p h p n j with a constant C independent of n. Proof of 1.15). We first prove that for every p 0,1] there exists m>0 such that 1.15) holds for every function ϕ such that ϕ S0 m [0, )), ϕ0) = 1. Indeed, it follows from Theorems 7.3, 8.7, Lemma 2.1, and the definition of atoms that H p A ψ n,k) f = j c j,n,k) a j,n,k), where a j,n,k) are H p A atoms and j c j,n,k) p C 1+2 n/2 x x n,k) ) l M n) ϕ,n fx) p L p dx), provided ϕ S0 m [0, )) with m large enough) and ϕ0) = 1. Therefore f = ψ n,k) f = c j,n,k) a j,n,k) n,k) n,k) j and c j,n,k) p C 1+2 n/2 x x n,k) ) l Mϕ,N fx) p L p dx). n,k) j n,k) Since n,k) 1+2n/2 x x n,k) ) lp is a bounded function for sufficiently large l cf. Lemmas 2.1 and 2.2), we have c j,n,k) p C Mϕ,Nf p L p, n,k) j which, by 4.15), gives c j,n,k) p C M ϕ f p L p. n,k) j

24 98 J. Dziubański Now let ϕ S[0, )) be such that ϕ0) 0. Then there are constants c 1,c 2,...,c m+1 such that ϕ 1) λ) = m+1 n=1 c nϕnλ) S0 m [0, )) and ϕ 1) 0) = 1. By the above f p H p atom C M ϕ 1) f p L C 1 M p ϕ f p L p, A which completes the proof of 1.15). [CGGP] [D1] [D2] [DHJ] [DZ] References M. Christ, D. Geller, P. G lowacki and L. Polin, Pseudodifferential operators on groups with dilations, Duke Math. J ), J. Dziubanski, Schwartz spaces associated with some non-differential convolution operators on homogeneous groups, Colloq. Math ), J. Dziubanski, A note on Schrödinger operators with polynomial potentials, Colloq. Math. to appear). J. Dziubanski, A. Hulanicki, and J.W. Jenkins, A nilpotent Lie algebra and eigenvalue estimates, Coll. Math ), J. Dziubanski and J. Zienkiewicz, Hardy space H 1 associated to Schrödinger operator with potential satisfying reverse Hölder inequality, Preprint). [DZ1], Hardy spaces associated with some Schrödinger operators, Preprint). [Fe] C. Fefferman, The uncertainty principle, Bull. Amer. Marth. Soc ), [FeS] C. Fefferman and E. Stein, H p spaces of several variables, Acta Math ), [FS] [G] G. Folland and E. Stein, Hardy Spaces on Homogeneous Groups, Princeton University Press, Princeton, New Jersey, P. G lowacki, Stable semigroups of measures as commutative approximate identities on nongraded homogeneous groups, Invent. Math ), [Go] D. Goldberg, A local version of real Hardy spaces, Duke Math. J ), [He] W. Hebisch, On operators satisfying Rockland condition, Preprint of the University of Wroc law). [HN] [H] [HJ] B. Helffer and J. Nourrigat, Une Inégalité L 2, Preprint). A. Hulanicki, A functional calculus for Rockland operators on nilpotent Lie groups, Studia Math ), A. Hulanicki and J.W. Jenkins, Almost everywhere summability on nilmanifolds, Trans. Amer. Math. Soc ), [S] E. Stein, Harmonic Analysis, Princeton University Press, Princeton, [Z] J. Zhong, The Sobolev estimates for some Schrödinger type operators, Ph.D. Thesis, Princeton University, Princeton, Institute of Mathematics University of Wroc law Plac Grunwaldzki 2/ Wroc law, POLAND E{mail: jdziuban@math.uni.wroc.pl Received: July 3rd, 1997; revised: November 2nd, 1997.