Research Article Boundedness of Singular Integrals on Hardy Type Spaces Associated with Schrödinger Operators

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1 Function Spaces Volume 05, Article ID 4095, pages Research Article Boundedness of Singular Integrals on Hardy Type Spaces Associated with Schrödinger Operators Jianfeng Dong, Jizheng Huang, andhepingiu 3 Department of Mathematics, Shanghai University, Shanghai 00444, hina ollege of Sciences, North hina University of Technology, Beijing 0044, hina 3 MAM, School of Mathematical Sciences, Peking University, Beijing 0087, hina orrespondence should be addressed to Jizheng Huang; hjzheng@63.com Received 9 May 04; Accepted August 04 Academic Editor: Jose uis Sanchez opyright 05 Jianfeng Dong et al. This is an open access article distributed under the reative ommons Attribution icense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. et = Δ+Vbe a Schrödinger operator on R n,n 3,whereV 0isanonnegative potential belonging to the reverse Hölder class B n/. The Hardy type spaces H p,n/(n+)<p,for some >0, are defined in terms of the maximal function with respect to the semigroup {e t } t>0. In this paper, we investigate the bounded properties of some singular integral operators related to, such as iγ and /, on spaces H p. We give the molecular characterization of Hp, which is used to establish the Hp -boundedness of singular integrals.. Introduction et = Δ + V be a Schrödinger operator on R n,n 3, where V 0 is a nonnegative potential belonging to the reverse Hölder class B for some n/; that is, there exists aconstant>0suchthat the reverse Hölder ineuality ( B V / (x) dx) ( B B V (x) dx) () B holds for every ball B in R n.itiswellknownthatifv B then V B +ε for some ε>0.alsoobviously,b B when >. Some singular integral operators related to,suchasthe imaginary power iγ, and the Riesz transform / have beenstudiedbyshen[]. Some of his results are following. The operator iγ is a alderón-zygmund operator for any γ R. / is a alderón-zygmund operator if n. When n/ < n, / is bounded on p for <pp 0, where /p 0 = / /n. The above range of p is optimal. Earlier results were given by Fefferman []andzhong[3]. The Hardy type spaces H p,n/(n+)<pfor some > 0,associatedwith, arestudiedbydziubański and Zienkiewicz [4, 5]. They establish the H p, atomic decomposition theorem and the Riesz transform characterization of H. Specifically, / is bounded from H to.we will investigate the bounded properties of the operators iγ and / on spaces H p.todothis,wegivethemolecular characterization of H p. Without loss of generalization, we assume that V B 0 for some 0 >n/and set =min( n/ 0,).When 0 >n, we set η= n/ 0.Throughoutthepaper,wewilluseAand to denote the positive constants, which are independent of main parameters and may be different at each occurrence. By B B,wemeanthatthereexistsaconstant>such that / B /B. et {T t } t>0 = {e t } t>0 be the semigroup of linear operators generated by and K t (x, y) their kernels. Since V is nonnegative, the Feynman-Kac formula implies that 0K t (x, y) K t (x y) = (4πt) n/ e (4t) x y, () where K t (x) is the convolution kernels of the heat semigroup {T t } t>0 = {e tδ } t>0.theestimate() canbeimprovedas

2 Function Spaces follows. We introduce the auxiliary function ρ(x, V) = ρ(x) defined by ρ (x) = sup {r > 0 : r n V(y)dy }. (3) B(x,r) It is known that 0<ρ(x)<.ForeveryN>0, K t (x, y) Nt n/ e (5t) x y ( + t ρ (x) + t N ρ(y) ), (4) (cf. [6, Theorem 4.0]). et 0< <;foreveryn>0and all t, K t (x + h, y) K t (x, y) N ( h t ) t n/ e At x y ( + t ρ (x) + t N ρ(y) ) (cf. [6, Proposition 4.]). We define the Hardy type spaces H p,n/(n+)<p,in terms of the maximal function with respect to the semigroup {T t } t>0. For p=,thehardyspaceh is defined, according to Dziubański and Zienkiewicz [4], by where (5) H ={f :M f }, (6) M f (x) = sup T t f (x). (7) t>0 The norm of a function f H is defined to be f H = M f. The Hardy spaces, H p,n/(n+)<p<, consist of some kind of distributions. But M f(x) may have no meaning for atempereddistributionf because K t (x, y) are not smooth. et f be a locally integrable function. B=B(x,r)is the ball of radius r centered at x.set f B = B f(y)dy, B f (B, V) ={ f (8) B, if r<ρ(x), 0, if r ρ(x). et n/(n + ) < p <,.Alocallyintegrable function f is said to be in the ampanato type space Λ /p, if f Λ /p, = sup { B /p / B R B f f(b, V) dx B ) } d <. (9) All spaces Λ /p, are mutually coincident with euivalent norms and will be simply denoted by Λ /p (cf. [7]). Due to (4)and(5), for every t>0, sup x R dk t (x, ) <t d/p Λ (0) /p (cf. [7, emma ]). Thus the semigroup maximal function Mf is well defined for distributions in (Λ /p ). We define the Hardy space, H p,n/(n+)<p<,by H p ={f (Λ /p ) :M f p }, () and set f p H = M f p. Similar to the classical case, the Hardy space H p admits an atomic decomposition. et n/(n+) < p, p =. Afunctiona is called an H p, -atom associated with a ball if () supp a, () a / /p, (3) if r < ρ(x 0 ), then a(x)dx = 0. Proposition (see [7,Theorem]). Given p, as above, then f H p ifandonlyif f can be written as f= j λ ja j,where a j are H p, -atoms and j λ j p <.ThesumconvergesinH p normandalsoin(λ /p ) when p<.moreover, f H p f H p,,a = inf { { { ( j } λ jp )/p }, () } where the infimum is taken over all decompositions of f into -atoms. H p, Now we state the main results in this paper. Theorem. For any γ R, the imaginary power iγ is bounded on H p for n/(n+)<p.when 0 >n,theriesz transform / is bounded on H p for n/(n + η) < p. Moreover, / is bounded on H whenever 0 >n/. Remark 3. When n/ < 0 <n, the kernel of Riesz transform / only satisfies the Hörmander condition with respect to the second variable, which is weaker than the smoothness condition of standard kernels. Thus we cannot expect, in general consideration, to deal with the boundedness of / for the case of p<. In order to prove Theorem, wegivethemolecular characterization of H p. et n/(n + ) < p, p =and ε>/p. Set a = /p + ε, b = / + ε. AfunctionM is called an H p,,ε -molecule with the center x 0 if () x nb M(x),

3 Function Spaces 3 () N(M) = μ b a M a/b x 0 nb M a/b, (3) if M /(a b) <μ ρ(x 0 ) n, then M(x)dx = 0, where μ isthevolumeoftheunitball. Theorem 4. Given p,, ε as above, then f H p if and only if f can be written as f = j λ j M j,wherem j are H p,,ε -molecules and j λ j p <.ThesumconvergesinH p norm and also in (Λ /p ) when p<,wherem j are H p - molecules. Moreover, f H p f H p,,ε,m = inf { { { ( j } λ jp )/p }, (3) } where the infimum is taken over all decompositions of f into -molecules. H p,,ε Remark 5. It is easy to verify that any H p, -atom is an Hp,,ε - molecule with a constant factor less than or eual to. We willseethattheimageofanh p, -atomundertheactionofa singular integral operator may not be an H p,,ε -molecule but is a sum of two H p,,ε -molecules up to constant factors. This is different from the classical case. This paper is organized as follows. In Section,wecollect some useful facts and results about the potential V, the auxiliary function ρ(x) and the kernels of operators iγ,and /, which will be used in the seuel. Most of these results are already known. In Section 3, weprovetheorem 4. The proof of Theorem isgiveninthelasttwosections.theh p - boundedness for p < is proved in Section 4 while H - boundedness is proved in Section 5.. Preliminaries Firstwelistsomeknownfactsandresultsaboutthepotential V, the auxiliary function ρ(x), and the kernels of operators iγ and /. emma 6. V(x)dx is a doubling measure; that is, there exists aconstant 0 >0such that B(x,r) V(y)dy 0 B(x,r) V(y)dy. (4) emma 7. onsider r n B(x,r) V(y)dy ( R n/ r ) 0 R n V(y)dy, B(x,R) emma 8. There exists m 0 >0such that 0<r<R<. (5) R n V (y) dy (+ R m0 B(x,R) ρ (x) ). (6) emma 9. There exists k 0 >0such that k0 (+x y ρ (x) ) ρ(y) k0/(k0+) ρ (x) (+x y ρ (x) ). In particular, ρ(y) ρ(x) if x y <ρ(x). (7) et F γ (x, y) and F γ(x, y) be the kernels of iγ and ( Δ) iγ,respectively,andr (x, y) and R(x, y) the kernels of / and ( Δ) /,respectively.set F γ (x, y) = F γ (x, y) F γ (x, y), R(x, y) = R (x, y) R(x, y). emma 0. iγ is a alderón-zygmund operator. It does not matter to assume that n/ < 0 < n.thekernelf γ (x, y) satisfies F γ (x, y+ h) F γ (x, y) eπ γ / h x y n+, and, for any N>0, In addition, F γ (x, y) Ne π γ / x y n ( + x y, (8) x y N ρ(y) ). (9) F γ (x, y) eπ γ / x y n ( x y ρ(y) ). (0) emma. When n/ < 0 <n, / is bounded on p for <pp 0,where/p 0 =/ 0 /n.thekernelr (x, y) satisfies, for any N>0, R (x, y) In addition, N x y n B(x, x y /4) (+ R(x,y) x y n B(x, x y /4) x y N ρ(y) ). z x n + x y ) () z x n + x y ( x y ρ(y) ) ). ()

4 4 Function Spaces emma. When 0 > n, / is a alderón-zygmund operator. The kernel R (x, y) satisfies and, for any N>0, R (x, y+ h) R (x, y) h η x y n+η, x y, (3) R (x, y) N x y n ( + x y N ρ(y) ). (4) In addition, for any <, R(x,y) x y n ( x y ρ(y) ). (5) For emmas 6, we refer readers to []. We also need the following estimates about F γ (x, y) and R(x, y). emma 3. When n/ < 0 <n, F γ (x, y+ h) F γ (x, y) eπ γ / x y n ( h ρ(y) ), When 0 >n,forany <, R(x,y+h) R(x,y) Proof. It is well known that x y. x y n ( h ρ(y) ), F γ (x, y+ h) F γ (x, y) R(x,y+h) R(x,y) Therefore, we also have the estimates x y. h x y n+, x y, h x y n+, F γ (x, y+ h) F γ (x, y) eπ γ / h x y n+, x y. (6) (7) (8) x y, (9) We may assume that x y < ρ(y).otherwise,emma 3 is obvious. We will use the following known facts (cf. []). et Γ (x,y,τ)and Γ(x, y, τ) denote, respectively, the fundamental solutions for the operators +iτand Δ + iτ in R n,where τ R. They satisfy the following estimates. For any k>0and x y /, Γ(x,y,τ) Γ (x,y+h,τ) Γ (x,y,τ) k k ( + τ / x y )k x y n, h ( + τ / x y )k x y ( + n + x y k ρ(y) ) (3) when n/ < 0 <n.set Γ(x, y, τ) = Γ (x,y,τ) Γ(x,y,τ). Then Γ(x, y, τ) is expressed as Thus, Γ(x,y,τ)= R n Γ (x, z, τ) V (z) Γ (z, y, τ) dz. (3) Γ(x,y+h,τ) Γ(x,y,τ) R n Γ (x, z, τ) V (z) Γ (z, y+ h, τ) Γ (z, y, τ) dz R n ( k h V (z) ( + ρ(y) z y ) k dz) ((+ τ / x z ) k ( + τ / z y )k x z n z y n + ) = ( ) + ( ) z x < x y / z y < x y / + ( ) z x x y /, z y x y / =I +I +I 3. (33) Note that V B 0 +ε for some ε>0.usinghölder ineuality and B 0 +ε condition, it is easy to see that, for 0σ, R(x,y+h) R(x,y) h η x y n+η, x y. (30) B(x,R) V(y) x y dy n +σ R n +σ V(y)dy. (34) B(x,R)

5 Function Spaces 5 Note that ρ(x) ρ(y) when x y < ρ(y). Makinguseof (34), we get I k h ( + τ / x y )k x y n + z x < x y / x z n k h ( + τ / x y )k x y n + z x < x y / x y n k h ( ( + τ / x y )k x y n ρ(y) ), (35) where we have used emma 7 in the last ineuality. Similarly, I k h ( + τ / x y )k x y n z y < x y / z y n + k h ( + τ / x y )k x y n + z y < x y / x y n Using Hölder ineuality and B 0 condition, we obtain x y / z y <ρ(y) z y n 4+ / 0 V(z) 0 dz) z y <ρ(y) dz ) z y x y / z y (n 4+) 0 x y n ρ(y). / 0 Using emma 6 and taking k sufficiently large, we get ρ(y) k z y ρ(y) z y n 4++k ρ(y) 4 n j(n 4++k) j= z y j ρ(y) ρ(y) 4 n j(n 4++k) j 0 j= z y ρ(y) ρ(y) n (38) x y n ρ(y). (39) k h ( ( + τ / x y )k x y n ρ(y) ). (36) Therefore, Γ(x,y+h,τ) Γ(x,y,τ) To estimate I 3,wewrite k h ( ( + τ / x y )k x y n ρ(y) ). (40) I 3 k h ( + τ / x y )k z y x y / ( + k h ( + τ / x y )k x y / z y <ρ(y) +ρ(y) k z y ρ(y) x y k ρ(y) ) z y n 4+ z y n 4+ z y n 4++k ). (37) We also have x Γ(x,y,τ)= R n x Γ (x, z, τ) V (z) Γ (z, y, τ) dz, (4) where x Γ(x, z, τ) satisfies the estimate xγ(x,y,τ) k ( + τ / x y )k x y n. (4) If 0 >n, by the same argument as (40), for any <, x Γ(x,y+h,τ) x Γ(x,y,τ) k h ( ( + τ / x y )k x y n ρ(y) ). (43)

6 6 Function Spaces By the functional calculus and making use of (40), we obtain F γ (x, y+ h) F γ (x, y) = π ( iτ) iγ ( Γ(x,y+h,τ) Γ(x,y,τ)) dτ R (44) eπ γ / x y n ( h ρ(y) ). This proves (6). Similarly, it follows from (43) that R(x,y+h) R(x,y) = π ( iτ) / ( x Γ(x,y+h,τ) x Γ(x,y,τ))dτ R x y n ( h ρ (y) ). This proves (7). 3. Molecular haracterization (45) Essentially, the proof of Theorem 4 is the same as the usual molecular theory. Proof of Theorem 4. By Proposition, itissufficienttoprove that for any H p,,ε -molecule M(x) admits an atomic decomposition M = j λ j a j, where a j are H p, -atoms and j λ j p <. We will give the proof in case =.Theproofissimilar in the case of =.SupposeM(x) is an H p,,ε -molecule centered at x 0.etσ = M /(a b),whereε > /p, a = /p + ε, b = / + ε.ifσ<μ ρ(x 0 ) n,wereturntheusual molecular theory (cf. [8]). Thus nothing needs to be proved. Suppose σ μ ρ(x 0 ) n.set B k ={x: x x 0 k μ /n σ /n }, k=0,,,..., (46) E 0 =B 0, E k =B k \B k, k=,,... Then M (x) = k=0 M (x) χ Ek (x) = k=0 M k (x). (47) Note that supp M k B k and k μ /n σ /n ρ(x 0 ), k = 0,,,....Alsowehave M 0 M =σa b = B 0 / /p, (48) M k (k )nb μ b σ b x 0 nb M Thus M k (x) = λ k a k (x), k = 0,,,..., wherea k are H p, - atoms and k=0 λ k p <. Originally, the sum in (47) converges pointwise. When p=, it is easy to see that the sum in (47)convergesin.If n/(n + ) < p <,foranyg Λ /p, ( + x nb ) g g (x) / dx) x <ρ(0) Therefore, + k= g (x) / dx) k ρ(0) x < k ρ(0) k=0 g Λ /p. kna g Λ /p (k )nb ρ(0) nb (50) Mg (+ x nb ) M (+ x nb ) g <. (5) It follows that the sum in (47) convergesin(λ /p ).The proof of Theorem 4 is completed. 4. H p -Boundedness In this section, we prove the boundedness of iγ on H p,n/(n+ )<p.when 0 >n, the boundedness of / on H p, n/(n+η)<p, can be proved by the same method. In fact, their kernels satisfy similar estimates. et a(x) be an H p, -atom associated with a ball B(x 0,r) for some suitable. Ifr ρ(x 0 ),wewillprovethat iγ a(x) is an H p,,ε -molecule up to a constant factor. If r < ρ(x 0 ), iγ a(x) may be not an H p,,ε -molecule up to a constant factor but ( Δ) iγ a(x)is (cf. [9]). We will prove that ( iγ ( Δ) iγ )a(x) is an H p,,ε -molecule up to a constant factor for some suitable ε. Thismeansthat iγ a(x) p H uniformly. Because the semigroup maximal function M f is subadditive, by Proposition, iγ is bounded on H p, n/(n+)<p. First, let r ρ(x 0 ).Because iγ a (x) a (x) B(x 0,r) a b, (5) (k )nb σ b M a/(a b) = (k )nb σ a b = nb kna B k / /p. (49) where ε > /p, a = /p + ε, b = / + ε,wehave iγ /(a b) a (x) ρ(x 0) n. (53)

7 Function Spaces 7 Thus there needs no the cancelation condition. We only need to estimate N( iγ a).write Next, suppose r < ρ(x 0 ). et us estimate ( iγ ( Δ) iγ )a(x). onsider R x x 0 nb iγ / a (x) dx) n / x x 0 nb iγ a (x) dx) / + x x 0 nb x x0 r iγ a (x) dx) (54) (iγ ( Δ) iγ / )a(x) R dx) n / (iγ ( Δ) iγ )a(x) dx) / + (iγ ( Δ) iγ )a(x) r x x dx) 0 <ρ(x 0) =I +I. It is obvious that I r nb iγ a (x) B(x 0,r) b a (x) B(x 0,r) a. (55) For y,ifρ(y) > r,byemma 9, ρ(y) ρ(x 0 ) r. Notethat x y x x 0 when x,y.usingemma 0,weget I = x x 0 nb x x 0 r / F γ (x,y)a(y) dy) dx) a(y) dy x x 0 nb x x 0 r a(y) rn dy / F γ (x, y) dx) / x x 0 (nb n N) dx) x x0 r r n n/p+n r nb n/ N B(x 0,r) a and provide N>nε. Therefore, (56) / + x x0 ρ(x (iγ ( Δ) iγ )a(x) dx) 0) =J +J +J 3. (59) Note that ρ(y) ρ(x 0 ),wheny and by emma 0, we have J = F / γ (x,y)a(y) dy) dx) B(x 0,r a(y) F / γ (x, y) dx) dy / a(y) ρ(x 0) dx x y ) dy (n ) / ρ(x 0 ) r n n/p dx B(0,3r) x ) (n ) ρ(x 0 ) r n/ n/p+ ρ(x 0 ) n(a b). (60) Here we choose such that < < n/(n ) and n/ n/p + > 0 or, euivalently, <<np/(n p). When r x x 0 <ρ(x 0 ), using the cancelation condition of a and emma 3,weobtain (iγ ( Δ) iγ )a(x) It follows that N ( iγ a) = μ b a x x 0 nb iγ a (x) B (x 0,r) a. (57) iγ a/b a (x) x x 0 nb iγ a (x) a/b. (58) = ( F γ (x, y) F γ (x, x 0 )) a (y) dy ρ(x 0) x x 0 n B(x a(y) y x 0 dy 0,r) ρ(x 0) r n n/p+ x x 0 n. (6)

8 8 Function Spaces It follows that / J ρ(x 0 ) r n n/p+ dx r x x 0 <ρ(x 0) x x 0 n ) ρ(x 0 ) r n/ n/p+ ρ(x 0 ) n(a b). When x x 0 ρ(x 0 ),by(9), we have (6) By (63), / H r n n/p+ x x 0 (nb n ) dx) x x0 ρ(x 0) r n n/p+ ρ(x 0 ) nb n/ ρ(x 0 ) na, (69) Then (iγ ( Δ) iγ )a(x) = ( F γ (x, y) F γ (x, x 0 ))a(y)dy x x 0 n+ a(y) y x 0 dy rn n/p+ x x 0 n+. J 3 r n n/p+ x x0 ρ(x 0) r n n/p+ ρ(x 0 ) n/ / dx x x ) 0 (n+) (63) (64) wherewehavetakenε such that /p <ε</n,which implies that (nb n ) + n < 0. The proof is complete. 5. H -Boundedness In this section we prove the boundedness of / on H when n/ < 0 <n. et a(x) be an H, -atom associated with a ball B(x 0,r) for some suitable. Astheabovesection,ifr ρ(x 0 ), we will prove that / a(x) is an H,,ε -molecule up to a constant factor. If r < ρ(x 0 ),wewillprovethat( / ( Δ) / )a(x) is an H,,ε -molecule up to a constant factor for some suitable ε.inanycasewehave / a(x) p H uniformly. Suppose r ρ(x 0 ).Itfollowsfromemma that ρ(x 0 ) n(a b). We have seen that (iγ ( Δ) iγ /(a b) )a(x) ρ(x 0) n. (65) As above, there needs no the cancelation condition. To finish the proof, we only need to prove N(( iγ ( Δ) iγ )a) or, euivalently, x x 0 nb ( iγ ( Δ) iγ )a(x) ρ(x 0) na. (66) Write R x x 0 nb iγ ( Δ) iγ / a (x) dx) n x x 0 nb iγ ( Δ) iγ / a (x) B(x 0,ρ(x 0 )) dx) + x x 0 nb iγ ( Δ) iγ / a (x) x x 0 ρ(x 0 ) dx) =H +H. (67) / a (x) a (x) B (x 0,r) a b, (70) provide <p 0,where/p 0 =/ 0 /n, a = ε > 0, b = / + ε. Thusthereneedsnothecancelation condition. Write R x x 0 nb / / a (x) dx) n / x x 0 nb / a (x) dx) / + x x 0 nb x x0 r / a (x) dx) =I +I. It is obvious that (7) It is clear that H ρ(x 0 ) nb iγ ( Δ) iγ a (x) ρ(x 0 ) na. (68) I r nb / a (x) B(x 0,r) b a (x) B(x 0,r) a. (7)

9 Function Spaces 9 On the other hand, we have I x x 0 nb x x 0 r / R (x, y) a(y) ) dx) dy a(y) x x 0 nb x x 0 r = G (y) a (y) dy. / R (x, y) dx) dy (73) j= B(x, x y /4) jn ( j r) n+nb+ / z x n ) dx) / B(x 0, j+ r) B(x, x y /4) z x n ) dx) j= j= jn ( j r) n+nb+ B(x 0, j+ r) jn ( j r) n+nb+ B(x 0, j+ r) V(x) s /s dx) /s Note that x y x x 0 when x x 0 rand y and by emma, G(y) ρ(y) N { { ( x x0 r B(x, x y /4) z x n ) { + x x 0 r / dx x x ) 0 (N nb+n ) / dx } x x ) 0 (N nb+n) }. } Since ρ(y) r for y,itisclearthat / ρ(y) N dx x x 0 r x x ) 0 (N nb+n) r nb n/ B(x 0,r) a (74) (75) provide N > na.wehavetaken such that < p 0, where /p 0 =/ 0 /n.et/ = /s /n.thens 0. Using the theorem on fractional integrals, B s condition, and emma 8,weobtain ρ(y) N ( x x 0 r B(x, x y /4) z x n ) / dx x x ) 0 (N nb+n ) r N j= j r x x 0 < j+ r x x 0 (n+n nb ) V (x) dx B(x 0, j+ r) = j= j(n m 0) ( j r) n+nb+n/s j= j(n m 0 na) r na B(x 0,r) a (76) provide N sufficiently large. Thus G(y) a.it follows that I G(y) a(y) dy B(x 0,r) a a B(x 0,r) a. (77) Therefore, x 0 nb / a a and N( / a). In case r < ρ(x 0 ), we need to prove that ( / ( Δ) / )a(x) is an H,,ε -molecule up to a constant factor for some suitable ε. First we give the estimate of ( / ( Δ) / )a(x).write ( / ( Δ) / / )a(x) R dx) n ( / ( Δ) / / )a(x) B(x 0,ρ(x 0 )) dx) + ( / ( Δ) / / )a(x) x x 0 ρ(x 0 ) dx) =J +J. (78)

10 0 Function Spaces We have (79) / J a(y) R(x,y) B(x 0,ρ(x 0 )) dx) dy = G(y) a(y) dy. By emma, G(y) B(x 0,ρ(x 0)) B(x, x y /4) x y (n ) / z x n ) dx) / +ρ(y) dx B(x 0,ρ(x 0 )) x y ) (n ) = G (y) + G (y). Note that ρ(y) ρ(x 0 ) when y.itisobviousthat (80) G (y) ρ(x 0 ) n/ =ρ(x 0 ) n(a b), (8) provide < < n/(n ). On the other hand, using the theorem on fractional integrals and B s condition with s 0, / = /s /n,and < < n/(n ),weget G (y) j=0 j+ ρ(x 0 ) x y < j+ ρ(x 0 ) j=0 x y (n ) / B(x, x y /4) z x n ) dx) ( j ρ(x 0 )) n+ x y < j+ ρ(x 0) B(x, x y /4) j=0 j=0 j=0 ( j ρ(x 0 )) n+ / z x n ) dx) x y < j+3 ρ(x 0 ) V(x) s /s dx) ( j ρ(x 0 )) n+n/s+ V (x) dx x y < j+3 ρ(x 0) ( j ρ(x 0 )) n+n/s j wherewehaveusedemma 7 in the last second ineuality. Thus, J G(y) B(x 0,r) a(y) dy ρ(x 0) n(a b). (83) Since x y x x 0 when x x 0 ρ(x 0 ) and y, R(x, y) / x y n,itiseasytoseethat / J a(y) R(x,y) x x 0 ρ(x 0 ) dx) dy / a(y) dx x x 0 ρ(x 0) x y n ) dy a(y) ρ(x 0) n(a b) dy ρ(x 0 ) n(a b). Therefore we have (84) ( / ( Δ) / )a(x) ρ(x 0 ) n(a b). (85) Asabove,thereneedsnothecancelationcondition. Write R x x 0 nb ( / ( Δ) / / )a(x) dx) n x x 0 nb B(x 0,ρ(x 0)) / ( / ( Δ) / )a(x) dx) + x x 0 nb / / a (x) x x 0 ρ(x 0 ) dx) + x x 0 nb / ( Δ) / a (x) x x 0 ρ(x 0 ) dx) =H +H +H 3. It is obvious that (86) H ρ(x 0 ) nb ( / ( Δ) / )a(x) ρ(x 0 ) na. (87) We have H a(y) x x 0 ρ(x 0 ) x x 0 nb / R (x, y) dx) dy (88) ρ(x 0 ) n(a b), (8) = G 0 (y) a(y) dy.

11 Function Spaces Since x y x x 0 and ρ(y) ρ(x 0 ) when x x 0 ρ(x 0 ) and y,byemma, G 0 (y) ρ(x 0 ) N { { x x0 ρ(x { 0) B(x, x y /4) z x n ) / dx x x ) 0 (N nb+n ) / dx } + x x 0 ρ(x 0 ) x x ) 0 (N nb+n) }. } (89) Similar to G(y) in the proof of (77), we obtain G 0 (y) ρ(x 0 ) na bythesameargument.itfollowsthat H G 0 (y) a(y) dy ρ(x 0) na. (90) Using the cancelation condition of a, H 3 a(y) x x 0 nb x x 0 ρ(x 0 ) a(y) R(x,y) R(x,x 0) / dx) dy National Natural Science Foundation of hina under Grants nos and 37036, and the Beijing Natural Science Foundation under Grant no References [] Z. W. Shen, p estimates for p operators with certain potentials, Annales de l Institut Fourier, vol.45,no.,pp , 995. [].. Fefferman, The uncertainty principle, American Mathematical Society,vol.9,no.,pp.9 06,983. [3] J. Zhong, Harmornic analysis for some Schrödinger type operators [Ph.D. thesis], Princeton University, 993. [4] J. Dziubański and J. Zienkiewicz, Hardy space H associated to Schrödinger operator with potential satisfying reverse Hölder ineuality, Revista Matemática Iberoamericana, vol.5,pp , 999. [5] J. Dziubański and J. Zienkiewicz, H p spaces for Schrödinger operators, Fourier Analysis and Ralated Topics, Banach enter Publications,vol.56,pp.45 53,00. [6] J. Dziubański and J. Zienkiewicz, H P spaces associated with Schrödinger operators with potentials from reverse Hölder classes, ollouium Mathematicum, vol. 98, no., pp. 5 38, 003. [7] J. Z. Huang and H. P. iu, ittlewood-paley theory and Hardy spaces related to Schrödinger operators, Submitted. [8] M.TaiblesonandG.Weiss, Themolecularcharacterizationof certain Hardy spces, Asterisue, vol. 77, pp. 67 5, 980. [9] J. Alvarez and M. Milman, H p continuty properties of alderon-zygmund operators, Mathematical Analysis and Applications,vol.8,pp.65 79,986. y x 0 / dx x x 0 ρ(x 0 ) x x ) dy 0 (n+ nb) a(y) ρ(x 0) na dy ρ(x 0 ) na, (9) wherewehavetakenε such that 0<ε</n.Thisprovesthat x x 0 nb ( / ( Δ) / )a(x) ρ(x 0) na. (9) It follows that N(( / ( Δ) / )a). Theproofis completed. onflict of Interests The authors declare that there is no conflict of interests regarding the publication of this paper. Acknowledgments This work was supported by Research Fund for the Doctoral Program of Higher Education of hina ( ), the

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