Thông tin tài liệu
Liu and Wang Journal of Inequalities and Applications 2014, 2014:194 http://www.journalofinequalitiesandapplications.com/content/2014/1/194 RESEARCH Open Access Boundedness for Riesz transform associated with Schrödinger operators and its commutator on weighted Morrey spaces related to certain nonnegative potentials Yu Liu* and Lijuan Wang * Correspondence: liuyu75@pku.org.cn School of Mathematics and Physics, University of Science and Technology Beijing, Beijing, 100083, China Abstract Let L = – + V be a Schrödinger operator, where is the Laplacian on Rn and the nonnegative potential V belongs to the reverse Hölder class Bq for q ≥ n/2 The Riesz transform associated with the operator L is denoted by T = ∇(– + V)– and the dual Riesz transform is denoted by T ∗ = (– + V)– ∇ In this paper, we establish the boundedness for the operator T ∗ and its commutator on the weighted Morrey spaces p,λ Lα ,V,ω (Rn ) related to certain nonnegative potentials belonging to the reverse Hölder class Bq for n/2 ≤ q < n, where p0 < p < ∞ and p10 = q1 – n1 Keywords: Morrey spaces; commutator; reverse Hölder class; Schrödinger operator; Riesz transform Introduction In this paper, we consider the Schrödinger operator L = – + V (x) on Rn , n ≥ , where V (x) is a nonnegative potential belonging to the reverse Hölder class Bq for q ≥ n/ The Riesz transform associated with the Schrödinger operator L is defined by T = ∇L– and the commutator operator [b, T](f )(x) = T(bf )(x) – b(x)Tf (x), x ∈ Rn , () where f is a suitable integral function Also, the dual Riesz transform associated with the Schrödinger operator L is defined by T ∗ = L– ∇ and the commutator operator b, T ∗ (f )(x) = T ∗ (bf )(x) – b(x)T ∗ f (x), x ∈ Rn () First, Tang and Dong established the boundedness of some Schrödinger type operators on the Morrey spaces related to the nonnegative potential V belonging to the reverse Hölder class in [] Furthermore, Liu and Wang investigated the boundedness of the dual ©2014 Liu and Wang; licensee Springer This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited Liu and Wang Journal of Inequalities and Applications 2014, 2014:194 http://www.journalofinequalitiesandapplications.com/content/2014/1/194 Page of 16 Riesz transforms and its commutators on the Morrey spaces related to the nonnegative potential V belonging to the reverse Hölder class in [] Recently, Pan and Tang established the boundedness of some Schrödinger type operators on weighted Morrey spaces related to the nonnegative potential V belonging to the reverse Hölder class in [] Motivated by [], our aim is to establish the boundedness for the dual Riesz transform associated with Schrödinger operators and its commutators on weighted Morrey spaces related to the certain nonnegative potentials, where the condition on the potential is weaker than that in [] Our result is a nontrivial generalization of the main results in [] A nonnegative locally Lq integrable function V (x) on Rn is said to belong to Bq ( < q < ∞) if there exists C > such that the reverse Hölder inequality |B| /q |B| ≤C V (x)q dx B V (x) dx () B holds for every ball B in Rn It is important that the Bq class has a property of ‘self improvement’; that is, if V ∈ Bq , then V ∈ Bq+ε for some ε > (see []) We assume the potential V ∈ Bq for q ≥ n/ throughout the paper We introduce the auxiliary function ρ(x, V ) = ρ(x) defined by ρ(x) = sup r : r> rn– V (y) dy ≤ , x ∈ Rn B(x,r) It is well known that < ρ(x) < ∞ for any x ∈ Rn (cf Lemma in Section ) A kind of new Morrey spaces is established by Tang and Dong in [] Furthermore, the weighted Morrey space is introduced by Pan and Tang in [] Let p ∈ [, ∞), α ∈ (–∞, ∞), p p,λ and λ ∈ [, ) For f ∈ Lloc (Rn ) and V ∈ Bq (q > ), we say f ∈ Lα,V ,ω (Rn ) (weighted Morrey spaces related to the nonnegative potential V ) provided that f p p,λ Lα,V ,ω (Rn ) = + sup B(x ,r)⊂Rn r ρ(x ) α ω B(x , r) –λ p f (x) ω(x) dx < ∞, B(x ,r) where B = B(x , r) denotes a ball with centered at x and radius r, and the weight functions ω ∈ Aρ,∞ (see Section ) p Now we are in a position to give the main results in this paper Theorem Suppose V ∈ Bq for n/ ≤ q < n, α ∈ (–∞, ∞), λ ∈ (, ), and /p = /q – /n Then, for p ≤ p < ∞ and ω ∈ Aρ,∞ , p/p T ∗f p,λ Lα,V ,ω (Rn ) ≤C f p,λ Lα,V ,ω (Rn ) , where C is independent of f Theorem Suppose V ∈ Bq for n/ ≤ q < n, b ∈ BMOρ , α ∈ (–∞, ∞), λ ∈ (, ), and p so that /p = /q – /n Then, for p ≤ p < ∞ and ω ∈ Aρ,∞ , p/p b, T ∗ f p,λ Lα,V ,ω (Rn ) ≤C f where C is independent of f p,λ Lα,V ,ω (Rn ) , Liu and Wang Journal of Inequalities and Applications 2014, 2014:194 http://www.journalofinequalitiesandapplications.com/content/2014/1/194 Page of 16 We will use C to denote a positive constant, which is not necessarily same at each occurrence and even is different in the same line, and may depend on the dimension n and the constant in () By A ∼ B, we mean that there exists a constant C such that /C ≤ A/B ≤ C Some lemmas In this section, we collect some known results proved in [] in order to prove the main results in this paper Lemma There exist constants C, k > such that |x – y| + C ρ(x) –k ≤ k /(k +) |x – y| ρ(y) ≤C + ρ(x) ρ(x) In particular, ρ(y) ∼ ρ(x) if |x – y| < Cρ(x) Lemma () For < r < R < ∞, rn– r V (y) dy ≤ C R B(x,r) rn– B(x,r) – nq V (y) dy Rn– B(x,R) and V (y) dy ∼ if and only if r ∼ ρ(x) () There exist C > and l > such that Rn– V (y) dy ≤ C + B(x,R) R ρ(x) l Let K be the kernel of T and K∗ be the kernel of T ∗ Lemma If V ∈ Bq for q ≥ n/, then for every N there exists a constant CN > such that K∗ (x, z) ≤ CN ( + |x–z| N ) ρ(x) |x – z|n– B(z,|x–z|/) V (u) du + n– |u – z| |x – z| () Moreover, the last inequality also holds with ρ(x) replaced by ρ(z) In this paper, we always write θ (B) = ( + r/ρ(x ))θ , where θ > ; x and r denote the center and radius of B, respectively A weight will always mean a nonnegative function which is locally integrable As in [], ρ,θ we say that a weight ω belongs to the class Ap for < p < ∞, if there is a positive constant C such that for the whole ball B = B(x, r) θ (B)|B| ω(y) dy B θ (B)|B| ω – p– p– (y) dy ≤ C B ρ,θ We also say that a nonnegative function ω satisfies the A positive constant C, for all balls B MVθ (ω)(x) ≤ Cω(x), a.e x ∈ Rn , condition if there exists a Liu and Wang Journal of Inequalities and Applications 2014, 2014:194 http://www.journalofinequalitiesandapplications.com/content/2014/1/194 Page of 16 where MVθ f (x) = sup x∈B f (y) dy θ (B)|B| B Since θ (B) ≥ , obviously, Ap ⊂ Aρ,θ p for ≤ p < ∞, where Ap denote the classical Muckenhoupt weights (see []) It follows from [] that Ap ⊂⊂ Aρ,θ p for ≤ p < ∞ For conρ,∞ venience, we always assume that (B) denotes θ (B), Aρ,∞ = θ > Aρ,θ p p , and A∞ = ρ,∞ p≥ Ap Lemma ([]) Let < θ < ∞, then: ρ,θ (i) If ≤ p < p < ∞, then Aρ,θ p ⊂ Ap (ii) ω ∈ Aρ,θ p if and only if ω – p– ∈ Aρ,θ p , where /p + /p = (iii) If ω ∈ Aρ,θ p for ≤ p < ∞, then there exists a constant C > such that for any λ > ω λB(x , r) ≤ C + (k +)θ λr ρ(x ) ω B(x , r) Lemma ([]) Let < θ < ∞, ≤ p < ∞ If ω ∈ Aρ,θ p , then there exist positive constants δ, η, and C such that |B| /(+δ) ω(y)+δ dy ≤C B |B| ω(y) dy + B r ρ(x ) η for all ball B(x , r) As a consequence of Lemma , we have the following result ρ,θ Corollary ([]) Let < θ < ∞, ≤ p < ∞ If ω ∈ Ap , then there exist positive constants q > , η, and C such that |E| ω(E) ≤C ω(B) |B| /q + r ρ(x ) η for any measurable subset E of a ball B(x , r) Bongioanni et al [] introduced a new space BMOθ (ρ) defined by f BMOθ (ρ) θ (B)|B| = sup B⊂Rn f (x) – fB dx < ∞, B θ where fB = |B| B f (y) dy and θ (B) = ( + r/ρ(x )) , B = B(x , r), and θ > In particularly, Bongioanni et al [] proved the following results for BMOθ (ρ) Proposition Let θ > and ≤ s < ∞ If b ∈ BMOθ (ρ), then |B| s b(x) – bB dx B /s ≤C b BMOθ (ρ) + r ρ(x ) θ holds for all B = B(x , r), with x ∈ Rn and r > , where θ = (k + )θ and k is the constant appearing in Lemma Liu and Wang Journal of Inequalities and Applications 2014, 2014:194 http://www.journalofinequalitiesandapplications.com/content/2014/1/194 Page of 16 Proposition Let b ∈ BMOθ (ρ), B = B(x , r), and ≤ s < ∞ Then |k B| s k B s b(y) – bB dy ≤C b BMOθ (ρ) k + θ k r ρ(x ) () for all k ∈ N, with θ = (k + )θ and the constant k is given as in Proposition Obviously, the classical BMO space is properly contained in BMOθ (ρ); for more examples please see [] For convenience, we let BMOρ = θ > BMOθ (ρ) From Corollary . in [], the following result holds true ρ,∞ Corollary If b ∈ BMOρ and ω ∈ A∞ , then there exist positive constants C and η such that for every ball B = B(x, r), we have ω(B) p b(x) – bB ω(x) dx ≤ + B where bB = |B| r ρ(x) η b p BMOρ , B b(y) dy The proof of our main results Proof of Theorem Without loss of generality, we may assume that α < and ω ∈ Aρ,θ p/p Pick any ball B = B(x , r), and write f (x) = f (x) + f (x), where f = χB(x ,r) f Hence, we have p T ∗ f (x) ω(x) dx /p B(x ,r) p T ∗ f (x) ω(x) dx ≤ /p p T ∗ f (x) ω(x) dx + B(x ,r) /p () B(x ,r) By the Lω boundedness of T ∗ (see Theorem in []), we obtain p p T ∗ f (x) ω(x) dx ≤ C + B(x ,r) r ρ(x ) –α ω B(x , r) λ f p p,λ Lα,V ,ω (Rn ) () Now, for x ∈ B(x , r) and using Lemma , we have T ∗ f (x) = |x –z|>r K∗ (x, z)f (z) dz ≤ I (x) + I (x), () where I (x) = CN |f (z)| |x –z|>r |x – z|n ( + |x–z| N ) ρ(x) dz Liu and Wang Journal of Inequalities and Applications 2014, 2014:194 http://www.journalofinequalitiesandapplications.com/content/2014/1/194 Page of 16 and |f (z)| I (x) = CN n– ( + |x–z| )N |x –z|>r |x – z| ρ(x) B(z,|x–z|/) V (u) du dz |u – z|n– Then p T ∗ f (x) ω(x) dx /p B(x ,r) /p p ≤ I (x) ω(x) dx /p p I (x) ω(x) dx + B(x ,r) () B(x ,r) By the proof of Theorem . in [], we have p I (x) ω(x) dx ≤ C + B(x ,r) –α r ρ(x ) λ ω B(x , r) |x –z| Next we deal with I (x) For x ∈ B(x , r), f ≤ |x – z| ≤ p p,λ Lα,V ,ω (Rn ) |x –z| We get p I (x) ω(x) dx B(x ,r) |f (z)| = CN B(x ,r) |x –z|>r |x – z|n– ( + |x–z| N ) ρ(x) ∞ ≤ × B(x ,r) B(x ,i+ r) B(x ,i+ r)\B(x ,i r) CN B(x ,r) i= p V (u) du dz |u – z|n– ∞ ≤ ( + i r ρ(x) (B(x , i r))|B(x , i r)| )Np i r ω(x) dx p –(n–)p B(x ,i r) p p f (z) I (V χB(x ,i r) ) dz ω(x) dx = + pv By the definition of Aρ,θ , p/p /v B(x ,i r) (B(x , i r))|B(x , i r)| ≤C ω(x) dx –z| N |x – z|n– ( + |xρ(x) ) Let p ≤ p < ∞ By simple computation, = p |f (z)| CN i= B(z,|x–z|/) V (u) du dz |u – z|n– ω–v/p (y) dy B(x ,i r) i (B(x , r))|B(x , i r)| ω –p +– B(x ,i r) v p ( v +–) (y) dy ω(y) dy p v( v +–) – p , () where /q = /s + /n and /p + /v + /s = Using Hölder’s inequality, (), and the boundedness of the fractional integral I : Lq → Ls with /q = /s + /n, for /p + /v + /s = , we have B(x ,i r) f (x) I (V χB(x ,i r) ) dx = B(x ,i r) f (x) ω/p ω–/p I (V χB(x ,i r) ) dx Liu and Wang Journal of Inequalities and Applications 2014, 2014:194 http://www.journalofinequalitiesandapplications.com/content/2014/1/194 ≤ B x , i r B x , i r v B(x ,i r) B(x ,i r) ≤C B(x ,i r) /s v B x , i r ≤C ω(x) dx B(x ,i r) /s I (V χB(x ,i r) ) dx B x , i r × ω B x , i r ≤C f (x) ω(x) dx –/p s B(x ,i r) /p p B(x ,i r) (B(x , i r))|B(x , i r)| × ω(x)–v/p dx I (V χB(x ,i r) ) dx B x , i r × f (x) ω(x) dx /v s × /p p (B(x , i r))|B(x , i r)| × Page of 16 –/p B x , i r × ω B x , i r /p+/v B x , i r B(x ,i r) I (V χB(x ,i r) ) –/p s /p+/v B x , i r /p p f (x) ω(x) dx p B(x ,i r) /p f (x) ω(x) dx V χB(x ,i r) q () For V ∈ Bq , using Lemma , we get V χB(x ,i r) q ≤ C i r –n/q ≤ C i r –n/q +n– ≤ C i r –n/q +n– V (x) dx B(x ,i r) i r –n+ V (x) dx B(x ,i r) + i r ρ(x ) l () +(n–)p+n+ pn = Furthermore, using Corollary , It is easy to check that –(n–)p– pn q v we have ω(B(x , r)) ≤ C i ω(B(x , i+ r)) – nq + i r ρ(x ) η () Therefore, by (), p I (x) ω(x) dx B(x ,r) |f (z)| = CN B(x ,r) ∞ ≤ C N i r i= |x –z|>r –(n–)p– |x – z|n– ( + pn pn +(n–)p+n+ v q |x–z| N ) ρ(x) B(z,|x–z|/) V (u) du dz |u – z|n– p ω(x) dx Liu and Wang Journal of Inequalities and Applications 2014, 2014:194 http://www.journalofinequalitiesandapplications.com/content/2014/1/194 i r (+ pv )θ+l p ) ρ(x ) ( + × B(x ,r) λ– CN ω B x , r i+ λ– CN ω B x , r ≤ i+ B(x ,i r) ( + f p,λ Lα,V ,ω (Rn ) ω B(x , r) ≤ CN i= ω(B(x , r)) ω(B(x , i+ r)) ω B(x , r) ∞ ≤ CN –in(–λ)/q ω B(x , r) ≤C f p p,λ Lα,V ,ω (Rn ) i r Np/(k +) ) ρ(x ) ( + ( + p,λ Lα,V ,ω (Rn ) i r Np/(k +) ) ρ(x ) i r Np/(k +) ) ρ(x ) f ω(x) dx p f i r –α+(+ pv )θ+l p λ ( + ρ(x ) ) i r –α+(+ pv )θ+l p+η λ ( + ρ(x ) ) i= i r Np ) ρ(x) i r –α+(+ pv )θ+l p ) ρ(x ) ( + –λ i r –α+(+ pv )θ+l p ) ρ(x ) ( + B(x ,r) ( + i= ∞ f (y) ω(y) dyω(x) dx p i= ∞ p i r Np ) ρ(x) ω(B(x , i r))( + ∞ ≤ Page of 16 f p p,λ Lα,V ,ω (Rn ) p p,λ Lα,V ,ω (Rn ) , () where we choose N large enough so that the above series converges From ()-(), we obtain T ∗f p,λ Lα,V ,ω (Rn ) ≤C f p,λ Lα,V ,ω (Rn ) Thus, Theorem is proved Proof of Theorem During the proof of Theorem , we always denote θ = (k + )θ Without loss of generality, we may assume that α < , b ∈ BMOθ (ρ), and ω ∈ Aρ,θ Pick any ball p/p B = B(x , r), and write f (x) = f (x) + f (x), where f = χB(x ,r) f Hence, we have p b, T ∗ f (x) ω(x) dx /p B(x ,r) p b, T ∗ f (x) ω(x) dx ≤ /p p b, T ∗ f (x) ω(x) dx + B(x ,r) /p () B(x ,r) By the Lω boundedness of [b, T ∗ ] (see Theorem in []), we obtain p p b, T ∗ f (x) ω(x) dx ≤ C + B(x ,r) Set bB = |B(x ,r)| B(x ,r) b(x) dx r ρ(x ) –α ω B(x , r) λ f p p,λ Lα,V ,ω (Rn ) () Write [b, T ∗ ]f = (b – bB )T ∗ f – T ∗ (f (b – bB )) Then p b, T ∗ f (x) ω(x) dx /p B(x ,r) p (b – bB )T ∗ f ω(x) dx ≤ B(x ,r) /p T ∗ f (b – bB ) + B(x ,r) p /p ω(x) dx () Liu and Wang Journal of Inequalities and Applications 2014, 2014:194 http://www.journalofinequalitiesandapplications.com/content/2014/1/194 Page of 16 By () in the proof of Theorem , we obtain p (b – bB )T ∗ f ω(x) dx B(x ,r) p B(x ,r) p p p p r B(x ,r) |x – z|n ( + |x–z| N ) ρ(x) ∞ ≤ B(x ,i+ r)\B(x ,i r) B(x ,r) i= ∞ ≤ |b – bB |p CN B(x ,r) i= ( + ∞ ≤ |b – bB |p CN B(x ,r) i= p |f (z)| |b – bB |p CN ω(x) dx dz i r i r Np ) ρ(x) |x –z| N ) ρ(x) i r i r Np ) ρ(x) f (z) dz ω(x) dx –np p × B(x ,i r) f (z) ω(z)/p ω(z)–/p dz ∞ ≤ |b – bB |p CN B(x ,r) i= × p B(x ,i r) ( + ω(z)– p dz p p |b – bB |p CN i= B(x ,r) × + i r ρ(x ) (+ p )θ p ≤ |b – bB | p CN i= B(x ,r) ( + i r Np ) ρ(x) B x , i r ∞ ω(x) dx i r –np p B(x ,i r) f (z) ω(z) dz ω(x) dx ∞ ≤ i r Np ) ρ(x) + ( + i r pθ ) ρ(x ) ( + i r Np ) ρ(x) i r –np p B(x ,i r) p p ω(x) dx p –np B(x ,i r) ( + |x – z|n ( + dz ω B x , i r ω B x , i r – – f (z) ω(z) dz ω(x) dx Liu and Wang Journal of Inequalities and Applications 2014, 2014:194 http://www.journalofinequalitiesandapplications.com/content/2014/1/194 × p B(x ,i r) f (z) ω(z) dzω(x) dx ∞ ≤ CN ω B x , r i+ λ– p f ∞ CN ω B x , r i+ λ– p,λ ω B(x , r) i= ∞ ≤ ω(B(x , r)) ω(B(x , i+ r)) CN i= × ( + i r –α+pθ +η ) ρ(x ) ( + i r Np/(k +) ) ρ(x ) b ≤ CN ( + i r –α+pθ +η ) ρ(x ) ( + i r Np/(k +) ) ρ(x ) ω B(x , r) p BMOρ ω B(x , r) f i r –α+pθ ) ρ(x ) ( + b i r Np ) ρ(x) p BMOρ f ω(x) dx p p,λ Lα,V ,ω (Rn ) λ p p,λ Lα,V ,ω (Rn ) i r –α+pθ +η λ ( + ρ(x ) ) i= ( + B(x ,r) –λ ∞ –in(–λ)/q |b – bB | p Lα,V ,ω (Rn ) i= ≤ Page 10 of 16 ( + i r Np/(k +) ) ρ(x ) b p BMOρ f p p,λ Lα,V ,ω (Rn ) , () where we choose N large enough so that the above series converges For I (x), we assume n/ < q < n due to Lemma Then, since x ∈ B(x , r), we also have |x –z| ≤ |x – z| ≤ |x –z| Then p |b – bB |p I (x) ω(x) dx B(x ,r) |f (z)| |b – bB |p ≤ CN n– ( + |x–z| )N |x –z|>r |x – z| ρ(x) B(x ,r) × B(z,|x–z|/) V (u) du dz |u – z|n– p ω(x) dx ∞ ≤ |b – bB |p CN B(x ,r) i= × B(x ,i+ r) V (u) du dz |u – z|n– ∞ ≤ CN i= |f (z)| B(x ,i+ r)\B(x ,i r) B(x ,r) ( + i r Np ) ρ(x) i r p |x – z|n– ( + ω(x) dx –(n–)p |b – bB |p p × B(x ,i r) f (z) I (V χB(x ,i r) ) dz ω(x) dx By () and () in the proof of Theorem , we obtain p |b – bB |p I (x) ω(x) dx B(x ,r) |b – bB |p ≤ CN B(x ,r) × B(z,|x–z|/) |f (z)| n– ( + |x–z| )N |x –z|>r |x – z| ρ(x) V (u) du dz |u – z|n– p ω(x) dx |x –z| N ) ρ(x) Liu and Wang Journal of Inequalities and Applications 2014, 2014:194 http://www.journalofinequalitiesandapplications.com/content/2014/1/194 ∞ ≤ CN ω B x , i+ r λ– p f p,λ Lα,V ,ω (Rn ) i= × |b – bB | p ( + i r –α+(+p/v)θ +l p ) ρ(x ) ( + B(x ,r) ∞ ≤ CN ω B x , i+ r Page 11 of 16 λ– ω(x) dx i r Np ) ρ(x) ω B(x , r) i= ( + × i r –α+(+p/v)θ +l p+η ) ρ(x ) ( + ∞ ≤ i= ( + × p,λ Lα,V ,ω (Rn ) λ p BMOρ p b i r Np/(k +) ) ρ(x ) ∞ ≤ p f ω B(x , r) i r –α+(+p/v)θ +l p+η ) ρ(x ) ( + p BMOρ –λ ω(B(x , r)) ω(B(x , i+ r)) CN b i r Np/(k +) ) ρ(x ) f p,λ Lα,V ,ω (Rn ) λ CN –in(–λ)/q ω B(x , r) i= ( + × i r –α+(+p/v)θ +l p+η ) ρ(x ) ( + i r Np/(k +) ) ρ(x ) b p BMOρ f p () p,λ Lα,V ,ω (Rn ) if we choose N large enough Now, for x ∈ B(x , r) and using Lemma , we have T ∗ f (b – bB ) = |x –z|>r K∗ (x, z)f (z)(b – bB ) dz ≤ I˜ (x) + I˜ (x), () where I˜ (x) = CN |f (z)(b – bB )| |x –z|>r |x – z|n ( + |x–z| N ) ρ(x) dz and I˜ (x) = CN |f (z)(b – bB )| n– ( + |x–z| )N |x –z|>r |x – z| ρ(x) B(z,|x–z|/) V (u) du dz |u – z|n– Then, T ∗ f (b – bB ) p /p ω(x) dx B(x ,r) p I˜ (x) ω(x) dx ≤ B(x ,r) /p p I˜ (x) ω(x) dx + B(x ,r) /p () Liu and Wang Journal of Inequalities and Applications 2014, 2014:194 http://www.journalofinequalitiesandapplications.com/content/2014/1/194 Page 12 of 16 Firstly, we consider I˜ (x) By Proposition and (), for /p + /v + /s = , we have B(x ,i r) ≤ f (x) b(x) – bB dx B(x ,i r) ≤ f (x) ω/p ω–/p b(x) – bB dx B x , i r B x , i r × (B(x , i r))|B(x , i r)| × (B(x , i r))|B(x , i r)| × i (B(x , r))|B(x , i r)| ≤C B x , i r B(x ,i r) /v B(x ,i r) (B(x B(x ,i r) , i r)| × i (B(x , r))|B(x , i r)| /p+/v B x , i r |B(x , i r)| ≤ Ci + i r ρ(x ) × ω B x , i r ≤ Ci + i r ρ(x ) × ω B x , i r –/p B(x ,i r) ω(x) dx B(x ,i r) ω B x , i r B(x ,i r) –/p /s s b(x) – bB dx (/p+/v)θ+θ b B(x ,i r) i r n /p p B(x ,i r) b f (x) ω(x) dx BMOρ (/p+/v)θ+θ –/p /p p B x , i r –/p /s s b(x) – bB dx /p p × B(x ,i r) f (x) ω(x) dx B x , i r f (x) ω(x) dx B(x ,i r) /p p , i r))|B(x (B(x , i r))|B(x , i r)| × /s b(x) – bB dx B x , i r × ≤C ω(x)–v/p dx s × /p p f (x) ω(x) dx f (x) ω(x) dx BMOρ () Then we get p I˜ (x) ω(x) dx B(x ,r) B(x ,r) |x –z|>r |x – z|n ( + ∞ ≤ |x–z| N ) ρ(x) dz ω(x) dx |f (z)(b – bB )| CN i= p |f (z)(b – bB )| = CN B(x ,r) B(x ,i+ r)\B(x ,i r) |x – z|n ( + |x –z| N ) ρ(x) p dz ω(x) dx Liu and Wang Journal of Inequalities and Applications 2014, 2014:194 http://www.journalofinequalitiesandapplications.com/content/2014/1/194 ∞ ≤ CN B(x ,r) i= ( + ∞ ≤ CN i i= × ω B x , i r – b λ– CN ip ω B x , i+ r p f × i r –α+(+p/v)θ +pθ ) ρ(x ) ( + B(x ,r) i r Np ) ρ(x) ∞ ≤ λ– CN ip ω B x , i+ r p BMOρ b p,λ Lα,V ,ω (Rn ) i= ( + f (z) ω(z) dz B(x ,i r) p BMOρ ω(x) dx ∞ ≤ ω(x) dx p i r Np ) ρ(x) ( + B(x ,r) B(x ,i r) f (z)(b – bB ) dz i r (+p/v)θ+pθ ) ρ(x ) ( + p p –np i r i r Np ) ρ(x) Page 13 of 16 ω(x) dx ω B(x , r) i= × ( + i r –α+(+p/v)θ +pθ ) ρ(x ) ( + i r Np/(k +) ) ρ(x ) ∞ ≤ ω(B(x , r)) ω(B(x , i+ r)) CN ip i= × ( + i r –α+(+p/v)θ +pθ ) ρ(x ) ( + i r Np/(k +) ) ρ(x ) p BMOρ b p p,λ Lα,V ,ω (Rn ) –λ ω B(x , r) p BMOρ b ∞ ≤ f CN ip –in(–λ)/q ω B(x , r) f λ p p,λ Lα,V ,ω (Rn ) λ i= × ( + i r –α+(+p/v)θ +pθ +η ) ρ(x ) ( + i r Np/(k +) ) ρ(x ) b p BMOρ f p p,λ Lα,V ,ω (Rn ) , () where we choose N large enough so that the above series converges For V ∈ Bq , then V ∈ Bq+ε for ε > Using Lemma , we get V χB(x ,i r) q+ε ≤ C i r –n/(q+ε) ≤ C i r –n/(q+ε) +n– ≤ C i r –n/(q+ε) +n– V (x) dx B(x ,i r) i r –n+ V (x) dx B(x ,i r) + i r ρ(x ) l Let p ≤ p < ∞ We choose u such that u = q(q+ε) and /p + /v + /u + /s = ε Let /(q + ε) = /s + /n By simple computation, p =p – + p q n =p – =p – – q(q + ε) s + + – q n q+ε q+ε p =+ v () Liu and Wang Journal of Inequalities and Applications 2014, 2014:194 http://www.journalofinequalitiesandapplications.com/content/2014/1/194 Page 14 of 16 Finally, we deal with I˜ (x) Using Hölder’s inequality, (), and the boundedness of the fractional integral I : Lq+ε → Ls , for /p + /v + /u + /s = , we have B(x ,i r) ≤ f (x) b(x) – bB I (V χB(x ,i r) ) dx B(x ,i r) ≤ f (x) ω/p ω–/p b(x) – bB I (V χB(x ,i r) ) dx /v B x , i r /v+/u B x , i r B(x ,i r) /v i (B(x , r))|B(x , i r)| × |B(x , i r)| × ≤C B(x ,i r) /v B x , i r |B(x , i r)| × ≤C B(x ,i r) /p+/v |B(x , i r)| × ≤ Ci + i r ρ(x ) × ω B x , i r ≤ Ci + B(x ,i r) i r ρ(x ) × ω B x , i r /u B(x ,i r) –/p ω B x , i r /u I (V χB(x ,i r) ) b(x) – bB dx (/p+/v)θ+θ /p+/v+/u B x , i r (/p+/v)θ+θ +l i r –/p /p f (x) ω(x) dx q+ε (n–) /p p B(x ,i r) s p B(x ,i r) V χB(x ,i r) BMOρ /s s I (V χB(x ,i r) ) dx /p+/v+/u /p b f (x) ω(x) dx ω(x) dx u –/p /p p B(x ,i r) B x , i r B(x ,i r) I (V χB(x ,i r) ) dx –/p f (x) ω(x) dx B(x ,i r) B(x ,i r) b(x) – bB dx p /s s /v+/u B x , i r u B x , i r × /u b(x) – bB dx i (B(x , r))|B(x , i r)| × ω(x)–v/p dx u B(x ,i r) /p p f (x) ω(x) dx f (x) ω(x) dx b BMOρ () Then p I˜ (x) ω(x) dx B(x ,r) |f (z)(b – bB )| = CN B(x ,r) |x –z|>r |x – z|n– ( + ∞ ≤ B(z,|x–z|/) |f (z)(b – bB )| CN i= |x–z| N ) ρ(x) V (u) du dz |u – z|n– B(x ,r) B(x ,i+ r)\B(x ,i r) |x – z|n– ( + |x –z| N ) ρ(x) p ω(x) dx Liu and Wang Journal of Inequalities and Applications 2014, 2014:194 http://www.journalofinequalitiesandapplications.com/content/2014/1/194 × B(x ,i+ r) V (u) du dz |u – z|n– ∞ ≤ CN B(x ,r) i= ( + i r Np ) ρ(x) p Page 15 of 16 ω(x) dx –(n–)p i r p × B(x ,i r) f (z)(b – bB ) I (V χB(x ,i r) ) dz ∞ ≤ CN ip ω B x , i+ r λ– f i= ( + × ∞ ≤ p,λ Lα,V ,ω (Rn ) i r –α+(+p/v)θ +pθ +l p ) ρ(x ) ( + B(x ,r) p CN ip ω B x , i+ r i r Np ) ρ(x) λ– ω(x) dx p BMOρ b ω(x) dx ω B(x , r) i= × ( + i r –α+(+p/v)θ +pθ +l p ) ρ(x ) ( + ∞ ≤ CN ip i= × ( + ω(B(x , r)) ω(B(x , i+ r)) f p p,λ Lα,V ,ω (Rn ) ω B(x , r) λ p BMOρ p i r –α+(+p/v)θ +pθ +l p ) ρ(x ) ( + p BMOρ –λ b i r Np/(k +) ) ρ(x ) ∞ ≤ b i r Np/(k +) ) ρ(x ) CN ip –in(–λ)/q ω B(x , r) f p,λ Lα,V ,ω (Rn ) λ i= × ( + i r –α+(+p/v)θ +pθ +l p+η ) ρ(x ) ( + i r Np/(k +) ) ρ(x ) b p BMOρ f p p,λ Lα,V ,ω (Rn ) , () where we choose N large enough so that the above series converges From ()-(), we obtain b, T ∗ f p,λ Lα,V ,ω (Rn ) ≤C f p,λ Lα,V ,ω (Rn ) Thus, we complete the proof of Theorem Competing interests The authors declare that they have no competing interests Authors’ contributions All authors contributed equally to the writing of this paper All authors read and approved the final manuscript Acknowledgements The authors would like to thank the referee for carefully reading which made the presentation more readable This paper was supported by the National Natural Science Foundation of China under grant No 10901018, the Fundamental Research Funds for the Central Universities, Program for New Century Excellent Talents in University and Beijing Natural Science Foundation under grant No 1142005 Received: March 2014 Accepted: 28 April 2014 Published: 14 May 2014 Liu and Wang Journal of Inequalities and Applications 2014, 2014:194 http://www.journalofinequalitiesandapplications.com/content/2014/1/194 References Tang, L, Dong, JF: Boundedness for some Schrödinger type operators on Morrey spaces related to certain nonnegative potentials J Math Anal Appl 355, 101-109 (2009) Liu, Y, Wang, LJ: Boundedness for Riesz transform associated with Schrödinger operators and its commutator on Morrey spaces related to certain nonnegative potentials Pan, GX, Tang, L: Boundedness for some Schrödinger type operators on weighted Morrey spaces J Funct Spaces 2014, Article ID 878629 (2014) Shen, ZW: Lp Estimates for Schrödinger operators with certain potentials Ann Inst Fourier (Grenoble) 45, 513-546 (1995) Bongioanni, B, Harboure, E, Salinas, O: Analysis and assessment of ab initio three-dimensional prediction, secondary structure, and contacts prediction J Math Anal Appl 373, 563-579 (2011) Stein, EM: Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals Princeton University Press, Princeton (1993) Tang, L: Weighted norm inequalities for Schrödinger type operators Forum Math (2013) doi:10.1515/forum-2013-0070 Bongioanni, B, Harboure, E, Salinas, O: Weighted inequalities for commutators of Schrödinger-Riesz transforms J Math Anal Appl 392, 6-22 (2012) Bongioanni, B, Harboure, E, Salinas, O: Commutators of Riesz transforms related to Schrödinger operators J Fourier Anal Appl 17, 115-134 (2011) 10.1186/1029-242X-2014-194 Cite this article as: Liu and Wang: Boundedness for Riesz transform associated with Schrödinger operators and its commutator on weighted Morrey spaces related to certain nonnegative potentials Journal of Inequalities and Applications 2014, 2014:194 Page 16 of 16
Ngày đăng: 01/11/2022, 09:02
Xem thêm:
