Research Article Boundedness of Oscillatory Integrals with Variable Calderón-Zygmund Kernel on Weighted Morrey Spaces

Transcription

1 Function Spaces and Applications Volume 203, Article ID , 5 pages Research Article oundedness of Oscillatory Integrals with Variable Calderón-Zygmund Kernel on Weighted Morrey Spaces Yali Pan, Changwen Li, and Xinsong Wang 2 School of Mathematical Sciences, Huaibei Normal University, Huaibei, Anhui , China 2 School of Science, Tianjin Chengjian University, Tianjin , China Correspondence should be addressed to Changwen Li; cwli2008@63.com Received 7 August 203; Accepted October 203 Academic Editor: Yoshihiro Sawano Copyright 203 Yali Pan et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Oscillatory integral operators play a key role in harmonic analysis. In this paper, the authors investigate the boundedness of the oscillatory singular integrals with variable Calderón-Zygmund kernel on the weighted Morrey spaces L p,k (ω). Meanwhile,the corresponding results for the oscillatory singular integrals with standard Calderón-Zygmund kernel are established.. Introduction and Main Results Suppose that k is the standard Calderón-Zygmund kernel. That is, k C (R n \{0})is homogeneous of degree n, and Σ k(x)dσ x =0,whereΣ={x R n : x =}.Theoscillatory integral operator T λ is defined by T λ f (x) =p V R n e iλφ(x,y) k (x y) φ (x, y) f (y) dy, () where λ R, φ C 0 (Rn R n ),wherec 0 (Rn R n ) is the space of infinitely differentiable functions on R n R n with compact supports, and Φ is a real-analytic function or a real-c (R n R n ) function satisfying that, for any (x 0,y 0 ) supp φ, there exists (j 0,k 0 ), j 0, k 0 n,suchthat 2 Φ(x 0,y 0 )/ x j0 y k0 does not vanish up to infinite order. These operators have arisen in the study of singular integrals supported on lower dimensional varieties and the singular Radontransform.In[], Pan proved that T λ are uniform in λ bounded on L p (R n )(<p<).luetal.[2] provedthe weighted L p boundedness of T λ defined by (). Let k(x, y) be a variable Calderón-Zygmund kernel. That means, for a.e. x R n, k(x, ) is a standard Calderón- Zygmund kernel and j k max j 2n,j Z y j =A<. (2) L (R n Σ) Define the oscillatory integral operator with variable Calderón-Zygmund kernel T λ by T λ f (x) =p V e iλφ(x,y) k (x, x y) φ (x, y) f (y) dy, R n (3) where λ, φ, andφ satisfy the same assumptions as those in the operator defined by (). Lu et al. [2] investigatedthel p and weighted L p boundedness about this class of oscillatory integral operators. The classical Morrey space L p,λ was first introduced by Morrey in [3] to study the local behavior of solutions to second order elliptic partial differential equations. In 2009, Komori and Shirai [4] first defined the weighted Morrey spaces L p,κ (ω) which could be viewed as an extension of weighted Lebesgue spaces. They studied the boundedness of the fractional integral operator, the Hardy-Littlewood maximal operator, and the Calderón-Zygmund singular integral operator on the space. The boundedness results about some operators on these spaces can be see in ([5 7]). Recently, Shi et al. [8] obtained the boundedness of a class of oscillatory integrals with Calderón-Zygmund kernel and polynomial phase on weighted Morrey spaces. Their results are stated as follows.

2 2 Function Spaces and Applications Let P(x, y) be a real valued polynomial defined on R n R n and let k satisfy the following hypotheses: k(x,y) C x y n, x=y, (4) xk(x,y) + yk(x,y) C x y, x n+ =y. We define Sf (x) =p V R n k (x, y) f (y) dy, Rf (x) =p V R n e ip(x,y) k (x, y) f (y) dy. Theorem A (see [8]). Let < p <, 0 < κ <,and ω A p.ifs is of type (L 2,L 2 ), then, for any real polynomial P(x, y), there exists a constantc >0such that Rf L p,κ (ω) C f L p,κ (ω). (6) The purpose of this paper is to generalize the above results to the case with real-c or analytic phase functions. Our mainresultsinthispaperareformulatedasfollows. Theorem. Let λ R, φ C 0 (Rn R n ),andφ areal- C (R n R n ) function satisfying that, for any (x 0,y 0 ) supp φ, there exists (j 0,k 0 ), j 0, k 0 n,suchthat 2 Φ(x 0,y 0 )/ x j0 y k0 does not vanish up to infinite order. Assume that k is a standard Calderón-Zygmund kernel and T λ is defined as in (). Thenforany<p<, 0<κ<,and ω A p, T λ is bounded on L p,κ (ω). Theorem 2. Let λ R, φ C 0 (Rn R n ),andφ areal- C (R n R n ) function satisfying that, for any (x 0,y 0 ) supp φ, there exists (j 0,k 0 ), j 0, k 0 n,suchthat 2 Φ(x 0,y 0 )/ x j0 y k0 does not vanish up to infinite order. Assume that k is a variable Calderón-Zygmund kernel and T λ is defined as in (3). Thenforany<p<, 0<κ<,and ω A p, T λ is bounded on Lp,κ (ω). 2. Notations and Preliminary Lemmas Let =(x 0,r)be the ball with the center x 0 and radius r. Given a ball and λ>0, λ denotes the ball with the same center as whose radius is λ times that of. The classical A p weighted theory was first introduced by Muckenhoupt in [9]. A weight ω is a locally integrable function on R n,whichtakesvaluesin(0, ) a.e. For a given weight function ω, we denote the Lebesgue measure of by and the weighted measure of E by ω(e); thatis,ω(e) = E ω(x)dx. Given a weight ω, wesaythatω satisfies the doubling condition if there exists a constant D>0such that, for any ball,wehaveω(2) Dω(). We say ω A p with <p<, if there exists a constant C>0,suchthat ( ω (x) dx) ( ω(x) /(p ) p dx) C, (7) (5) for every ball R n.whenp=, ω A if there exists C>0,suchthat ω (x) dx Cess infω (x), (8) x for almost every x R n. We define A = p A p.aweight function ω is said to belong to the reverse Hölder class RH r if there exist two constants r>0and C>0suchthat the following reverse Hölder inequality holds: ( ω(x) r /r dx) C( ω (x) dx), (9) for every ball R n. It is well known that, if ω A p with p<,then there exists r>such that ω RH r. Lemma 3 (see [20]). Let ω A p, p,andr>0.thenfor any ball and λ>, ω (2) Cω(), ω (λ) Cλ np ω (), where C does not depend on nor on λ. (0) Lemma 4 (see [2]). Let ω RH r with r>. Then there exists aconstantc such that (r )/r ω (E) ω () C( E ), () for any measurable subset E of a ball. The weighted Morrey spaces were defined as follows. Definition 5 (see [4]). Let p<, 0<κ<,andω a weight function. Then the weighted Morrey space is defined by where L p,κ (ω) ={f L p loc (ω) : f L p,κ (ω) <}, (2) f L p,κ (ω) = sup ( ω() κ f (x) p /p ω (x) dx), (3) and the supremum is taken over all balls in R n.thespace L p loc (ω) is defined by L p loc (ω) ={f:fχ K L p (ω), (4) for every compact set K R n }. Our argument is based heavily on the following results. Lemma 6 (see [2]). Assume that T λ is defined as in (). Then for any <p<and ω A p,onehas T λf L p (ω) C(n,p,Φ,φ,C p,ω ) f L p (ω), (5) where C(n, p, Φ, φ, C p,ω ) is independent of λ, k,andf and = k C (Σ).

3 Function Spaces and Applications 3 Lemma 7 (see [2]). Assume that T λ is defined as in (3).Then for any <p<and ω A p,onehas T λ f L p (ω) C(n,p,Φ,φ,C p,ω )A f L p (ω), (6) where C(n, p, Φ, φ, C p,ω ) is independent of λ, k, andf. A is defined in (2). Definition 8 (see [4]). The Hardy-Littlewood maximal operator M is defined by Mf (x) = sup f(y) dy, f L loc (R n ). (7) x We now estimate I 2.Wecanwrite T λf 2 (x) = (2) c eiλφ(x,y) k (x y) φ (x, y) f (y) dy. (2) Now by an argument similar to the proof of Lemma 6 in [2], we choose φ C 0 (Rn ) such that φ (x),when x, andφ (x) 0 when x > 2. Letφ 2 = φ and N N which is large enough and will be determined later. Write k (x) =k λ (x) +k2 λ (x), (22) Lemma 9 (see [4]). If <p<, 0<κ<,andω A p then the Hardy-Littlewood maximal operator M is bounded on L p,κ (ω). where k j λ (x) =k(x) φ j (λ /N x), j =, 2. (23) Lemma 0 (see [22]). Denote by H m the spaces of spherical harmonic functions of degree m.then (a) L 2 (Σ) = m=0 H m,andg m = dim H m C(n)m n 2 for any m N; (b) for any m = 0,, 2,..., there exists an orthogonal system {Y jm } g m of H j= m such that Y jm L (Σ) C(n)m n/2, Y jm =( m) n (m+n 2) n Λ n Y jm, j=,...,g m,andλ is the eltrami-laplace operator on Σ. Then T λ f 2 (x) =p V (2) c eiλφ(x,y) k λ (x y) φ(x,y)f(y)dy +p V eiλφ(x,y) k 2 (2) c λ (x y) φ(x,y)f(y)dy (24) In the following the letter C will denote a constant which may vary at each occurrence. 3. Proof of Theorems Proof of Theorem. It is sufficient to prove that there exists a constant C>0such that ω() κ T λf (x) p ω (x) dx C f p L p,κ (ω). (8) Fix a ball =(x 0,r ) and decompose f=f +f 2,with f =fχ 2.Thenwehave ω() κ T λf (x) p ω (x) dx C{ ω() κ T λf (x) p ω (x) dx + ω() κ T λf 2 (x) p ω (x) dx} =C{I +I 2 }. Using Lemmas 3 and 6,weget I C ω() κ f (x) p ω (x) dx 2 C f p L p,κ (ω) ω(2)κ ω() κ C f p L p,κ (ω). (9) (20) := T λ f 2 (x) +T 2 λ f 2 (x). Let us first estimate T λ f 2(x). Todoso,usingTaylor s expansion and the compactness of supp φ,we write Φ(x,y)=Φ(x, x) +P(x,y)+r N (x, y) (25) for(x, y) supp φ, where P(x, y) is a polynomial with deg P<Nand r N (x, y) C x y N with C independent of x and y.define Rf (x) =p V eiλp(x,y) k (2) c λ (x y)φ(x,y)f(y)dy. (26) Therefore e iλφ(x,x) T λ f 2 (x) Rf(x) = x y 2λ /N e iλp(x,y) [e iλr N(x,y) ] = k λ (x y) φ (x, y) f (y) dy 2 j λ /N < x y 2 j+ λ /N e iλp(x,y) [e iλr N(x,y) ] k λ (x y)φ(x,y)f(y)dy T λ,j f 2 (x). (27)

4 4 Function Spaces and Applications On T λ,j f 2(x), by the properties of r N and k,wehave So we have T λ,j f 2 (x) C2 jn Mf (x). (28) T λ f 2 (x) C 2 jn Mf (x) +C Rf (x). (29) y Theorem A and Lemma 9,wehave T λ f 2 L C p,κ (ω) f L p,κ (ω). (30) Now, let us turn to estimate T 2 λ f 2(x). We consider the following two cases. Case (λ ). Similar to that estimate of T 2 λ in Lemma 6 in [2], we have T2 λ f 2 (x) CM(f)(x). (3) y Lemma 9 we have T2 λ f 2 L p,κ (ω) C f L p,κ (ω). (32) Case 2 (λ >). We choose φ 0 C 0 (Rn ) such that supp φ 0 {x R n :< x 2}, Let Then φ 2 (x) = φ 0 (2 j x). (33) k 2 λ,j (x) =k(x) φ 0 (2 j λ /N x). (34) T 2 λ f 2 (x) = (2) c eiλφ(x,y) k 2 λ (x y)φ(x,y)f(y)dy, eiλφ(x,y) k 2 (2) c λ,j (x y) φ (x, y) f (y) dy T 2 λ,j f 2 (x). For T 2 λ,j, by its definition, we can get (35) T2 λ,j f 2 (x) C 2 j λ /N < x y 2 j+ λ /N (36) c x y n f(y) dy CM (f) (x). The inequality (36) alsocanbeseenin[2]; we omit the details here. y Lemma 9,wehave T2 λ f 2 L C p,κ (ω) f L p,κ (ω). (37) Therefore I 2 C f p L p,κ (ω). (38) This finishes the proof of Theorem. Proof of Theorem 2. It is sufficient to prove that there exists a constant C>0such that ω() κ T λ f (x) p ω (x) dx C f p L p,κ (ω). (39) Fix a ball =(x 0,r ) and decompose f=f +f 2,with f =fχ 2.Thenwehave ω() κ T λ f (x) p ω (x) dx C{ ω() κ T λ f (x) p ω (x) dx + ω() κ T λ f 2 (x) p ω (x) dx} =C{J +J 2 }. Using Lemmas 3 and 7,weget J C ω() κ f (x) p ω (x) dx 2 C f p L p,κ (ω) ω(2)κ ω() κ C f p L p,κ (ω). We now estimate J 2. For each m N and j=,...,g m,weget (40) (4) a jm (x) = Ω (x, z) Y jm (z) dσ z, (42) Σ where Ω(x, z) = z n k(x, z).thenfora.e.x R n, g m m=j= Ω (x, z) = a jm (x) Y jm (z ), (43) where z = z/ z for any z R n \{0}.yLemma 0,wehave that, for any x R n, a jm (x) =m n (m+n 2) n Σ Ω (x, z) Λ n Y jm (z) dσ z =m n (m+n 2) n Λ n Ω (x, z) Y jm (z) dσ z Σ C(n) Am 2n. (44) y Lemma 0 again, we can verify that, for any ε>0, N N,anda.e.x R n,if y x ε,then N g m e iλφ(x,y) a jm (x) Y jm ((x y) ) m=j= x y n φ(x,y)f 2 (y) (45) C(ε) A f 2 (y).

5 Function Spaces and Applications 5 Therefore, from (43), (45), and the Lebesgue dominated convergence theorem, it follows that T λ f 2 (x) We write = lim e iλφ(x,y) k(x,x y)φ(x,y)f 2 (y) dy ε 0 x y ε g m ε 0 m=j= = lim g m ε 0 m=j= = lim R jm f 2 (x) = x y ε e iλφ(x,y) a jm (x) Y jm ((x y) ) x y ε x y n a jm (x) φ(x,y)f 2 (y) dy x y ε e iλφ(x,y) Y jm ((x y) ) x y n φ (x, y) f 2 (y) dy. e iλφ(x,y) Y jm ((x y) ) x y n φ(x,y)f 2 (y) dy. (46) (47) It is easy to see that R jm f 2 (x) is the oscillatory integral operator defined by (). y Theorem we have that R jm is bounded on weighted Morrey spaces. Therefore, by (44) and the above discussion we have This finishes the proof of Theorem 2. Acknowledgments J 2 C f p L p,κ (ω). (48) This work is supported by the National Natural Science Foundation of China (Grant no. 0000), Natural Science Foundation from the Education Department of Anhui Province (nos. KJ20266, KJ203A235). References [] Y. Pan, Uniform estimates for oscillatory integral operators, JournalofFunctionalAnalysis,vol.00,no.,pp ,99. [2]S.Lu,D.Yang,andZ.Zhou, Onlocaloscillatoryintegrals with variable Calderón-Zygmund kernels, Integral Equations and Operator Theory,vol.33,no.4,pp ,999. [3] C.. Morrey, Jr., On the solutions of quasi-linear elliptic partial differential equations, Transactions of the American Mathematical Society,vol.43,no.,pp.26 66,938. [4] Y. Komori and S. Shirai, Weighted Morrey spaces and a singular integral operator, Mathematische Nachrichten, vol. 282, no. 2, pp , [5] F. Chiarenza and M. Frasca, Morrey spaces and Hardy- Littlewood maximal function, Rendiconti di Matematica e delle sue Applicazioni,vol.7,no.3-4,pp ,987. [6] J. Peetre, On the theory of L p,λ spaces, JournalofFunctional Analysis,vol.4,no.,pp.7 87,969. [7] S.Z.Lu,Y.Ding,andD.Y.Yan,Singular Integrals and Related Topics, World Scientific Publishing, River Edge, NJ, USA, [8] E. Nakai, Hardy-Littlewood maximal operator, singular integral operators and the Riesz potentials on generalized Morrey spaces, Mathematische Nachrichten,vol.66,pp.95 03,994. [9] Y. Sawano and H. Tanaka, Morrey spaces for non-doubling measures, Acta Mathematica Sinica, vol. 2, no. 6, pp , [0]H.WangandH.P.Liu, Someestimatesforochner-Riesz operators on the weighted Morrey spaces, Acta Mathematica Sinica,vol.55,no.3,pp ,202. [] H. Wang and H. P. Liu, Weak type estimates of intrinsic square functions on the weighted Hardy spaces, Archiv der Mathematik,vol.97,no.,pp.49 59,20. [2] R. Ch. Mustafayev, On boundedness of sublinear operators in weighted Morrey spaces, Azerbaijan Mathematics, vol.2,no.,pp.66 79,202. [3]X.F.YeandX.S.Zhu, Estimatesofsingularintegralsand multilinear commutators in weighted Morrey spaces, Journal of Inequalities and Applications,vol.202,article302,202. [4] H. Wang, The boundedness of some operators with rough kernel on the weighted Morrey spaces, Acta Mathematica Sinica, Chinese Series,vol.55,no.4,pp ,202(Chinese). [5] H. Wang, The boundedness of fractional integral operators with rough kernels on the weighted Morrey spaces, Acta Mathematica Sinica, Chinese Series, vol.56,no.2,pp.75 86, 203 (Chinese). [6] S. He, The boundedness of some multilinear operator with rough kernel on the weighted Morrey spaces, submitted, [7] H. Wang, Intrinsic square functions on the weighted Morrey spaces, Mathematical Analysis and Applications,vol. 396, no., pp , 202. [8] S. G. Shi, Z. W. Fu, and S. Z. Lu, oundedness of oscillatory integral operators and their commutators on weighted Morrey spaces, Scientia Sinica Mathematica, vol. 43, pp , 203 (Chinese). [9]. Muckenhoupt, Weighted norm inequalities for the Hardy maximal function, Transactions of the American Mathematical Society,vol.65,pp ,972. [20] J. García-Cuerva and J. L. Rubio de Francia, Weighted Norm Inequalities and Related Topics,vol.6ofNorth-Holland Mathematics Studies,North-Holland,Amsterdam,TheNetherlands, 985. [2] R. F. Gundy and R. L. Wheeden, Weighted integral inequalities for the nontangential maximal function, Lusin area integral, and Walsh-Paley series, Studia Mathematica, vol. 49, pp , 974. [22] E. M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton University Press, Princeton, NJ, USA, 97.

6 Advances in Operations Research Advances in Decision Sciences Applied Mathematics Algebra Probability and Statistics The Scientific World Journal International Differential Equations Submit your manuscripts at International Advances in Combinatorics Mathematical Physics Complex Analysis International Mathematics and Mathematical Sciences Mathematical Problems in Engineering Mathematics Discrete Mathematics Discrete Dynamics in Nature and Society Function Spaces Abstract and Applied Analysis International Stochastic Analysis Optimization