JFSA Journal of Function Spaces and Applications 1758-4965 0972-6802 Hindawi Publishing Corporation 946435 10.1155/2013/946435 946435 Research Article Boundedness of Oscillatory Integrals with Variable Calderón-Zygmund Kernel on Weighted Morrey Spaces http://orcid.org/0000-0002-3179-0151 Pan Yali 1 Li Changwen 1 Wang Xinsong 2 Sawano Yoshihiro 1 School of Mathematical Sciences Huaibei Normal University Huaibei, Anhui 235000 China hbcnc.edu.cn 2 School of Science, Tianjin Chengjian University Tianjin 300384 China tjuci.edu.cn 2013 13 11 2013 2013 17 08 2013 11 10 2013 2013 Copyright © 2013 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 Lp,k(ω). Meanwhile, the corresponding results for the oscillatory singular integrals with standard Calderón-Zygmund kernel are established.

1. Introduction and Main Results

Suppose that k is the standard Calderón-Zygmund kernel. That is, kC(n{0}) is homogeneous of degree n, and Σk(x)dσx=0, where Σ={xn:|x|=1}. The oscillatory integral operator Tλ is defined by (1)Tλf(x)=p·v·neiλΦ(x,y)k(x-y)φ(x,y)f(y)dy, where λ,φC0(n×n), where C0(n×n) is the space of infinitely differentiable functions on n×n with compact supports, and Φ is a real-analytic function or a real-C(n×n) function satisfying that, for any (x0,y0)suppφ, there exists (j0,k0), 1j0, k0n, such that 2Φ(x0,y0)/xj0yk0 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 Radon transform. In , Pan proved that Tλ are uniform in λ bounded on Lp(n)(1<p<). Lu et al.  proved the weighted Lp boundedness of Tλ defined by (1).

Let k(x,y) be a variable Calderón-Zygmund kernel. That means, for a.e. xn,k(x,·) is a standard Calderón-Zygmund kernel and (2)max|j|2n,j|j|kyjL(n×Σ)=A<. Define the oscillatory integral operator with variable Calderón-Zygmund kernel Tλ* by (3)Tλ*f(x)=p·v·neiλΦ(x,y)k(x,x-y)φ(x,y)f(y)dy, where λ, φ, and Φ satisfy the same assumptions as those in the operator defined by (1).

Lu et al.  investigated the Lp and weighted Lp boundedness about this class of oscillatory integral operators.

The classical Morrey space Lp,λ was first introduced by Morrey in  to study the local behavior of solutions to second order elliptic partial differential equations. In 2009, Komori and Shirai  first defined the weighted Morrey spaces Lp,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 (). Recently, Shi et al.  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.

Let P(x,y) be a real valued polynomial defined on n×n and let k satisfy the following hypotheses: (4)|k(x,y)|C|x-y|n,xy,|xk(x,y)|+|yk(x,y)|C|x-y|n+1,xy. We define (5)Sf(x)=p·v·nk(x,y)f(y)dy,Rf(x)=p·v·neiP(x,y)k(x,y)f(y)dy.

Theorem A (see [<xref ref-type="bibr" rid="B18">18</xref>]).

Let 1<p<, 0<κ<1, and ωAp. If S is of type (L2,L2), then, for any real polynomial P(x,y), there exists a constant C>0 such that (6)RfoLp,κ(ω)CfLp,κ(ω).

The purpose of this paper is to generalize the above results to the case with real-C or analytic phase functions. Our main results in this paper are formulated as follows.

Theorem 1.

Let λ, φC0(n×n), and Φ a real-C(n×n) function satisfying that, for any (x0,y0)suppφ, there exists (j0,k0), 1j0, k0n, such that 2Φ(x0,y0)/xj0yk0 does not vanish up to infinite order. Assume that k is a standard Calderón-Zygmund kernel and Tλ is defined as in (1). Then for any 1<p<, 0<κ<1, and ωAp, Tλ is bounded on Lp,κ(ω).

Theorem 2.

Let λ, φC0(n×n), and Φ a real-C(n×n) function satisfying that, for any (x0,y0)suppφ, there exists (j0,k0), 1j0, k0n, such that 2Φ(x0,y0)/xj0yk0 does not vanish up to infinite order. Assume that k is a variable Calderón-Zygmund kernel and Tλ* is defined as in (3). Then for any 1<p<, 0<κ<1, and ωAp, Tλ* is bounded on Lp,κ(ω).

2. Notations and Preliminary Lemmas

Let B=B(x0,r) be the ball with the center x0 and radius r. Given a ball B and λ>0, λB denotes the ball with the same center as B whose radius is λ times that of B.

The classical Ap weighted theory was first introduced by Muckenhoupt in . A weight ω is a locally integrable function on n, which takes values in (0,) a.e. For a given weight function ω, we denote the Lebesgue measure of B by |B| and the weighted measure of E by ω(E); that is, ω(E)=Eω(x)dx. Given a weight ω, we say that ω satisfies the doubling condition if there exists a constant D>0 such that, for any ball B, we have ω(2B)Dω(B).

We say ωAp with 1<p<, if there exists a constant C>0, such that (7)(1|B|Bω(x)dx)(1|B|Bω(x)-1/(p-1)dx)p-1C, for every ball Bn. When p=1, ωA1 if there exists C>0, such that (8)1|B|Bω(x)dxCessinfxBω(x), for almost every xn. We define A=p1Ap. A weight function ω is said to belong to the reverse Hölder class RHr if there exist two constants r>0 and C>0 such that the following reverse Hölder inequality holds: (9)(1|B|Bω(x)rdx)1/rC(1|B|Bω(x)dx), for every ball Bn.

It is well known that, if ωAp with 1p<, then there exists r>1 such that ωRHr.

Lemma 3 (see [<xref ref-type="bibr" rid="B20">20</xref>]).

Let ωAp, p1, and r>0. Then for any ball B and λ>1, (10)ω(2B)Cω(B),ω(λB)Cλnpω(B), where C does not depend on B nor on λ.

Lemma 4 (see [<xref ref-type="bibr" rid="B21">21</xref>]).

Let ωRHr with r>1. Then there exists a constant C such that (11)ω(E)ω(B)C(|E||B|)(r-1)/r, for any measurable subset E of a ball B.

The weighted Morrey spaces were defined as follows.

Definition 5 (see [<xref ref-type="bibr" rid="B4">4</xref>]).

Let 1p<, 0<κ<1, and ω a weight function. Then the weighted Morrey space is defined by (12)Lp,κ(ω)={fLlocp(ω):foLp,κ(ω)<}, where (13)ofLp,κ(ω)=supB(1ω(B)κB|f(x)|pω(x)dx)1/p, and the supremum is taken over all balls B in n. The space Llocp(ω) is defined by (14)Llocp(ω)={f:fχKLp(ω),foreverycompactsetKn}. Our argument is based heavily on the following results.

Lemma 6 (see [<xref ref-type="bibr" rid="B2">2</xref>]).

Assume that Tλ is defined as in (1). Then for any 1<p< and ωAp, one has (15)TλfoLp(ω)C(n,p,Φ,φ,Cp,ω)BfoLp(ω), where C(n,p,Φ,φ,Cp,ω) is independent of λ, k, and f and B=kC1(Σ).

Lemma 7 (see [<xref ref-type="bibr" rid="B2">2</xref>]).

Assume that Tλ* is defined as in (3). Then for any 1<p< and ωAp, one has (16)Tλ*foLp(ω)C(n,p,Φ,φ,Cp,ω)AfoLp(ω), where C(n,p,Φ,φ,Cp,ω) is independent of λ, k, and f. A is defined in (2).

Definition 8 (see [<xref ref-type="bibr" rid="B4">4</xref>]).

The Hardy-Littlewood maximal operator M is defined by (17)Mf(x)=supBx1|B|B|f(y)|dy,fLloc(n).

Lemma 9 (see [<xref ref-type="bibr" rid="B4">4</xref>]).

If 1<p<, 0<κ<1, and ωAp then the Hardy-Littlewood maximal operator M is bounded on Lp,κ(ω).

Lemma 10 (see [<xref ref-type="bibr" rid="B22">22</xref>]).

Denote by m the spaces of spherical harmonic functions of degree m. Then

L2(Σ)=m=0m, and gm=dimmC(n)mn-2 for any m;

for any m=0,1,2,, there exists an orthogonal system {Yjm}j=1gm of m such that YjmL(Σ)C(n)mn/2-1, Yjm=(-m)-n(m+n-2)-nΛnYjm, j=1,,gm, and Λ is the Beltrami-Laplace operator on Σ.

In the following the letter C will denote a constant which may vary at each occurrence.

3. Proof of Theorems Proof of Theorem <xref ref-type="statement" rid="thm1.1">1</xref>.

It is sufficient to prove that there exists a constant C>0 such that (18)1ω(B)κB|Tλf(x)|pω(x)dxCfLp,κ(ω)p.

Fix a ball B=B(x0,rB) and decompose f=f1+f2, with f1=fχ2B. Then we have (19)1ω(B)κB|Tλf(x)|pω(x)dxC{1ω(B)κB|Tλf1(x)|pω(x)dx+1ω(B)κB|Tλf2(x)|pω(x)dx}=C{I1+I2}.

Using Lemmas 3 and 6, we get (20)I1C1ω(B)κ2B|f(x)|pω(x)dxCfLp,κ(ω)p·ω(2B)κω(B)κCfLp,κ(ω)p. We now estimate I2. We can write (21)|Tλf2(x)|=|(2B)ceiλΦ(x,y)k(x-y)φ(x,y)f(y)dy|.

Now by an argument similar to the proof of Lemma  6 in , we choose ϕ1C0(n) such that ϕ1(x)1, when |x|1, and ϕ1(x)0 when |x|>2. Let ϕ2=1-ϕ1 and N which is large enough and will be determined later. Write (22)k(x)=kλ1(x)+kλ2(x), where (23)kλj(x)=k(x)ϕj(λ1/Nx),j=1,2. Then (24)Tλf2(x)=p·v·(2B)ceiλΦ(x,y)kλ1(x-y)×φ(x,y)f(y)dy+p·v·(2B)ceiλΦ(x,y)kλ2(x-y)×φ(x,y)f(y)dy:=Tλ1f2(x)+Tλ2f2(x).

Let us first estimate Tλ1f2(x). To do so, using Taylor’s expansion and the compactness of suppφ, we write (25)Φ(x,y)=Φ(x,x)+P(x,y)+rN(x,y) for(x,y)suppφ, where P(x,y) is a polynomial with deg P<N and |rN(x,y)|C|x-y|N with C independent of x and y. Define (26)Rf(x)=p·v·(2B)ceiλP(x,y)kλ1(x-y)φ(x,y)f(y)dy. Therefore (27)e-iλΦ(x,x)Tλ1f2(x)-Rf(x)=|x-y|2λ-1/NeiλP(x,y)[eiλrN(x,y)-1]×kλ1(x-y)φ(x,y)f(y)dy=j=02-jλ-1/N<|x-y|2-j+1λ-1/NeiλP(x,y)[eiλrN(x,y)-1]×kλ1(x-y)φ(x,y)f(y)dyj=0Tλ,j1f2(x). On Tλ,j1f2(x), by the properties of rN and k, we have (28)|Tλ,j1f2(x)|C2-jNMf(x).

So we have (29)|Tλ1f2(x)|Cj=02-jN|Mf(x)|+C|Rf(x)|. By Theorem A and Lemma 9, we have (30)Tλ1f2Lp,κ(ω)CfLp,κ(ω).

Now, let us turn to estimate Tλ2f2(x). We consider the following two cases.

Case 1 (λ1). Similar to that estimate of Tλ2 in Lemma 6 in , we have (31)|Tλ2f2(x)|CM(f)(x).

By Lemma 9 we have (32)Tλ2f2Lp,κ(ω)CfLp,κ(ω).

Case 2 (λ>1). We choose φ0C0(n) such that (33)suppφ0{xn:1<|x|2},ϕ2(x)=j=0φ0(2-jx). Let (34)kλ,j2(x)=k(x)φ0(2-jλ1/Nx). Then (35)Tλ2f2(x)=(2B)ceiλΦ(x,y)kλ2(x-y)φ(x,y)f(y)dy,j=0(2B)ceiλΦ(x,y)kλ,j2(x-y)φ(x,y)f(y)dyj=0Tλ,j2f2(x). For Tλ,j2, by its definition, we can get (36)|Tλ,j2f2(x)|C2jλ-1/N<|x-y|2j+1λ-1/N×1|x-y|n|f(y)|dyCM(f)(x). The inequality (36) also can be seen in ; we omit the details here.

By Lemma 9, we have (37)Tλ2f2Lp,κ(ω)CfLp,κ(ω). Therefore (38)I2CfLp,κ(ω)p. This finishes the proof of Theorem 1.

Proof of Theorem <xref ref-type="statement" rid="thm1.2">2</xref>.

It is sufficient to prove that there exists a constant C>0 such that (39)1ω(B)κB|Tλ*f(x)|pω(x)dxCfLp,κ(ω)p. Fix a ball B=B(x0,rB) and decompose f=f1+f2, with f1=fχ2B. Then we have (40)1ω(B)κB|Tλ*f(x)|pω(x)dxC{1ω(B)κB|Tλ*f1(x)|pω(x)dx+1ω(B)κB|Tλ*f2(x)|pω(x)dx}=C{J1+J2}. Using Lemmas 3 and 7, we get (41)J1C1ω(B)κ2B|f(x)|pω(x)dxCfLp,κ(ω)p·ω(2B)κω(B)κCfLp,κ(ω)p. We now estimate J2.

For each m and j=1,,gm, we get (42)ajm(x)=ΣΩ(x,z)Yjm(z)dσz, where Ω(x,z)=|z|nk(x,z). Then for a.e. xn, (43)Ω(x,z)=m=1j=1gmajm(x)Yjm(z), where z=z/|z| for any zn{0}. By Lemma 10, we have that, for any xn, (44)|ajm(x)|=m-n(m+n-2)-n|ΣΩ(x,z)ΛnYjm(z)dσz|=m-n(m+n-2)-n|ΣΛnΩ(x,z)Yjm(z)dσz|C(n)Am-2n.

By Lemma 10 again, we can verify that, for any ϵ>0, N, and a.e. xn, if |y-x|ϵ, then (45)|m=1Nj=1gmeiλΦ(x,y)ajm(x)Yjm((x-y))|x-y|nφ(x,y)f2(y)|C(ϵ)A|f2(y)|.

Therefore, from (43), (45), and the Lebesgue dominated convergence theorem, it follows that (46)Tλ*f2(x)=limϵ0|x-y|ϵeiλΦ(x,y)k(x,x-y)φ(x,y)f2(y)dy=limϵ0m=1j=1gm|x-y|ϵeiλΦ(x,y)ajm(x)Yjm((x-y))|x-y|n×φ(x,y)f2(y)dy=limϵ0m=1j=1gmajm(x)|x-y|ϵeiλΦ(x,y)Yjm((x-y))|x-y|n×φ(x,y)f2(y)dy. We write (47)Rjmf2(x)=|x-y|ϵeiλΦ(x,y)Yjm((x-y))|x-y|nφ(x,y)f2(y)dy. It is easy to see that Rjmf2(x) is the oscillatory integral operator defined by (1). By Theorem 1 we have that Rjm is bounded on weighted Morrey spaces. Therefore, by (44) and the above discussion we have (48)J2CfLp,κ(ω)p.

This finishes the proof of Theorem 2.

Acknowledgments

This work is supported by the National Natural Science Foundation of China (Grant no. 11001001), Natural Science Foundation from the Education Department of Anhui Province (nos. KJ2012B166, KJ2013A235).

Pan Y. Uniform estimates for oscillatory integral operators Journal of Functional Analysis 1991 100 1 207 220 10.1016/0022-1236(91)90108-H MR1124299 Lu S. Yang D. Zhou Z. On local oscillatory integrals with variable Calderón-Zygmund kernels Integral Equations and Operator Theory 1999 33 4 456 470 10.1007/BF01291837 MR1682811 Morrey, C. B. Jr. On the solutions of quasi-linear elliptic partial differential equations Transactions of the American Mathematical Society 1938 43 1 126 166 10.2307/1989904 MR1501936 Komori Y. Shirai S. Weighted Morrey spaces and a singular integral operator Mathematische Nachrichten 2009 282 2 219 231 10.1002/mana.200610733 MR2493512 Chiarenza F. Frasca M. Morrey spaces and Hardy-Littlewood maximal function Rendiconti di Matematica e delle sue Applicazioni 1987 7 3-4 273 279 MR985999 Peetre J. On the theory of Lp,λ spaces Journal of Functional Analysis 1969 4 1 71 87 2-s2.0-0002667555 Lu S. Z. Ding Y. Yan D. Y. Singular Integrals and Related Topics 2007 River Edge, NJ, USA World Scientific Publishing Nakai E. Hardy-Littlewood maximal operator, singular integral operators and the Riesz potentials on generalized Morrey spaces Mathematische Nachrichten 1994 166 95 103 10.1002/mana.19941660108 MR1273325 Sawano Y. Tanaka H. Morrey spaces for non-doubling measures Acta Mathematica Sinica 2005 21 6 1535 1544 10.1007/s10114-005-0660-z MR2190025 Wang H. Liu H. P. Some estimates for Bochner-Riesz operators on the weighted Morrey spaces Acta Mathematica Sinica 2012 55 3 551 560 MR2977591 Wang H. Liu H. P. Weak type estimates of intrinsic square functions on the weighted Hardy spaces Archiv der Mathematik 2011 97 1 49 59 10.1007/s00013-011-0264-z MR2820587 Mustafayev R. Ch. On boundedness of sublinear operators in weighted Morrey spaces Azerbaijan Journal of Mathematics 2012 2 1 66 79 MR2967285 Ye X. F. Zhu X. S. Estimates of singular integrals and multilinear commutators in weighted Morrey spaces Journal of Inequalities and Applications 2012 2012, article 302 10.1186/1029-242X-2012-302 MR3017333 Wang H. The boundedness of some operators with rough kernel on the weighted Morrey spaces Acta Mathematica Sinica, Chinese Series 2012 55 4 589 600 Wang H. The boundedness of fractional integral operators with rough kernels on the weighted Morrey spaces Acta Mathematica Sinica, Chinese Series 2013 56 2 175 186 He S. The boundedness of some multilinear operator with rough kernel on the weighted Morrey spaces submitted, http://arxiv.org/abs/1111.5463 Wang H. Intrinsic square functions on the weighted Morrey spaces Journal of Mathematical Analysis and Applications 2012 396 1 302 314 10.1016/j.jmaa.2012.06.021 MR2956964 Shi S. G. Fu Z. W. Lu S. Z. Boundedness of oscillatory integral operators and their commutators on weighted Morrey spaces Scientia Sinica Mathematica 2013 43 147 158 Muckenhoupt B. Weighted norm inequalities for the Hardy maximal function Transactions of the American Mathematical Society 1972 165 207 226 MR0293384 García-Cuerva J. Rubio de Francia J. L. Weighted Norm Inequalities and Related Topics 1985 116 Amsterdam, The Netherlands North-Holland North-Holland Mathematics Studies MR807149 Gundy R. F. Wheeden R. L. Weighted integral inequalities for the nontangential maximal function, Lusin area integral, and Walsh-Paley series Studia Mathematica 1974 49 107 124 MR0352854 Stein E. M. Weiss G. Introduction to Fourier Analysis on Euclidean Spaces 1971 Princeton, NJ, USA Princeton University Press