L p Bounds for the parabolic singular integral operator

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1 RESEARCH L p Bounds for the parabolic singular integral operator Yanping Chen *, Feixing Wang 2 and Wei Yu 3 Open Access * Correspondence: yanpingch@26. com Department of Applied Mathematics, School of Mathematics and Physics, University of Science and Technology Beijing, Beijing 00083, China Full list of author information is available at the end of the article Abstract Let <p< and n 2. The authors establish the L p R n+ ) boundedness for a class of parabolic singular integral operators with rough kernels. MR2000) Subject Classification: 42B20; 42B25. Keywords: parabolic singular integral operator, rough kernel, surfaces of revolution Introduction Let a,..., a n be fixed real numbers, a i. For fixed x Î R n, the function F x, ρ) n i 2 x i ρ 2α i is a decreasing function in r >0. We denote the unique solution of the equation Fx, r) byrx). Fabes and Rivière [] showed that rx) isametricon R n, and R n, r) is called the mixed homogeneity space related to {α i } n i. λ α 0 For l >0, let A λ.... Suppose that Ωx) is a real valued and measurable 0 λ α n function defined on R n. We say is Ωx) is homogeneous of degree zero with respect to A l, if for any l >0 and x Î R n A λ x) x). :) Moreover, Ωx) satisfies the following condition S n x ) J x ) dσ x ) 0, :2) where Jx ) is a function defined on the unit sphere S n- in R n, which will be defined in Section 2. In 966, Fabes and Rivière [] proved that if Ω Î C S n- ) satisfying.) and.2), then the parabolic singular integral operator T Ω is bounded on L p R n )for<p<, where T Ω is defined by y ) T f x) p.v. ρ y ) α f x y ) n dy and α α i. R n i 202 Chen et al; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

2 Page 2 of 9 In 976, Nagel et al. [2] improved the above result. They showed T Ω is still bounded on L p R n )for<p< if replacing Ω Î C S n- ) by a weaker condition Ω Î L log + L S n- ). Recently, Chen et al. [3] improve Theorem A, the result is Theorem A. If Ω Î H S n- ) satisfies.) and.2); then the operator T Ω is bounded on L p R n ) for <p<. For a suitable function j on [0, ), and Γ {y, jry)): y Î R n }. Define the singular integral operator T j,ω in R n+ along Γ by Tφ, f ) x, x n+ ) p.v. f x y, xn+ φ ρ y )) y ) ρ y ) α dy, R n where x, x n+ ) Î R n R R n+. On the other hand, we note that if a...a n, then rx) x, a n and R n, r) R n, ). In this case, T j,ω is just the classical singular integral operator along surfaces of revolution, which was studied by the authors of [4-7]. The purpose of this article is to investigate the L p boundedness of the parabolic singular integral operator T j,ω along Γ when Ω Î F b S n- ). For a b >0,F b S n- ) denotes the set of all Ω which are integrable over S n- and satisfies ) +β sup θ) ln dθ<. ξ S n θ ξ :3) S n Condition.3) was introduced by Grafakos and Stefanov [8]. The examples in [8] show that there is the following relationship between F b S n- ) and H S n- ): F β S n ). β>0 F β S n ) H S n ) β>0 We shall state our main results as follows: Theorem Let m Î N. Suppose that j is a polynomial of degree m and d α i φ t) dt α t0 0, where α i s are the all positive integers which is less than m in {a,..., i a n }. In addition, let Ω Î F b S n- ) for some b >0 and satisfies.) and.2), then T j,ω ) 2+2β is bounded on L p R n+ ) for p +2β,2+2β. Corollary Let m Î N. Suppose that j is a polynomial and dα i φ t) dt α t0 0, where i α i s are the all positive integers which is less than m in {a,..., a n }. In addition, let β>0 F ) β S n and satisfies.) and.2), then T j, Ω is bounded on L p R n+ ) for <p <. 2 Notations and lemmas In this section, we give some notations and lemmas which will be used in the proof of Theorem. For any x Î R n, set x ρ α cos ϕ cos ϕ n 2 cos ϕ n x 2 ρ α 2 cos ϕ cos ϕ n 2 sin ϕ n... x n ρ α n cos ϕ sin ϕ 2 x n ρ α n sin ϕ.

3 Page 3 of 9 Then dx r a- J,..., n- )drds, whereα n α i, ds is the element of area of S n- and r a- J,..., n- ) is the Jacobian of the above transform. In [], it was shown there exists a constant L such that J,..., n- ) L and J,..., n- ) Î C 0, 2π) n-2 0,π)). So, it is easy to see that J is also a C function in the variable y Î S n-. For simplicity, we denote still it by Jy ). In order to prove our theorems, we need the following lemmas: Lemma 2.. [9]) Let d Î N. Suppose that g t): R + R d satisfies ) γ t) γ t) M for a fixed matrix M, and assume gt) doesn t lie in an affine hyperplane. t Then 2 e iγt) η dt η /d. Lemma 2.2. [9]) Suppose that λ j s and α j s are fixed real numbers, jt) is a polynomial and Ɣ t) λ t α,..., λ n t α n, φ t)) is a function from R + to R n+. For suitable f, the maximal function associated to the homogeneous curve Γ is defined by M Ɣ f ) x) sup h h h 0 i f x Ɣ t)) dt, h > 0. 2:) Then for <p, there is a constant C >0, independent of λ j s, the coefficient of j t) and f, such that M Ɣ f ) L p f L p. 2:2) Lemma 2.3. Let L : R n+ R n be a linear transformation. Suppose that {s k } kîz is a sequence of uniformly bounded measures on R d satisfying { σ k ξ) Cmin A Lξ, ln A Lξ )) β} 2:3) for ξ Î R n+ and k Î Z. For any <p 0 < and A >0 ) /2 ) /2 σ k g k 2 A g k 2 2:4) L p 0 k Z L p 0 holds for arbitrary functions {g k } kîz on R n+.thenfor p k Z ) 2+2β +2β,2+2β there exists a constant C p Cp, n) which is independent of L such that σ k f p f L p 2:5) L p k Z and ) /2 σ k f 2 k Z L p f L p 2:6) for every f Î L p R n+ ).

4 Page 4 of 9 Proof. The main idea of the proof is taken from [7,8], we assume that Lξ ξ,..., ξ n ) ζ for ξ ξ,..., ξ n, ξ n+ ) Î R n+. Choose a ψ C 0 R) such that 0 ψ, suppψ) /4, 4), and [ ψ 2 j t )] 2 2:7) j Z For each j, we define F j in R n by j ζ ) ψ 2 j ρ ζ ) ) f ζ ) for ξ ξ,..., ξ n+ ) Î R n+. If we set Tf k Z σ k f, 2:8) and let δ represent the Dirac delta on R, then by 2.7), for any Schwartz function f, Tf j Z T j f, where T j f k Z j+k δ ) σ k j+k δ ) f. By using 2.4) and Littlewood-Paley theory as in [3]), one obtains that for any <p 0 <, T j f L p 0R n+ ) p 0 f L p 0R n+ ). 2:9) On the other hand, by using Plancherel s theoremand2.3),ifj>0, using the estimate σ k ξ) C A ζ we have T j f ) L 2 R n+ ) f ξ) 2 A ζ 2 dξ k 2 j k ρζ ) 2 j k+ 2 fξ) k 2 j k ρζ) 2 j k+ 2 2kα ρζ ) 2α ζ ) kα n ρζ ) 2α n ζ )2) dξ n 2 2j min{α j} 2 f ξ) k 2 j k ρζ ) 2 j k+ ζ ) ζ )2) dξ n 2 2j 2 dξ k C2 2j f L 2 R n+ ). 2 j k ρζ ) 2 j k+ f ξ) 2:0)

5 Page 5 of 9 Similar to the proof of 2.0), using Plancherel s theorem and 2.3), if j<0 we get In short T j f L2 R n+ ) + j ) +β) f L2 R n+ ). 2:) T j f L2 R n+ ) + j ) +β) f L2 R n+ ), forj Z. 2:2) By interpolating between 2.9) and 2.2), we obtain T j f Lp R n+ ) + j ) +β) f Lp R n+ ). 2:3) for p ) 2+2β +2β,2+2β and some b >0. Thus, 2.5) follows from 2.3). One may then use a randomization argument to derive 2.6). Lemma 2. is proved. 3 Proof of Theorem The main idea of the proof of Theorem is taken from [0]and []. Let Ω satisfies.),.2), and.3) for some b >0. Let Fy) y, jry))), where φ t) m j0 a jt j, m Î N. Let {y Î R n : <ry) + } and define the family of measures s k on R n+ by ) ))) y ) f y, yn+ dσk f y, φ ρ y ρ y ) α dy, 3:) R n+ and s* fx) sup kîz s k * f x). It is easy to see that σ k y ) ρ y ) α dy S n + y ) J y ) dρ ρ dσ y ). 3:2) In light of 3.2) and Lemma 2.3, it suffices to show that s k satisfies 2.3) and 2.4). For ξ, ξ n+ ) Î R n R, y Î S n-, and l Î Z. Let I λ ξ, ξn+, y ) 2 e i[a λρξ y +ξ n+ φλρ)] dρ. Set Λ {a i : a i is the positive integers which is less than m in {a,..., a n }and {, 2,..., m} \.Then dα i φ t) dt α t0 0,wherea i Î Λ, and is not a subset of i {a,..., a n }. Therefore, we get A λρ ξ y + ξ n+ φ λρ) ρ α λ α ξ y ρα n λ α n ξ n y n + ξ n+ a j λρ) j. Without loss of generality, we may assume Λ consists of r distinct numbers and let {i, i 2,..., i m r } If α j s are all distinct, by Lemma 2., we get immediately j

6 Page 6 of 9 I λ ξ, ξn+, y ) λ α ξ y + + λα n ξ n y n + m r) λξ n+ ) /n+m r) λ α ξ y + + λα n ξ n y n ) /n+m r) Aλ ξ y /n+m r). 3:3) If {aj} only consists of s distinct numbers, we suppose that α α 2 α l, α l + α l +l 2,... α l + +l s + α n, where s is a positive integer with s n, l, l 2,..., l s are positive integers such that l + l l s n and α, α l +l 2,, α l + +l s,α n are distinct. Obviously, γ t) t α, t α l +l 2,..., t α l + +l s, t α n, t i, t i 2,..., t i ) m r does not lie in an affine hyperplane in R s+m-r. Then using Lemma 2. again, there exists C>0 such that for any vector h h,..., h n ) Î R n, 2 e 2iη + +η l )t α l +ηl ++ +η l +l 2 )t α l +l 2 + +ηl + +l s ++ +η n)t αn +λξ n+ j tj dt η + + η l 2 + η l η l +l η l + +l s η n 2 + m r) λξ n+ 2) /2s+m r) η + + η l + η l η l +l η l + +l s η n ) /s+m r) n /s+m r) η j. j Let η j λ α jξ j y j, we have I λ ξ, ξn+, y ) λ α ξ y + + λα n ξ n y n ) /s+m r) λ α ξ y + + λα n ξ n y n ) /s+m r) Aλ ξ y /s+m r). 3:3a) On the other hand, it is easy to see that I λ ξ, ξn+, y ). 3:4) From 3.3), 3.3 ) and 3.4), we get I λ ξ, ξn+, y ) [ ln / η y )] +β ln A λ ξ ) +β, for A λ ξ 2, where η A λξ. Thus, by.3), we get A λ ξ I λ ξ, ξn+, y ) y ) dσ y ) ln A λ ξ ) +β). S n

7 Page 7 of 9 Therefore, σ k ξ, ξ n+ ) e iξ y+ξn+φρy))) y ) ρ y ) α dy + eiξ Aρy +ξn+φρ)) y ) J y ) y ) dρ ρ dσ S n 2 e i ξ A ρ y +ξ n+ φ ρ) ) y ) J y ) dρ S n ρ dσ y ) ξ, ξn+, y ) y ) dσ y ) I 2k 3:5) S n ln A ξ ) +β). On the other hand, by.2), we can obtain σ k ξ, ξ n+ ) e iξ y+ξn+φρy))) y ) ρ y ) α dy + iξ A e ρ y +ξ n+ φρ)) y ) J y ) dσ y ) dρ ρ S n + e iξ A ρy +ξ n+ φρ)) ) e iξ n+ φρ) y ) J y ) dσ y ) dρ ρ S n S n S n e iξ Aρy +ξn+φρ)) e iξn+φρ) y ) J y ) dσ y ) dρ ρ ξ A ρ y y ) J y ) dσ y ) dρ ρ A 2kξ y y ) J y ) dσ y ) dρ ρ 3:6) S n A ξ A ξ. + S n y ) J y ) A +ξ A +ξ y dσ y ) dρ

8 Page 8 of 9 Clearly, 3.5) and 3.6) imply 2.3) holds. Finally, we shall show that 2.4) holds. σk ) f x) sup σk f ) x) k R f x y )) y ) ρ y ) α dy + f x Aρ y )) y ) dρ ρ dσ y ) S n y ) + x Aρ y )) dρ dσ y ) S n S n f y ) ) M f x) dσ y ). By Lemma 2.2, we obtain M F f) p f p,wherec>0 is independent of k, the coefficient of jt) andf, sinceω is integrable on S n-,thus s*f) p f p.this shows 2.4) holds. This completes the proof of the Theorem. Acknowledgements The research was supported by the NSF of China Grant No ), NCET of China Grant No. NCET--0574), and the Fundamental Research Funds for the Central Universities. Author details Department of Applied Mathematics, School of Mathematics and Physics, University of Science and Technology Beijing, Beijing 00083, China 2 Department of Information and Computer Science, School of Mathematics and Physics, University of Science and Technology Beijing, Beijing 00083, China 3 Department of Mathematics and Mechanics, University of Science and Technology Beijing, Beijing 00083, China Authors contributions YC carried out the parabolic singular integral operator studies and drafted the manuscript. WY participated in the study of Littlewood-Paley theory. FW conceived of the study, and participated in its design and coordination. All authors read and approved the final manuscript. Competing interests The authors declare that they have no competing interests. Received: 7 March 202 Accepted: 30 May 202 Published: 30 May 202 References. Fabes, E, Rivière, N: Singular integrals with mixed homogeneity. Studia Math. 27, ) 2. Nagel, A, Riviere, N, Wainger, S: On Hilbert transforms along curves, II. Am Math J. 98, ). doi:0.2307/ Chen, Y, Ding, Y, Fan, D: A parabolic singular integrals with rough kernel. J Aust Math Soc. 84, ) 4. Fan, D, Guo, K, Pan, Y: A note of a rough singular integral operator. Math Inequal Appl. 738, ) 5. Lu, S, Pan, Y, Yang, D: Rough singular integrals associated to surfaces of revolution. P Am Math Soc. 29, ). doi:0.090/s Kim, W, Wainger, S, Wright, J, Ziesler, S: Singular integrals and maximal functions associated to surfaces of revolution. Bull Lond Math Soc. 28, ). doi:0.2/blms/ Cheng, LC, Pan, Y: L p bounds for singular integrals associated to surfaces of revolution. J Math Anal Appl. 265, ). doi:0.006/jmaa Grafakos, L, Stefanov, A: Convolution Calderón-Zygmund singular integral operators with rough kernel. Indiana Univ math J. 47, ) 9. Stein, EM, Wainger, S: Problems in harmonic analysis related to curvature. Bull Am Math Soc. 84, ). doi:0.090/s Xue, Q, Ding, Y, Yabuta, K: Parabolic Littlewood-Paley g-function with rough kernels. Acta Math Sin Engl Ser). 24, ). doi:0.007/s

9 Page 9 of 9. Duoandikoetxea, J, Rubio de Francia, JL: Maximal and singular integral operators via Fourier transform estimates. Invent Math. 84, ). doi:0.007/bf doi:0.86/ x Cite this article as: Chen et al.: L p Bounds for the parabolic singular integral operator. Journal of Inequalities and Applications :2. Submit your manuscript to a journal and benefit from: 7 Convenient online submission 7 Rigorous peer review 7 Immediate publication on acceptance 7 Open access: articles freely available online 7 High visibility within the field 7 Retaining the copyright to your article Submit your next manuscript at 7 springeropen.com