Báo cáo hóa học: " Research Article Hypersingular Marcinkiewicz Integrals along Surface with Variable Kernels on Sobolev Space and Hardy-Sobolev Space"

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Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2011, Article ID 479576, 14 pages doi:10.1155/2011/479576 Research Article Hypersingular Marcinkiewicz Integrals along Surface with Variable Kernels on Sobolev Space and Hardy-Sobolev Space Wei Ruiying and Li Yin School of Mathematics and Information Science, Shaoguan University, Shaoguan 512005, China Correspondence should be addressed to Wei Ruiying, weiruiying521@163.com Received 30 June 2010; Revised 5 December 2010; Accepted 20 January 2011 Academic Editor: Andrei Volodin Copyright q 2011 W. Ruiying and L. Yin. 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. ≥ Let α 0, the authors introduce in this paper a class of the hypersingular Marcinkiewicz integrals along surface with variable kernels deﬁned by μΦ,α f x Ω Ên Ën−1 1/2 ∞ 2 Ω x, y /|y|n−1 f x − Φ |y| y dy| dt/t3 ∈ L∞ | , where Ω x, z × Lq 2α |y |≤t 0 with q > max{1, 2 n − 1 / n 2α }. The authors prove that the operator μΦ,α is bounded from Ω Sobolev space Lα Ên to Lp Ên space for 1 < p ≤ 2, and from Hardy-Sobolev space Hα Ên to p p L Ê space for n/ n α < p ≤ 1. As corollaries of the result, they also prove the Lα R − L2 Rn ˙ p n 2 n boundedness of the Littlewood-Paley type operators μΦ,α,S and μ∗,Φ which relate to the Lusin Ω Ω,α,λ ∗ area integral and the Littlewood-Paley gλ function. 1. Introduction Let Ên n ≥ 2 be the n-dimensional Euclidean space and Ën−1 be the unit sphere in Ên equipped with the normalized Lebesgue measure dσ dσ · . For x ∈ Ên \ {0}, let x x/|x|. Before stating our theorems, we ﬁrst introduce some deﬁnitions about the variable kernel Ω x, z . A function Ω x, z deﬁned on Ên × Ên is said to be in L∞ Ên × Lq Ën−1 , q ≥ 1, if Ω x, z satisﬁes the following two conditions: Ω x, z , for any x, z ∈ Ên and any λ > 0; 1 Ω x, λz 1/q y, z |q dσ z Ω supr ≥0, y∈Ên Ën−1 |Ω rz < ∞. Ên ×Lq Ën−1 2 L∞ In 1955, Calderon and Zygmund 1 investigated the Lp boundedness of the singular ´ integrals TΩ with variable kernel. They found that these operators connect closely with the
2 Journal of Inequalities and Applications problem about the second-order linear elliptic equations with variable coeﬃcients. In 2002, Tang and Yang 2 gave Lp boundedness of the singular integrals with variable kernels associated to surfaces of the form {x Φ |y| y }, where y y/|y| for any y ∈ Ên \ {0} n ≥ Φ 2 . That is, they considered the variable Calderon-Zygmund singular integral operator TΩ ´ deﬁned by Ω x, y Φ p·v· f x − Φ y y dy. TΩ f x 1.1 n Ên y On the other hand, as a related vector-valued singular integral with variable kernel, the Marcinkiewicz singular with rough variable kernel associated with surfaces of the form {x Φ |y| y } is considered. It is deﬁned by ∞ 1/2 2 dt μΦ f x Φ FΩ,t x , 1.2 Ω t3 0 where Ω x, y Φ f x − Φ y y dy, FΩ,t x 1.3 n−1 y |y|≤t Ω x, z d σ z 0. 1.4 Ën−1 |y|, we put μΦ If Φ | y | μΩ . Historically, the higher dimension Marcinkiewicz Ω integral operator μΩ with convolution kernel, that is Ω x, z Ω z , was ﬁrst deﬁned and studied by Stein 3 in 1958. See also 4–6 for some further works on μΩ with convolution kernel. Recently, Xue and Yabuta 7 studied the L2 boundedness of the operator μΦ with Ω variable kernel. Theorem 1.1 see 7 . Suppose that Ω x, y is positively homogeneous in y of degree 0, and satisﬁes 1.4 and 1/q supy∈Ên Ën−1 |Ω y, z |q dσ z < ∞, for some q > 2 n − 1 /n. Let Φ be a positive 2 and monotonic (or negative and monotonic) C1 function on 0, ∞ and let it satisfy the following conditions: i δ ≤ |Φ t / tΦ t | ≤ M for some 0 < δ ≤ M < ∞; ii Φ t is monotonic on 0, ∞ . Then there is a constant C such that μΦ f ≤ C f 2 , where constant C is independent of f . Ω 2 Since the condition 2 implies 2 , so the L2 Ên boundedness of μΦ holds if Ω ∈ Ω L Ê × Lq Ën−1 with q > 2 n − 1 /n. ∞ n Our aim of this paper is to study the hypersingular Marcinkiewicz integral μΦ,α along Ω surfaces with variable kernel Ω, and with index α ≥ 0, on the homogeneous Sobolev space
Journal of Inequalities and Applications 3 Lα Ên for 1 < p ≤ 2 and the homogeneous Hardy-Sobolev space Hα Ên p p for some n/ n α < p ≤ 1. Let FΩ,t x be as above, we then deﬁne the operators μΦ,α by Φ Ω ∞ 1/2 dt 2 μΦ,α f x Φ α ≥ 0. FΩ,t x , 1.5 Ω t3 2α 0 Our main results are as follows. Theorem 1.2. Suppose that α ≥ 0, Ω x, y satisﬁes 1.4 and Ω ∈ L∞ Ên × Lq Ën−1 with q > max{1, 2 n − 1 / n 2α }. Let Φ be a positive and increasing C1 function on 0, ∞ and let it satisfy the following conditions: i Φt tΦ t ; ii 0 ≤ Φ t ≤ W on 0, ∞ . Then there is a constant C such that μΦ,α f Ên ≤ C f Ên , where constant C is independent Ω L2 L2 α of f . Theorem 1.3. Suppose 0 < α < n/2, and that Ω ∈ L∞ Ên × Lq Sn−1 , with q > max{1, 2 n − 1 / n 2α }, and satisﬁes 1.4 . Let Φ be a positive and increasing C1 function on 0, ∞ and let it satisfy the following conditions: i Φt tΦ t ; ii 0 < Φ t ≤ 1, Φ 0 0. α < p ≤ 1, there is a constant C such that μΦ,α f Ên ≤ C f Ên , Then, for n/ n p Ω Hα Lp Ê Ê p where constant C is independent of any f ∈ ∩S n n . Hα Furthermore, our result can be extended to the Littlewood-Paley type operators μΦ,α,S Ω μ∗,Φ with variable kernels and index α ≥ 0, which relate to the Lusin area integral and and Ω,α,λ ∗ Φ the Littlewood-Paley gλ function, respectively. Let FΩ,t x be as above, we then deﬁne the Ên operators μΦ,α,S and μ∗,Φ for f ∈ S , respectively by Ω Ω,α,λ 1/2 dydt 2 μΦ,α,S Φ fx FΩ,t y , Ω tn 3 2α Γx ⎛ ⎞1/2 1.6 λn dydt ⎠ t 2 ⎝ μ∗,Φ f x Φ FΩ,t y , Ω,α,λ x−y Ê tn 3 2α t n1 Ên with λ > 1, where Γ x { y, t ∈ : |x − y| < t}. As an application of Theorem 1.2, we 1 have the following conclusion. Theorem 1.4. Under the assumption of Theorem 1.2, then Theorem 1.2 still holds for μΦ,α,S and Ω μ∗,Φ . Ω,α,λ By Theorems 1.2 and 1.3 and applying the interpolation theorem of sublinear operator, p we obtain the Lα − Lp boundedness of μΦ,α . Ω
4 Journal of Inequalities and Applications Corollary 1.5. Suppose 0 < α < n/2, and that Ω ∈ L∞ Ên × Lq Sn−1 , q > max{1, 2 n − 1 / n 2α }, and satisﬁes 1.4 . Let Φ be given as in Theorem 1.3. Then, for 1 < p ≤ 2, there exists an absolute positive constant C such that μΦ,α f ≤C f Ên , 1.7 Ên p Ω Lα Lp Ên Ên p for all f ∈ Lα ∩S . Remark 1.6. It is obvious that the conclusions of Theorem 1.2 are the substantial improve- ments and extensions of Stein’s results in 3 about the Marcinkiewicz integral μΩ with convolution kernel, and of Ding’s results in 8 about the Marcinkiewicz integral μΩ with variable kernels. Remark 1.7. Recently, the authors in 9 proved the boundedness of hypersingular p Marcinkiewicz integral with variable kernels on homogeneous Sobolev space Lα Rn for 1 < p ≤ 2 and 0 < α < 1 without any smoothness on Ω. So Corollary 1.5 extended the results in 9, Theorem 5 . Throughout this paper, the letter C always remains to denote a positive constant not necessarily the same at each occurrence. 2. The Bounedness on Sobolev Spaces Before giving the deﬁnition of the Sobolev space, let us ﬁrst recall the Triebel-Lizorkin space. Fix a radial function ϕ x ∈ C∞ satisfying supp ϕ ⊆ {x : 1/2 < |x| ≤ 2} and 0 ≤ ϕ x ≤ 1, and ϕ x > c > 0 if 3/5 ≤ |x| ≤ 5/3. Let ϕj x ϕ 2j x . Deﬁne the function ψj x by F ψj ξ ϕj ξ , such that F ψj ∗ f ξ F f ξ ϕj ξ . For 0 < p, q < ∞, and α ∈ Ê, the homogeneous Triebel-Lizorkin space Fp is the set of ˙ α,q all distributions f satisfying ⎧ ⎫ ⎪ ⎪ ⎨ ⎬ 1/q q Ên Ên ˙ α,q 2−αk ψk ∗ f f ∈S
Journal of Inequalities and Applications 5 For 0 < p ≤ 1, we deﬁne the homogeneous Hardy-Sobolev space Hα Ên by Hα Ên p p Ê . It is well known that H p Ên Fp ,2 Ên for 0 < p ≤ 1, one can refer 10 for the Fp 2 ˙ α, ˙0 n details. Next, let us give the main lemmas we will use in proving theorems. Lemma 2.1 see 11 . Suppose that n ≥ 2 and f ∈ L1 Ên ∩ L2 Ên has the form f x f0 |x| P x where P x is a solid spherical harmonic polynomial of degree m. Then the Fourier transform of f has the form F f x F0 |x| P x , where ∞ 2πi−m r − n 2m−2 /2 2πrs s n 2m /2 F0 r f0 s J n ds, 2.4 2m−2 /2 0 |ξ|, Jm s is the Bessel function. and r Lemma 2.2 see 12 . For λ n − 2 /2, and −λ ≤ α ≤ 1, there exists C > 0 such that for any h ≥ 0 and m 1, 2, . . ., h Jm λ t C dt ≤ λ α . 2.5 tλ α m 0 Lemma 2.3. Let α ≥ 0, λ n − 2 /2, Φ is a C1 function on 0, ∞ and let it satisfy the conditions (i) and (ii) in Theorem 1.2. ∞ 1/2 |Nε f x |2 dε/ε1 2α Denote gα f x , if 0 Φ ε |ξ | Jm λ t F Nε f ξ dt · F f ξ . 2.6 tλ 1 0 ≤ C/mλ 1α Then there exists a constant C independent of m, such that gα f for every f L2 L2 integer m ∈ Æ , m > α. α |x | Proof. Let η |x| t /tλ 1 Jm dt, then we have λ 0 ∞ dε 2 2 gα f Nε f x dx 2 Ê ε1 2α n 0 ∞ dε 2 η Φ ε |ξ | F f ξ dξ Ê ε1 2α n 0 2.7 ∞ 2 β dβ 2α ηΦ |ξ | F f ξ |ξ| dξ 1 2α |ξ | Ên β 0 ∞ 2 β dβ 2 |ξ|2α dξ. ηΦ Ff ξ |ξ | |ξ | Ê β1 2α n 0 ∞ 1α2 2 So it suﬃces to show η Φ β/|ξ| |ξ| ≤ C/mλ dβ/β1 2α . 0 ∞ ∞ m/2 Decompose this integral into two parts : I1 I2 . 0 0 m/2
6 Journal of Inequalities and Applications For I2 , by using Lemma 2.2 and Φ t tΦ t , we can get 2 ∞ Φ β/|ξ | |ξ | dβ Jm λ t I2 dt β1 2α tλ 1 m/2 0 ∞ dβ C 2.8 ≤ β1 2α m2λ 2 m/2 C ≤ . m2λ 2 2α For the other part I1 , applying Stirling’s formula, we have √ √ 2πxx−1/2 e−x ≤ Γ x ≤ 2 2πxx−1/2 e−x . 2.9 Also in 13 , the authors proved the following inequality t/2 ν |J ν t | ≤ . 2.10 Γν 1 So by 2.9 and 2.10 , 0 ≤ α < α 1 ≤ m, and noting that Φ t ≤ Wt, we have 2 Φ β/|ξ | |ξ | m/2 dβ Jm λ t I1 dt β1 2α tλ 1 0 0 2 Φ β/|ξ | |ξ | m/2 |J m λ t | dβ ≤ dt β1 2α tλ 1 0 0 2 Φ β/|ξ | |ξ | m λ m/2 dβ 1 t ≤ dt 2λ Γ2 m β1 2α 22m tλ 1 λ 1 0 0 2m m/2 β dβ 1 ≤ Φ |ξ | 2λ Γ2 m β1 2α 22m 2.11 λ 1 0 2m m/2 e2m 2λ 2 β dβ ≤ Φ |ξ| β1 2α−2m 2m 2λ 1 2π 22m 2λ m λ 1 0 m/2 e2m 2λ 2 dβ ≤C 2α−2m β1 2m 2λ 1 2π 22m 2λ m λ 1 0 e 1 2m 2λ 2 ≤C m2α 2λ 2 4 C ≤ . m2α 2λ 2 So far we can deduce the desired conclusion of Lemma 2.3.
Journal of Inequalities and Applications 7 Proof of Theorem 1.2. The basic idea of proof can go back to 14 , for recently papers, one see 8, 15 . By the same argument as in 1 , let {Ym,j } m ≥ 1, j 1, 2, . . . , Dm denote the complete system of normalized surface spherical harmonics. See 14 for instance, we can decompose Ω x, y as following: ∞ Dm Ω x, y am,j x Ym,j y is a ﬁnite sum. 2.12 m 1j 1 Denote ⎛ ⎞1/2 Dm am,j x ⎝ 2⎠ am x am,j x , bm,j x , 2.13 am x j1 then we get ∞ Dm Dm Ω x, y 2 bm,j x 1, am x bm,j x Ym,j y . 2.14 j1 m1 j1 Then, applying Holder inequality twice, we have for any 0 < ε < 1 that ¨ 2 ∞ ∞ Ym,j y dt 2 μΦ,α f x f x − Φ y y dy bm,j x Ω n−1 t3 2α |y|≤t m 1 y 0 ∞ ∞ a2 x m−ε 1 ≤ 2α mε 1 2α m m1 m1 2 ∞ Dm Ym,j y dt × f x − Φ y y dy bm,j x n−1 t3 2α y |y|≤t j 1 0 ⎛ ⎞ ∞ ∞ ∞ Dm ⎝ bm,j x ⎠ a2 x m−ε 1 ≤ 2α mε 1 2α 2 2.15 m 0 m1 m1 j1 2 Dm Ym,j y dt × f x − Φ y y dy n−1 t3 2α |y|≤t y j1 ∞ ∞ a2 x m−ε 1 2α mε 1 2α m m1 m1 2 ∞ Dm Ym,j y dt × f x − Φ y y dy . n−1 t3 2α |y|≤t y 0 j1
8 Journal of Inequalities and Applications By 14, page 230, equation 4.4 , we can observe that the series in the ﬁrst parenthesis on the right-hand side of the inequality above, for each x ﬁxed, is equal to Ω x, · 2 2 Ën−1 , L −γ where L2 γ Ën−1 is the Sobolev space on Ën−1 with γ ε 1/2 α for any 0 < ε < 1. So if we − take ε suﬃciently close to 1, then by the Sobolev imbedding theorem Lq ⊂ L2 γ , we have − 1/2 a2 x m−ε 1 ≤C Ω Ên ×Lq Ën−1 : C Ω 2α 2.16 L∞ m m with q > max{1, 2 n − 1 / n 2α }. By Fourier transform and 2.16 , we get 2 ∞ ∞ Dm Ym,j y dt 2 μΦ,α f 2 ≤C Ω f x − Φ y y dy dx mε 1 2α Ω n−1 Ê t3 2α 2 |y|≤t y n 0 m1 j1 2 ∞ ∞ Dm Ym,j y dt 2 ≤C Ω F f · − Φ y y dy ε 1 2α m ξ dξ n−1 Ê t3 2α |y|≤t y n 0 m1 j1 ∞ Dm 2 μΦ,j,α f 2 :C Ω mε 1 2α . Ω 2 m1 j1 2.17 For μΦ,j,α f , we have Ω 2 ∞ Ym,j y dt 2 μΦ,j,α f f x − Φ y y e−2πix·ξ dx dy dξ Ω n−1 Ên Ên t3 2α 2 |y|≤t y 0 ∞ Ym,j y e−2πiΦ |y| y ·ξ n−1 Ên |y|≤t y 0 2 dt f x − Φ y y e−2πi x−Φ |y| y ·ξ × dx dy dξ 2.18 Ê t3 2α n 2 ∞ Ym,j y dt e−2πiΦ |y| y ·ξ dy 2 Ff ξ dξ n−1 Ê t3 2α |y|≤t y n 0 2 ∞ Ym,j y 1 dt −2πiΦ |y| y ·ξ 2 Ff ξ e dy dξ. n−1 Ê t1 α t |y|≤t y n 0
Journal of Inequalities and Applications 9 For the integral on the right hand side of the above inequality, by changing of variable, we can get Ym,j y 1 e−2πiΦ |y| y ·ξ dy n−1 t1 α |y|≤t y t 1 Ym,j y e−2πiΦ s y ·ξ dy ds Ën−1 t1 α 0 2.19 Φt 1 Ym,j y e−2πiγ y ·ξ Φ−1 γ dy dγ Ën−1 t1 α 0 Ym,j y 1 e−2πiy·ξ Φ−1 y dy. n−1 t1 α y |y|≤Φ t So we have 2 ∞ Ym,j y 1 dt 2 μΦ,j,α −2πiy·ξ −1 2 Φ Ff ξ f e y dy dξ. Ω n−1 Ên t1 α t 2 |y|≤Φ t y 0 2.20 Φ,m,j Pm,j x · |x|−n−m 1 χ|x|≤Φ t x Φ−1 |x| t−1−α , Ym,j x |x|m and ϕt,α Put Pm,j x x we can deduce from Lemma 2.1 that Φ,m,j Ym,j ξ · |ξ|m F0 |ξ| , F ϕt,α Pm,j |ξ| · F0 |ξ| ξ 2.21 where Φt 2πi−m r − n/2 −m t−1−α s−n−m Φ−1 s 1 1 2πrs s n/2 m F0 r J n/2 ds m−1 0 Φt 2πi−m r − n/2 −m 1 t−1−α s− n/2 2πrs d Φ−1 s 1 J n/2 m−1 0 2.22 2πr Φ β t J n/2 m−1 −m − n/2 −m 1 −1−α 2πi r t dβ n/2 −1 Φβ 0 2πr Φ β −α t J n/2 n/2 −m −m t m−1 2π i r dβ. n/2 −1 t 2πr Φ β 0
10 Journal of Inequalities and Applications Hence, we have 2 μΦ,j,α f Ω 2 ∞ dt 2 Φ,m,j ∗f x ϕt,α dx Ên t 0 ∞ dt 2 Φ,m,j F ϕt,α ∗f ξ dξ Ên t 0 2 ∞ 2π |ξ|Φ β −α t J n/2 n/2 t dt m−1 Ym,j ξ |ξ|m i−m |ξ|−m 2π 2 ≤ Ff ξ dβ dξ n/2 −1 Ê t t 2 π |ξ |Φ β n 0 0 2 ∞ 2 π |ξ |Φ β t J n/2 1 dt m−1 2 ≤C Ff ξ Ym,j ξ dβ dξ. n/2 −1 Ê t1 2α t 2π |ξ|Φ β n 0 0 2.23 |Ym,j z |2 ∼ mn−2 . Dm By 14 , we know that j1 So we can get 2 ∞ 2π |ξ|Φ β Dm t J n/2 1 dt 2 m−1 μΦ,j,α f ≤ Cmn−2 2 Ff ξ dβ dξ. Ω n/2 −1 Ên t1 2α t 2 π |ξ |Φ β 2 0 0 j1 2.24 n/2 − 1, ρ 2π |ξ|Φ β and note that Φ t tΦ t , we can deduce that Set λ 2π |ξ|Φ β t J n/2 1 m−1 U: dβ n/2 −1 t 2 π |ξ |Φ β 0 2π |ξ |Φ t Jm ρ 1 1 λ 2.25 dρ 2π |ξ|Φ Φ−1 ρ/2π |ξ| ρλ t 0 2π |ξ |Φ t Jm ρ ρ 1 λ Φ−1 dρ. 2π |ξ| ρλ 1 t 0 Noting that Φ t is increasing, by using the second mean-value theorem, we get, for some 0 ≤ η < 2π |ξ|Φ t , 2π |ξ |Φ t Jm ρ 1 −1 λ |U | ≤ Φ Φt dρ ρλ 1 t η 2.26 2π |ξ |Φ t Jm ρ λ ≤ dρ . ρλ 1 0
Journal of Inequalities and Applications 11 From 2.26 , it follows that 2 ∞ 2π |ξ |Φ t Dm Jm ρ dt 2 λ μΦ n−2 ≤ Cm dρ · F f ξ f dξ. 2.27 m,j,α Ên t1 2α ρλ 1 2 0 0 j1 Thus using Lemma 2.3, we can deduce the desired conclusion of Theorem 1.2. Proof of Theorem 1.4. First, we know that μΦ,α,S f x ≤ 2λn μ∗,Φ f x . On the other hand, Ω Ω,α,λ 2 μ∗,Φ f Ω,α,λ 2 λn 2 Ω x, z t 1 dzdt f x − Φ |z| z dz dx n−1 x−y Ê tn 1 2α |z| t t |z|≤t n1 n R ⎛ ⎞ λn 2 ∞ Ω x, z ⎝1 t 1 dzdt dx⎠ f x − Φ |z| z dz n−1 x−y Ê tn Ên t1 2α |z| t t |z|≤t n 0 2 ≤ C μΦ,α f . Ω 2 2.28 Thus, using Theorem 1.2, we can ﬁnish Theorem 1.4. 3. The Bounedness on Hardy-Sobolev Spaces In order to prove the boundedness for operator μΦ,α on Hardy-Sobolev spaces and prove Ω Theorem 1.3, we ﬁrst introduce a new kind of atomic decomposition for Hardy-Sobolev space as following which will be used next. Deﬁnition 3.1 see 16 . For α ≥ 0, the function a x is called a p, 2, α atom if it satisﬁes the following three conditions: 1 supp a ⊂ B with a ball B ⊂ Ên ; ≤ |B| 1/2 − 1/p ; 2 a L2 α 0, for any polynomial P x of degree ≤ N n 1/p − 1 α . 3 Ên a x P x Ên p By 16 , we have that every f ∈ Hα can be written as a sum of p, 2, α atoms aj x , that is, f λj aj . 3.1 j
12 Journal of Inequalities and Applications Proof of Theorem 1.3. Similar to the argument of Lemma 3.3 in 17 and using above atomic decomposition, it suﬃces to show that p μΦ,α a ≤ C, 3.2 Ω Lp with the constant C independent of any p, 2, α atom a. Assume supp a ⊂ B 0, R . We ﬁrst note that p p p μΦ,α a μΦ,α a x μΦ,α a x ≤ dx dx Ω Ω Ω Lp |x|≤8R |x|>8R 3.3 : U1 U2 . For U1 , using Theorem 1.2, it is not diﬃcult to deduce that p U1 ≤ C μΦ,α a Rn 1− p/2 ≤ C a p Rn 1− p/2 Ω L2 L2 α 3.4 p/2 −1 Rn 1− p/2 ≤ C. ≤ CRn For U2 , we ﬁrst consider the case n/ n α < p < 1, according to 15, Lemma 5.5 , for 0 < α < n/2 and p, 2, α atom a with support B B 0, R , one has |a x |dx ≤ CRn− n/p α . 3.5 B Using Minkowski inequality and Holder inequality for integrals, and 3.5 , we can get ¨ p μΦ,α a x U2 dx Ω |x|>8R ⎛ ⎞p/2 2 ∞ Ω x, y dt ⎠ ⎝ a x − Φ y y dy dx 3.6 n−1 t3 2α |y|≤t y |x|>8R 0 p Ω x, y ≤ a x−Φ y y dy dx. nα Ên y |x|>8R
Journal of Inequalities and Applications 13 For the integral on the right hand side of the above inequality, by changing of variable and noting that 0 < Φ t ≤ 1, Φ 0 0, we can get Ω x, y a x−Φ y y dy nα Ên y Ω x, y R a x−Φ r y dr dy Ën−1 r1 α 0 ΦR Ω x, y 1 a x − γy dγ dy Φ−1 Φ Ën−1 1α Φ−1 γ γ 0 Φ−1 γ ΦR Ω x, y a x − γy dγ dy Ën−1 1α γ Φ−1 γ 0 3.7 ΦR Ω x, y a x − γy dγ dy α Ën−1 Φ−1 γ γ 0 Ω x, y a x − y dy n α Φ−1 y y |y|≤Φ R Ω x, x − y a y dy n α Φ−1 x − y x−y |x−y|≤Φ R Ω x, x − y ≤ a y dy. nα x−y |x−y|≤Φ R By 3.7 , we can get p ∞ Ω x, x − y U2 ≤ a y dy dx nα Ên x−y 2j R
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