Báo cáo hóa học: "Research Article Boundedness on Hardy-Sobolev Spaces for Hypersingular Marcinkiewicz Integrals with Variable Kernels"

Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2008, Article ID 835938, 17 pages doi:10.1155/2008/835938 Research Article Boundedness on Hardy-Sobolev Spaces for Hypersingular Marcinkiewicz Integrals with Variable Kernels Xiangxing Tao,1 Xiao Yu,1, and Songyan Zhang1 Department of Mathematics, Faculty of Science, Ningbo University, Ningbo, Zhejiang Province 315211, China Department of Mathematics, Faculty of Science, Zhejiang University, Hangzhou, Zhejiang Province 315211, China Correspondence should be addressed to Songyan Zhang, zhangsongyan@nbu.edu.cn Received August 2008; Accepted 20 October 2008 Recommended by Nikolaos Papageorgiou The existence and boundedness on Sobolev spaces and Hardy-Sobolev spaces for the hypersingular Marcinkiewicz integrals with variable kernels are derived Copyright q 2008 Xiangxing Tao 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 Introduction The function Ωx, z defined on Rn × Rn is said to belong to L∞ Rn  × Lq Sn−1 , if it satisfies the following two conditions: 1 Ωx, λz  Ωx, z, for any x, z ∈ Rn and any λ > 0;  1/q 2 ΩL∞ Rn ×Lq Sn−1  : supr≥0, y∈Rn  Sn−1 |Ωrz  y, z |q dσz  < ∞ Let α ≥ and q > max{1, 2n − 1/n  2α}, and let Ω ∈ L∞ Rn  × Lq Sn−1  satisfy the following cancellation property:  Sn−1       Ω x, z Ym z dσ z  0, 1.1 for all spherical harmonic polynomials Ym z  with degree ≤ α and for any x ∈ Rn We consider the hypersingular Marcinkiewicz integral μΩ,α fx defined by Journal of Inequalities and Applications  ∞    μΩ,α fx   Ωx, x − y |x−y|≤t |x − y|n−1 1/2 2  dt fydy 32α , t 1.2 for f ∈ SRn , the Schwartz class In general, the way in which the integrals were given ∞ 1/2 has to be interpreted Let H  {ht : hH   |ht|2 dt/t < ∞}, and let hf t, x   n−1 −1−α t Ωx, x − y/|x − y| fydy We have to show hf ·, x ∈ H for every x ∈ Rn and |x−y|≤t f ∈ S, and so that μΩ,α fx  hf ·, xH can be regarded as a well-defined Hilbert-valued function To see this, applying the Taylor’s expansion, we can write fx − y   κ |κ|≤α :  Cκ y D f x  κ Cκ y |κ|α1 aκ xy  bx, y|y| κ κ 1   1 − sα Dκ f x − syds 1.3 α1 |κ|≤α with some bounded functions aκ x and bx, y, where κ denotes the multi-indices By cancellation property 1.1 of Ω, we have hf t, x  t−1−α t  Sn−1 t −1−α t α1 s ≤ Ct  Sn−1 1α−α       Ω x, y f x − sy dσ y ds       Ω x, y b x, sy dσ y ds 1.4 Simultaneously, by choosing r so that < r < q and r suciently closing to 1, the Holder ă inequality for integrals implies hf t, x ≤ t −1−α   Ωx, yr  fLr  |y|≤t |y|n−1r 1/r dy ≤ Ct−α−nn/r , 1.5 where 1/r  1/r   Hence, noting  α − α > and −α − n  n/r < 0, one can easily check that hf ·, xH ≤ C for any test function f and every x ∈ Rn , and so μΩ,α fx is well defined for f ∈ SRn  and α ≥ Historically, as high dimension corresponding to the Littlewood-Paley g-function, Stein 1 first introduced the classical Marcinkiewicz integral with convolution kernel Ωx, z ≡ Ωz and in the special case α  The Lp H p  boundedness, < p ≤ 2, or p ≤ but is close to 1, for the classical Marcinkiewicz integrals was extended by many authors to various kernels, for example, 1–4 On the other hand, because of connecting with the problems about the partial differential equations with variable coefficients, more attentions are focused on the variable kernels A fresh look at this problem is due to Ding et al 5–7, where they showed the following theorem Xiangxing Tao et al Theorem 1.1 see 7 Let α  0, and Ωx, z  ∈ L∞ Rn  × Lq Sn−1 , q > 2n − 1/n, satisfy cancellation property 1.1 Then, the operator μΩ,0 is bounded from L2 Rn  to L2 Rn  Moreover, if Ω satisfies the L1 -Dini condition, then μΩ,0 is bounded from Hardy space H Rn  to L Rn  Our main results are as follows Theorem 1.2 Let α > 0, and Ωx, z  ∈ L∞ Rn  × Lq Sn−1 , q > max{1, 2n − 1/n  2α}, satisfy cancellation property 1.1 Then, μΩ,α f n ≤ Cf n Lα R  L R  1.6 with the constant C independent of any f ∈ L2α Rn  ∩ SRn  Theorem 1.3 Let α > 0, and Ωx, z  ∈ L∞ Rn  × Lq Sn−1 , q > max{1, 2n − 1/n  2α}, satisfy cancellation property 1.1 Then, for n/n  α < p ≤ 1, μΩ,α f p n ≤ Cf p n Hα R  L R  1.7 p with the constant C independent of any f ∈ Hα Rn  ∩ SRn  By Theorems 1.2 and 1.3 and applying the interpolation theorem of sublinear operator, p we obtain the Lp − Lα boundedness of μΩ,α Corollary 1.4 Let α > 0, and Ωx, z  ∈ L∞ Rn  × Lq Sn−1 , q > max{1, 2n − 1/n  2α}, satisfy cancellation property 1.1 Then, μΩ,α f p n ≤ Cf p n Lα R  L R  for < p ≤ 1.8 p with the constant C independent of any f ∈ Lα Rn  ∩ SRn  p Further, we can derive the boundedness of μΩ,α on Hα for some p ≤ n/n  α under an additional assumption, the L1,β -Dini condition, on Ω as follows: 1 δ dδ < ∞, 1β δ 1.9 where δ is the integral modulus of continuity of Ω defined by  δ  sup r>0, |O| max{1, 2n−1/n2α}, satisfy cancellation property 1.1 and the L1,β -Dini condition 1.9 with β ≥ Then, for max{n/n  α  β, 2n/2n  2α  1} < p ≤ n/n  α, < α < n/2, one has μΩ,α f p n ≤ Cf p n Hα R  L R  1.11 p with the constant C independent of any f ∈ Hα Rn  ∩ SRn  We will give the definitions for Sobolev spaces and Hardy-Sobolev spaces, and prove the key lemma Lemma 2.2 for Bessel functions in the next section The proof of Theorem 1.2 will be given in Section 3, and Theorems 1.3 and 1.5 will be proved in Section Finally, as an application of Theorem 1.2, in Section we will get the L2α − L2 boundedness for a class of the Littlewood-Paley type operators μΩ,α,S and μ∗Ω,α,λ with variable kernels and index α ≥ 0, which relate to the Lusin area integral and the Littlewood-Paley gλ∗ function, respectively Here we point out that, by the same arguments in the paper, our theorems are also hold for the following hypersingular parametric Marcinkiewicz integrals ρ μΩ,α fx 1/2  ∞   2  dt 1 Ωx, x − y     ρ n−ρ fydy  12α t t |x−y|≤t |x − y| 1.12 with the complex parameter ρ with Reρ > Throughout the paper, C always denotes a positive constant not necessarily the same at each occurrence we use a ∼  b to mean the equivalence of a and b; that is, there exists a positive constant C independent of a, b such that C−1 a ≤ b ≤ Ca Some notations and lemmas Let us first recall the Triebel-Lizorkin space Fix a radial function ϕx ∈ C∞ satisfying suppϕ ⊆ {x : 1/2 < |x| ≤ 2} and ≤ ϕx ≤ 1, and ϕx > c > if 3/5 ≤ |x| ≤ 5/3 Let ϕj x  ϕ2j x Define the function ψj x by ψ j ξ  ϕj ξ For < p, q < ∞ and α ∈ R, α,q the homogeneous Triebel-Lizorkin space F˙ p is the set of all distributions f satisfying  α,q  F˙ p Rn     f ∈S R n  : fF˙ pα,q  1/q   q −αk   ψk ∗f  depending only on λ such that  t   Jmλ ρ   ≤ C dρ   mλ λ ρ for < t < ∞ , m  1, 2, 2.5 Proof Let us write ν  m  λ In case < t ≤ ν, since Jν ρ > for < ρ < ν, it follows from 10, inequality 6.1 that  t    t  Jν ρ   Jν ρ  Cν C    dρ ≤ ν dρ ≤ 1λ ≤ λ  λ 1λ m m ρ ρ 2.6 In case ν < t ≤ 2ν, the second mean value theorem and 10, inequality 6.2 yield t Jν ρ dρ  ν−λ λ ν ρ ν  h O ν  ν < h ≤ t < 2ν  ν −λ1  Jν ρdρ h Jν νρdρ   1 0, one can easily see that I1 ≤ t 2ν ρ dρ ≤ ρ2 − ν  λ−1 t dρ C ≤ λ 1λ ρ ν 2ν 2.12 −1 If λ  0, then ν  m is an integer Since |Jm ρ| ≤ see 9, page 137, and ρ1 − ν/ρ2  decreasing for ρ ≥ 2ν, we apply the second mean value theorem again to obtain  t     Jm ρ  dρ I1   −1 2 2ν ρ ρ − ν      t2   ≤ J ρdρ m   3ν 2ν ≤  2ν ≤ t2 ≤ t  is 2.13     Jν t2 − Jν 2ν ≤ C 3ν Now we have gotten  t       Jν ρ   2ν Jν ρ   t Jν ρ   ≤  ≤ C , dρ dρ dρ       mλ λ λ λ ρ ρ ρ 0 2ν 2.14 which implies the lemma The boundedness on Sobolev spaces In this section, we derive the boundedness for the Marcinkiewicz integral μΩ,α on Sobolev spaces, and give the proof of Theorem 1.2 Xiangxing Tao et al As in 9, let {Ym,1 , Ym,2 , , Ym,Nm } denote an orthonormal basis of the space of normalized spherical harmonics of degree m, then by using 1.1 and the spherical harmonic development, see 11 for instance, we can decompose Ωx, z  as follows: ∞   Ω x, z  Nm   am,j xYm,j z , 3.1       Ω x, z Ym,j z dσ z 3.2 mα1 j1 where Nm ∼  mn−2 and  am,j x  Sn−1 Denote 1/2 Nm  2   am,j x am x  , bm,j x  j1 then Nm j1 am,j x , am x 3.3 bm,j x  We also let  ∞    μm,j,α fx   Ym,j x − y |x − y|n−1 |x−y|≤t 1/2 2  dt  fydy 32α t 3.4 Then, applying the Holder inequality twice, we have for any < < ă   μΩ,α fx2  2  ∞  Nm ∞  dt Ym,j x − y   am x bm,j x fydy 32α  n−1  t  |x − y| |x−y|≤t mα1 j1  ∞ ≤  a2m xm−ε12α mα1 ∞ mε12α mα1 2  ∞  Nm  dt Ym,j x − y   · bm,j x fydy 32α  n−1  t  |x − y| |x−y|≤t j1  ∞ ≤  a2m xm−ε12α mα1   j1 ∞ mα1 mε12α mα1  ∞ Nm   ·  ∞ Ym,j x − y |x−y|≤t 2  dt fydy 32α t |x − y|n−1  ∞ a2m xm−ε12α mα1 mε12α Nm   μm,j,α fx2 j1 3.5 Journal of Inequalities and Applications By 11, we can observe that the series in the first parenthesis on the right-hand side of the inequality above is equal to Ωx, ·2L2 Sn−1  , where L2−γ Sn−1  is the Sobolev space on −γ Sn−1 with γ  ε1/2  α for any < ε < So if we take ε sufficiently close to 1, then by the Sobolev imbedding theorem Lq ⊂ L2−γ , we have m a2m xm−ε12α ≤ CΩ2L∞ Rn ×Lq Sn−1  3.6 with q > max{1, 2n − 1/n  2α} Therefore, to prove the theorem, it remains to show that for ε sufficiently close to 1, we have ∞ mε12α Nm  Rn j1 mα1   μm,j,α fx2 dx ≤ Cf2 Lα 3.7 Put Pm,j x  Ym,j x|x|m and ϕm,j,t x  t−1−α Pm,j x|x|−n−m1 χ{|x|≤t} One can deduce from Lemma 2.1 that       m    ϕ m,j,t ξ  F0 |ξ| Pm,j |ξ|  Ym,j ξ |ξ| F0 |ξ| , 3.8 where, by the changing of variable, F0 r  i −m 2π n/2−1 −m−1 −1−α r  2πrt t Jmn−2/2 ρ dρ ρn−2/2 3.9 Hence by Plancherel theorem, we have Nm  j1 Rn Nm   μm,j,α fx2 dx  j1  R  ∞     n Nm  ∞  j1   Nm  ∞  j1 R     n 2  dt dx ϕm,j,t x − yfydy t Rn 2  dt ϕm,j,t x − yfydy dx t Rn        2 F0 |ξ|2 dξ dt Ym,j ξ 2 |ξ|2m fξ t Rn 3.10 Xiangxing Tao et al By 11, we know that Nm j1 |Ym,j z |2 ∼  mn−2 , thus 3.10 is bounded by  2π|ξ|t Jmn−2/2 ρ dt Cm dρ dξ 12α n−2/2 2π|ξ|t t n ρ 0 R   m/2  r   dr Jmn−2/2 ρ 2 |ξ|2α fξ ≤ Cmn−2 dρ dξ 12α n−2/2 r r n ρ 0 R   ∞  r   dr Jmn−2/2 ρ 2 |ξ|2α fξ  Cmn−2 dρ dξ 12α n−2/2 r ρ m/2 r Rn  n−2   2 fξ  ∞ 3.11 : I1  I2 r Let ηr  1/r Jmλ ρ/ρλ dρ Using 12, Lemma 2.1 the proof of Lemma 2.1b and the estimation in 12, page 565, we have ηr ≤ Jmλ r/r λ and |Jν r| ≤ r/2ν /Γν  1 Thus, by Stirling’s formula, we have for m ≥ α  that  m/2 ηr2 dr r 12α ≤  m/2 r 2m−1−2α  2mλ Γm  λ  1 m/2mα ≤C m Γm  λ  1 ≤ C m2λ2−2α 2 dr 3.12 Now we let λ  n − 2/2 and use the inequality above to see that I1 ≤ Cm−2−2α  Rn   2 |ξ|2α dξ ≤ Cm−2−2α f2 fξ Lα 3.13 r To estimate I2 , using Lemma 2.2 that | Jmλ t/tλ dt| ≤ C/mλ , we obtain    2 |ξ|2α fξ I2 ≤ Cmn−2 Rn   ∞ m/2 C mn−2/2 dr dξ r 32α   2 |ξ|2α m−2−2α dξ fξ ≤C 3.14 Rn ≤ Cm−2−2α f2L2α Combining the estimates of I1 and I2 , we get from 3.10 and 3.11 that ∞ mα1 mε12α Nm  j1 Rn ∞   μm,j,α fx2 dx ≤ C mε12α−2−2α f2L2α , mα1 which implies 3.7 Then, we complete the proof of the theorem 3.15 10 Journal of Inequalities and Applications The boundedness on Hardy-Sobolev spaces In order to show the boundedness for the operator μΩ,α on Hardy-Sobolev spaces, and prove Theorems 1.3 and 1.5, we first recall the atomic decomposition for Hardy-Sobolev spaces Definition 4.1 see 13 For α ≥ 0, the function ax is called a p, 2, α atom if it satisfies the following three conditions: 1 suppa ⊂ B with a ball B ⊂ Rn ; 2 aL2α ≤ |B|1/2−1/p ;  3 Rn axP x  0, for any polynomial P x of degree ≤ N  n1/p − 1α p By 13, we know that every f ∈ Hα Rn  can be written as a sum of p, 2, α atoms  p n aj x, that is, f  j λj aj which converges in Hα R  and in the sense of distributions p ∼ Moreover, fH p  j |λj |p α To consider the hypersingular Marcinkiewicz integral μΩ,α f on Hardy-Sobolev p p spaces Hα Rn , we can start with f in a nice dense class of function, say f ∈ Hα Rn  ∩ SRn  Then, by the Lebesgue dominated convergence theorem, we have   λj  · ·, x hf ·, x ≤ j H H 4.1 j Recalling μΩ,α fx  hf ·, xH , so we obtain for < p ≤  p μΩ,α fp p n ≤ λj  μΩ,α aj p p n L R  L R  4.2 j Now we are in the position to prove the following theorem Proof of Theorem 1.3 Because of 4.2, it is sufficient to show that μΩ,α ap p ≤ C L 4.3 with the constant C independent of any p, 2, α atom a By translation argument, we can assume that suppa ⊂ B0, ρ We first note that μΩ,α ap p ≤ L  |x|≤8ρ   μΩ,α ap dx   |x|>8ρ   μΩ,α ap dx : J1  J2 4.4 By Theorem 1.2 and size condition 2 of a, it is not difficult to deduce that p p J1 ≤ CμΩ,α aL2 ρn1−p/2 ≤ CaL2 ρn1−p/2 ≤ C α Next we estimate J2 according to three cases 4.5 Xiangxing Tao et al 11 Case < α < n/2 In this case, one may refer to 14, Lemma 5.5 to get    axdx ≤ Cρn−n/pα 4.6 B Since < p ≤ 1, the Minkowski inequality and the Holder inequality give that ă  J2   ∞     Ωx, x − y p/2 2  dt  aydy 32α dx t |x − y|n−1       p  Ωx, x − y      ≤  n |x − y|nα ay dy dx |x|>8ρ R  p   ∞    Ωx, x − y    nα |ay|dy  dx  j j1 Rn |x − y| j3 ρ 8ρ, |y| ≤ ρ, we have |x−y| ∼  |x| ∼  |x|2ρ, so the mean value theorem −2−2α −2−2α −3−2α − |x − y| | ≤ Cρ|x − y| Thus, by the Minkowski inequality for gives ||x|  2ρ integrals, V1 ≤ Cρ p/2 ∞  j3      p  Ωx, x − y   aydy dx   n1/2α 2j ρ≤|x| 8ρ So by the cancelation of the atom ax, we have  V2  |x|>8ρ  ∞     |x|2ρ |x−y|≤t Ωx, x − y |x − y|  n−1 Ωx, x |x − y|n−1 − − Ωx, x |x − y|n−1 Ωx, x |x|n−1 p/2 2  dt  dx · aydy 32α t  p/2     ∞   2 dt   Ωx, x − y  Ωx, x    aydy ≤ −   t32α  dx    n−1 |x − y|n−1 |x|>8ρ  |x|2ρ |x−y|≤t |x − y|   p/2     ∞   2 dt   Ωx, x  Ωx, x    aydy  −   t32α  dx    n−1 |x|n−1 |x|>8ρ  |x|2ρ |x−y|≤t |x − y|  : V21  V22 4.17 14 Journal of Inequalities and Applications Applying the Minkowski inequality, decomposition of integral domain, the Holder ă inequality, and the Fubini theorem successively, we can obtain V21 ≤ ≤ j3      p  Ωx, x − y − Ωx, x   aydy dx   |x − y|nα 2j ρ≤|x| 2ρ, we have n−1 − 1/|x|n−1 | ≤ C|y|/|x|n  |x| ∼  |x − y| ∼  |x|  2ρ, and by mean value theorem, |1/|x − y| Therefore, by Minkowski’s inequality for integrals, the inequality 4.6 and the condition Xiangxing Tao et al 15 n/n   α < p < 1, we can obtain  p    Ωx, x    Ωx, x      ay  ≤C − dy dx  n−1 n−1  |x|1α |x − y| |x| |x|>8ρ B       Ωx, x · |y| p      ay ≤C dy dx  |x|n1α |x|>8ρ B   p   Ωx, xp   p aydy ≤ Cρ dx n1αp B |x|>8ρ |x|  V22 4.21 ≤ ρp ρpn−n/pα ρ−n1αpn ΩL∞ ×Lp ≤ C Combining 4.13, 4.14, 4.16, 4.20, and 4.21, we have proved that μΩ,α ap p ≤ W  V1  V21  V22 ≤ C, L 4.22 which is our desired inequality The proof of the theorem is finished Final remark By similar argument as that for μΩ,α , we can consider the Littlewood-Paley type operators μΩ,α,S and μ∗Ω,α,λ with variable kernels and index α ≥ 0, which relate to the Lusin area integral and the Littlewood-Paley gλ∗ function, as follows:  μΩ,α,S fx  μ∗Ω,α,λ fx  Γx      Rn1  Ωy, y − z |y−z|≤t |y − z|n−1 t t  |x − y| λn     1/2 2  dy dt fzdz n32α , t Ωy, y − z |y−z|≤t |y − z|n−1 1/2 2  dy dt  fzdz n32α t 5.1 with λ > 1, where Γx  {y, t ∈ Rn1 : |x − y| < t}  As an application of Theorem 1.2, we have the following conclusion Theorem 5.1 Let α ≥ 0, and assume that Ωx, z  ∈ L∞ Rn  × Lq Sn−1 , q > max{1, 2n − 1/n  2α}, satisfy 1.1 Then, μΩ,α,S f n ≤ 2λn μ∗ f ≤ CfL2α Rn  Ω,α,λ L R  L2 Rn  with the constant C independent of any f ∈ L2α Rn  ∩ SRn  5.2 16 Journal of Inequalities and Applications The proof of the theorem is based on the following lemma Lemma 5.2 Let λ > 1, there exists a positive constant Cλ,n such that for any nonnegative and locally integrable function ϕ,  2 μ∗Ω,α,λ fx ϕxdx  Rn  ≤ Cλ,n 2 μΩ,α fx Mϕxdx,  Rn 5.3 where M denotes the Hardy-Littlewood maximal operator Proof Directly from the definition, we have  Rn   λn   FΩ,t y2 dy dt ϕxdx tn32α Rn Rn1   ∞  λn 2 dt  t   FΩ,t y 32α sup ≤ ϕx n dx dy t t t>0 Rn t  |x − y| Rn  2  ≤ Cλ,n μΩ,α fy Mϕydy, 2 μ∗Ω,α,λ fx ϕxdx   t t  |x − y| Rn 5.4 which implies the lemma Now, if we take the supremum in inequality 5.3 over all ϕ satisfying ϕL∞ ≤ 1, and apply Theorem 1.2, we can see ∗ μ Ω,α,λ f L2 Rn  ≤ Cλ,n ≤ sup MϕL∞ ϕL∞ ≤1  2 μΩ,α fx dx  Rn 5.5 Cλ,n f2L2α Rn  Moreover, by the observation that 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