ON WEIGHTED INEQUALITIES FOR PARAMETRIC MARCINKIEWICZ INTEGRALS H. M. AL-QASSEM Received 25 February 2005; Revised 30 May 2005; Accepted 3 July 2005 We establish a weighted L p boundedness of a parametric Marcinkiewicz integral operator ᏹ ρ Ω,h if Ω isallowedtobeintheblockspaceB (0,−1/2) q (S n−1 )forsomeq>1andh satis- fies a mild integrabilit y condition. We apply this conclusion to obtain the weighted L p boundedness for a class of the parametric Marcinkiewicz integral operators ᏹ ∗,ρ Ω,h,λ and ᏹ ρ Ω,h,S related to the Littlewood-Paley g ∗ λ -function and the area integral S, respectively. It is known that the condition Ω ∈ B (0,−1/2) q (S n−1 )isoptimalfortheL 2 boundedness of ᏹ 1 Ω,1 . Copyright © 2006 H. M. Al-Qassem. 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. 1. Introduction Suppose that S n−1 istheunitsphereofR n (n ≥ 2) equipped with the normalized Lebesgue measure dσ = dσ(·). Let Ω be a function defined on S n−1 with Ω ∈ L 1 (S n−1 ) and satisfies the vanishing condition  S n−1 Ω(x) dσ(x) = 0. (1.1) For γ>1, let Δ γ (R + ) denote the set of all measurable functions h on R + such that sup R>0 1 R  R 0   h(t)   γ dt < ∞. (1.2) It is easy to see that the following inclusions hold and are proper: L ∞  R +  ⊂ Δ β  R +  ⊂ Δ α  R +  for α<β. (1.3) Throughout this paper, we let x  denote x/|x| for x ∈ R n \{0} and p  denote the con- jugate index of p; that is, 1/p+1/p  = 1. Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2006, Article ID 91541, Pages 1–17 DOI 10.1155/JIA/2006/91541 2 Weighted marcinkiewicz integrals Suppose that Γ(t)isastrictlymonotonicC 1 function on R + and h : R + → C is a mea- surable function. Define the parametric Marcinkiewicz integral operator ᏹ ρ Ω,Γ,h by ᏹ ρ Ω,Γ,h f (x) =   ∞ 0   F ρ Ω,Γ,h f (t,x)   2 dt t  1/2 , (1.4) where F ρ Ω,Γ,h f (t,x) = 1 t ρ  |u|≤t f  x − Γ  | u|)u   Ω  u   |u| n−ρ h  | u|  du, (1.5) ρ = σ + iτ (σ, τ ∈ R with σ>0), f ∈ ᏿(R n ), the space of Schwartz functions. For the sake of simplicity, we denote ᏹ ρ Ω,Γ,h = ᏹ ρ Ω,h if Γ(t) ≡ t. It is well-known that ᏹ 1 Ω,1 is the classical Marcinkiewicz integral operator of higher di- mension, corresponding to the Littlewood-Paley g-function, introduced by Stein in [17]. Stein showed that ᏹ 1 Ω,1 is bounded on L p (R n )forp ∈ (1,2] if Ω ∈ Lip α (S n−1 )(0<α≤ 1). Subsequently, Benedek et al. proved that ᏹ 1 Ω,1 is bounded on L p (R n )forp ∈ (1,∞)if Ω ∈ C 1 (S n−1 ) (see [3]). Later on, the case of rough kernels (Ω satisfies only size and cance- lation conditions but no regularity is assumed) became the interest of many authors. For a sample of past studies, see ([1, 2, 4, 5]). In [2], Al-Qassem and Al-Salman showed that ᏹ 1 Ω,1 is bounded on L p (R n )forp ∈ (1,∞)ifΩ belongs to the block space B (0,−1/2) q (S n−1 ) and that the condition Ω ∈ B (0,−1/2) q (S n−1 ) is optimal in the sense that there exists an Ω which lies in B (0,υ) q (S n−1 )forall−1 <υ<−1/2suchthatᏹ 1 Ω,1 is not bounded on L 2 (R n ). In H ¨ ormander [10] defined the parametric Marcinkiewicz operator ᏹ ρ Ω,1 for ρ>0and proved that ᏹ 1 Ω,1 is bounded on L p (R n )forp ∈ (1,∞)ifΩ ∈ Lip α (S n−1 )(0<α≤ 1). Sakamoto and Yabuta [15]studiedtheL p -boundedness of the more general parametric Marcinkiewicz integral operator ᏹ ρ Ω,1 if ρ is complex and proved that ᏹ ρ Ω,1 is bounded on L p (R n )forp ∈ (1,∞)ifRe(ρ) = σ>0andΩ ∈ Lip α (S n−1 )(0<α≤ 1). Recently, in [1] the author of this paper gave that the more general operator ᏹ ρ Ω,Γ,h is bounded on L p (R n )forp satisfying |1/p− 1/2|≤min{1/2,1/γ  } if Re(ρ) = σ>0, Γ satisfies a convex- ity condition, Ω ∈ B (0,−1/2) q (S n−1 )andh ∈ Δ γ (R + )forsomeq,γ>1. This is an essential improvement and extension of the results mentioned above. On the other hand, the weighted L p boundedness of ᏹ 1 Ω,h has also attracted the atten- tion of many authors in the recent years. Indeed, Torchinsky and Wang in [19]provedthat if Ω ∈ Lip α (S n−1 ), (0 <α≤ 1), then ᏹ 1 Ω,1 is bounded on L p (ω)forp ∈ (1,∞)andω ∈ A p (The Muckenhoupt’s weight class, see [9] for the definition). In Sato in [16]improvedthe weighted L p boundedness of Torchinsky-Wang by proving that ᏹ 1 Ω,h is bounded on L p (ω) for p ∈ (1,∞)providedthath ∈ L ∞ (R + ), Ω ∈ L ∞ (S n−1 )andω ∈ A p (R n ). Subsequently, in Ding et al. in [5] were able to show that ᏹ 1 Ω,h is bounded on L p (ω)forp ∈ (1,∞) provided that h ∈ L ∞ (R + ), Ω ∈ L q (S n−1 ), q>1andω q  ∈ A p (R n ). In a recent paper, Lee and Lin in [13] showed that ᏹ 1 Ω,h is bounded on L p (ω)forp ∈ (1,∞)ifh ∈ L ∞ (R + ), Ω ∈ H 1 (S n−1 )andω ∈  A I p (R n ), where H 1 (S n−1 )istheHardyspaceontheunitsphere and  A I p (R n ) is a special class of r adial weig hts introduced by Duoandikoetxea [6] whose definition will recalled in Section 2. H. M. Al-Qassem 3 In this paper, we will investigate the weighted L p (ω) boundedness of the paramet- ric Marcinkiewicz operator ᏹ ρ Ω,Γ,h for ω ∈  A I p (R n ) and under the natural condition Ω ∈ B (0,−1/2) q (S n−1 ). To state our results, we will need the following definitions from [8]. Definit ion 1.1. We say that a function Γ satisfies “hypothesis I”if (a) Γ is a nonnegative C 1 function on (0,∞), (b) Γ is strictly increasing, Γ(2t) ≥ ηΓ(t)forsomefixedη>1andΓ(2t) ≤ cΓ(t)for some constant c ≥ η>1. (c) Γ  (t) ≥ αΓ(t)/t on (0,∞)forsomefixedα ∈ (0,log 2 c]andΓ  (t) is monotone on (0, ∞). Definit ion 1.2. We say that Γ satisfies “hypothesis D”if (a  ) Γ is a nonnegative C 1 function on (0,∞), (b  ) Γ is strictly decreasing, Γ(t) ≥ ηΓ(2t)forsomefixedη>1andΓ(t) ≤ cΓ(2t)for some constant c ≥ η>1. (c  ) |Γ  (t)|≥αΓ(t)/t on (0,∞)forsomefixedα ∈ (0, log 2 c]andΓ  (t)ismonotoneon (0, ∞). Model functions for the Γ satisfy hypothesis I are Γ(t) = t d with d>0, and their linear combinations with positive coefficients. Model functions for the Γ satisfy hypothesis D are Γ(t) = t r with r<0, and their linear combinations with positive coefficients. Theorem 1.3. Let h ∈ Δ γ (R + ) for some γ>1. Assume that Γ satisfies either hypothesis I or hypothesis D and Ω ∈ B (0,−1/2) q (S n−1 ) for some q>1. Then   ᏹ ρ Ω,Ψ,h ( f )   L p (R n ) ≤ C p   Ω   B (0,−1/2) q (S n−1 )   f   L p (R n ) (1.6) is bounded on L p (R n ) for |1/p− 1/2| < min{1/γ  ,1/2}. Theorem 1.4. Let h ∈ Δ γ (R + ) for some γ ≥ 2, 1 <p<∞. Assume that Γ satisfies either hypothesis I or hypothesis D and Ω ∈ B (0,−1/2) q (S n−1 ) for some q>1. Then there exists C p > 0 such that the following inequality holds:   ᏹ ρ Ω,Γ,h ( f )   L p (ω) ≤ C p   Ω   B (0,−1/2) q (S n−1 )   f   L p (ω) (1.7) for γ  <p<∞ and ω ∈  A I p/γ  (R + ). Remark 1.5. (a) In order to make a comparison among the above mentioned results, we remark that on S n−1 ,foranyq>1, 0 <α≤ 1and−1 <υ, the following inclusions hold and are pro per: C 1  S n−1  ⊂ Lip α  S n−1  ⊂ L q  S n−1  ⊂ L  log + L  S n−1  ⊂ H 1  S n−1  ,  r>1 L r  S n−1  ⊂ B (0,υ) q  S n−1  . (1.8) With regard to the relationship between B (0,υ) q (S n−1 )andH 1 (S n−1 )(forυ>−1) remains open. (b) We point out that the result in Theorem 1.3 extends the result of Al-Qassem and Al-Salman [2] who obtained Theorem 1.3 in the special case h ≡ 1andΓ(t) ≡ t and 4 Weighted marcinkiewicz integrals also improves substantially the result of Sakamoto and Yabuta [15]. We remark also that Theorem 1.4 represents an improvement and extension of [5, Theorem 1] in the case ω ∈  A I p (R + ). (c) The method employed in this paper is based in part on ideas from [1, 2, 7, 8, 16], among others. The paper is organized as follows. In Section 2 we give some definitions and we estab- lish the main estimates needed in the proofs of our main results. The proofs of T heorems 1.3 and 1.4 will be given in Section 3. Additional results can be found in Section 4. Throughout the rest of the paper the letter C will denote a positive constant whose value may change at each occurrence. 2. Definitions and lemmas Let us begin by recalling the definition of some special classes of weights and some of their important properties. Definit ion 2.1. Let ω(t) ≥ 0andω ∈ L 1 loc (R + ). For 1 <p<∞,wesaythatω ∈ A p (R + )if there is a positive constant C such that for any interval I ⊂ R + ,  | I| −1  I ω(t)dt  | I| −1  I ω(t) −1/(p−1) dt  p−1 ≤ C<∞. (2.1) A 1 (R + )istheclassofweightsω for which M satisfies a weak-type estimate in L 1 (ω), where M( f )istheHardy-Littlewoodmaximalfunctionof f . It is well-known that the class A 1 (R + ) is also characterized by all weights ω for which Mω(t) ≤ Cω(t)fora.e.t ∈ R + and for some positive constant C. Definit ion 2.2. Let 1 ≤ p<∞.Wesaythatω ∈  A p (R + )if ω(x) = ν 1  | x|  ν 2  | x|  1−p , (2.2) where either ν i ∈ A 1 (R + ) is decreasing or ν 2 i ∈ A 1 (R + ), i = 1,2. Let A I p (R n ) be the weight class defined by exchanging the cubes in the definitions of A p for all n-dimensional intervals with sides parallel to coordinate axes (see [12]). Let  A I p =  A p ∩ A I p .Ifω ∈  A p ,itfollowsfrom[6] that the classical Hardy-Littlewood maximal function Mf is bounded on L p (R n ,ω(|x|)dx). Therefore, if ω(t) ∈  A p (R + ), then ω(|x|) ∈ A p (R n ). By following the same argument as in the proof of the elementary properties of A p weight class (see, e.g., [9]) we get the following lemma. Lemma 2.3. If 1 ≤ p<∞, then the weight class  A I p (R + ) has the following properties: (i)  A I p 1 ⊂  A I p 1 ,if1 ≤ p 1 <p 2 < ∞; (ii) For any ω ∈  A I p ,thereexistsanε>0 such that ω 1+ε ∈  A I p ; (iii) For any ω ∈  A I p and p>1,thereexistsanε>0 such that p − ε>1 and ω ∈  A I p −ε . The block spaces originated in the work of Taibleson and Weiss on the convergence of the Fourier series in connection with the developments of the real Hardy spaces. Below H. M. Al-Qassem 5 we will recall the definition of block spaces on S n−1 . For further background information about the theory of spaces generated by blocks and its applications to harmonic analysis, see the book [14]. Definit ion 2.4. A q-block on S n−1 is an L q (1 <q≤∞) function b(x) that satisfies (i) supp(b) ⊂ I; (ii) b L q ≤|I| −1/q  , (2.3) where |I|=σ(I), and I = B(x  0 ,θ 0 ) ={x  ∈ S n−1 : |x  − x  0 | <θ 0 } is a cap on S n−1 for some x  0 ∈ S n−1 and θ 0 ∈ (0,1]. Jiang and Lu introduced (see [14]) the class of block spaces B (0,υ) q (S n−1 )(forυ>−1) with respect to the study of homogeneous singular integral operators. Definit ion 2.5. The block space B (0,υ) q (S n−1 )isdefinedby B (0,υ) q  S n−1  =  Ω ∈ L 1  S n−1  : Ω = ∞  μ=1 η μ b μ , M (0,υ) q  η μ  < ∞  , (2.4) where each η μ is a complex number; each b μ is a q-block supported on a cap I μ on S n−1 , υ> −1and M (0,υ) q  η μ  = ∞  μ=1   η μ    1+log (υ+1)    I μ   −1  . (2.5) Let Ω B (0,υ) q (S n−1 ) = inf{M (0,υ) q ({η μ }):Ω =  ∞ μ=1 η μ b μ and each b μ is a q-block function supported on a cap I μ on S n−1 }.Then· B (0,υ) q (S n−1 ) is a norm on the space B (0,υ) q (S n−1 ) and (B (0,υ) q (S n−1 ),· B (0,υ) q (S n−1 ) )isaBanachspace. In their investigations of block spaces, Keitoku and Sato in [11] showed that these spaces enjoy the following properties: B (0,υ 2 ) q  S n−1  ⊂ B (0,υ 1 ) q  S n−1  if υ 2 >υ 1 > −1; B (0,υ) q 2  S n−1  ⊂ B (0,υ) q 1  S n−1  if 1 <q 1 <q 2 ,foranyυ>−1;  q>1 B (0,υ) q  S n−1    q>1 L q  S n−1  for any υ>−1. (2.6) Definit ion 2.6. For a suitable C 1 function Γ on R + , a measurable function h : R + → C and a suitable function  b μ on S n−1 we define the family of measures {σ  b μ ,t : t ∈ R + } and the maximal operator σ ∗  b μ on R n by  R n fdσ  b μ ,t = 1 t ρ  (1/2)t<|y|≤t f  Γ  | y|  y   h  | y|   b μ (y  ) |y| n−ρ dy, σ ∗  b μ f (x) = sup t∈R +       σ  b μ ,t    ∗ f (x)    , (2.7) 6 Weighted marcinkiewicz integrals where |σ  b μ ,t | is defined in the same way as σ  b μ ,t ,butwith  b μ replaced by |  b μ | and h replaced by |h|. For k ∈ Z, μ ∈ N ∪{0},andacapI μ on S n−1 with |I μ | <e −2 ,weletθ μ = [log|I μ | −1 ] and ω μ = 2 θ μ ,where[·] denotes the greatest integer function. Now set a k,μ = Γ(ω k μ )ifΓ satisfies hypothesis I and a k,μ = (Γ(ω k μ )) −1 if Γ satisfies hy pothesis D. Then by the con- ditions of Γ,itiseasytoseethat {a k,μ } is a lacunary sequence of positive numbers with inf k∈Z (a k+1,μ /a k,μ ) ≥ η θ μ > 1. Lemma 2.7. Let μ ∈ N ∪{0} and h ∈ Δ γ (R + ) for some γ with 1 <γ≤ 2.Let  b μ be a function on S n−1 satisfying (i)  S n−1  b μ (y)dσ(y) = 0; (ii)   b μ  q ≤|I μ | −1/q  for some q>1 and for some cap I μ on S n−1 with |I μ | <e −2 ;and(iii)   b μ  1 ≤ 1. Then there exist constants C and 0 <υ<1/q  such that if Γ satisfies hypothesis I,   σ  b μ ,t   ≤ C; (2.8)  ω k+1 μ ω k μ    σ  b μ ,t (ξ)   2 dt t ≤ Cθ μ  a k,μ  −2υ/γ  θ μ |ξ| −2υ/γ  θ μ ; (2.9)  ω k+1 μ ω k μ    σ  b μ ,t (ξ)   2 dt t ≤ Cθ μ  a k,μ  2υ/γ  θ μ |ξ| 2υ/γ  θ μ , (2.10) and if Γ satisfies hypothesis D,   σ  b μ ,t   ≤ C;  ω k+1 μ ω k μ     σ  b μ ,t (ξ)    2 dt t ≤ Cθ μ  a k,μ  −2υ/γ  θ μ   ξ   2υ/γ  θ μ ;  ω k+1 μ ω k μ     σ  b μ ,t (ξ)    2 dt t ≤ Cθ μ  a k,μ  2υ/γ  θ μ |ξ| −2υ/γ  θ μ , (2.11) where σ  b μ ,t  stands for the total variation of σ  b μ ,t . The constant C is independent of k, μ, ξ and Γ( ·). Proof. We will only present the proof of the lemma if Γ satisfies hypothesis I, since the proofforthecasethatΓ satisfies hypothesis D will be essentially the same. By (iii) and the definition of σ  b μ ,t , one can easily see that (2.8) holds with a constant C independent of t and μ.Nextweprove(2.9). By definition, σ  b μ ,t (ξ) = 1 t ρ  t (1/2)t  S n−1 e −iΨ(s)ξ·x  b μ (x) h(s) s 1−ρ dσ(x)ds. (2.12) H. M. Al-Qassem 7 By H ¨ older’s inequality, a change of variable and since |  S n−1 e −iΨ(s)ξ·x  b μ (x) dσ(x)|≤1, we obtain    σ  b μ ,t (ξ)   ≤   t (1/2)t   h(s)   γ ds s  1/γ   t (1/2)t      S n−1 e −iΨ(s)ξ·x  b μ (x) dσ(x)     γ  ds s  1/γ  ≤ C   t (1/2)t      S n−1 e −iΨ(s)ξ·x  b μ (x) dσ(x)     2 ds s  1/γ  = C   S n−1 ×S n−1  b μ (x)  b μ (y)I μ,t (ξ,x, y)dσ(x)dσ(y)  1/γ  , (2.13) where I μ,t (ξ,x, y) =  1 1/2 e −iΓ(ts)ξ·(x−y) ds s . (2.14) Write I μ,t (ξ,x, y)as I μ,t (ξ,x, y) =  1 1/2 Y  t (s) ds s , (2.15) where Y t (s) =  s 1/2 e −iΓ(tw)ξ·(x− y) dw,1/2 ≤ s ≤ 1. (2.16) Now, using the assumptions on Γ,weobtain d dw  Γ(tw)  = tΓ  (tw) ≥ α Γ(tw) w ≥ α Γ(t/2) s ≥ α c Γ(t) s for 1/2 ≤ w ≤ s ≤ 1. (2.17) Thus by van der Corput’s lemma, |Y t (s)|≤(c/α)|Γ(t)ξ/s| −1 |ξ  · (x − y)| −1 .Byintegration by parts, we get   I μ,t (ξ,x, y)   ≤ C   Γ(t)ξ   −1   ξ  · (x − y)   −1 , (2.18) which when combined with the trivial estimate |I μ,t (ξ,x, y)|≤log2 and choosing τ such that 0 <τ<1/q  yields   I μ,t (ξ,x, y)   ≤   Γ(t)ξ   −τ   ξ  · (x − y)   −τ . (2.19) By H ¨ older’s inequality and (ii) we get     σ  b μ ,t (ξ)    ≤ C   Γ(t)ξ   −τ/γ      b μ    2/γ  q ×   S n−1 ×S n−1   ξ  · (x − y)   −τq  dσ(x)dσ(y)  1/(q  γ  ) ≤ C   Γ(t)ξ   −τ/γ    I μ   −2/(q  γ  ) . (2.20) 8 Weighted marcinkiewicz integrals Therefore,  ω k+1 μ ω k μ    σ  b μ ,t (ξ)   2 dt t ≤ Cmin  log    I μ   −1  ,log    I μ   −1    Γ  ω k μ  ξ   −2τ/γ    I μ   −2/(q  γ  )  ≤ C log    I μ   −1    Γ  ω k μ  ξ   −2τ/γ  log(|I μ | −1 ) , (2.21) which proves (2.9). To prove (2.10), we use the cancellation condition of  b μ to get     σ  b μ ,t (ξ)    ≤  S n−1  t t/2   e −iΓ(s)ξ·x − 1     h(s)       b μ (x)    ds s dσ(x). (2.22) Hence, by (iii) and since Γ is increasing we get     σ  b μ ,t (ξ)    ≤ C   Γ(t)ξ   . (2.23) By using the same argument as above we get (2.10). The lemma is proved.  Lemma 2.8. Let μ ∈ N ∪{0}, h ∈ Δ γ (R + ) for some γ>1, γ  <p<∞ and ω ∈  A p/γ  (R + ). Assume that  b μ ∈ L 1 (S n−1 ) and Γ satisfies either hypothesis I or hypothesis D. Then there exists a posit ive constant C p such that     σ ∗  b μ ( f )     L p (ω) ≤ C p     b μ    L 1  S n−1   f  L p (ω) . (2.24) Proof. By H ¨ older’s inequality, we have     σ  b μ ,t   ∗ f (x)   ≤   t (1/2)t   h(s)   γ ds s  1/γ   t (1/2)t      S n−1  b μ (y  ) f  x − Γ(s)y   dσ(y  )     γ  ds s  1/γ  ≤ C    b μ   1/γ L 1 (S n−1 )   t (1/2)t  S n−1     b μ (y  )      f  x − Γ(s)y     γ  dσ(y  ) ds s  1/γ  . (2.25) Thus σ ∗  b μ f (x) ≤ C     b μ    1/γ L 1 (S n−1 )   S n−1     b μ (y  )    M Γ,y   | f | γ   (x) dσ(y  )  1/γ  , (2.26) where M Γ,y  f (x) = sup t∈R +      t t/2 f  x − Γ(s)y   ds s     . (2.27) H. M. Al-Qassem 9 Let w = Γ(s). Assume first that Γ satisfies hypothesis I.BytheassumptionsonΓ,wehave ds/s ≤ dw/αw. So, by a change of variable we have M Γ,y  f (x) ≤ sup t∈R +   Γ(t) Γ(t/2)   f (x − wy  )   dw w  ≤ sup t∈R +   cΓ(t/2) Γ(t/2)   f (x − wy  )   dw w  ≤ CM y  f (x), (2.28) where M y  f (x) = sup R>0 R −1  R 0   f (x − wy  )   dw (2.29) is the Hardy-Littlewood maximal function of f in the direction of y  . On the other hand, if Γ satisfies hypothesis D, as above we have ds/s ≤−dw/αw and M Γ,y  f (x) ≤ 1 α sup t∈R +   Γ(t/2) Γ(t)   f (x − wy  )   dw w  ≤ 1 α sup t∈R +   Γ(t/2) (1/c)Γ(t/2)   f  x − wy  )   dw w  ≤ CM y  f (x). (2.30) By (2.26)–(2.30) and Minkowski’s inequality for integrals we get    σ ∗  b μ ( f )    L p (ω) ≤ C     b μ    1/γ L 1 (S n−1 )   S n−1     b μ (y  )      M y   | f | γ     L p/γ  (ω) dσ(y  )  1/γ  . (2.31) By [6, equation (8)] and since ω ∈  A p/γ  (R + )wehave   M y  f   L p/γ  (ω) ≤ C f  L p/γ  (ω) (2.32) with C independent of y  .Thus,by(2.31)–(2.32)weget(2.24). This completes the proof of the lemma.  Lemma 2.9. Let μ ∈ N∪{0}, h ∈ Δ γ (R + ) for some γ ≥ 2, γ  <p<∞ and ω ∈  A p/γ  (R + ). Assume that  b μ ∈ L 1 (S n−1 ) and Γ satisfies either hypothesis I or hypothesis D. Then there exists a posit ive constant C p such that the inequality        k∈Z  ω k+1 μ ω k μ    σ  b μ ,t ∗ g k    2 dt t  1/2      L p (ω) ≤ C p  log   I μ   −1  1/2     b μ    L 1 (S n−1 )        k∈Z   g k   2  1/2      L p (ω) , (2.33) holds for any sequence of functions {g k } k∈Z on R n . 10 Weighted marcinkiewicz integrals Proof. Let γ  <p<∞. By a change of variable, we have   k∈Z  ω k+1 μ ω k μ    σ  b μ ,t ∗ g k    2 dt t  1/2 ≤   k∈Z  ω μ 1    σ  b μ ,ω k μ t ∗ g k    2 dt t  1/2 . (2.34) By H ¨ older’s inequality and following a similar arguments as in the proof of (2.25)weget    σ  b μ ,t ∗ g k (x)    γ  ≤ C     b μ    L 1 (S n−1 )   t t/2  S n−1     b μ (y  )      g k  x − Γ(s)y     γ  dσ(y  ) ds s  . (2.35) Let d = p/γ  . By duality, there is a nonnegative function f ∈ L d  (ω 1−d  ,R n ) satisfying  f  L d  (ω 1−d  ) ≤ 1suchthat        k∈Z  ω μ 1    σ  b μ ,ω k μ t ∗ g k    γ  dt t  1/γ       γ  L p (ω) =  R n  k∈Z  ω μ 1    σ  b μ ,ω k μ t ∗ g k (x)    γ  dt t f (x)dx. (2.36) Therefore, by (2.35)–(2.36) and a change of variable we get        k∈Z  ω μ 1    σ  b μ ,ω k μ t ∗ g k    γ  dt t  1/γ       γ  L p (ω) ≤ C  log   I μ   −1      b μ    L 1 (S n−1 )  R n  k∈Z   g k (x)   γ  M ∗ μ f (x)dx, (2.37) where M ∗ μ f (x) = sup t∈R +  (1/2)t<|y|≤t f  x + Γ  | y|  y       b μ (y  )    | y| −n dy. (2.38) By H ¨ older’s inequality, we obtain        k∈Z  ω μ 1    σ  b μ ,ω k μ t ∗ g k    γ  dt t  1/γ       γ  L p (ω) ≤ C  log   I μ   −1      b μ    L 1 (S n−1 )        k∈Z   g k   γ   1/γ       γ  L p (ω)    M ∗ μ f    L d  (ω 1−d  ) . (2.39) It is easy to verify that ω ∈  A d (R + )ifandonlyifω 1−d  ∈  A d  (R + ). By the same argument as in the proof of Lemma 2.8,wehave   M ∗ μ f   L d  (ω 1−d  ) ≤ C p     b μ    L 1 (S n−1 ) (2.40) [...]... (Rn ) for γ < p < ∞ (3.21) By Lemma 2.3, for any ω ∈ AIp/γ (R+ ), there is an ε > 0 such that ω1+ε ∈ AIp/γ (R+ ) Thus T j,μ ( f ) L p (ω1+ε ) ≤ C p log Iμ −1 1/2 f L p (ω1+ε ) for γ < p < ∞ (3.22) By interpolating with change of measures between (3.20) and (3.21) we get (3.19) 4 Further results As an application of Theorem 1.4, we get the weighted L p boundedness for a class of ∗,ρ ρ parametric Marcinkiewicz. .. (1962), 356–365 [4] J Chen, D Fan, and Y Pan, A note on a Marcinkiewicz integral operator, Mathematische Nachrichten 227 (2001), no 1, 33–42 [5] Y Ding, D Fan, and Y Pan, Weighted boundedness for a class of rough Marcinkiewicz integrals, Indiana University Mathematics Journal 48 (1999), no 3, 1037–1055 [6] J Duoandikoetxea, Weighted norm inequalities for homogeneous singular integrals, Transactions of the... gk k ∈Z 2 L p (Rn ) holds for arbitrary functions {gk (·)}k∈Z on Rn The constant C p is independent of μ 12 Weighted marcinkiewicz integrals 3 Proofs of main results We will only present the proof of Theorems 1.3 and 1.4 for the case Γ satisfies hypothesis I, since the proofs for the case Γ satisfies hypothesis D are essentially the same As(0, sume that Ω ∈ Bq −1/2) (Sn−1 ) for some q > 1 and satisfies... μ ,Γ,h (f) L p (Rn ) ≤ C p log Iμ −1 1/2 f L p (Rn ) (3.7) f L p (ω) (3.8) for p satisfying |1/ p − 1/2| < min{1/γ ,1/2}; and ρ ᏹb μ ,Γ,h (f) L p (ω) ≤ C p log Iμ −1 1/2 for all ω ∈ AIp/γ (R+ ) and γ < p < ∞ Proof of (3.7) Since Δγ (R+ ) ⊆ Δ2 (R+ ) for γ ≥ 2, we may assume that 1 < γ ≤ 2 Therefore, it suffices to prove (3.7) for p satisfying |1/ p − 1/2| < 1/γ Let {φk,μ }∞ be a smooth −∞ 1 partition... Francia, Maximal and singular integral operators via Fourier transform estimates, Inventiones Mathematicae 84 (1986), no 3, 541–561 [8] D Fan, Y Pan, and D Yang, A weighted norm inequality for rough singular integrals, The Tohoku Mathematical Journal Second Series 51 (1999), no 2, 141–161 [9] J Garc´a-Cuerva and J L Rubio de Francia, Weighted Norm Inequalities and Related Topics, ı North-Holland Mathematics... ( f ) L p (ω) ≤ f L p (ω) (4.6) for all ω(x) = |x|α and α ∈ (−1, p/γ − 1) Similar results hold regarding the operators ρ ρ,∗ ᏹΩ,Ψ,h,S and ᏹΩ,Ψ,h,λ A proof of this theorem can be obtained by Theorem 4.3 and noticing that |x|α ∈ AIp (R+ ) for α ∈ (−1, p − 1) Acknowledgment I would like to thank Professor Chin-Cheng Lin for sending me a preprint of his paper [13] and for his encouraging comments References... Chin-Cheng Lin for sending me a preprint of his paper [13] and for his encouraging comments References [1] H M Al-Qassem, L p estimates for rough parametric Marcinkiewicz integrals, SUT Journal of Mathematics 40 (2004), no 2, 117–131 [2] H M Al-Qassem and A J Al-Salman, A note on Marcinkiewicz integral operators, Journal of Mathematical Analysis and Applications 282 (2003), no 2, 698–710 ´ [3] A Benedek, A.-P... ∈ Rn : ξ ∈ ᏶k,μ (3.16) 14 Weighted marcinkiewicz integrals By Lemma 2.7 we have T j,μ ( f ) L2 (Rn ) ≤ C log Iμ −1 1/2 −β| j | η f L2 (Rn ) (3.17) Next, let us compute the L p boundedness of the operator T j,μ For |1/ p − 1/2| < 1/γ , we have T j,μ ( f ) L p (Rn ) ≤ C p log Iμ 1/2 −1 1/2 2 Υk+ j,μ ∗ f k ∈Z ≤ C p log Iμ −1 1/2 f L p (Rn ) (3.18) L p (Rn ) The last two inequalities are obtained by... (Rn ,ω(x)dx) Therefore, we can interpolate (2.43) and (2.44) (See [9, page 481] for the vector-valued interpolation) to get (2.33) The lemma is proved By following the same argument as in the proof of [1, Lemma 3.4], we get the following lemma Lemma 2.10 Let μ ∈ N∪{0}, h ∈ Δγ (R+ ) for some γ ∈ (1,2] Assume that bμ ∈ L1 (Sn−1 ) and Γ satisfies either hypothesis I or hypothesis D Then, for any p satisfying... Let J = {μ ∈ N :|Iμ | < e−2 } Let b0 = Ω− the following holds for all μ ∈ J ∪ {0}: Sn−1 Sn−1 (3.1) Then for some positive constant C, bμ (u)dσ(u) = 0, bμ ≤ C Iμ q bμ 1 −1/q (3.2) , (3.3) ≤ C, Ω= (3.4) λμ bμ , (3.5) μ∈J∪{0} where I0 is a cap on Sn−1 with |I0 | = e−3 By (3.5) we have ρ ᏹΩ,Γ,h ( f ) ≤ ρ μ∈J∪{0} λ μ ᏹb μ ,Γ,h ( f ) (3.6) Therefore, Theorems 1.3 and 1.4 are proved if we can show that ρ ᏹb . ON WEIGHTED INEQUALITIES FOR PARAMETRIC MARCINKIEWICZ INTEGRALS H. M. AL-QASSEM Received 25 February 2005; Revised 30 May 2005; Accepted 3 July 2005 We establish a weighted L p boundedness. mild integrabilit y condition. We apply this conclusion to obtain the weighted L p boundedness for a class of the parametric Marcinkiewicz integral operators ᏹ ∗,ρ Ω,h,λ and ᏹ ρ Ω,h,S related to. lies in B (0,υ) q (S n−1 )forall−1 <υ<−1/2suchthatᏹ 1 Ω,1 is not bounded on L 2 (R n ). In H ¨ ormander [10] defined the parametric Marcinkiewicz operator ᏹ ρ Ω,1 for ρ>0and proved that