Boundedness of Marcinkiewicz integrals with mixed homogeneity along compound surfaces

In this note we establish the Lp boundedness of Marcinkiewicz integrals with mixed homogeneity along compound surfaces, which improve and extend some previous results. The main ingredient is to present a systematic treatment with several singular integral operators. MSC:42B20, 42B15, 42B25.


Introduction
Let R n , n ≥ , be the n-dimensional Euclidean space and S n- denote the unit sphere in R n equipped with the induced Lebesgue measure dσ . Let α j ≥  (j = , . . . , n) be fixed real numbers. Define the function F : R n × (, ∞) − → R by F(x, ρ) = n j= x  j ρ -α j , x = (x  , x  , . . . , x n ). It is clear that, for each fixed x ∈ R n , the function F(x, ρ) is a decreasing function in ρ > . We let ρ(x) denote the unique solution of the equation F(x, ρ) = . Fabes and Rivière [] showed that (R n , ρ) is a metric space, which is often called the mixed homogeneity space related to {α j } n j= . For λ > , we let A λ be the diagonal n × n matrix A λ = diag{λ α  , . . . , λ α n }. Let R + := (, ∞) and ϕ : R + → R + , we denote A ϕ(ρ(y)) y by A ϕ (y) for y ∈ R n , where y = A ρ(y) - y ∈ S n- .
Let be integrable on S n- and satisfy When α  = · · · = α n = , we denote M by μ . Clearly, if n = d and (y) = y, the operator μ reduces to the classical Marcinkiewicz integral operator denoted by μ , which was introduced by Stein [] and investigated by many authors (see [-] for example). In particular, Ding et al. [] proved that if ∈ H  (S n- ), then μ is bounded on L p (R n ) for  < p < ∞. Subsequently, Chen et al. [] showed that μ is bounded on L p (R n ) for β/(β -) < p < β if ∈ F β (S n- ) for some β > . Here The functions class F β (S n- ) was introduced by Grafakos and Stefanov [] in the study of L p boundedness of singular integral operator with rough kernels. It follows from Later on, Al-Salman et al. [] proved that μ is bounded on L p (R n ) for  < p < ∞ provided that ∈ L(log + L) / (S n- ). It is well known that L(log + L) / (S n- ) and H  (S n- ) do not contain each other. When n = d and (y) = P(|y|)y with P(y) being a real polynomial on R sat-http://www.journalofinequalitiesandapplications.com/content/2014/1/265 isfying P() = , Wu [] proved that μ is bounded on L p (R n ) for  + /(β) < p <  + β provided that ∈ F β (S n- ) for some β > /. The L p boundedness for the Marcinkiewicz integral operator associated to polynomial mappings has also been obtained (see [, ]). When α j ≥  (j = , . . . , n), n = d and (y) = y, we denote M by M . In , Ding et al. [] proved that M is bounded on L p (R n ) for  < p < ∞, provided that ∈ L q (S n- ) for fixed q > . Chen and Ding [] extended the above result to the case ∈ L(log L) / (S n- ). Later on, Chen and Lu [] proved that M is bounded on L p (R n ) for β/(β -) < p < β, provided that ∈ F β (S n- ) for some β > . This result was recently refined by Liu and Wu [], who extended the range of β to the case β > / and the range of p to the case +/(β) < p < +β. When n = d and (y) = A ϕ (y), Al-Salman [] obtained the following result.
(i) If ϕ(t) = P(t) with P being a real polynomial on R, then M are bounded on L p (R n ) for β/(β -) < p < β. The bounds are independent of the coefficients of P.
It is natural to ask whether Theorem A also holds if the range of β is relaxed to β > / and the range of p is relaxed to  + /(β) < p <  + β. In this paper, we will give an affirmative answer to this question. Our main results can be stated as follows.
Theorem . Let n = d and (y) = A P N (ϕ) (y) with ϕ ∈ F and P N (t) = N i= a i t i and P N (t) >  if t = . Suppose that ∈ F β (S n- ) for some β > / satisfying (.)-(.). Then M are bounded on L p (R n ) for +/(β) < p < +β. The bounds are independent of the coefficients of P N but depend on N and ϕ.
Remark . It is clear that Theorem . implies Theorem .. When α  = · · · = α n = , ϕ(t) = t and P  (t) = · · · = P n (t) = N i= a i t i , Theorem . implies the result of []. In fact, Theorem . with ϕ(t) = t extends the result of [] to the mixed case. Comparing Theorem A with Theorem ., the range of β is extended to the case β > / and the range of p http://www.journalofinequalitiesandapplications.com/content/2014/1/265 is enlarged to the case  + /(β) < p <  + β. Thus Theorem . essentially improves and generalizes the corresponding results in Theorem A. In addition, Theorem . implies the result [, Theorem .] when P N (t) = ϕ(t) = t.
When n = , we have the following result.
The bounds are independent of the coefficients of P N but depend on ϕ and N .
The third type of surfaces we consider are polynomial compound subvarieties. To state the rest of our result, we need to recall some notations. Let A(n, m) be the set of polynomials on R n which have real coefficients and degrees not exceeding m, and let V (n, m) be the collection of polynomials in A(n, m) which are homogeneous of degree m.
It should be pointed out that the condition (.) was introduced by Al-Salman and Pan [] (also see []) in a study of the L p boundedness of singular integrals with rough kernels. It is easy to check that F(n, The rest of the results can be stated as follows. Theorem . Let P = (P  , . . . , P d ) with P j : R n → R being a polynomial for  ≤ j ≤ d. Let (y) = P(ϕ(ρ(y))y ) and ϕ ∈ F. Suppose that satisfies (.)-(.) and ∈ ∞ s= F(n, s, β) for some β > /. Then M are bounded on L p (R d ) for  + /(β) < p <  + β. The bounds are independent of the coefficients of P j for all  ≤ j ≤ d but depend on max ≤j≤d deg(P j ) and ϕ.
The rest of this paper is organized as follows. After recalling some preliminary notations and lemmas in Section , we will prove our results in Section . We would like to remark that the main methods employed in this paper is a combination of ideas and arguments from [, , ]. The main ingredient in our proofs is to give a systematic treatment with these operators mentioned above.
Throughout this paper, we let p satisfy /p + /p = . The letter C, sometimes with additional parameters, will stand for positive constants, not necessarily the same one at each occurrence, but independent of the essential variables.

Preliminaries
Lemma . Let {σ j,t } be a family of measures. Suppose that sup j∈Z sup t> |σ j,t | * g p ≤ C g p holds for some p >  and g ∈ L p (R n ). Then there exists a constant C >  such that Proof By the assumption, we have On the other hand, by the dual argument, there exists a function h ∈ L p (R n ) satisfying whereh(x) = h(-x). Thus, Lemma . follows from the standard interpolation arguments.
Let {a k } k∈Z be a sequence of real positive numbers with satisfying inf k∈Z a k+ /a k = a > . Let {λ k } k∈Z be a collection of C ∞ (, ∞) functions satisfying the following conditions: where C is independent of t and k. Let M ∈ N\{} and L : R n → R M be a linear transformation. For each k ∈ Z, we define the multiplier operators S k in R n by By an argument which is similar to those used in [, Proposition .], one can easily get the following lemma. The details are omitted here.
Lemma . Let S k be as in (.) and {g j,k,t } arbitrary functions on R n . Then (i) for each fixed  < p <  and  < q < p, The following lemma is our main ingredient in the proof of our main results.
Lemma . Let {τ k,t : k ∈ Z, t ∈ R + } be a family of uniformly bounded Borel measures on R n . Let {a k } k∈Z be a sequence of real numbers with satisfying inf k∈Z a k+ /a k = a > . Let M ∈ N\{} and L : R n → R M be a linear transformation. Suppose that for all  < q < ∞. Then for p ∈ (+/(β), +β) and β > /, there exists a constant C(a) >  such that Proof Let S k be as in (.). Then we can write Case .  + /(β) < p < . It follows from (.) and (.) that For each fixed k ∈ Z, we set
This, combined with (.), implies which, together with (.), completes the proof of Lemma .. is bounded on L p (R d ) for  < p < ∞. The bound is independent of the coefficients of P j for all  ≤ j ≤ d and f but depends on ϕ.
Then we can write where L i : R n → R n is the linear transformation given by L i (ξ ) = (a i, ξ  , . . . , a i,n ξ n ).
For each j ∈ Z, t ∈ R + and  ≤ s ≤ N , we define the measures {σ s j,t } and {|σ s j,t |} by We get from (.) On the other hand, by a change of variable, we have By Lemma ., we have Combining the trivial inequality |I j,t,s,ξ (θ )| ≤ C with the fact that t/(log t) β is increasing in (e β , ∞), we have where η = L s (ξ )/|L s (ξ )|. This, together (.) with the fact that ∈ F β (S n- ), implies Now we can choose a function ψ ∈ C ∞  (R) such that ψ(t) ≡  for |t| ≤ / and ψ(t) ≡  for |t| > . For  ≤ s ≤ N , j ∈ Z and t ∈ R + , we define the measures {τ s j,t } by Here we use the convention j∈∅ a j = . It is easy to see that It follows from (.), (.), and the trivial estimate | σ s j,t (ξ )| ≤ C that, for  ≤ s ≤ N , (.) http://www.journalofinequalitiesandapplications.com/content/2014/1/265 By the definition of σ s j,t and (.), we can write On the other hand, by a change of variable we have where M P,ϕ is as in Lemma . and P(t) = (P (s)  (t)y  , . . . , P (s) n (t)y n Then the rest of the proof of Theorem . follows from an argument which is similar to those in the proof of Theorem . and (.)-(.). We omit the details.
Since ∈ ∞ s= F(n, s, β), Q s,ξ ∈ V (n, s) and Q s,ξ = , we immediately obtain Then the rest proof of Theorem . follows from similar arguments to the proof of Theorem . and (.)-(.). Details will be omitted.

Competing interests
The authors declare that they have no competing interests.