Fractional type Marcinkiewicz integral operators associated to surfaces

In this paper, we discuss the boundedness of the fractional type Marcinkiewicz integral operators associated to surfaces, and extend a result given by Chen, Fan and Ying in 2002. They showed that under certain conditions the fractional type Marcinkiewicz integral operators are bounded from the Triebel-Lizorkin spaces $\dot F_{pq}^{\alpha}({\mathbb R}^n)$ to $L^p({\mathbb R}^n)$. Recently the second author, together with Xue and Yan, greatly weakened their assumptions. In this paper, we extend their results to the case where the operators are associated to the surfaces of the form $\{x=\phi(|y|)y/|y|\} \subset {\mathbb R}^n \times ({\mathbb R}^n \setminus \{0\})$. To prove our result, we discuss a characterization of the homogeneous Triebel-Lizorkin spaces in terms of lacunary sequences.


Y. Sawano and K. Yabuta
where we write B(r) := {|x| < r} ⊂ R n for r > 0 here and below. This is a kind of singular integral operators. In this paper, we shall prove that this operator is bounded under a certain highly weak integrability assumption. To this end, we plan to employ a modified Littlewood-Paley decomposition adapted to our situation. It turns out that we can relax the integrability assumption on Ω and that the integral operator itself can be generalized to a large extent. Let S n−1 be the unit sphere in the n-dimensional Euclidean space R n (n ≥ 2), with the induced Lebesgue measure dσ = dσ(x ′ ) and Ω ∈ L 1 (S n−1 ). In the sequel, we often suppose that Ω satisfies the cancellation condition S n−1 Ω(y ′ ) dσ(y ′ ) = 0. (1.2) Here, for the symbols x ′ and y ′ , we adopt the following convention: Sometimes they stand for points in S n−1 . But for x ∈ R n \ {0}, we abbreviate x/|x| to x ′ in the present paper. We make this slight abuse of notation since no confusion is likely to occur.
Here and below a tacit understanding in the present paper is that the letter C is used for constants that may change from one occurrence to another, that is, the letter C will denote a positive constant which may vary from line to line but will remain independent of the relevant quantities.
Fractional type Marcinkiewicz integral operators 3 About the fractional type Marcinkiewicz integral operator, in 2002, J. Chen, D. Fan and Y. Ying obtained a result [10], which we recall now.
For 1 < p < ∞, we say that a nonnegative function ω ∈ L 1 loc (R n ) is in A p (R n ) if there is a constant C > 0 such that for any n-dimensional cubes Q with sides parallel to coordinate axes If there is a constant C > 0 such that where M ω denotes the standard Hardy-Littlewood maximal function of ω on R n , then we say ω is in A 1 (R n ). These are the usual Muckenhoupt weight classes on R n . Next, we recall a class of weights on R + = (0, ∞). Suppose that a nonnegative function ω is in L 1 loc (R + ). For 1 < p < ∞, we say that ω is in A p (R + ) if there is a constant C > 0 such that for any interval I ⊂ R + , If there is a constant C > 0 such that where ω * denotes the standard Hardy-Littlewood maximal function of ω on R + , then we say ω is in A 1 (R + ). For 1 < p < ∞, we define the weight classes A p (R n ) as follows: For p = 1, we define We know by [14] that the Hardy-Littlewood maximal operator M is bounded on L p (ω) for ω ∈ A p (R n ), and thus The space L p (ω) denotes the weighted L p -space associated to the weight ω defined by We formulate our main theorem. Here and below we write R + := (0, ∞). Then: If Ω satisfies the cancellation condition (1.2), then for all f ∈ S ∞ (R n ), If φ satisfies the following additional condition and Ω satisfies then for all f ∈ S ∞ (R n ), If Ω ∈ L log L(S n−1 ) and it satisfies the cancellation condition (1.2), then for all f ∈ S ∞ (R n ), Note that (1.14) is referred to as the doubling condition. We can prove Theorem 2 by modifying the proof of Theorem 3 in the case where γ = ∞, as in [29].
We state our main result in full generality. Theorem 2 is almost a direct consequence of the next theorem; Theorem 3. Suppose that we are given parameters and the functions First assume γ > 1 2 max(p,q). (1.22) Consider three cases: Assume in addition that Ω satisfies the following conditions in the table: Case α1 the cancellation condition (1.2) Case α2 (1.26) for some β ∈ (0, 1) Case α3 Ω ∈ L log L(S n−1 ), (1.2) Fractional type Marcinkiewicz integral operators 7 Here (1.26) is; (1.26) Assume also that φ satisfies (1.14) and (1.15) in Theorem 2. Finally assume that w is such that and that w satisfies one of the following conditions in the table below: Here the conditions (1.28)-(1.31) stand for
5. When q = 2, we can cover the results for homogeneous Sobolev spaces.
Let h be an even C ∞ (R)-function satisfying χ [2,3] So, choosing ε > 0 sufficiently small, and taking A > 0 sufficiently large, we see that The case when γ = ∞ requires a simpler condition on weights.
Theorem 4. Suppose that we are given parameters Suppose Ω satisfies the same condition as in Theorem 3. Assume also that φ satisfies (1.14) and (1.15) in Theorem 2. Finally assume that w is such that for some a > 0. In the case 1 < p < 2, we assume furthermore Theorem 2 Cases α1 and α3 are direct consequences of the corresponding assertions of Theorem 4. Indeed, to obtain Cases α1 and α3, we have only to apply Theorem 4 with b ≡ 1.
We rely upon the modified Littlewood-Paley decomposition for the proofs of Theorem 3 and 4, which we shall describe now. Let {a k } k∈Z be a lacunary sequence of positive numbers in the sense that a k+1 /a k ≥ a > 1 (k ∈ Z). A sequence Fractional type Marcinkiewicz integral operators . (1.36) We admit that Proposition 1 below is true and we prove Theorem 3 first. We postpone the proof of Proposition 1 until the end of the paper.
and, in this case, f Ḟ α,{Φ k } k∈Z pq (w) is equivalent to the usual weighted homogeneous Triebel-Lizorkin space norm f Ḟ α pq (w) . We prepare one more lemma. See Section 5 for a generalization. Lemma 1.1. Let 1 ≤ p 1 , p 2 , q 1 , q 2 < ∞, and α ∈ R. Define p and q by 1/p = Proof. Let I α be the operator defined by I α (ξ) = |ξ| −α . Then by the assumption the operator T •I α is a bounded linear operator fromḞ 0 10 Y. Sawano and K. Yabuta 2 Preparation for the proof of Theorem 3

A setup
For t > 0, a function b on R + and a homogeneous kernel Ω on R n , assume For ρ > 0 and a nice function, we define the family {σ t ; t ∈ R + } of measures and the maximal operator σ * on R n by (1.14) and (1.15). Therefore, if we consider the measure σ † t by then the above diffeomorphism induces σ t . So, about the absolute value of σ t , we have We need the following lemma whose proof is in [29].

For all admissible parameters,
2. If in addition Ω satisfies (1.2), then we have Meanwhile, thanks to Lemma 2.2 below and the Minkowski inequality, for From the monotonicity, the doubling property (1.14) and (2.4) we get Fractional type Marcinkiewicz integral operators 11 for α ∈ R and k ∈ Z.
As for the maximal operator σ * given by (2.2), we invoke the following lemma in [12, Lemma 3.2]: We define the directional Hardy-Littlewood maximal function of F for a fixed vector y ′ ∈ S n−1 by By the orthogonal decomposition R n = H ⊕Ry ′ , we can prove that M y ′ is bounded on L p (R n ) for all 1 < p < ∞ and that the bound is uniform over y ′ . By combining the Hölder inequality and the change of variables to polar coordinates, we can also prove; The proof is in [ Note that this is where the number 2 appears in the right-hand side of (1.34).

Construction of partition of unity
For our purpose, we introduce a partition of unity and a characterization of the homogeneous Triebel-Lizorkin spaces associated to φ satisfying (1.14) and (1.15). It is easily seen from (1.15) that {φ(2 j )} j∈Z is a lacunary sequence of positive numbers satisfying (see e.g. [12, Lemma 2.8]). We denote a j = 1/φ(2 −j ) and a = 2 1/ ϕ L ∞ (R + ) . Then {a j } j∈Z is also a lacunary sequence of the same lacunarity as {φ(2 j )} j∈Z . Take a nonincreasing C ∞ R)-function η such that

Y. Sawano and K. Yabuta
We define functions ψ j on R n by Then observe that and that (2.14) That is, {ψ j } j∈Z is a smooth partition of unity adapted to {a j } j∈Z .
Let Ψ j be defined on R n by Ψ j (ξ) = ψ j (ξ). By Proposition 1, we have for is bounded. Note that this condition is satisfied in our case, i.e.

A reduction by using the scaling invariance
As we did in [29], for our purpose, it is sufficient to estimate the modified operator Fractional type Marcinkiewicz integral operators If we use (2.1), then we havẽ Now we proceed to the final preparation for the proof of Theorem 3. Let for each j. Using the partition of unity (2.10) and the triangle inequality as we did in [29], we then havẽ We need the L 2 (R n )-estimate ofμ Ω,ρ,φ,α,q,j f . Note that we do not need the weighted estimate here. (2.20) Next, we treat the L p (w)-estimate ofμ Ω,ρ,φ,α,q,j f . In Section 3 we plan to distinguish three cases to prove; Lemma 2.5. Assume that the parameters p, q, γ and the weight w satisfies as well as either one of the following conditions in the table below: If Ω ∈ L 1 (S n−1 ), then we have Let us set  Here we do not need the cancellation property and hence we can consider its absolute value, which means that σ t may be assumed to be positive.
(1) Let us consider the case that 1 < q < r < γq. Write Let us set s = (r/q) ′ = r/(r − q). By the duality L r/q (w 1−s )-L s (w 1−s ), we can take a nonnegative function h ∈ L s (w 1−s ) with h L s (w 1−s ) = 1 such that Denote by σ t the total mass of σ t . Hence, by the Hölder inequality Now, since w 1−s ∈ A s/γ ′ (R n ), by (2.5) and Hölder's inequality we get Thus, by the doubling property of φ, we have (3.1) (2) Let us consider the case that Then we have By using the Hölder inequality for sequences, we have Since the total measure σ t of σ t is bounded by 2 n−ρ Ω L 1 (S n−1 ) b ∆1 , by using the Hölder inequality, we get Fractional type Marcinkiewicz integral operators By using the maximal operator σ * , we obtain from the estimate for σ * (h)(x) (see Lemma 2.2): Recall again that Thus, we have Y. Sawano and K. Yabuta For the later use, we note that if γ = ∞, our assumption in this case means (3) Finally let us consider the case that In case 1 < r = q < ∞, it is easy to see by using (2.5) that for r > γ ′ and
Here and below in this subsection, we let j ≤ 0.
Case W 0 First, we shall obtain a general estimate which follows from (1.27).
Hence, by, (2.20) and (2.21), we get for j ≤ 0 Cases W 1, W 4, W 5, W 6 We place ourselves in the interior of the small square of the following 1/p − 1/q diagram: We shall begin with an arithmetic. The proof is in [29].

Y. Sawano and K. Yabuta
After interpolations in Steps 1 and 2, we are led to . An arithmetic shows that Thus, taking we obtain the desired estimate (3.2). What remains to do is to verify that our assumptions postulated on each domain are suitable to justify our Steps 1 and 2.
On the square W 4 Step 1 As we did on W 1, we obtain (3.10).
Step 2 This is the same as the estimate on W 1.

Fractional type Marcinkiewicz integral operators 23
On the square W 6 Step 1 This is the same as the estimate on W 5.
Step 2 This is the same as the estimate on W 4.
Cases W 2, W 3, W 7, W 8 This is a degenerate case: In the case p = 2 or q = 2, we can get the desired estimate (3.4) more simply, by applying interpolation once.
Remark 3. The proof of Theorem 4 is similar to the above cases, so we omit the details of the proof.

The proof of Theorem 3 Case α3
Below we prove Theorem 3 Case α3. By the Schwarz inequality, Recall that we have set (3.12) Then, by (2.8), we have (3.13) By (3.13) and the support property of ψ j−k , we have

Y. Sawano and K. Yabuta
(3.14) See [29] as well. As for the L p -estimate, since α < 0, it follows that for j ≤ 0. Hence we get, as in the L p (w)-estimate in Case α1, for any 1 < q, r < ∞ withr < γq, and j ∈ Z It follows that, for there still exists δ > 0 such that by using (3.15) in the case j ≤ 0, and interpolating (3.14) and (3.15) This completes the proof of Theorem 3 Case α3.
Fractional type Marcinkiewicz integral operators 25

The proof of Theorem 3 Case α2
This is close to the corresponding assertion in [29] using the technique in [3].

Proof of Proposition 1
The part is an appendix of the present paper, where we prove Proposition 1. For the sake of completeness, we supply a whole proof. Let ψ ∈ S(R n ) be chosen so that Notice that supp ϕ k ⊂ {ξ ∈ R n ; a k−1 ≤ |ξ| ≤ a 1/3 a k } (k ∈ Z) and that ϕ k (ξ) = 1 on Then, we see that {Φ k } k∈Z is a partition of unity adapted to {a k } k∈Z . Similarly, taking ψ so that we obtain another partition of unity {Ψ k } k∈Z adapted to {a k } k∈Z satisfying supp Ψ k ⊂ {ξ ∈ R n ; a k /a 1/3 ≤ |ξ| ≤ a k+1 } (k ∈ Z) and Ψ k (ξ) = 1 on {a k ≤ |ξ| ≤ a k+1 /a 1/3 }. Note that {a k ≤ |ξ| ≤ a 2/3 a k } ⊂ {a k ≤ |ξ| ≤ a k+1 /a 1/3 }. Let us take a function Θ ∈ S so that supp(FΘ) ⊂ B(a 1/3 /2 − 1/2). Consider Then we have where e 1 := (1, 0, . . . , 0). It follows that supp Y. Sawano and K. Yabuta Since the two norms are assumed equivalent, we obtain a k+1 a k ≤ C 0 for some C 0 > 1. Since a k+1 a k ≥ a, we have C 0 ≥ a. Thus we have proved the first part of our proposition. We proceed to the second part. Let {a k } k∈Z be a lacunary sequence of positive numbers with 1 < a ≤ a k+1 /a k ≤ C 0 (k ∈ Z), and {Φ k } k∈Z be a partition of unity adapted to {a k } k∈Z . Now we can define the classical weighted homogeneous Triebel-Lizorkin spaces as follows: Let ψ ∈ S(R n ) be chosen so that Let us prove Combining with a m k +1 ≤ aa k ≤ a k+1 , we get m k+1 ≥ m k + 1. And combining with a k+1 /a k ≤ C 0 , we have m k+1 − m k ≤ 1 + log a C 0 . Furthermore we have Consequently, we obtain .
Noting m k+1 − m k ≥ 1, m k+1 − m k ≤ 1 + log a C 0 and that a m k−1 +1 ≤ a m k ≤ a k < a m k +1 ≤ a m k−1 +1+[log a C0] , we conclude Let us prove the reverse inequality. For each k ∈ Z, we can choose ℓ k ∈ Z so that a ℓ k ≤ a k−1 ≤ a k+1 ≤ a ℓ k +3 .

Y. Sawano and K. Yabuta
As we did in [29], we have sup l∈Z ♯{k : ℓ k = l} ≤ 3 log a C 0 because a l+3 ≤ C 0 3 a l . Thus, it follows that .
Again by the weighted Fefferman-Stein vector-valued maximal inequality (see [5,19]), we obtain This completes the proof of our proposition.

Appendix: Complex interpolation of homogeneous weighted Triebel-Lizorkin spaces
In this section, for a weight function w we use the following notations:Ḟ s,w p,q = F s,w p,q (R n ) and L p,w = L p,w (R n ) in place ofḞ s p,q (w) and L p (w), respectively, because we use ℓ q -valued L p spaces L p,w (ℓ q ) etc.
We shall prove the following result: ||f ||Ḟ s,w p,q .

Atomic decomposition
Here and below Q ν,m = 2 −ν m + [0, 2 −ν ] n for m ∈ Z n and χ Qν,m denotes the characteristic function on Q ν,m . 3. The function f can be decomposed as follows: where the convergence is in S ′ (R n ).
Denote by S ′ ∞ (R n ) the topological dual. Then we know the following result. .

Y. Sawano and K. Yabuta
See [20,34,24] for these propositions. We let D denote the set of all dyadic cubes: Let ν ∈ Z, k ∈ Z, l ∈ N 0 , m ∈ Z n and M ≥ n. A function m ν,m is said to be a We say that {m ν,m } (ν,m)∈Z×Z n is a family of smooth molecules forḞ s,w p,q (R n ) if it is a family of (N + ǫ, s + 1 + ǫ, M )-smooth molecules, where N := n min{1, p, q} − n − s, for some constant ǫ > 0, and M is a sufficiently large constant.
The number M needs to be chosen sufficiently large.