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Fractional type Marcinkiewicz integral operators associated to surfaces
Journal of Inequalities and Applications volume 2014, Article number: 232 (2014)
Abstract
In this paper, we discuss the boundedness of the fractional type Marcinkiewicz integral operators associated to surfaces, and we extend a result given by Chen et al. (J. Math. Anal. Appl. 276:691-708, 2002). They showed that under certain conditions the fractional type Marcinkiewicz integral operators are bounded from the Triebel-Lizorkin spaces to . 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 . To prove our result, we discuss a characterization of the homogeneous Triebel-Lizorkin spaces in terms of lacunary sequences.
MSC:42B20, 42B25, 47G10.
1 Introduction
The fractional type Marcinkiewicz operator is defined by
where we write for here and below. The operator is the so called singular integral operator. 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 be the unit sphere in the n-dimensional Euclidean space (), with the induced Lebesgue measure and . In the sequel, we often suppose that Ω satisfies the cancellation condition
Here, for the symbols and , we adopt the following convention: Sometimes they stand for points in . But for , we abbreviate to in the present paper. We make this slight abuse of notation since no confusion is likely to occur.
In the present paper we deal with operators of Marcinkiewicz type. Define
As a special case, by letting , , , we recapture the Marcinkiewicz integral operator that Stein introduced in 1958 [1]. In 1960, Hörmander considered the parametric Marcinkiewicz integral operator [2]. Since then, about Marcinkiewicz type integral operators, many works appeared. A nice survey is given by Lu [3].
We formulate our results in the framework of Triebel-Lizorkin spaces of homogeneous type. For and , we let be the Triebel-Lizorkin space defined in [4, 5]. Note that the space given by
is dense in as long as and . If , then define and . 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. Our main theorem in the simplest form reads as follows:
Theorem 1 Let , and .
-
(i)
If and Ω satisfies the cancellation condition (1.2), then
(1.4)
for all .
-
(ii)
If and
(1.5)
for some , then
for all .
-
(iii)
If and satisfies the cancellation condition (1.2), then
(1.7)
for all .
In any case, by density we can extend (1.4), (1.6) and (1.7) and have them for all .
In 2002, Chen et al. obtained a result about the fractional type Marcinkiewicz integral operator [6], which we recall now.
Theorem A Let and . Suppose satisfies the cancellation condition (1.2). If and , then
for all .
Si, Wang and Jiang discussed ones of somewhat different type [7]. About Theorems 1 and A, a couple of remarks may be in order.
Remark 1 If , and , it is easily seen that the condition (1.5) is satisfied. In this case
So, our result includes completely Theorem A, where they assumed that . Let and define
Then it is also easily checked that Ω is in and satisfies (1.5) for any .
In the case , and , the conclusion in Theorem 1(iii) is shown to hold even when in [8].
Remark 2 We can relax the condition on α: suffices. Indeed, one can get by direct computation.
By reexamining their proof, we can parametrize Theorem A: we can prove
provided . Comparing (1.9) with Theorem 1, one concludes that our theorem outranges Theorem A in view of the case when . In our earlier paper [9], we improved Theorem A by relaxing the conditions postulated on Ω.
Our method is also applicable even in more generalized settings. For , and , we define the fractional type Marcinkiewicz integral operator by (1.1) and the fractional type Marcinkiewicz integral operator associated to surfaces by
Theorem 1 extends further to the case when the operator is equipped with a function space with . Regarding to Calderón-Zygmund singular integral and Marcinkiewicz integral operators, many authors discussed those operators with modified kernel in place of , where b belongs to the class of all measurable functions satisfying (), see [10–14], etc. We note that
and that all these inclusions are proper. We refer to [15–17] for extension and generalization of the space .
We define the modified fractional type Marcinkiewicz operator by
We can recover Theorem 1 by letting in the next theorem.
Theorem 2 Suppose that we are given and parameters p, q, α, γ, ρ satisfying
-
(i)
Let . If and Ω satisfies the cancellation condition (1.2), then
(1.12)
for all .
-
(ii)
Assume with . If and
(1.13)
then
for all .
-
(iii)
Assume . If , and Ω satisfies the cancellation condition (1.2), then
(1.15)
for all .
In any case, by density we can extend (1.12), (1.14) and (1.15) and have them for all .
Remark 3 In Theorem 1(ii) a modification of the proof changes 4β into 2β. We cannot estimate directly the Fourier transform of the measure in Section 3, and we use the idea given by Duoandikoetxea and Rubio de Francia [[11], p.551] as in Chen et al. [6].
If , and , it is easily seen that the condition (1.13) is satisfied. In this case
In the case , and , it is again known in [18] that the conclusion in Theorem 2(iii) holds even when .
In the earlier paper [9], in Theorem 1(ii) (respectively, in Theorem 2(ii)), we needed to postulate the additional conditions (respectively, ) and the cancellation condition on Ω. However, these are no longer necessary in the new theorems.
Remark In [19], instead of , the following quantity is proposed:
In addition to the factor of b, we can even distort the convolution. For , , a kernel Ω and a positive function ϕ on , we define the operator and the modified one by
and
Now we formulate our main theorem. Here and below we write .
Theorem 3 Let , and . Let and . Suppose that is a nonnegative increasing -function such that
and that
Define
Then:
-
(i)
Let
(1.20)
If Ω satisfies the cancellation condition (1.2), then
for all .
-
(ii)
Let
If ϕ satisfies the following additional condition:
and Ω satisfies
for some , then
for all .
-
(iii)
Let . If and it satisfies the cancellation condition (1.2), then
(1.25)
for all .
In any case, by density we can extend (1.21), (1.24) and (1.25) and have them for all .
Note that (1.18) is referred to as the doubling condition. Thanks to the useful conversation with Professor XX Tao and Miss S He in the Zhejiang University of Science and Technology, we could improve our results.
We state our main result in full generality. Theorem 3 is almost a direct consequence of the next theorem.
Theorem 4 Suppose that we are given , and parameters p, q, α, γ, ρ satisfying
in addition to (1.18) and (1.19) in Theorem 3. Then:
-
(i)
Assume that
(1.26)
If and Ω satisfies the cancellation condition (1.2), then
for all .
-
(ii)
Assume for some . If and
(1.28)
then
for all .
-
(iii)
Assume . If , and it satisfies the cancellation condition (1.2), then
(1.30)
for all .
In any case, by density we can extend (1.27), (1.29) and (1.30) and have them for all .
Theorem 3(i) and (iii) are direct consequences of Theorem 4. Indeed, assuming (1.20) and choosing , we have (1.26). So, to obtain (i) we can apply Theorem 4 for such γ with . Theorem 3(iii) is a direct conseuqence of Theorem 4(iii). Note that in Theorems 3(ii) and 4(ii), the conditions of α is slightly improved.
Our strategy is to employ the Littlewood-Paley decomposition as Ding et al. did in [20]. However, we distort things via the sequence . We rely upon the modified Littlewood-Paley decomposition for the proof of Theorem 4, which we shall describe now. Let be a lacunary sequence of positive numbers in the sense that (). A sequence of -functions is said to be a partition of unity adapted to if
and
for any multiindex β.
Denote by the set of all polynomials in . Let and . For , we define the norm by
We admit that Proposition 1 below is true and we prove Theorem 4 first. We postpone the proof of Proposition 1 until the end of the paper.
Proposition 1 Let and . Let be a lacunary sequence of positive numbers with (). If and are equivalent for any two partitions of unity, and , adapted to , then there exists such that
and, in this case, is equivalent to the usual homogeneous Triebel-Lizorkin space norm .
In Sections 3-5, we shall prove Theorems 3 and 4 as well as Proposition 1, respectively.
2 A strategy of the proof of Theorem 4
2.1 A setup
For , a function b on and a homogeneous kernel Ω on , assume
For and a nice function ϕ, we define the family of measures and the maximal operator on by
Note that the mapping is a -diffeomorphism, since satisfies (1.18) and (1.19). Therefore, if we consider the measure by
then the above diffeomorphism induces . So, as regards the absolute value of , we have
Denote by the total mass of . A direct consequence of this alternative definition of is that we have
If we use (2.1), then we can write
Lemma 2.1 Let .
-
(1)
For all admissible parameters,
(2.5) -
(2)
If in addition Ω satisfies (1.2), then we have
(2.6)
Proof
-
(1)
From the definition of the Fourier transform, we have an expression of :
(2.7)
From (2.7) we get (2.5).
-
(2)
Using the cancellation property (1.2) of Ω, we have another expression of :
(2.8)
From the monotonicity of ϕ, (1.18) and (2.8) we obtain
So we are done.
□
As for the maximal operator given by (2.2), we invoke the following lemma in [[21], Lemma 3.2]: We define the directional Hardy-Littlewood maximal function of F for a fixed vector by
By the orthogonal decomposition , we can prove that is bounded on for all and that the bound is uniform over . By combining the Hölder inequality and the change of variables to polar coordinates, we can prove the following.
Lemma 2.2 Let . Then there exists such that
for all .
Thanks to Lemma 2.2 and the Minkowski inequality, for there exists such that
From the monotonicity, (1.18) and (2.6) we get, for , ,
Using (1.18) and (1.19), we have the following.
Lemma 2.3 For any ,
for . is the quantity defined in (1.28).
For a precise proof, see the proof of [[21], Lemma 2.4].
2.2 Properties of ϕ
We denote and . Then is also a lacunary sequence of the same lacunarity as . From the assumption (1.18), it follows that
for . It is easily seen from (1.19) that is a lacunary sequence of positive numbers satisfying
for and . See e.g. [[21], Lemma 2.8] for details.
Note also that, for satisfying (1.18), the condition (1.19) implies
for some . Indeed, assuming (1.18), there exists
by the mean value theorem, proving (2.15).
If in addition ϕ is concave, then (2.15) implies (1.19). Indeed,
2.3 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 (1.18) and (1.19).
Take a nonincreasing -function η such that for all (see Figure 1).
We define functions on by
Then observe that
and that
That is, is a smooth partition of unity adapted to .
Let be defined on by for . By Proposition 1, we have
if () for some .
This condition is satisfied in our case, i.e. .
2.4 A reduction by using the scaling invariance
Now, using the definition of and the triangle inequality, via change of variables , we obtain
Hence
So, in the case we have
So, in the case , from (2.22), we have
Notice that b and satisfy the same condition due to the scaling invariance of . Likewise ϕ and satisfy the same conditions (1.18) and (1.19) with constants independent of k. Hence, for our purpose, it is sufficient to consider the modified operator given by
for .
Now we proceed to the proof of Theorem 4. Let
for each j. Using the partition of unity (2.16) and the triangle inequality, we then have
Next, we treat the -estimate of .
Let us set
In Section 4 we plan to distinguish three cases to prove.
Lemma 2.4 Assume either one of the following three conditions:
-
1.
(see Figure 2).
-
2.
(see Figure 3).
-
3.
.
If , then we have
However, in case 3, we just interpolate cases 1 and 2. So we concentrate on cases 1 and 2 in Section 4.
Note that cases 1-3 do not cover all the cases as the above images show.
We also need to prove the following.
Lemma 2.5 Let ϕ satisfy the same conditions (1.18) and (1.19). Assume that satisfies the cancellation condition (1.2). Then
By using the strong decay of (2.25), interpolate (2.25) and (2.26) to have (2.25) again for any admissible p and q. Thus, in conclusion, (2.24) is summable over j by virtue of (2.25).
3 Proof of Theorem 4
In this section, we prove Theorem 4. One can obtain Theorem 4 by observing carefully the proof of [[6], Theorem 6], but for the sake of completeness we shall give its detailed proof, modifying their one.
3.1 Proof of Lemma 2.4
Here we do not need the cancellation property of Ω and hence we can consider its absolute value of .
-
(1)
In the case , let
Let us set . By the duality -, we can take a nonnegative function with such that
Denote by the total mass of . By the Hölder inequality
By virtue of (2.3), we have
Since , we have . So, by (2.10) and Hölder’s inequality, we conclude
Thus, we have
-
(2)
In case and , it follows that . By duality, there is a sequence of functions such that
and such that
Then we have
By using the Hölder inequality for sequences, we have
By the properties of ϕ and Proposition 1, we conclude
In the same way as in [[6], p.705], using (2.10), we can check
if . Hence we have, for ,
So we are done.
3.2 Proof of Lemma 2.5
By virtue of the Plancherel theorem and the Fubini theorem, we have
By (2.11), (3.3) and the support property of , we have
For , from (2.14), it follows that , and for , from (2.13) we get . Likewise, we have .
We need to control the integrand; first of all,
When , we use
and
When , we use
and
So, if , we have
Hence, after incorporating a similar estimate for , we get (2.26).
3.3 Interpolation and the conclusion of the proof of (i)
Let
By interpolating (2.26) and (2.25), we claim that there exists such that
When , then (3.4) is correct by virtue of (2.25) () and (2.26) (). We check next the case and . For , by (2.25) we may take . For , we take and satisfying
Note that we have
We choose so that
and then determine , by (3.5), (3.6). As in the Figure 4, we can arrange that
We shall see that this choice is possible. Recall that . Then some arithmetic shows that
and that
Assuming that , we conclude that the parameters and are increasing on with respect to and as functions in and , respectively. Hence
Therefore, since
and , we get (3.7) by choosing sufficiently near 2γ if and sufficiently near 2γ if , and by choosing similarly according to or .
Now, interpolating (2.26) and (2.25) with , , we get
We then interpolate (2.25) and (3.8) with , . As a consequence, we have
An arithmetic together with (3.7) shows that
Thus, taking , we obtain the desired estimate (3.4).
In the case or , we can get the desired estimate more simply, by applying interpolation once.
Thus by (2.24) and (3.4) we obtain
This completes the proof of Theorem 4(i).
3.4 The proof of (ii)
Below we shall prove Theorem 4(ii). By the Schwarz inequality, we have
Recall
Then, by (2.12) and the doubling condition of ϕ, we have
By (3.3), (3.11) and the support property of , we have
As in the case (i), we have
for , and
for . Similarly, we have
for and
for . So, as in the -estimate in (i), we obtain
As for the -estimate, since , we use for and for . Hence we get, as in the -estimate in (i), for any with and
It follows that, for
there still exists such that
by using (3.13) in the case , and interpolating (3.12) and (3.13) in the case , as in the case (i).
Thus by (2.24) and (3.14) we obtain
This completes the proof of Theorem 4(ii).
3.5 Proof of (iii)
We proceed to show (iii). Let . We normalize Ω to have . Then, as in [[8], pp.698-699], there is a subset and a sequence of functions satisfying and the following conditions:
Indeed, we just let
and define
For details we refer to [22].
Now for , by observing the proof of the case (i), we choose and very close to
so that for small . For large m, setting , we obtain
Next, from it follows that satisfies the condition in Theorem 4(ii) for any . Fix and with
Let also
in the proof of the case (ii). Then we obtain
Since and , an interpolation between (3.18) and (3.19) yields
Thus, summing up the above estimate, we obtain
Combining (3.20) with (3.16) and (3.17) and the definition of , we obtain the desired estimate
Thus, we are done.
4 Proof of Theorem 3
Here we shall relax the condition on α by taking advantage of a new condition on ϕ. We use the notations in the proof of Theorem 4, by setting and . Using (1.18) and (1.19), we apply Theorem 4(i) and we obtain the conclusion of Theorem 3(i).
We go to the proof of (ii). First
With a change of variables we get
Suppose now that is increasing on . Then is also increasing. So by applying the second mean value theorem to the real part of the expression (4.2), we see that there exists u with such that
Since is increasing, we have
After estimating in a similar manner, we obtain
In the case is decreasing or is monotonic, we get the same estimate (4.3) in a similar way. Clearly, we have , and hence for any , . By (4.1) we get
Now the rest of the proof is the same as that of the case (i).
This completes the proof of Theorem 3.
5 Proof of Proposition 1
The part is an appendix of the present paper, where we prove Proposition 1. Let (see Figure 5) be chosen so that
Define
Notice that () and that on . Let
Then we see that is a partition of unity adapted to . Similarly, taking ψ so that
and setting
we obtain another partition of unity adapted to satisfying () and on . Note that . Let us take a function so that . Consider
Then we have
where . It follows that . Hence we have
where
for .
and
Since the two norms are assumed equivalent, we obtain
for some . Since , we have .
Thus we have proved the first part of our proposition. We proceed to the second part. Let be a lacunary sequence of positive numbers with (), and let be a partition of unity adapted to .
Now we can define the classical homogeneous Triebel-Lizorkin spaces as follows: Let be chosen so that . Define
Notice that on .
Define
Let us prove
For each , we choose so that
Combining with , we get . And combining with , we have . Furthermore we have
Consequently, we obtain
We now invoke the Plancherel-Polya-Nikolskij inequality: We have
Using Plancherel’s theorem, the assumption for all β and that , we get
Hence, it follows that
By the Fefferman-Stein vector-valued maximal inequality (see [23]), we obtain
If we use , then we obtain
Noting , and that , we conclude
Let us prove the reverse inequality. For each , we can choose so that
Then we have
Notice that
because . Thus, it follows that
Again by the Fefferman-Stein vector-valued maximal inequality (see [23]), we obtain
This completes the proof of our proposition.
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Acknowledgements
This work was partially supported by Grant-in-Aid for Scientific Research (C) No. 23540228, Japan Society for the Promotion of Science and Grant-in-Aid for Young Scientists (B) No. 24740085, Japan Society for the Promotion of Science.
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Sawano, Y., Yabuta, K. Fractional type Marcinkiewicz integral operators associated to surfaces. J Inequal Appl 2014, 232 (2014). https://doi.org/10.1186/1029-242X-2014-232
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DOI: https://doi.org/10.1186/1029-242X-2014-232