Fractional type Marcinkiewicz integral operators associated to surfaces
© Sawano and Yabuta; licensee Springer. 2014
Received: 1 March 2014
Accepted: 6 May 2014
Published: 4 June 2014
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.
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.
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.
As a special case, by letting , , , we recapture the Marcinkiewicz integral operator that Stein introduced in 1958 . In 1960, Hörmander considered the parametric Marcinkiewicz integral operator . Since then, about Marcinkiewicz type integral operators, many works appeared. A nice survey is given by Lu .
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:
- (i)If and Ω satisfies the cancellation condition (1.2), then(1.4)
- (ii)If and(1.5)
- (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 , which we recall now.
for all .
Si, Wang and Jiang discussed ones of somewhat different type . About Theorems 1 and A, a couple of remarks may be in order.
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 .
Remark 2 We can relax the condition on α: suffices. Indeed, one can get by direct computation.
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 , we improved Theorem A by relaxing the conditions postulated on Ω.
We can recover Theorem 1 by letting in the next theorem.
- (i)Let . If and Ω satisfies the cancellation condition (1.2), then(1.12)
- (ii)Assume with . If and(1.13)
- (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 [, p.551] as in Chen et al. .
In the case , and , it is again known in  that the conclusion in Theorem 2(iii) holds even when .
In the earlier paper , 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.
Now we formulate our main theorem. Here and below we write .
- (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.
- (i)Assume that(1.26)
- (ii)Assume for some . If and(1.28)
- (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.
for any multiindex β.
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.
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
- (1)For all admissible parameters,(2.5)
- (2)If in addition Ω satisfies (1.2), then we have(2.6)
- (1)From the definition of the Fourier transform, we have an expression of :(2.7)
- (2)Using the cancellation property (1.2) of Ω, we have another expression of :(2.8)
So we are done.
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.
for all .
Using (1.18) and (1.19), we have the following.
for . is the quantity defined in (1.28).
For a precise proof, see the proof of [, Lemma 2.4].
2.2 Properties of ϕ
for and . See e.g. [, Lemma 2.8] for details.
by the mean value theorem, proving (2.15).
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).
That is, is a smooth partition of unity adapted to .
if () for some .
This condition is satisfied in our case, i.e. .
2.4 A reduction by using the scaling invariance
Next, we treat the -estimate of .
In Section 4 we plan to distinguish three cases to prove.
(see Figure 2).
(see Figure 3).
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.
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 [, Theorem 6], but for the sake of completeness we shall give its detailed proof, modifying their one.
3.1 Proof of Lemma 2.4
- (1)In the case , let
- (2)In case and , it follows that . By duality, there is a sequence of functions such that
So we are done.
3.2 Proof of Lemma 2.5
For , from (2.14), it follows that , and for , from (2.13) we get . Likewise, we have .
Hence, after incorporating a similar estimate for , we get (2.26).
3.3 Interpolation and the conclusion of the proof of (i)
and , we get (3.7) by choosing sufficiently near 2γ if and sufficiently near 2γ if , and by choosing similarly according to or .
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.
This completes the proof of Theorem 4(i).
3.4 The proof of (ii)
by using (3.13) in the case , and interpolating (3.12) and (3.13) in the case , as in the case (i).
This completes the proof of Theorem 4(ii).