Weighted Norm Inequalities for Multilinear Fourier Multipliers with Mixed Norm
In this paper, weighted norm inequalities for multilinear Fourier multipliers satisfying Sobolev regularity with mixed norm are discussed. Our result can be understood as a generalization of the result by Fujita and Tomita by using the -based Sobolev space, with mixed norm.
For , the -linear Fourier multiplier operator is defined by for , where , and . Let be such that and set where is as in (2) with . We denote by the smallest constant satisfying
See Section 2 for the definition of function spaces.
In the unweighted case, Tomita  proved a Hörmander-type multiplier theorem for multilinear operators, namely, if then for satisfying , where is the -based Sobolev space of usual type. Grafakos and Si  extended this result to the case by using the -based Sobolev space, . For further results in this direction, see [3–7]. Let and . In the weighted case, Fujita and Tomita  proved that if , , and for all , then where and is the -based Sobolev space of product type. This result can also be obtained from another approach of . See [10, 11] for the endpoint cases.
The following is our main result which can be understood as a generalization of the result by Fujita and Tomita . Taking for all in (8), we have (6). Si  obtained some weighted estimates for multilinear Fourier multipliers with the -based Sobolev regularity, .
Theorem 1. Let , , , , , and for all . Assume Then, where and is the Sobolev space of product type with mixed norm.
Let be the fixed dimension of the Euclidean space, and is defined by . The Lebesgue measure on is denoted by (see, for example, Chapters 1 and 2 of ). Let be a natural number, . An operator acting on -tuples of functions defined on is called the -linear operator. For two nonnegative quantities and , the notation means that for some unspecified constant independent of and , and the notation means that and . If , we denote by . Let and be the Schwartz class of all rapidly decreasing smooth functions and tempered distributions, respectively. We define the Fourier transform and the inverse Fourier transform of by
See, for example, Chapter 1 of . To distinguish linear and multilinear operators, for , we denote the linear Fourier multiplier operator by defined by for , where . Let and . The weighted Lebesgue space consists of all measurable functions on such that
Let . We say that a weight belongs to the Muckenhoupt class if where the supremum is taken over all balls in , is the Lebesgue measure of , and is the conjugate exponent of , that is, . It is well known that the Hardy-Littlewood maximal operator is bounded on if and only if (see Theorem 7.3 of ).
2.2. Function Spaces
To distinguish spaces of usual type and mixed type concerning integrable indices, we use and , respectively.
We recall the definition of -spaces with mixed norm . Let . The Lebesgue spaces with mixed norm consists of all measurable functions on such that where and is the Lebesgue measure with respect to the variable for all . In particular, if each is equal to , then we have . For and , the norm of the Sobolev space of product type with mixed norm for is defined by where for and is the inverse Fourier transform of . Taking for all , we obtain the -based Sobolev space of product type , namely, . It should be remarked that if for all , where is the -based Sobolev space of usual type, that is to say, , where .
For and , the norm of the weighted Lebesgue space with mixed norm for is also defined by where and for all . For accuracy, we will frequently write instead of in the proof.
For , , we shall agree that if is a relation between numbers and , then means that holds for each .
2.3. Cut-Off Functions
We collect cut-off functions which will be used later on . Let be a -function on satisfying
We set . For , we define the function on by where and . Note that
The following lemmas will be used in the proof of Theorem 1.
Lemma 2 (see Lemma 3.1 of ). Let be the same as in (18). Then, the following are true: (1)For , (2)For and , then there exists a constant such that for all .(3)If for some and for all with , then . If for some with , then .
Lemma 3 (see Chapter 7 of ). Let and . Then, there exists such that .
Lemma 4 (see ). Let be such that for some . If and , then where .
Lemma 5 (see ). Let and . Then,
Lemma 6 (see Proposition 2.7 of ). Let be a function which is positive, radial, decreasing (as a function on ), and integrable. Set for . Then, for .
Lemma 9. Let , , , , and . Then, the estimate holds, where .
Proof. We consider only the case . Let be such that
Since , we see that . Then, it is easy to see that
For fixed , by Minkowski’s inequality for integrals, it follows that
Since , for fixed , we obtain
where we have used the fact .
For the first term on the right-hand side of (31), by Hölder’s inequality and a change of variables, we have where . Thus, we see that For the second term on the right-hand side of (31), by Young’s inequality, we obtain By (33) and (34), it follows that where we have used the fact that By the same way for , we have the desired estimate with .☐
The following is a key lemma in the proof of Theorem 1. Fujita and Tomita (Proposition A.2 of ) proved (6) using the fact that is a multiplication algebra when for all . Instead of this, we shall use the following lemma.
Lemma 10. Let , , , , , and for all . Then, the estimate holds for all , , and .
Proof. We consider only the case . By the change of variables and satisfies , we see that where is defined by (3). By Lemma 9 and Hausdorff-Young’s inequality with mixed type (Lemma 8), it follows that where we have used the fact . This completes the proof with .☐
The following lemma is known, but we shall give a proof for the reader’s convenience.
Lemma 11. Let , , and for all . Then, the estimate holds for all , , with and .
Proof. We consider only the case . For all and , by Fubini’s theorem and the change of variables, we see that where is the inverse Fourier transform of . For fixed , by Hölder’s inequality and Lemma 6 with , it follows that where we have used the fact that . By the same way for and the change of variables, we have the desired estimate with .☐
4. Proof of Theorem 1
In this section, we prove Theorem 1. Let , , , , , and for all . Assume and for all and set . We also assume that satisfies
4.1. Estimate for Type
We first consider the case where satisfies . Without loss of generality, we may assume that . We simply write instead of . Note that by Lemma 2 (3),
It is easy to see that if , then .
By Fubini’s theorem and the Fourier inversion formula, it is easy to see that
We shall prove that we can find functions and independent of such that
Once this is proven, setting we have where . Let satisfying . We take functions such that on and such that on . Hence, we obtain (47).