Boundedness Criteria and Norm of Some Multilinear Hilbert-Type Operators

Abstract

We consider two families of multilinear Hilbert-type operators for which we give exact relations between the parameters so that they are bounded. We also find the exact norm of these operators.

1. Introduction

Let , and let be a real number. We denote by or simply the weighted Lebesgue space . When , we simply write for the corresponding space. We will be using the notation and we recall that if and only if the above quantity is finite. For , we simply write for . All over the text, for we denote by its conjugate exponent, that is, the unique extended real number satisfying .

We recall that the Hilbert operator is defined by The boundedness of this operator has been heavily studied in the literature; in particular people have been very interested in the norm estimate of this operator and its siblings (see, e.g., the following and the references therein [1–6]). In [7], we considered a more general family of this operator for which we provided boundedness criteria and some sharp norm estimates. More precisely, for , real parameters we considered the family of operators defined for compactly supported functions byThis family as shown in [7] can be related to Bergman-type projections. This family can be extended in two different ways into -linear operators on . For the first family, we let be real parameters and put and . Put , where , , are compactly supported functions on . Consider the operators defined by Following the idea in [7], one can also relate this family to multilinear Bergman projections. To define the second family, we let be real parameters and put . We define the operators as follows: This last family appears in [8] in the study of the Laplace representation of some mixed norm Bergman spaces in relation to the question of the boundedness of the Bergman projection in tube domains over symmetric cones. Many authors have provided the norm of the operators in some particular cases and some variations [9–14]. The fact is that, in these papers, the relations between the parameters are directly given without indicating how they are obtained and the authors are only interested in finding the exact norm of the operators or proving the corresponding Hardy-Hilbert inequality.

We aim in this note to provide exact relations between the parameters so that the above operators are bounded from to when , being real numbers withWe also find the exact norm of these operators, extending the results in [9, 12]. Note that, in the above relations, we allow some (not all) of the exponents s to be infinite, a situation which has not been considered before as far as we know.

2. Statement of the Results

We give in this section all our results. We denote, by , the Beta-function of and to be defined in the next section. We start by the following.

Theorem 1. Let be real numbers. Let , , let be real numbers, and assume that (6) and (7) hold. Then the following conditions are equivalent. (i)The operator is bounded from to .(ii)The parameters satisfyIn this case, if we denote by the operator norm of , then

It is easy to see that condition (9) provides that, for any ,hence As , , we also have that Now observing that (8) can be written as we conclude that or equivalently that .

We also obtain the following result.

Theorem 2. Let be real numbers. Let , , let be real numbers, and assume that (6) and (7) hold. Then the following conditions are equivalent. (i)The operator is bounded from to .(ii)The parameters satisfyIn this case, if we denote by the operator norm of , then

We note that the norm of the operator for and was computed in [9] for the unweighted case and [12] for the weighted case.

If in relation (6) we allow only some (but not all) of the exponents to be finite while all the other exponents are equal to infinity, then we obtain a kind of mixed endpoints version of the previous results.

Theorem 3. Let be real numbers. Let , , , and assume that . Let be real numbers and assume that . Then the following conditions are equivalent. (i)The operator is bounded from to .(ii)The parameters satisfy In this case, if we denote by the operator norm of , then

We also obtain the following result.

Theorem 4. Let be real numbers. Let , , , and assume that . Let be real numbers and assume that . Then the following conditions are equivalent.(i)The operator is bounded from to .(ii)The parameters satisfy In this case, if we denote by the operator norm of , then

We are essentially motivated by the need of explaining the right relations between the parameters that make these operators bounded and the generalization of the previous results on the norm estimates of these operators. In the proofs of the necessary parts, we appeal to duality and use appropriate local test functions. Note that as our operators are -linear, each of them has adjoints. For the computation of the norm of each operator, to simplify our presentation, we give an upper estimate and a lower estimate. The proof of the lower estimate appeals to a clever choice of test functions, a good decomposition of multiple integrals to find the right lower bound.

3. Some Useful Tools

We recall that the Beta-function of the cone is defined by We note that this integral converges if and only if . Recall that where is the classical Gamma-function. The following can be obtained by induction (see, e.g., [10, Lemma ]):where , .

4. The Norm of and

4.1. The Norm of

We have the following upper estimate of the norm of .

Lemma 5. Let be real numbers. Let , , and let be real numbers and assume that relations (6) and (7) hold. Assume that Then the operator is bounded from to . Moreover,

Proof. For , , we write again . Using an easy change of variables and Minkowski’s inequality, we obtain Using (6), (7), and Hölder’s inequality, we easily obtain that It follows that The proof is complete.

We also have the following result.

Lemma 6. Let be real numbers. Let , , , and assume that . Let be real numbers and assume that . Suppose that the parameters satisfy Then the operator is bounded from to . In this case, if we denote by the operator norm of , then

Proof. For , , and , we write again . Proceeding as at the beginning of the proof of the previous lemma, we obtain The proof is complete.

Let us now prove the lower bound for the norm of .

Lemma 7. Let be real numbers. Let , , and let be real numbers and assume that relations (6) and (7) hold. Suppose that the parameters satisfy Then

Proof. For simplicity, let us put For any vector , , (for , replace by ), we have hence for any (with replaced by if ), Let and defineThen Substituting these into (36), we obtain Observing that , we obtain that where We have thatwhere Let us write Thenwith It follows that if is an even integer, then where ; and if is an odd number, then Taking (48) into (39), we obtain that if is even, then Thus Taking (49) into (39) and doing the same type of calculations as above, we obtain that if is an odd integer, then Letting in (51) or (52) we obtain that . That is, The proof is complete.

4.2. The Norm of

Let us prove the following upper bound for the norm of .

Lemma 8. Let be real numbers. Let , , and let be real numbers, and assume that relations (6) and (7) hold. Assume that Then the operator is bounded from to . In this case, if we denote by the operator norm of , then

Proof. The proof is essentially the same as above. Let us give it here for completeness. For , , we write again . We remark that and . Using an easy change of variables and Minkowski’s inequality, Hölder’s inequality, and equality (23), we obtain The proof is complete.

Let us prove the following result.

Lemma 9. Let be real numbers. Let , , , and assume that . Let be real numbers and assume that . Suppose that the parameters satisfy Then the operator is bounded from to . In this case, if we denote by the operator norm of , then

Proof. For , , and , we write again . We observe that and . We proceed as in the proof of Lemma 8; we use a change of variables and Minkowski’s inequality, Hölder’s inequality, and (23) to obtain The proof is complete.

We next obtain a lower bound of the norm of .

Lemma 10. Let be real numbers. Let , , and let be real numbers with and assume that relations (6) and (7) hold. Suppose that the parameters satisfy Then

Proof. Let us put Then for any vector , , (for , replace by ), hence for any (with replaced by if ), Let and defineThen Substituting these into (64), we obtain Put Then where We have where It follows that, for even, and, for odd, Taking (73) into (67), we obtain that if is even, then ThusTaking (74) into (67), we obtain that if is odd, then Thus for odd, Letting in (76) or (78) we obtain that ; that is, The proof is complete.

5. Necessity for Boundedness of and

5.1. Necessity for Boundedness of

Lemma 11. Let be real numbers. Let , , and let be real numbers and assume that relations (6) and (7) hold. If the operator is bounded from to , then the parameters satisfy

Proof. For simplicity, we put and . Let be a real number. Given a function , we denote, by , the function defined by . One easily checks that, for , For a vector , we write . Using some easy changes of variables, we obtain It comes that Recall that the boundedness of means that there exists a constant such that, for any , , It follows using (81) and (83) that or equivalently, As (86) holds for any and any real number , we necessarily have that which combined with (7) gives that

Lemma 12. Let be real numbers. Let , , and let be real numbers and assume that relations (6) and (7) hold. Then if the operator is bounded from to , then the parameters satisfy

Proof. Assume that is bounded from to . Then its -th adjoint defined by where , is bounded from to . Let us take and , . We observe that, for , while . It follows for this choice of functions that, for any , It follows from the boundedness of that there is a constant such that From the properties of the Beta-function, we know that this implies that and . These two inequalities are equivalent to The proof is complete.

The proof of the following result follows the same steps as in the proof of the above lemma.

Lemma 13. Let be real numbers. Let , and let be real numbers and assume that relations (6) and (7) hold with . If the operator is bounded from to , then the parameters satisfy

5.2. Necessity for Boundedness of

Lemma 14. Let be real numbers. Let , , and let be real numbers and assume that relations (6) and (7) hold. If the operator is bounded from to , then the parameters satisfy

Proof. This is obtained exactly the same way as in the proof of Lemma 11. We leave it to the interested reader.

Lemma 15. Let be real numbers. Let , , and let be real numbers and assume that relations (6) and (7) hold. If the operator is bounded from to , then and .

Proof. Let us take , . We observe that, for , . It follows for this choice of functions that, for any , It follows from the boundedness of that there is a constant such that Clearly, this is only possible if and . The proof is complete.

Lemma 16. Let be real numbers. Let , , and let be real numbers and assume that relations (6) and (7) hold. If the operator is bounded from to , then the parameters satisfy

Proof. Assume that is bounded from to , Then its -th adjoint is defined by where again is bounded from to . Let us take and , . We observe that, for , , . It follows for this choice of functions that, for any , It follows from the boundedness of that there is a constant such that This as in the proof of Lemma 12 implies that The proof is complete.

The following is obtained as above.

Lemma 17. Let be real numbers. Let , and let be real numbers and assume that relations (6) and (7) hold with . If the operator is bounded from to , then the parameters satisfy and

5.3. Necessity for the Other Cases

Let us start by proving the following

Lemma 18. Let be real numbers. Let , , , and assume that . Let be real numbers and assume that . If the operator is bounded from to , then the parameters satisfy

Proof. Suppose that is bounded from to Then for , the -th adjoint, , of is defined as in the proof of Lemmas 12 and 13. Hence the proof of the first inequalities follows from the proof of Lemmas 12 and 13. To prove the second inequalities, we observe that, for , -th adjoint, , of is defined bywhere is bounded from to . Hence the second inequality can now be established similarly as in the proof of Lemma 12.

Let us finish with the following.

Lemma 19. Let be real numbers. Let , , , and assume that . Let be real numbers and assume that . If the operator is bounded from to , then the parameters satisfy

Proof. Suppose that is bounded from to Then for , the -th adjoint, , of is defined as in the proof of Lemmas 16 and 17. Hence the proof of the first inequality follows from the proof of Lemmas 16 and 17. To prove the second inequality, we observe that, for , , of , is given by where again is bounded from to . Hence the second inequality can now be established similarly as in the proof of Lemma 16.

6. Proof of the Results

Theorem 1 follows from Lemmas 5, 7, 11, 12, and 13. Theorem 2 follows from Lemmas 8, 10, 14, 15, 16, and 17. Theorem 3 follows from Lemmas 6, 7, 11, and 18. Theorem 4 follows from Lemmas 9, 10, 14, 15, and 19. The proof is complete.

Competing Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.