## Final Report Summary - PARAPRODUCTS (Time-frequency analysis of paraproducts)

Description of the project objectives

The main purpose of the project Time-Frequency Analysis of Paraproducts is the development of a complete theory of boundedness of Singular Integral Operators in the product setting. This family of operators can be represented in the following way:

T(f)(x) =∫Rn f(y)K(x,x−y) dy,

where the integral kernel K satisfies standard bounds of tensor product type. These are the natural type of bounds exhibited by the product of several different one- parameter kernels. They reflect the product singularities of such kernels, which cannot be treated by the one-parameter theory . In particular, the goal of the project is to state and prove boundedness conditions for operators given by such kernels in the integrable Lebesgue spaces Lp (Rn).

In the development of this programme, we distinguish three stages: the L2-theory, the extension to Lp spaces, and the endpoint estimates.

As in the case of one-parameter operators, the starting point of the project is the establishment of a suitable L2-theory which characterizes or at least provides sufficient conditions for boundedness of singular integrals in square integrable Lebesgue spaces by means of a suitable decomposition into paraproducts. This strategy is analogous to that of the classical David and Journeʼs T(1) theorem, although the decomposition is much more subtle in the product setting.

Once the bounds in L2 are achieved, the following natural step is to extend such boundedness results to the integrable Lebesgue Lp spaces with 1 < p < ∞. This process is quite standard in the classical setting but not in our setting of multiparameter operators, where new methods are required.

Finally, we also address the issue of endpoint estimates, that is, the behaviour of the operator when the integrability exponent reaches the values p = 1 or p = ∞.

Description of the work performed

The principal strategy to achieve these objectives has been the appropriate combination of the use of classical tools like Calderon-Zygmund decomposition or square functions, with the application of the recent `Time-frequency Analysisʼ-techniques, which have been developed by several authors during the last decade. A more detailed account of our approach can be found in the `Work progress and achievementsʼ section.

Description of the main results

The main result obtained is a very natural generalization of the classical T(1) theory for L2- boundedness to the multiparameter setting. In this sense, we have improved a classical theorem of J.L. Journe which was until now the best extension of Calderoń -Zygmund theory to the multiparameter case. The advantage of our approach lies in the fact that we do not make use of any vector-valued methods. Thus our new criterium is more general in the sense that it applies to a wider class of operators, and simpler, since the kernel conditions that the operators need to satisfy are easier to check in practice.

The second interesting novelty of our work has been the method employed to achieve the Lp extension: we had to define a new type of square function which, to our knowledge, has never appeared in the literature. After proving some of the essential properties, we managed to use this new operator to prove boundedness of the product singular integrals.

In fact, this method has been so successful that minor modifications of the previous ideas allowed us also to even obtain the endpoint estimates.

Moreover, as a side effect of our efforts, we have also obtained several new results about Haar multipliers and paraproducts, which naturally appear in our product T(1) theorem, an interpolation result, and a new covering lemma, which are of independent interest.

Expected final results and their impact

We expect that our work will offer a substantial contribution to the theory of multiparameter singular integrals. In the same way as the vector-valued approach of Journeʼs theorem was adopted by many other authors in later developments of singular integration in product spaces, we expect our papers to become a reference for future developments in the study of multiplier and pseudodifferential operators in several variables. More generally, our new results imply an advancement of Singular Integration theory and thus can have a noticeable impact in many other fields. As an example of this potential for applications, we give a stronger version of a previous theorem of R. Fefferman and E. Stein about convolution operators in the product setting. Moreover, we believe that our endpoint estimates can be applied to the study of some generalizations of the Cauchy integral to the setting of several complex variables. Our results on Lp-Lq multipliers in the product setting have implications for more general Lp - Lq boundedness problems.

The main purpose of the project Time-Frequency Analysis of Paraproducts is the development of a complete theory of boundedness of Singular Integral Operators in the product setting. This family of operators can be represented in the following way:

T(f)(x) =∫Rn f(y)K(x,x−y) dy,

where the integral kernel K satisfies standard bounds of tensor product type. These are the natural type of bounds exhibited by the product of several different one- parameter kernels. They reflect the product singularities of such kernels, which cannot be treated by the one-parameter theory . In particular, the goal of the project is to state and prove boundedness conditions for operators given by such kernels in the integrable Lebesgue spaces Lp (Rn).

In the development of this programme, we distinguish three stages: the L2-theory, the extension to Lp spaces, and the endpoint estimates.

As in the case of one-parameter operators, the starting point of the project is the establishment of a suitable L2-theory which characterizes or at least provides sufficient conditions for boundedness of singular integrals in square integrable Lebesgue spaces by means of a suitable decomposition into paraproducts. This strategy is analogous to that of the classical David and Journeʼs T(1) theorem, although the decomposition is much more subtle in the product setting.

Once the bounds in L2 are achieved, the following natural step is to extend such boundedness results to the integrable Lebesgue Lp spaces with 1 < p < ∞. This process is quite standard in the classical setting but not in our setting of multiparameter operators, where new methods are required.

Finally, we also address the issue of endpoint estimates, that is, the behaviour of the operator when the integrability exponent reaches the values p = 1 or p = ∞.

Description of the work performed

The principal strategy to achieve these objectives has been the appropriate combination of the use of classical tools like Calderon-Zygmund decomposition or square functions, with the application of the recent `Time-frequency Analysisʼ-techniques, which have been developed by several authors during the last decade. A more detailed account of our approach can be found in the `Work progress and achievementsʼ section.

Description of the main results

The main result obtained is a very natural generalization of the classical T(1) theory for L2- boundedness to the multiparameter setting. In this sense, we have improved a classical theorem of J.L. Journe which was until now the best extension of Calderoń -Zygmund theory to the multiparameter case. The advantage of our approach lies in the fact that we do not make use of any vector-valued methods. Thus our new criterium is more general in the sense that it applies to a wider class of operators, and simpler, since the kernel conditions that the operators need to satisfy are easier to check in practice.

The second interesting novelty of our work has been the method employed to achieve the Lp extension: we had to define a new type of square function which, to our knowledge, has never appeared in the literature. After proving some of the essential properties, we managed to use this new operator to prove boundedness of the product singular integrals.

In fact, this method has been so successful that minor modifications of the previous ideas allowed us also to even obtain the endpoint estimates.

Moreover, as a side effect of our efforts, we have also obtained several new results about Haar multipliers and paraproducts, which naturally appear in our product T(1) theorem, an interpolation result, and a new covering lemma, which are of independent interest.

Expected final results and their impact

We expect that our work will offer a substantial contribution to the theory of multiparameter singular integrals. In the same way as the vector-valued approach of Journeʼs theorem was adopted by many other authors in later developments of singular integration in product spaces, we expect our papers to become a reference for future developments in the study of multiplier and pseudodifferential operators in several variables. More generally, our new results imply an advancement of Singular Integration theory and thus can have a noticeable impact in many other fields. As an example of this potential for applications, we give a stronger version of a previous theorem of R. Fefferman and E. Stein about convolution operators in the product setting. Moreover, we believe that our endpoint estimates can be applied to the study of some generalizations of the Cauchy integral to the setting of several complex variables. Our results on Lp-Lq multipliers in the product setting have implications for more general Lp - Lq boundedness problems.