##
**Summability of multi-dimensional Fourier series and Hardy spaces.**
*(English)*
Zbl 1306.42003

The book under review is mainly devoted to the study of multi-dimensional classical and dyadic martingale Hardy spaces and of some of the summability methods for Walsh-, trigonometric- or spline-Fourier series in the multi-dimensional case. The main objective is, for different summability methods, to get a.e. convergence results, weak type inequalities or the boundedness on \(L^p\)-spaces or Hardy spaces for the corresponding maximal operators.

The book is aimed to be self-contained, and it is probably the first one containing the theory of several-parameter Hardy spaces. It becomes very technical in many occasions, but it is clearly written and contains most of the basic results achieved in multidimensional summability in the last decades. In particular the atomic decomposition of Hardy spaces plays a very important role.

The methods of proofs are different from those used in the one-variable case, and problems here are usually much more difficult to handle. Sometimes it is not even clear which should be the analogue to the one-variable case, and in many occasions there is no such an analogue and several natural extensions can be considered and different behaviours can be seen according to the chosen point of view.

The first two chapters are devoted to introducing Hardy spaces and to obtaining their formulations in terms of atomic decompositions. Also, interpolation and boundedness of operators are dealt with. Chapter I is devoted to the dyadic martingale Hardy spaces. They are introduced by using maximal or square functions. The difference between the diagonal and the general case is very well clarified and leads to different spaces; for instance one can use either \(f^*= \sup_{(n_1,\dots ,n_d)} |f_{(n_1,\dots ,n_d)}|\) or \(f_\square^*= \sup_{n} |f_{(n,\dots ,n)}|\) to define different Hardy-Lorentz spaces. It is also shown that different kinds of atoms are needed in each case, and several atomic decompositions are obtained for the \(H^p\) spaces in the case \(0<p\leq 1\).

Interpolation of Hardy-Lorentz spaces is also studied. It is well known that, for certain operators, Calderón-Zygmund decomposition makes it possible to get weak type inequalities. Since this cannot be applied in the multi-dimensional case the author uses the atomic decompositon and interpolation to get weak-type inequalities for a special class of operators.

It is worth mentioning that sometimes not only the results for one and several parameters are different but also there can be essential differences when passing from two to more parameters. Chapter II contains the same type of results as in Chapter I, but for the classical Hardy spaces.

The rest of the book is concerned with summability of Fourier series in different multi-dimensional settings.

In Chapter III the author considers Walsh-Fourier series of dyadic martingales and studies the convergence of Cèsaro (or conjugate Cèsaro) and Riesz (or conjugate Riesz) means of such series. Also, the boundedness on Hardy spaces and weak type inequalities of the corresponding restricted (provided \(2^{-\tau}\leq n_k/n_j\leq 2^{\tau}\) for some \(\tau>0\) and all \((n_1,\dots ,n_d)\)) and non-restricted maximal operators are obtained by using first some estimates for the corresponding Cèsaro and Riesz kernels.

Chapter IV is devoted to the analogue of the Lebesgue differentiation theorem for dyadic functions in the multi-dimensional case. The dyadic derivative \(d_{(n_1,\dots ,n_d)}(f)\) and dyadic integral \(I(f)\) are introduced for integrable functions in \(L^1([0,1)^d)\). As in previous chapters, the boundedness of the maximal operators \(I_* f= \sup\{ |d_{(n_1,\dots ,n_d)}(If)|\} \) and \(I_\square f= \sup\{ |d_{(n_1,\dots ,n_d)}(If)|\colon |n_i-n_j|\leq \tau\}\) on Lebesgue and Hardy spaces is analyzed.

Chapters V and VI are concerned with a very general method of summability for \(d\)-dimensional trigonometric Fourier series or Fourier transforms. The \(\theta\)-means of Fourier series \(\sigma_{(n_1,\dots,n_d)}^\theta (f)\) or transforms \(U_{(T_1,\dots ,T_d)}^\theta (f)\) give a common formulation for a big number of summability methods. As in previous chapters the author deals with the associated restricted and non-restricted maximal operators \(\sigma_\square^\theta(f)= \sup\{|\sigma_{(n_1,\dots ,n_d)}^\theta(f)|\colon 2^{-\tau}\leq n_k/n_j\leq 2^{\tau}\}\) and \(\sigma_*^\theta(f)= \sup\{|\sigma_{(n_1,\dots ,n_d)}^\theta(f)|\}\), and several boundedness results are provided.

In Chapter VII Hardy spaces associated to spline systems are considered. Boundedness of maximal operators of partial sums and a.e. convergence results are obtained for spline-Fourier series. It is also shown that spline systems are unconditional and equivalent bases in the multidimensional Hardy spaces.

The book is aimed to be self-contained, and it is probably the first one containing the theory of several-parameter Hardy spaces. It becomes very technical in many occasions, but it is clearly written and contains most of the basic results achieved in multidimensional summability in the last decades. In particular the atomic decomposition of Hardy spaces plays a very important role.

The methods of proofs are different from those used in the one-variable case, and problems here are usually much more difficult to handle. Sometimes it is not even clear which should be the analogue to the one-variable case, and in many occasions there is no such an analogue and several natural extensions can be considered and different behaviours can be seen according to the chosen point of view.

The first two chapters are devoted to introducing Hardy spaces and to obtaining their formulations in terms of atomic decompositions. Also, interpolation and boundedness of operators are dealt with. Chapter I is devoted to the dyadic martingale Hardy spaces. They are introduced by using maximal or square functions. The difference between the diagonal and the general case is very well clarified and leads to different spaces; for instance one can use either \(f^*= \sup_{(n_1,\dots ,n_d)} |f_{(n_1,\dots ,n_d)}|\) or \(f_\square^*= \sup_{n} |f_{(n,\dots ,n)}|\) to define different Hardy-Lorentz spaces. It is also shown that different kinds of atoms are needed in each case, and several atomic decompositions are obtained for the \(H^p\) spaces in the case \(0<p\leq 1\).

Interpolation of Hardy-Lorentz spaces is also studied. It is well known that, for certain operators, Calderón-Zygmund decomposition makes it possible to get weak type inequalities. Since this cannot be applied in the multi-dimensional case the author uses the atomic decompositon and interpolation to get weak-type inequalities for a special class of operators.

It is worth mentioning that sometimes not only the results for one and several parameters are different but also there can be essential differences when passing from two to more parameters. Chapter II contains the same type of results as in Chapter I, but for the classical Hardy spaces.

The rest of the book is concerned with summability of Fourier series in different multi-dimensional settings.

In Chapter III the author considers Walsh-Fourier series of dyadic martingales and studies the convergence of Cèsaro (or conjugate Cèsaro) and Riesz (or conjugate Riesz) means of such series. Also, the boundedness on Hardy spaces and weak type inequalities of the corresponding restricted (provided \(2^{-\tau}\leq n_k/n_j\leq 2^{\tau}\) for some \(\tau>0\) and all \((n_1,\dots ,n_d)\)) and non-restricted maximal operators are obtained by using first some estimates for the corresponding Cèsaro and Riesz kernels.

Chapter IV is devoted to the analogue of the Lebesgue differentiation theorem for dyadic functions in the multi-dimensional case. The dyadic derivative \(d_{(n_1,\dots ,n_d)}(f)\) and dyadic integral \(I(f)\) are introduced for integrable functions in \(L^1([0,1)^d)\). As in previous chapters, the boundedness of the maximal operators \(I_* f= \sup\{ |d_{(n_1,\dots ,n_d)}(If)|\} \) and \(I_\square f= \sup\{ |d_{(n_1,\dots ,n_d)}(If)|\colon |n_i-n_j|\leq \tau\}\) on Lebesgue and Hardy spaces is analyzed.

Chapters V and VI are concerned with a very general method of summability for \(d\)-dimensional trigonometric Fourier series or Fourier transforms. The \(\theta\)-means of Fourier series \(\sigma_{(n_1,\dots,n_d)}^\theta (f)\) or transforms \(U_{(T_1,\dots ,T_d)}^\theta (f)\) give a common formulation for a big number of summability methods. As in previous chapters the author deals with the associated restricted and non-restricted maximal operators \(\sigma_\square^\theta(f)= \sup\{|\sigma_{(n_1,\dots ,n_d)}^\theta(f)|\colon 2^{-\tau}\leq n_k/n_j\leq 2^{\tau}\}\) and \(\sigma_*^\theta(f)= \sup\{|\sigma_{(n_1,\dots ,n_d)}^\theta(f)|\}\), and several boundedness results are provided.

In Chapter VII Hardy spaces associated to spline systems are considered. Boundedness of maximal operators of partial sums and a.e. convergence results are obtained for spline-Fourier series. It is also shown that spline systems are unconditional and equivalent bases in the multidimensional Hardy spaces.

### MSC:

42-02 | Research exposition (monographs, survey articles) pertaining to harmonic analysis on Euclidean spaces |

42B05 | Fourier series and coefficients in several variables |

42B30 | \(H^p\)-spaces |

42B35 | Function spaces arising in harmonic analysis |

46E15 | Banach spaces of continuous, differentiable or analytic functions |