Boundary values and convolution in ultradistribution spaces. (English) Zbl 1155.46002
Series on Analysis, Applications and Computation 1. Hackensack, NJ: World Scientific (ISBN 978-981-270-769-7/hbk). xii, 217 p. (2007).
This book analyses various spaces of ultradistributions considered as boundary values of analytic functions with appropriate growth estimates and is of interest for researchers in this and related areas.
The preparatory Chapter 1 deals with the Cauchy and Poisson kernels corresponding to tube domains. The spaces of ultradifferentiable test functions and ultradistributions are introduced in Chapter 2. The authors consider spaces of ultradifferentiable functions of Beurling and of Roumieu type defined by assuming conditions on the sequence of derivatives, following Komatsu’s point of view [see H. Komatsu, J. Fac. Sci. Univ. Tokyo, Sect. IA 20, 25–105 (1973; Zbl 0258.46039)]. They focus on the spaces \({\mathcal D}(\ast,L^s)\) defined as usual with \(L^s\) norms instead of supremum norms. Here, \(\ast = \{M_p\}\) (Roumieu case) or \(\ast = (M_p)\) (Beurling case). Theorems 2.3.1 and 2.3.2 characterize the dual space, denoted \({\mathcal D}^\prime(\ast,L^s).\) Section 2.5 studies spaces of Gelfand–Shilov type whose elements are ultradifferentiable functions of ultrapolynomial growth. They are used as the test spaces for spaces \({\mathcal S}^{\prime \ast}\) of tempered ultradistributions.
Chapter 3 contains a characterization of bounded sets in \({\mathcal D}^\prime((M_p),L^t)\) and in \({\mathcal S}^{\prime \ast}.\) Two types of theorems are presented: under the assumption of conditions (M1) and (M3’) on the sequence \((M_p)\) or under the conditions (M1), (M2) and (M3). The reader should be warned that in this chapter, and at some point in subsequent chapters (see, for instance, Theorem 5.4.2), \({\mathcal D}^\prime((M_p),L^t)\) is not the dual space of \({\mathcal D}((M_p),L^t)\) but the dual space of \({\mathcal D}^\prime((M_p),L^s)\), \(\frac{1}{s} + \frac{1}{t} = 1\).
Chapters 4 and 5 constitute the most interesting part of the book. With the aim of extending previous work by Luszczki and Zielezny in the one-dimensional case and by Tillmann in the \(n\)-dimensional case for analytic functions on special tubes, about the representation of the Schwartz distributions in \({\mathcal D}^\prime_{L^s}({\mathbb R}^n)\) as boundary values of holomorphic functions (\(1 < s < \infty\)), the authors consider tube domains \(T^C:= {\mathbb R}^n + i C\), with \(C\) being a regular cone in \({\mathbb R}^n.\) To be more precise, let \(K\) and \(Q\) denote the Cauchy and Poisson kernels corresponding to this tube. Here is the statement of Theorems 4.1.1 and 4.1.2: Let the sequence \((M_p)\) satisfy (M1) and (M\(3'\)). Then, for \(z\in {\mathbb R}^n + i C,\) we have \(K(z-\cdot)\in {\mathcal D}^\prime((M_p),L^s)\) for \(1 < s \leq \infty\) and \(Q(z,\cdot)\in {\mathcal D}^\prime((M_p),L^s)\) for \(1 \leq s \leq \infty\). After defining the Cauchy integral of \(U\in {\mathcal D}^\prime(\ast,L^s)\) as \(C(U,z):=\langle U, K(z-\cdot)\rangle\), \(z\in T^C,\) and proving that it is an analytic function in \(T^C\) (\(1 < s < \infty\)), a norm growth estimate for the Cauchy integral is obtained for \(2 \leq s < \infty\) (Theorem 4.2.3). The calculation of the boundary value of the Cauchy integral is the content of Theorems 4.2.5 and 4.2.6 (\(2 \leq s < \infty\)). As a consequence, one obtains an analytic decomposition theorem for elements in \({\mathcal D}^\prime(\ast L^s)\), \(2 \leq s < \infty\) (Theorem 4.2.7). The last section of Chapter 4 analyses the boundary value property of the Poisson integral of an arbitrary ultradistribution \(U\in {\mathcal D}^\prime(\ast, L^s)\), \(1 < s < \infty.\)
Chapter 5 begins with a study of generalizations of the Hardy spaces of analytic functions on tubes associated to a sequence \((M_p).\) Theorem 5.1.1 gives Fourier–Laplace representations of the functions in \(H^r_{(M_p)}(T^C)\) for \(1 < r \leq 2\). Theorem 5.2.1 shows that the elements of \(H^r_{(M_p)}(T^C)\) satisfying certain growth conditions define ultradistributional boundary values in \({\mathcal D}^\prime((M_p),L^1)\). Theorem 5.2.2 gives ultradistributional boundary values in \({\mathcal D}^\prime((M_p),L^r)\) for \(1 < r \leq 2\). After restricting to special types of cones, some results of Sections 5.1 and 5.2 are extended, in Section 5.3, to the case \(2 < r < \infty\). Section 5.4 gives another approach to the boundary values via almost analytic extension. Only the case \(n = 1\) is considered and the authors claim that the cases \(p =1,\infty\) are still open. For \(1 \leq s < \infty\) and \(f\in {\mathcal H}^\ast{L^s}\), a boundary value \(Tf\in {\mathcal D}^\prime(\ast, L^r)\) exists and \(\langle Tf,\varphi\rangle\) can be expressed in terms of an almost analytic extension of \(\varphi\). This is the content of Theorem 5.4.1. For \(r > 1\) and \(s = \frac{r}{r-1}\), the surjectivity of the boundary value map \(T:{\mathcal H}(\ast,L^s) \to {\mathcal D}^\prime(\ast, L^r)\) is proved in Theorem 5.4.2. The cases \(s = \infty\) and \(s = 1\) are discussed in Section 5.5. The methods of the previous section cannot be applied because the function \(t\mapsto \frac{1}{t - z}\), \(\operatorname{Im}z \neq 0,\) is not in \(L^1\) and, consequently, only the existence of the boundary value operator is proved but nothing is done about its surjectivity.
At this point, it is important to remark that C. Fernández, C. Gómez–Collado and the reviewer [RACSAM, Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat. 97, No. 2, 243–255 (2003; Zbl 1084.46036)] obtained a complete description of the elements in the dual of \({\mathcal D}_{L^s}({\mathbb R}^n)\) as boundary values of spaces of analytic functions. In particular, the surjectivity of the corresponding boundary value operator was obtained in the several variables case and for every value of \(1 \leq s \leq \infty,\) which is the content of the Theorem 1 in our paper. Moreover, Theorem 2 in that paper gives the corresponding result for ultradistributions of Beurling type in the sense of R. W. Braun, R. Meise and B. A. Taylor [Result. Math. 17, No. 3, 206–237 (1990; Zbl 0735.46022)], a class that includes Komatsu’s classes in the most relevant cases.
Chapter 6 in the book under review is devoted to the convolution of ultradistributions. The authors do not restrict their attention to the case when one of the ultradistributions is compactly supported. Several definitions of convolution for ultradistributions are given and their equivalence is proved in Sections 6.2 and 6.3. Analogous results for \({\mathcal S}^{\prime (M_p)}\)-convolution are the content of Sections 6.4 and 6.5. The authors also analyse sufficient conditions for the existence of the convolution, expressed in terms of the supports of the ultradistributions or conditions which rely on distinguishing subspaces of ultradistributions.
The final Chapter 7 studies various integral transforms on spaces of tempered ultradistributions. Theorem 2.2.2 provides properties of \({\mathcal S}^\ast\) in terms of Hermite expansions and its proof is postponed until Section 7.5. Section 7.3 gives characterizations of the spaces \({\mathcal S}^\ast\) in terms of several integral transforms, while Sections 7.6 and 7.7 analyse the Hilbert transform and more general singular integral operators on spaces of tempered ultradistributions.
Summarizing, this book contains a large amount of information on ultradistributions as boundary values of analytic functions, which will be useful for researchers on this topic and its applications to linear partial differential operators.
The preparatory Chapter 1 deals with the Cauchy and Poisson kernels corresponding to tube domains. The spaces of ultradifferentiable test functions and ultradistributions are introduced in Chapter 2. The authors consider spaces of ultradifferentiable functions of Beurling and of Roumieu type defined by assuming conditions on the sequence of derivatives, following Komatsu’s point of view [see H. Komatsu, J. Fac. Sci. Univ. Tokyo, Sect. IA 20, 25–105 (1973; Zbl 0258.46039)]. They focus on the spaces \({\mathcal D}(\ast,L^s)\) defined as usual with \(L^s\) norms instead of supremum norms. Here, \(\ast = \{M_p\}\) (Roumieu case) or \(\ast = (M_p)\) (Beurling case). Theorems 2.3.1 and 2.3.2 characterize the dual space, denoted \({\mathcal D}^\prime(\ast,L^s).\) Section 2.5 studies spaces of Gelfand–Shilov type whose elements are ultradifferentiable functions of ultrapolynomial growth. They are used as the test spaces for spaces \({\mathcal S}^{\prime \ast}\) of tempered ultradistributions.
Chapter 3 contains a characterization of bounded sets in \({\mathcal D}^\prime((M_p),L^t)\) and in \({\mathcal S}^{\prime \ast}.\) Two types of theorems are presented: under the assumption of conditions (M1) and (M3’) on the sequence \((M_p)\) or under the conditions (M1), (M2) and (M3). The reader should be warned that in this chapter, and at some point in subsequent chapters (see, for instance, Theorem 5.4.2), \({\mathcal D}^\prime((M_p),L^t)\) is not the dual space of \({\mathcal D}((M_p),L^t)\) but the dual space of \({\mathcal D}^\prime((M_p),L^s)\), \(\frac{1}{s} + \frac{1}{t} = 1\).
Chapters 4 and 5 constitute the most interesting part of the book. With the aim of extending previous work by Luszczki and Zielezny in the one-dimensional case and by Tillmann in the \(n\)-dimensional case for analytic functions on special tubes, about the representation of the Schwartz distributions in \({\mathcal D}^\prime_{L^s}({\mathbb R}^n)\) as boundary values of holomorphic functions (\(1 < s < \infty\)), the authors consider tube domains \(T^C:= {\mathbb R}^n + i C\), with \(C\) being a regular cone in \({\mathbb R}^n.\) To be more precise, let \(K\) and \(Q\) denote the Cauchy and Poisson kernels corresponding to this tube. Here is the statement of Theorems 4.1.1 and 4.1.2: Let the sequence \((M_p)\) satisfy (M1) and (M\(3'\)). Then, for \(z\in {\mathbb R}^n + i C,\) we have \(K(z-\cdot)\in {\mathcal D}^\prime((M_p),L^s)\) for \(1 < s \leq \infty\) and \(Q(z,\cdot)\in {\mathcal D}^\prime((M_p),L^s)\) for \(1 \leq s \leq \infty\). After defining the Cauchy integral of \(U\in {\mathcal D}^\prime(\ast,L^s)\) as \(C(U,z):=\langle U, K(z-\cdot)\rangle\), \(z\in T^C,\) and proving that it is an analytic function in \(T^C\) (\(1 < s < \infty\)), a norm growth estimate for the Cauchy integral is obtained for \(2 \leq s < \infty\) (Theorem 4.2.3). The calculation of the boundary value of the Cauchy integral is the content of Theorems 4.2.5 and 4.2.6 (\(2 \leq s < \infty\)). As a consequence, one obtains an analytic decomposition theorem for elements in \({\mathcal D}^\prime(\ast L^s)\), \(2 \leq s < \infty\) (Theorem 4.2.7). The last section of Chapter 4 analyses the boundary value property of the Poisson integral of an arbitrary ultradistribution \(U\in {\mathcal D}^\prime(\ast, L^s)\), \(1 < s < \infty.\)
Chapter 5 begins with a study of generalizations of the Hardy spaces of analytic functions on tubes associated to a sequence \((M_p).\) Theorem 5.1.1 gives Fourier–Laplace representations of the functions in \(H^r_{(M_p)}(T^C)\) for \(1 < r \leq 2\). Theorem 5.2.1 shows that the elements of \(H^r_{(M_p)}(T^C)\) satisfying certain growth conditions define ultradistributional boundary values in \({\mathcal D}^\prime((M_p),L^1)\). Theorem 5.2.2 gives ultradistributional boundary values in \({\mathcal D}^\prime((M_p),L^r)\) for \(1 < r \leq 2\). After restricting to special types of cones, some results of Sections 5.1 and 5.2 are extended, in Section 5.3, to the case \(2 < r < \infty\). Section 5.4 gives another approach to the boundary values via almost analytic extension. Only the case \(n = 1\) is considered and the authors claim that the cases \(p =1,\infty\) are still open. For \(1 \leq s < \infty\) and \(f\in {\mathcal H}^\ast{L^s}\), a boundary value \(Tf\in {\mathcal D}^\prime(\ast, L^r)\) exists and \(\langle Tf,\varphi\rangle\) can be expressed in terms of an almost analytic extension of \(\varphi\). This is the content of Theorem 5.4.1. For \(r > 1\) and \(s = \frac{r}{r-1}\), the surjectivity of the boundary value map \(T:{\mathcal H}(\ast,L^s) \to {\mathcal D}^\prime(\ast, L^r)\) is proved in Theorem 5.4.2. The cases \(s = \infty\) and \(s = 1\) are discussed in Section 5.5. The methods of the previous section cannot be applied because the function \(t\mapsto \frac{1}{t - z}\), \(\operatorname{Im}z \neq 0,\) is not in \(L^1\) and, consequently, only the existence of the boundary value operator is proved but nothing is done about its surjectivity.
At this point, it is important to remark that C. Fernández, C. Gómez–Collado and the reviewer [RACSAM, Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat. 97, No. 2, 243–255 (2003; Zbl 1084.46036)] obtained a complete description of the elements in the dual of \({\mathcal D}_{L^s}({\mathbb R}^n)\) as boundary values of spaces of analytic functions. In particular, the surjectivity of the corresponding boundary value operator was obtained in the several variables case and for every value of \(1 \leq s \leq \infty,\) which is the content of the Theorem 1 in our paper. Moreover, Theorem 2 in that paper gives the corresponding result for ultradistributions of Beurling type in the sense of R. W. Braun, R. Meise and B. A. Taylor [Result. Math. 17, No. 3, 206–237 (1990; Zbl 0735.46022)], a class that includes Komatsu’s classes in the most relevant cases.
Chapter 6 in the book under review is devoted to the convolution of ultradistributions. The authors do not restrict their attention to the case when one of the ultradistributions is compactly supported. Several definitions of convolution for ultradistributions are given and their equivalence is proved in Sections 6.2 and 6.3. Analogous results for \({\mathcal S}^{\prime (M_p)}\)-convolution are the content of Sections 6.4 and 6.5. The authors also analyse sufficient conditions for the existence of the convolution, expressed in terms of the supports of the ultradistributions or conditions which rely on distinguishing subspaces of ultradistributions.
The final Chapter 7 studies various integral transforms on spaces of tempered ultradistributions. Theorem 2.2.2 provides properties of \({\mathcal S}^\ast\) in terms of Hermite expansions and its proof is postponed until Section 7.5. Section 7.3 gives characterizations of the spaces \({\mathcal S}^\ast\) in terms of several integral transforms, while Sections 7.6 and 7.7 analyse the Hilbert transform and more general singular integral operators on spaces of tempered ultradistributions.
Summarizing, this book contains a large amount of information on ultradistributions as boundary values of analytic functions, which will be useful for researchers on this topic and its applications to linear partial differential operators.
Reviewer: Antonio Galbis (Valencia)
MSC:
| 46-02 | Research exposition (monographs, survey articles) pertaining to functional analysis |
| 46F20 | Distributions and ultradistributions as boundary values of analytic functions |
| 46F05 | Topological linear spaces of test functions, distributions and ultradistributions |
| 46F12 | Integral transforms in distribution spaces |
