##
**Tempered homogeneous function spaces.**
*(English)*
Zbl 1336.46004

EMS Series of Lectures in Mathematics. Zürich: European Mathematical Society (EMS) (ISBN 978-3-03719-155-2/pbk; 978-3-03719-655-7/ebook). xii, 131 p. (2015).

In this interesting book, motivated by applications from Navier-Stokes equations, the author mainly works on tempered homogeneous quasi-Banach function spaces of Besov-Sobolev type within the framework of \((S(\mathbb R^n),S'(\mathbb R^n))\) based on Gauss-Weierstrass semi-groups. Recall that a quasi-Banach space \(A(\mathbb R^n)\), which is continuous embedded into the tempered space \(\mathcal S'(\mathbb R^n)\), is called homogeneous if there exists some \(\sigma\in\mathbb R\) such that, for all \(\lambda\in(0,\infty)\) and \(f\in A(\mathbb R^n)\),
\[
\|f(\lambda\cdot)|A(\mathbb R^n)\|=\lambda^\sigma\|f|A(\mathbb R^n)\|.
\]
This book consists of three chapters.

Chapter 1 is devoted to the motivation why the author chooses such a way to deal with these homogeneous function spaces, and also to preliminaries on heat kernels, types of norms and homogeneity.

In Chapter 2, the author collects definitions and properties of the homogeneous spaces \(\dot A_{p,q}^{s}(\mathbb R^n)\), where \(A\in\{B,F\}\), in Sections 2.1 through 2.5, which are complemented in Section 2.6 by sketchy proposals to study further types of tempered homogeneous spaces (anisotropic spaces, hybrid spaces, spaces with dominating mixed smoothness, weighted spaces, radial spaces). Moreover, the ambiguities caused by polynomials in such homogeneous spaces are pointed out in this chapter.

The heart of this book is Chapter 3. In this chapter, a new approach to introduce homogeneous spaces \({\overset{\ast}{A^s_{p,q}}}(\mathbb R^n)\) within the framework of \((S(\mathbb R^n),S'(\mathbb R^n))\) is developed. Let \(s\in(-\infty,0)\) and \(p, q\in(0,\infty]\). Then the space \({\overset{\ast}{B^s_{p,q}}}(\mathbb R^n)\) is defined to be the set of all \(f\in\mathcal S'(\mathbb R^n)\) such that \[ \left\|f|{\overset{\ast}{B^s_{p,q}}}(\mathbb R^n)\right\| :=\left\{\int_0^\infty t^{-sq/2}\|W_tf|L_p(\mathbb R^n)\|^q\,\frac{dt}t\right\}^{1/q} \] is finite (usual modification if \(q=\infty\)) and the space \({\overset{\ast}{F^s_{p,q}}}(\mathbb R^n)\) (with \(p\in(0,\infty)\)) is defined to be the set of all \(f\in\mathcal S'(\mathbb R^n)\) such that \[ \left\|f|{\overset{\ast}{F^s_{p,q}}}(\mathbb R^n)\right\| :=\left\|\left\{\int_0^\infty t^{-sq/2}|W_tf(\cdot)|^q\, \frac{dt}t\right\}^{1/q}|L_p(\mathbb R^n)\right\| \] is finite (usual modification if \(q=\infty\)), where, for all \(h\in\mathcal S'(\mathbb R^n)\), \(t\in(0,\infty)\) and \(x\in \mathbb R^n\), \(W_th(x):=\frac1{(4\pi t)^{n/2}}(h,e^{-\frac{|x-\cdot|^2}{4t}})\). The definitions of these two spaces with positive smoothness are given in Definition 3.9. Theorems 3.3, 3.5, 3.11, 3.20 and 3.24 can be considered as the basic and interesting assertions of this book, in which the Fatou properties, homogeneity, equivalent domestic or admissible quasi-norms, embedding properties, dense subspaces are established. One of the important advantages of such a new approach is that there is no longer a need for concern over ambiguities caused by polynomials.

Altogether, this book is meticulously presented and makes a careful distinction between several types of norms including admissible, regional, domestic and community norms. Moreover, it provides a new way to avoid the usual ambiguities modulo polynomials when homogeneous function spaces are considered in the context of homogeneous tempered distributions, which may be useful for dealing with (nonlinear) heat and Navier-Stokes equations in homogeneous function spaces. This is surely a useful book for those who are interested in function spaces and their applications in partial differential equations such as the heat and Navier-Stokes equations.

Chapter 1 is devoted to the motivation why the author chooses such a way to deal with these homogeneous function spaces, and also to preliminaries on heat kernels, types of norms and homogeneity.

In Chapter 2, the author collects definitions and properties of the homogeneous spaces \(\dot A_{p,q}^{s}(\mathbb R^n)\), where \(A\in\{B,F\}\), in Sections 2.1 through 2.5, which are complemented in Section 2.6 by sketchy proposals to study further types of tempered homogeneous spaces (anisotropic spaces, hybrid spaces, spaces with dominating mixed smoothness, weighted spaces, radial spaces). Moreover, the ambiguities caused by polynomials in such homogeneous spaces are pointed out in this chapter.

The heart of this book is Chapter 3. In this chapter, a new approach to introduce homogeneous spaces \({\overset{\ast}{A^s_{p,q}}}(\mathbb R^n)\) within the framework of \((S(\mathbb R^n),S'(\mathbb R^n))\) is developed. Let \(s\in(-\infty,0)\) and \(p, q\in(0,\infty]\). Then the space \({\overset{\ast}{B^s_{p,q}}}(\mathbb R^n)\) is defined to be the set of all \(f\in\mathcal S'(\mathbb R^n)\) such that \[ \left\|f|{\overset{\ast}{B^s_{p,q}}}(\mathbb R^n)\right\| :=\left\{\int_0^\infty t^{-sq/2}\|W_tf|L_p(\mathbb R^n)\|^q\,\frac{dt}t\right\}^{1/q} \] is finite (usual modification if \(q=\infty\)) and the space \({\overset{\ast}{F^s_{p,q}}}(\mathbb R^n)\) (with \(p\in(0,\infty)\)) is defined to be the set of all \(f\in\mathcal S'(\mathbb R^n)\) such that \[ \left\|f|{\overset{\ast}{F^s_{p,q}}}(\mathbb R^n)\right\| :=\left\|\left\{\int_0^\infty t^{-sq/2}|W_tf(\cdot)|^q\, \frac{dt}t\right\}^{1/q}|L_p(\mathbb R^n)\right\| \] is finite (usual modification if \(q=\infty\)), where, for all \(h\in\mathcal S'(\mathbb R^n)\), \(t\in(0,\infty)\) and \(x\in \mathbb R^n\), \(W_th(x):=\frac1{(4\pi t)^{n/2}}(h,e^{-\frac{|x-\cdot|^2}{4t}})\). The definitions of these two spaces with positive smoothness are given in Definition 3.9. Theorems 3.3, 3.5, 3.11, 3.20 and 3.24 can be considered as the basic and interesting assertions of this book, in which the Fatou properties, homogeneity, equivalent domestic or admissible quasi-norms, embedding properties, dense subspaces are established. One of the important advantages of such a new approach is that there is no longer a need for concern over ambiguities caused by polynomials.

Altogether, this book is meticulously presented and makes a careful distinction between several types of norms including admissible, regional, domestic and community norms. Moreover, it provides a new way to avoid the usual ambiguities modulo polynomials when homogeneous function spaces are considered in the context of homogeneous tempered distributions, which may be useful for dealing with (nonlinear) heat and Navier-Stokes equations in homogeneous function spaces. This is surely a useful book for those who are interested in function spaces and their applications in partial differential equations such as the heat and Navier-Stokes equations.

Reviewer: Dachun Yang (Beijing)

### MSC:

46-02 | Research exposition (monographs, survey articles) pertaining to functional analysis |

46E35 | Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems |

42B35 | Function spaces arising in harmonic analysis |

42C40 | Nontrigonometric harmonic analysis involving wavelets and other special systems |