*(English)*Zbl 1157.46001

This nice book may be considered as a continuation of the authors’ first monograph [“Theory of multipliers in spaces of differentiable functions” (Monographs and Studies in Mathematics 23; Boston–London–Melbourne: Pitman) (1985; Zbl 0645.46031)]. The main purpose of the present volume is to give a comprehensive exposition of the theory of pointwise multipliers acting in pairs of spaces of differentiable functions. The theory was essentially developed by the authors during the past thirty years and the present volume is mainly based on their results.

A multiplier acting from one function space ${S}_{1}$ into another ${S}_{2}$ is a function which defines a bounded linear mapping of ${S}_{1}$ into ${S}_{2}$ by pointwise multiplication. Thus, with any pair of spaces ${S}_{1}$ and ${S}_{2}$, there exists an associated third one, the space of multipliers $M({S}_{1}\to {S}_{2})$ endowed with the norm of the operator of multiplication. The spaces ${S}_{1}$ and ${S}_{2}$ can be any of Sobolev spaces, Bessel potential spaces, Besov spaces and the like.

Sobolev multipliers arise in various problems of analysis and theories of partial differential and integral equations. Coefficients of differential operators can be naturally considered as multipliers. The same is true for symbols of more general pseudodifferential operators. Multipliers also appear in the theory of differentiable mappings preserving Sobolev spaces. Solutions of boundary value problems can be sought in classes of multipliers. Because of their algebraic properties, multipliers are suitable objects for generalizations of the basic facts of calculus (theorems on implicit functions, traces and extensions, point mappings and their compositions, etc.). Moreover, some basic operators of harmonic analysis, such as the classical maximal and singular integral operators, act in certain classes of multipliers.

The calculus of Sobolev multipliers may provide an adequate language for future work in the theory of linear and nonlinear differential and pseudodifferential equations under minimal restrictions on the coefficients, domains, and other data.

The book consists of two parts. Part I includes nine chapters. In Chapter 1, “Trace Inequalities for Functions in Sobolev Spaces”, the authors characterize the best constant $C$ in the so-called trace inequality

with an arbitrary measure $\mu $ on the left-hand side. When the domain is not indicated in the notation of a space or a norm, then it is assumed to be ${\mathbb{R}}^{n}$. Another variant of (1) will be with ${W}_{p}^{m}$ replaced by ${w}_{p}^{m}$. Here, ${W}_{p}^{k}$ and ${w}_{p}^{k}$ are completions of the space ${C}_{0}^{\infty}$ with respect to the norms $\parallel {\nabla}_{k}{u\parallel}_{{L}_{p}}+{\parallel u\parallel}_{{L}_{p}}$ and $\parallel {\nabla}_{k}{u\parallel}_{{L}_{p}}$, ${\nabla}_{k}=\{{\partial}^{k}/\partial {x}_{1}^{{\alpha}_{1}}\cdots \partial {x}_{n}^{{\alpha}_{n}}\}$. Two-sided estimates for $C$ are given in different terms for $p\in [1,\phantom{\rule{0.166667em}{0ex}}\infty )$. The case of $p$ on the left-hand side of (1) replaced by $q\ne p$ is also considered in this chapter.

In Chapter 2, “Multipliers in Pairs of Sobolev Spaces”, the authors study multipliers acting in pairs of spaces ${W}_{p}^{k}$ and ${w}_{p}^{k}$, where $k$ is a nonnegative integer. The concepts introduced in this chapter prove to be prototypes for the subsequent study of multipliers in other pairs of spaces. Using the results of Chapter 1, the authors derive necessary and sufficient conditions for a function to belong to the space of multipliers $M({W}_{p}^{m}\to {W}_{p}^{l})$ and $M({w}_{p}^{m}\to {w}_{p}^{l})$, where $m\ge l\ge 0$ and $p\in [1,\phantom{\rule{0.166667em}{0ex}}\infty )$. The case of the half-space ${\mathbb{R}}_{+}^{n}$ is treated. The authors give conditions for the inclusion $\gamma \in M({W}_{p}^{m}\to {W}_{p}^{-k}),\phantom{\rule{4pt}{0ex}}k>0$, present a brief description of the space $M({W}_{p}^{m}\to {W}_{p}^{l})$ and establish certain properties of multipliers. They also give a description of multipliers preserving spaces of functions with bounded variation. As usual, the authors omit ${\mathbb{R}}^{n}$ in notations of spaces, norms and integrals.

In Chapter 3, “Multipliers in Pairs of Potential Spaces”, the authors study the space of multipliers $M({H}_{p}^{m}\to {H}_{p}^{l})$ and $M({h}_{p}^{m}\to {h}_{p}^{l}),\phantom{\rule{4pt}{0ex}}m\ge l\ge 0$, where ${H}_{p}^{s}$ and ${h}_{p}^{s}$ are the space of Bessel and Riesz potentials of order $s$ with densities in ${L}_{p}$. The authors first provide some information on Bessel linear and nonlinear potentials, on capacity and on imbedding theorems. A characterization of the spaces $M({H}_{p}^{m}\to {H}_{p}^{l})$ and $M({h}_{p}^{m}\to {h}_{p}^{l})$ is given in this chapter. Then, the authors obtain either necessary or sufficient conditions for a function to belong to $M({H}_{p}^{m}\to {H}_{p}^{l})$, formulated in terms of different classes of functions. Furthermore, certain properties of elements of $M({H}_{p}^{m}\to {H}_{p}^{l})$ are studied. In particular, the authors consider the imbedding of $M({H}_{p}^{m}\to {H}_{p}^{l})$ into $M({H}_{p}^{m-j}\to {H}_{p}^{l-j})$. Descriptions of the point, residual, and continuous spectra of multipliers in ${H}_{p}^{l}$ and ${H}_{{p}^{\text{'}}}^{-l}$ are also given. Finally, the authors establish a characterization of positive homogeneous elements of the spaces $M({H}_{p}^{m}\to {H}_{p}^{l})$ and $M({h}_{p}^{m}\to {h}_{p}^{l})$.

In Chapter 4, “The Space $M({B}_{p}^{m}\to {B}_{p}^{l})$ with $p>1$”, the authors give necessary and sufficient conditions for a function to be a multiplier acting from one Besov space ${B}_{p}^{m}\left({\mathbb{R}}^{n}\right)$ into another ${B}_{p}^{l}\left({\mathbb{R}}^{n}\right)$, where $0<l\le m$ and $p\in (1,\infty )$.

In Chapter 5, “The Space $M({B}_{1}^{m}\to {B}_{1}^{l})$”, a complete description of the space $M({B}_{1}^{m}\left({\mathbb{R}}^{n}\right)\to {B}_{1}^{l}\left({\mathbb{R}}^{n}\right))$, $0\le l\le m$, is given for integer and non-integer $l$. The authors also survey some results on multipliers in Besov, $BMO$, and related function spaces.

Let $A$ be a subset of a Banach function space. Then $A$ is called a multiplication algebra if for all $u$ and $v$ in $A$, their product $uv$ belongs to $A$ and there exists a non-negative constant $c$ such that $\parallel uv\parallel \le c\parallel u\parallel \parallel v\parallel $. In Chapter 6, “Maximal Algebras in Spaces of Multipliers”, the authors show that the maximal Banach algebra ${A}_{p}^{m,l}$, imbedded in the space of multipliers $M({W}_{p}^{m}\to {W}_{p}^{l})$ which map the Sobolev space ${W}_{p}^{m}$ to ${W}_{p}^{l}$ with non-integer $m$ and $l,m>l$ and $p\in [1,\infty )$, is isomorphic to $M({W}_{p}^{m}\to {W}_{p}^{l})\cap {L}_{\infty}$. Moreover, the authors prove that the maximal Banach algebra ${\mathcal{A}}_{p}^{m\phantom{\rule{0.166667em}{0ex}}l}$, imbedded in the space of multipliers acting between Bessel potential spaces $M({H}_{p}^{m}\to {H}_{p}^{l})$, is isomorphic to $M({H}_{p}^{m}\to {H}_{p}^{l})\cap {L}_{\infty}$. Finally, the imbeddings ${A}_{p}^{m,l}\subset {A}_{p}^{\mu ,\lambda}$ and ${\mathcal{A}}_{p}^{m,l}\subset {\mathcal{A}}_{p}^{\mu ,\lambda}$ are given.

For $\gamma \in M({W}_{p}^{m}\to {W}_{p}^{l})$, denote by ${\text{ess}\phantom{\rule{0.166667em}{0ex}}\parallel \gamma \parallel}_{M({W}_{p}^{m}\to {W}_{p}^{l})}$ the essential norm of the operator of multiplication by $\gamma $. Sharp two-sided estimates for ${\text{ess}\phantom{\rule{0.166667em}{0ex}}\parallel \gamma \parallel}_{M({W}_{p}^{m}\to {W}_{p}^{l})}$ with $m>l$ are given in Chapter 7, “Essential Norm and Compactness of Multipliers”. As a corollary, the authors obtain characterizations of the space $\stackrel{\circ}{M}({W}_{p}^{m}\to {W}_{p}^{l}),\phantom{\rule{4pt}{0ex}}m>l$, of compact multipliers.

In Chapter 8, “Traces and Extensions of Multipliers”, let ${\mathbb{R}}_{+}^{n}$ denote the upper half-space $\{z=(x,\phantom{\rule{0.166667em}{0ex}}y):x\in {\mathbb{R}}^{n-1},\phantom{\rule{4pt}{0ex}}y>0\}$, $m\ge l>0,\phantom{\rule{4pt}{0ex}}s=\left[l\right]+1,\phantom{\rule{4pt}{0ex}}p\in (1,\infty )$, $\alpha =1-\left\{l\right\}-1/p$, $\beta =1-\left\{m\right\}-1/p$ and $t=\left[m\right]+1$, where for any $\alpha \in \mathbb{R}$, $\left[\alpha \right]$ and $\left\{\alpha \right\}$, respectively, denote the integer and fractional parts. The authors prove that the multiplier space $M({W}_{p}^{m}\left({\mathbb{R}}^{n-1}\right)\to {W}_{p}^{l}\left({\mathbb{R}}^{n-1}\right))$ is the space of traces on ${\mathbb{R}}^{n-1}$ of functions in $M({W}_{p}^{t,\beta}\left({\mathbb{R}}_{+}^{n}\right)\to {W}_{p}^{s,\alpha}\left({\mathbb{R}}_{+}^{n}\right))$. The traces of multipliers on the smooth boundary of a domain are also discussed. Finally, the authors obtain three trace and extension theorems for multipliers preserving a certain Sobolev-type space.

In Chapter 9, “Sobolev Multipliers in a Domain, Multiplier Mappings and Manifolds”, the authors deal with multipliers in pairs of Sobolev spaces in a domain. Let $G=\{(x,y):x\in {\mathbb{R}}^{n},\phantom{\rule{4pt}{0ex}}y>\phi \left(x\right)\}$ be a special Lipschitz domain, where $\phi $ is a function satisfying a Lipschitz condition. The authors find necessary and sufficient conditions for a function to belong to the space $M({W}_{p}^{m}\left(G\right)\to {W}_{p}^{l}\left(G\right))$, where $m$ and $l$ are integers with $0\le l\le m$. Then the authors show that the Stein extension operator maps $M({W}_{p}^{m}\left(G\right)\to {W}_{p}^{l}\left(G\right))$ continuously into $M({W}_{p}^{m}\left({\mathbb{R}}^{n}\right)\to {W}_{p}^{l}\left({\mathbb{R}}^{n}\right))$. Analogous results for the space $M({W}_{p}^{m}\left({\Omega}\right)\to {W}_{p}^{l}\left({\Omega}\right))$, where ${\Omega}$ is a bounded domain with boundary in the Lipschitz class ${C}^{0,1}$, are also obtained. A description of the space $M{L}_{p}^{1}\left({\Omega}\right)$ is given, where ${L}_{p}^{1}\left({\Omega}\right)=\{u\in {L}_{p,\text{loc}}\left({\Omega}\right):\nabla u\in {L}_{p}\left({\Omega}\right)\}$ and ${\Omega}$ is an arbitrary domain. Further, the authors introduce classes of mappings ($(p,l)$-diffeomorphisms) which preserve the space ${W}_{p}^{l}$, as well as classes of nonsmooth manifolds on which the space ${W}_{p}^{l}$ is well defined. These definitions of mappings and manifolds involve spaces of multipliers. In conclusion, a change of variables ${T}_{p}^{m,l}$ acting in the pair of Sobolev spaces ${W}_{p}^{m}\left(V\right)\to {W}_{p}^{l}\left(U\right)$ is defined and investigated. Moreover, a modification of the classical implicit function theorem, which involves multipliers in its statement, is obtained. Finally, the authors give a description of the space $M({\stackrel{\circ}{W}}_{p}^{m}\left({\Omega}\right)\to {W}_{p}^{l}\left({\Omega}\right))$.

Part II of this volume comprises a further seven chapters. In Chapter 10, “Differential Operators in Pairs of Sobolev Spaces”, the authors give estimates for the norms of general differential operators performing a mapping between two Sobolev spaces, formulated in terms of their coefficients as multipliers. These estimates involve multiplier norms of the coefficients, and for some values of integrability and smoothness parameters they are two-sided. The authors also describe a class of differential operators for which their continuity in pairs of Sobolev spaces is equivalent to the inclusion of the coefficients into classes of multipliers without any additional conditions on the indices. Further, a counterexample showing that in general the inclusion of the coefficients into the natural classes of multipliers is not necessary for the continuity of differential operators. Moreover, an estimate for the essential norms of general differential operators is given. By the example of a Schrödinger operator in ${\mathbb{R}}^{n}$, the authors outline the role of the essential norm of a multiplier in the Fredholm theory of elliptic differential operators. Finally, this chapter deals with a characterization of pairs of differential operators with constant coefficients which obey the dominance property between ${L}_{2}$ and its weighted counterpart.

In Chapter 11, “Schrödinger Operator and $M({w}_{2}^{1}\to {w}_{2}^{-1})$”, a characterization is given for the class of measurable functions (or, more generally, real- or complex-valued distributions) $V$ such that the Schrödinger operator $H=-{\Delta}+V$ maps the energy space ${w}_{2}^{1}\left({\mathbb{R}}^{n}\right)$ to its dual ${w}_{2}^{-1}\left({\mathbb{R}}^{n}\right)$. Similar results are obtained for the inhomogeneous Sobolev space ${W}_{2}^{1}\left({\mathbb{R}}^{n}\right)$. In other words, this chapter finds a complete solution to the problem of the relative form-boundedness of the potential energy operator $V$ with respect to the Laplacian $-{\Delta}$, which is fundamental to quantum mechanics. Relative compactness criteria for the corresponding quadratic forms are established as well. Analogous boundedness and compactness criteria for Sobolev spaces on domains ${\Omega}\subset {\mathbb{R}}^{n}$ are obtained under mild restrictions on $\partial {\Omega}$.

In Chapter 12, “Relativistic Schrödinger Operator and $M({W}_{2}^{1/2}\to {W}_{2}^{-1/2})$”, the authors give necessary and sufficient conditions for the boundedness of the relativistic Schrödinger operator $\mathscr{H}=\sqrt{-{\Delta}}+Q$ from the Sobolev space ${W}_{2}^{1/2}\left({\mathbb{R}}^{n}\right)$ to its dual ${W}_{2}^{-1/2}\left({\mathbb{R}}^{n}\right)$, for an arbitrary real- or complex-valued potential $Q$ on ${\mathbb{R}}^{n}$. In other words, a complete characterization of the space $M({W}_{2}^{1/2}\left({\mathbb{R}}^{n}\right)\to {W}_{2}^{-1/2}\left({\mathbb{R}}^{n}\right))$ is obtained.

In Chapter 13, “Multipliers as Solutions to Elliptic Equations”, solutions of second-order linear and quasi-linear elliptic differential equations and systems are considered as multipliers in certain spaces of differentiable functions in a domain ${\Omega}$. On the one hand, this can be of interest for the theory of functions, since it leads to new characterizations of multipliers and, on the other hand, for the theory of partial differential equations, since it allows to obtain a priori information about the solutions in spaces different from the usual ones. In this chapter, moreover, the authors derive coercive estimates in multiplier spaces for solutions of linear elliptic systems in a half-space and the regularity of solutions to higher order semilinear elliptic equations.

The purpose of Chapter 14, “Regularity of the Boundary in ${L}_{p}$-Theory of Elliptic Boundary Value Problems”, is to give applications of the theory of multipliers. The authors consider an operator $\{P,\text{tr}\phantom{\rule{0.166667em}{0ex}}{P}_{1},\cdots ,\text{tr}\phantom{\rule{0.166667em}{0ex}}{P}_{h}\}$ of the general elliptic boundary value problem with smooth coefficients in a bounded domain ${\Omega}\subset {\mathbb{R}}^{n}$. We assume that $\text{ord}\phantom{\rule{0.166667em}{0ex}}P=2h\le l$ and $\text{ord}\phantom{\rule{0.166667em}{0ex}}{P}_{j}={k}_{j}<l$, where $l$ is an integer. The trace operator on the boundary $\partial {\Omega}$ is denoted by $\text{tr}$. It is well-known that the mapping

where $1<p<\infty $, is Fredholm, i.e., it has a finite index and a closed range, provided that the boundary is sufficiently smooth. In this chapter, the authors show that the mapping (2) is Fredholm in the case $p(l-1)\le n$, provided that the boundary $\partial {\Omega}$ belongs to the class ${M}_{p}^{l-1/p}\left(\delta \right)$, and in the case $p(l-1)>n$, provided that $\partial {\Omega}$ belongs to the class ${W}_{p}^{l-1/p}$. In this chapter, the authors also consider two variants of the first boundary value problem for a strongly elliptic operator $P$ in divergence form. One variant is to look for a solution $u\in {W}_{p}^{l}\left({\Omega}\right)$ of the equation

satisfying the condition

where $g$ is a given function in ${W}_{p}^{l}\left({\Omega}\right)$. The authors show that this problem has a unique solution if $\partial {\Omega}$ is in the class ${M}_{p}^{l+1-h-1/p}\left(\delta \right)$ for $p(l-h)\le n$ and $\partial {\Omega}$ belongs to the class ${W}_{p}^{l+1-h-1/p}$ for $p(l-h)>n$. Another variant is to consider a stronger formulation, which means that the boundary data are prescribed by means of some differential operators ${P}_{j}$, $1\le j\le h$. In this case, the authors prove that such a problem is solvable for $h>1$ if $\partial {\Omega}$ belongs to the class ${M}_{p}^{l-1/p}$. In the case $p(l-1)>n$, this condition is equivalent to $\partial {\Omega}\in {W}_{p}^{l-1/p}$. In this chapter, the authors also give an analytic description of the class ${M}_{p}^{l-1/p}\left(\delta \right)$ involving a capacity and obtain some simpler conditions for the inclusion of $\partial {\Omega}$ into ${M}_{p}^{l-1/p}\left(\delta \right)$.

In Chapter 15, “Multipliers in the Classical Layer Potential theory for Lipschitz Domains”, the authors give applications of Sobolev multipliers to the question of higher regularity in fractional Sobolev spaces of solutions to boundary integral equations generated by the classical boundary value problems for the Laplace equation in- and outside a Lipschitz domain. Since the mere Lipschitz graph property of $\partial {\Omega}$ does not guarantee higher regularity of solutions, the authors are forced to select an appropriate subclass of Lipschitz domains whose descriptions involve a space of multipliers. For domains of this subclass, the authors develop a solvability and regularity theory analogous to the classical one for smooth domains. The authors also show that the chosen subclass of Lipschitz domains proves to be the best possible in a certain sense. This chapter ends with a brief discussion of boundary integral equations of linear elastostatics.

In Chapter 16, “Applications of Multipliers to the Theory of Integral Operators”, it is shown that Sobolev multipliers are useful for the study of integral operators. First, the authors consider an arbitrary convolution operator acting in a pair of weighted ${L}_{2}$-spaces and collect corollaries of the theory of multipliers providing criteria of boundedness and compactness of the convolutions and a characterization of their spectra. Next, the authors turn to classical singular integral operators acting in Sobolev spaces. A calculus of these operators is developed under the assumptions that their symbols belong to classes of multipliers in Sobolev spaces. Finally, sharp conditions for continuity of the singular integral operators acting from ${W}_{2}^{m}$ to ${W}_{2}^{l}$ are found. These conditions are formulated in terms of certain classes of multipliers.

This book is well written and its content is very rich and well organized. It should provide an extremely useful reference to those mathematicians working in functional analysis and in the theories of partial differential, integral, and pseudodifferential operators. Prerequisites for reading this book are undergraduate courses in these subjects.

##### MSC:

46-02 | Research monographs (functional analysis) |

42B15 | Multipliers, several variables |

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

35-02 | Research monographs (partial differential equations) |