##
**Interpolation of operators.**
*(English)*
Zbl 0647.46057

From the preface: “This is a book about the real method of interpolation. Our goal has been to motivate and develop the entire theory from its classical origin, that is, through the theory of spaces of measurable functions although the influence of Riesz, Thorin, and Marcinkiewicz is everywhere evident, the work of G. H. Hardy, J. E. Littlewood, and G. Pólya on rearrangements of functions also play a seminal role. It is through the Hardy-Littlewood-Pólya relation that spaces of measurable functions and interpolation of operators come together, in a simple blend which has the capacity for great generalization. Interpolation between \(L^ 1\) and \(L^{\infty}\) is thus prototype for interpolation between more general pairs of Banach spaces. This theme airs constantly throughout the book.

The theory and applications of interpolation are as diverse as language itself. Our goal is not a dictionary, or an encyclopedia, but instead a brief biography of interpolation, with a beginning and end, and (like interpolation itself) some substance in between.”

Chapter 1 is devoted to Banach function spaces. Such spaces are defined with the help of Banach function norms. Let \({\mathcal M}^+\) be the set of measurable function on a measure space (R,\(\mu)\) that take values in \([0,+\infty]\). A mapping \(\rho: {\mathcal M}^+\to [0,+\infty]\) is called a Banach function norm if the following properties hold:

(P1) \(\rho(f)=0 \Leftrightarrow\) \(f=0\) a.e.; \(\rho(af)=a\rho(f)\), \(a\geq 0\); \(\rho(f+g)\leq \rho(f)+\rho(g);\)

(P2) \(0\leq g\leq f\) a.e. \(\Rightarrow\rho\) (g)\(\leq \rho (f);\)

(P3) \(0\leq f_ n\uparrow f\) a.e. \(\Rightarrow\rho (f_ n)\uparrow \rho (f);\)

(P4) \(\mu(E)<\infty\Rightarrow\rho (\chi_ E)<\infty;\)

(P5) \(\mu(E)<\infty\Rightarrow\int_{E}f d\mu \leq const\cdot \rho (f).\)

If \(\rho\) is a function norm, then the set \(X(\rho)\) of measurable functions such that \(\rho(| f|)<\infty\) is called a Banach function space, it is equipped with the norm \(\| f\|_ X=\rho (| f|)\). Basic properties of Banach function spaces are studied, the concept of the associate space is introduced and relationship between the associate space and the dual space are studied.

Chapter 2 deals with rearrangement-invariant Banach function spaces. Given a function f on a measure space, the decreasing rearrangement \(f^*\) is introduced as a function on \([0,\infty)\) that is equimeasurable with \(| f|\). A Banach function norm \(\rho\) is called rearrangement-invariant if \(\rho(f)=\rho(g)\) for any equimeasurable f and g in \({\mathcal M}^+\). In this case the Banach space \(X(\rho)\) is called a rearrangement-invariant space. The Luxemburg representation theorem is proved which claims that each rearrangement- invariant Banach space X(\(\rho)\) on a measure space is generated by a rearrangement invariant function norm \({\bar\rho}\) on \({\mathbb{R}}^+\) (with Lebesgue measure), i.e. \(\rho (f)={\bar \rho}(f^*).\) Given a rearrangement-invariant space X, the fundamental function \(\rho_ X\) is defined by \[ \phi_ X(t)=\| \chi_ E\|_ X,\quad t>0,\quad \mu (E)=t. \] With a rearrangement-invariant space X two Lorentz spaces \(\Lambda(X)\) and M(X) are associated that have the same fundamental function \(\phi_ X(t)\). The space \(\Lambda(X)\) is the minimal space among the rearrangement-invariant spaces whose fundamental function equals \(\phi_ X(t)\), while the space M(X) is maximal. The rearrangement-invariant spaces \(L^ 1+L^{\infty}\) and \(L^ 1\cap L^{\infty}\) play a special role since any rearrangement-invariant space is between them. In the last section of Chapter 2 a theorem of Ryff is established which asserts that given a non-negative measurable function f on a finite non-atomic measure space (R,\(\mu)\), there exists a measure- preserving transformation \(\sigma: R\to (0,\mu(R))\) such that \(f=f^*\circ \sigma\mu\)-a.e.

Chapter 3 concerns interpolation of operators on rearrangement-invariant spaces. In Section 2 a fundamental theorem is given which claims that a Banach function space is an interpolation space between \(L^ 1\) and \(L^{\infty}\) if and only if it is rearrangement-invariant. Two important operators are considered: the Hardy-Littlewood maximal operator M and the Hilbert transform H. The important concept of joint weak type \((p_ 0,q_ 0;p_ 1,q_ 1)\) is introduced for quasilinear operators. It is shown that both M and H are of joint weak type \((1,1;\infty,\infty)\). An important result of Calderón is presented that describes pairs of rearrangement-invariant Banach spaces X and Y such that every operator of joint weak type \((p_ 0,q_ 0;p_ 1,q_ 1)\) acts from X to Y. In the case \(p_ i=q_ i\) and \(X=Y\) Calderón’s theorem can be formulated in terms of the Boyd indices. To each rearrangement invariant space there correspond the so-called Boyd indices \({\underline\alpha}_ X\) and \({\bar\alpha}_ X\) that are defined in terms of the dilation operators on the corresponding rearrangement- invariant space on \({\mathbb{R}}^+\). Then Boyd’s theorem claims that in order that every linear (or quasilinear) operator of joint weak type (p,p;q,q) be bounded on a rearrangement-invariant space X it is necessary and sufficient that \(1/q<{\underline \alpha}_ X\leq {\bar\alpha}_ X<1/p\). The operators M and H play an essential role here. Namely, M is bounded on a rearrangement-invariant space X on \({\mathbb{R}}^ n\) if and only if \({\bar\alpha}_ X<1\) (Lorentz-Shimogaki theorem), while H is bounded on a rearrangement-invariant X space on \({\mathbb{R}}\) if and only if \(0<{\underline \alpha}_ X\leq {\bar\alpha}_ X<1\) (Boyd theorem).

Chapter 3 concludes with two results important in the sequel. The first one is a theorem of Lorentz-Shimogaki which claims that for nonnegative integrable functions f and g on a totally \(\sigma\)-finite measure space the following inequality holds: \[ f^*-g^* \prec f-g \] (the notation \(\phi\prec\psi\) means that \(\int^{t}_{0}\phi^*(s)ds \leq \int^{t}_{0}\psi^*(s)ds).\) The second one provides a splitting of measurable functions with respect to the Hardy-Littlewood-Pólya relation.

Chapter 4 concerns classical interpolation theorems. In Section 2 the Riesz-Thorin theorem is established. In Section 3 Stein’s interpolation theorem for analytic families of operators and Stein’s interpolation theorem for spaces with different weights are established. In Section 4 a very important interpolation theorem due to Marcinkiewicz is presented. In Section 5 operators of restricted weak type (p,q) are studied, i.e. the operators that satisfy the weak type (p,q) inequality when applied to characteristic functions. It is proved that in the case \(p<\infty\) and \(q>1\) the operators of restricted weak type (p,q) are of weak type (p,q). An interpolation theorem due to Stein and Weiss for restricted weak type operators is established. Section 5 concludes with an important theorem of Stein about the weak type of the maximal operator associated with a sequence of translation invariant operators. Section 6 deals with the spaces L log L and \(L_{\exp}\). In the final section of Chapter 4 Orlicz spaces which constitute an important subclass of rearrangement-invariant spaces is considered.

Chapter 5 is devoted to the K-method of interpolation. The notions of K- functional and J-functional are introduced. Given a compatible pair of Banach spaces \((X_ 0,X_ 1)\) the notion of Gagliardo completion of \(X_ 0\) and \(X_ 1\) is introduced. The interpolation spaces \((X_ 0,X_ 1)_{\vartheta,q}\) are defined. Interpolation spaces of a more general form are defined with the help of the so-called Riesz-Fischer spaces. These are spaces defined by a rearrangement-invariant functional \({\bar \rho}\) that satisfies properties (P1), (P2), (P4), (P5) of Chapter 1 and such that \(\rho\) (\(\sum_{n}f_ n)\leq \sum_{n}{\bar \rho}(f_ n)\) for all nonnegative sequences \(\{f_ n\}\). A Riesz-Fischer space is called monotone if \[ g\prec f,\quad f\in X\quad \Rightarrow \quad g\in X,\quad \| g\|_ X\leq \| f\|_ X. \] To each monotone Riesz-Fischer norm \(\rho\) there corresponds the interpolation space \((X_ 0,X_ 1)_{\rho}\). These spaces include the spaces \((X_ 0,X_ 1)_{\vartheta,q}.\)

The K-functional for the pair \((L^ 1,L^{\infty})\) is computed explicitly: \(K(f,t,L^ 1,L^{\infty})= \int^{t}_{0}f^*(s)ds.\) It is proved that the exactinterpolation spaces between \(L^ 1\) and \(L^{\infty}\) (with respect to a resonant measure space, i.e. either non-atomic or completely atomic with atoms having equal masses) are precisely the monotone Riesz-Fischer spaces.

In Section 2 the reiteration theorem for the K-method is proved. The section concludes with an important theorem of T. Wolff that allows a patching together of two interpolation scales.

Section 3 considers the question of which interpolation spaces for a given couple of Banach spaces are generated by the K-method. The crucial here is the so-called monotonicity property.

In Section 4 Sobolev and Besov spaces are defined, a description of the K-functional for the pair \((L^ p,W^ p_ r)\) is obtaind and the K- interpolation spaces are identified with Besov spaces. Interpolation spaces for certain pairs of Besov and Sobolev spaces are also described.

In Section 5 the K-functional for the pair \((W^ 1_ k,W^{\infty}_ k)\) is evaluated and the interpolation spaces \((W^ 1_ k,W^{\infty}_ k)_{\vartheta,q}\) are described. The main technical tool here is the Whitney covering lemma.

Section 6 is devoted to the study of the Hardy space \(H^ 1.\) Several characterizations of the space Re\(H^ 1\) are obtained and the space BMO of functions having bounded mean oscillation is identified with the dual of Re H 1.

The space \(BMO(Q_ 0)\) of functions on a fixed cube \(Q_ 0\) in \({\mathbb{R}}^ n \)is studied in Section 7. Given a function f on \(Q_ 0\), the sharp function \(f^{\#}_{Q_ 0}(x)\) is introduced which measures oscillation at \(x\in Q_ 0\). The space \(BMO(Q_ 0)\) is not rearrangement-invariant but its rearrangement-invariant hull admits a nice characterization. Let (R,\(\mu)\) be a totally \(\sigma\)-finite measure space. A measurable function f on R is said to belong to \(W=W(R)\) if \(\sup_{t>0}(f^{**}(t)-f\) \(*(t))<\infty\), where \(f^{**}(t)=^{def}(1/t)\int^{t}_{0}f\) *(s)ds, \(t>0\). The set W is not linear. It turns out that the rearrangement-invariant hull of \(BMO(Q_ 0)\) coincides with \(W(Q_ 0)\). It is shown that the space W can be considered as a substitute for weak \(L^{\infty}.\)

In Section 8 the K-functional for the pair \((L^ 1,BMO)\) is evaluated in terms of the sharp function \(f^{\#}\) which permits one to describe the (\(\vartheta\),q)-interpolation spaces as the Lorentz \(L^{p,q}\)-spaces.

In Section 9 P. Jones’ solution of the equation \({\bar \partial}f=\mu\) is given for Carleson measures \(\mu\). This solution is used in Section 10 to describe the K-functional for the pair \((H^ 1,H^{\infty})\) in terms of the nontangential maximal function. It is proved that an intermediate space Y is interpolation between \(H^ 1\) and \(H^{\infty}\) if and only if Y is equivalent to the Hardy space H(X) for some monotone Riesz- Fischer space X.

The theory and applications of interpolation are as diverse as language itself. Our goal is not a dictionary, or an encyclopedia, but instead a brief biography of interpolation, with a beginning and end, and (like interpolation itself) some substance in between.”

Chapter 1 is devoted to Banach function spaces. Such spaces are defined with the help of Banach function norms. Let \({\mathcal M}^+\) be the set of measurable function on a measure space (R,\(\mu)\) that take values in \([0,+\infty]\). A mapping \(\rho: {\mathcal M}^+\to [0,+\infty]\) is called a Banach function norm if the following properties hold:

(P1) \(\rho(f)=0 \Leftrightarrow\) \(f=0\) a.e.; \(\rho(af)=a\rho(f)\), \(a\geq 0\); \(\rho(f+g)\leq \rho(f)+\rho(g);\)

(P2) \(0\leq g\leq f\) a.e. \(\Rightarrow\rho\) (g)\(\leq \rho (f);\)

(P3) \(0\leq f_ n\uparrow f\) a.e. \(\Rightarrow\rho (f_ n)\uparrow \rho (f);\)

(P4) \(\mu(E)<\infty\Rightarrow\rho (\chi_ E)<\infty;\)

(P5) \(\mu(E)<\infty\Rightarrow\int_{E}f d\mu \leq const\cdot \rho (f).\)

If \(\rho\) is a function norm, then the set \(X(\rho)\) of measurable functions such that \(\rho(| f|)<\infty\) is called a Banach function space, it is equipped with the norm \(\| f\|_ X=\rho (| f|)\). Basic properties of Banach function spaces are studied, the concept of the associate space is introduced and relationship between the associate space and the dual space are studied.

Chapter 2 deals with rearrangement-invariant Banach function spaces. Given a function f on a measure space, the decreasing rearrangement \(f^*\) is introduced as a function on \([0,\infty)\) that is equimeasurable with \(| f|\). A Banach function norm \(\rho\) is called rearrangement-invariant if \(\rho(f)=\rho(g)\) for any equimeasurable f and g in \({\mathcal M}^+\). In this case the Banach space \(X(\rho)\) is called a rearrangement-invariant space. The Luxemburg representation theorem is proved which claims that each rearrangement- invariant Banach space X(\(\rho)\) on a measure space is generated by a rearrangement invariant function norm \({\bar\rho}\) on \({\mathbb{R}}^+\) (with Lebesgue measure), i.e. \(\rho (f)={\bar \rho}(f^*).\) Given a rearrangement-invariant space X, the fundamental function \(\rho_ X\) is defined by \[ \phi_ X(t)=\| \chi_ E\|_ X,\quad t>0,\quad \mu (E)=t. \] With a rearrangement-invariant space X two Lorentz spaces \(\Lambda(X)\) and M(X) are associated that have the same fundamental function \(\phi_ X(t)\). The space \(\Lambda(X)\) is the minimal space among the rearrangement-invariant spaces whose fundamental function equals \(\phi_ X(t)\), while the space M(X) is maximal. The rearrangement-invariant spaces \(L^ 1+L^{\infty}\) and \(L^ 1\cap L^{\infty}\) play a special role since any rearrangement-invariant space is between them. In the last section of Chapter 2 a theorem of Ryff is established which asserts that given a non-negative measurable function f on a finite non-atomic measure space (R,\(\mu)\), there exists a measure- preserving transformation \(\sigma: R\to (0,\mu(R))\) such that \(f=f^*\circ \sigma\mu\)-a.e.

Chapter 3 concerns interpolation of operators on rearrangement-invariant spaces. In Section 2 a fundamental theorem is given which claims that a Banach function space is an interpolation space between \(L^ 1\) and \(L^{\infty}\) if and only if it is rearrangement-invariant. Two important operators are considered: the Hardy-Littlewood maximal operator M and the Hilbert transform H. The important concept of joint weak type \((p_ 0,q_ 0;p_ 1,q_ 1)\) is introduced for quasilinear operators. It is shown that both M and H are of joint weak type \((1,1;\infty,\infty)\). An important result of Calderón is presented that describes pairs of rearrangement-invariant Banach spaces X and Y such that every operator of joint weak type \((p_ 0,q_ 0;p_ 1,q_ 1)\) acts from X to Y. In the case \(p_ i=q_ i\) and \(X=Y\) Calderón’s theorem can be formulated in terms of the Boyd indices. To each rearrangement invariant space there correspond the so-called Boyd indices \({\underline\alpha}_ X\) and \({\bar\alpha}_ X\) that are defined in terms of the dilation operators on the corresponding rearrangement- invariant space on \({\mathbb{R}}^+\). Then Boyd’s theorem claims that in order that every linear (or quasilinear) operator of joint weak type (p,p;q,q) be bounded on a rearrangement-invariant space X it is necessary and sufficient that \(1/q<{\underline \alpha}_ X\leq {\bar\alpha}_ X<1/p\). The operators M and H play an essential role here. Namely, M is bounded on a rearrangement-invariant space X on \({\mathbb{R}}^ n\) if and only if \({\bar\alpha}_ X<1\) (Lorentz-Shimogaki theorem), while H is bounded on a rearrangement-invariant X space on \({\mathbb{R}}\) if and only if \(0<{\underline \alpha}_ X\leq {\bar\alpha}_ X<1\) (Boyd theorem).

Chapter 3 concludes with two results important in the sequel. The first one is a theorem of Lorentz-Shimogaki which claims that for nonnegative integrable functions f and g on a totally \(\sigma\)-finite measure space the following inequality holds: \[ f^*-g^* \prec f-g \] (the notation \(\phi\prec\psi\) means that \(\int^{t}_{0}\phi^*(s)ds \leq \int^{t}_{0}\psi^*(s)ds).\) The second one provides a splitting of measurable functions with respect to the Hardy-Littlewood-Pólya relation.

Chapter 4 concerns classical interpolation theorems. In Section 2 the Riesz-Thorin theorem is established. In Section 3 Stein’s interpolation theorem for analytic families of operators and Stein’s interpolation theorem for spaces with different weights are established. In Section 4 a very important interpolation theorem due to Marcinkiewicz is presented. In Section 5 operators of restricted weak type (p,q) are studied, i.e. the operators that satisfy the weak type (p,q) inequality when applied to characteristic functions. It is proved that in the case \(p<\infty\) and \(q>1\) the operators of restricted weak type (p,q) are of weak type (p,q). An interpolation theorem due to Stein and Weiss for restricted weak type operators is established. Section 5 concludes with an important theorem of Stein about the weak type of the maximal operator associated with a sequence of translation invariant operators. Section 6 deals with the spaces L log L and \(L_{\exp}\). In the final section of Chapter 4 Orlicz spaces which constitute an important subclass of rearrangement-invariant spaces is considered.

Chapter 5 is devoted to the K-method of interpolation. The notions of K- functional and J-functional are introduced. Given a compatible pair of Banach spaces \((X_ 0,X_ 1)\) the notion of Gagliardo completion of \(X_ 0\) and \(X_ 1\) is introduced. The interpolation spaces \((X_ 0,X_ 1)_{\vartheta,q}\) are defined. Interpolation spaces of a more general form are defined with the help of the so-called Riesz-Fischer spaces. These are spaces defined by a rearrangement-invariant functional \({\bar \rho}\) that satisfies properties (P1), (P2), (P4), (P5) of Chapter 1 and such that \(\rho\) (\(\sum_{n}f_ n)\leq \sum_{n}{\bar \rho}(f_ n)\) for all nonnegative sequences \(\{f_ n\}\). A Riesz-Fischer space is called monotone if \[ g\prec f,\quad f\in X\quad \Rightarrow \quad g\in X,\quad \| g\|_ X\leq \| f\|_ X. \] To each monotone Riesz-Fischer norm \(\rho\) there corresponds the interpolation space \((X_ 0,X_ 1)_{\rho}\). These spaces include the spaces \((X_ 0,X_ 1)_{\vartheta,q}.\)

The K-functional for the pair \((L^ 1,L^{\infty})\) is computed explicitly: \(K(f,t,L^ 1,L^{\infty})= \int^{t}_{0}f^*(s)ds.\) It is proved that the exactinterpolation spaces between \(L^ 1\) and \(L^{\infty}\) (with respect to a resonant measure space, i.e. either non-atomic or completely atomic with atoms having equal masses) are precisely the monotone Riesz-Fischer spaces.

In Section 2 the reiteration theorem for the K-method is proved. The section concludes with an important theorem of T. Wolff that allows a patching together of two interpolation scales.

Section 3 considers the question of which interpolation spaces for a given couple of Banach spaces are generated by the K-method. The crucial here is the so-called monotonicity property.

In Section 4 Sobolev and Besov spaces are defined, a description of the K-functional for the pair \((L^ p,W^ p_ r)\) is obtaind and the K- interpolation spaces are identified with Besov spaces. Interpolation spaces for certain pairs of Besov and Sobolev spaces are also described.

In Section 5 the K-functional for the pair \((W^ 1_ k,W^{\infty}_ k)\) is evaluated and the interpolation spaces \((W^ 1_ k,W^{\infty}_ k)_{\vartheta,q}\) are described. The main technical tool here is the Whitney covering lemma.

Section 6 is devoted to the study of the Hardy space \(H^ 1.\) Several characterizations of the space Re\(H^ 1\) are obtained and the space BMO of functions having bounded mean oscillation is identified with the dual of Re H 1.

The space \(BMO(Q_ 0)\) of functions on a fixed cube \(Q_ 0\) in \({\mathbb{R}}^ n \)is studied in Section 7. Given a function f on \(Q_ 0\), the sharp function \(f^{\#}_{Q_ 0}(x)\) is introduced which measures oscillation at \(x\in Q_ 0\). The space \(BMO(Q_ 0)\) is not rearrangement-invariant but its rearrangement-invariant hull admits a nice characterization. Let (R,\(\mu)\) be a totally \(\sigma\)-finite measure space. A measurable function f on R is said to belong to \(W=W(R)\) if \(\sup_{t>0}(f^{**}(t)-f\) \(*(t))<\infty\), where \(f^{**}(t)=^{def}(1/t)\int^{t}_{0}f\) *(s)ds, \(t>0\). The set W is not linear. It turns out that the rearrangement-invariant hull of \(BMO(Q_ 0)\) coincides with \(W(Q_ 0)\). It is shown that the space W can be considered as a substitute for weak \(L^{\infty}.\)

In Section 8 the K-functional for the pair \((L^ 1,BMO)\) is evaluated in terms of the sharp function \(f^{\#}\) which permits one to describe the (\(\vartheta\),q)-interpolation spaces as the Lorentz \(L^{p,q}\)-spaces.

In Section 9 P. Jones’ solution of the equation \({\bar \partial}f=\mu\) is given for Carleson measures \(\mu\). This solution is used in Section 10 to describe the K-functional for the pair \((H^ 1,H^{\infty})\) in terms of the nontangential maximal function. It is proved that an intermediate space Y is interpolation between \(H^ 1\) and \(H^{\infty}\) if and only if Y is equivalent to the Hardy space H(X) for some monotone Riesz- Fischer space X.

Reviewer: V.V.Peller

### MSC:

46M35 | Abstract interpolation of topological vector spaces |

46E30 | Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) |

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

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

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