Theory of multipliers in spaces of differentiable functions.

*(English)*Zbl 0645.46031
Monographs and Studies in Mathematics, 23. Pitman Advanced Publishing Program. Boston - London - Melbourne: Pitman Publishing Inc. XIII, 344 p. (1985).

The subject of this book is the theory of multipliers in certain classes of differentiable functions which often occur in analysis. It is intended for students and researchers who are interested in function spaces and their relation to partial differential equations and operator theory. We discuss the description of multipliers, their properties and applications. The theory of Fourier multipliers is not dealt with in this book. A knowledge of basic Sobolev space theory is assumed.

The monograph contains seven chapters, of which the first three concern multipliers in pairs of integer and fractional Sobolev spaces, Bessel potential spaces etc. Here we present conditions for a function to belong to different classes of multipliers. Topics related to the spaces of multipliers, which are also treated in Chapters 1-3, include imbedding and composite function theorems and the spectral properties of multipliers. The applications considered include the calculus of singular integral operators whose symbols are multipliers, as well as coercive estimates for solutions of elliptic boundary value problems in spaces of multipliers. In Chapter 4 we study the essential norm of a multiplier. The trace and extension theorems for multipliers in Sobolev spaces are proved in Chapter 5. The next chapter deals with multipliers in a domain. In particular, we discuss the change of variables in norms of Sobolev spaces and present some implicit function theorems for multipliers. Chapter 7 presents applications of these results to \(L_ p\) theory of elliptic boundary value problems in domains with non-smooth boundaries.

The monograph contains seven chapters, of which the first three concern multipliers in pairs of integer and fractional Sobolev spaces, Bessel potential spaces etc. Here we present conditions for a function to belong to different classes of multipliers. Topics related to the spaces of multipliers, which are also treated in Chapters 1-3, include imbedding and composite function theorems and the spectral properties of multipliers. The applications considered include the calculus of singular integral operators whose symbols are multipliers, as well as coercive estimates for solutions of elliptic boundary value problems in spaces of multipliers. In Chapter 4 we study the essential norm of a multiplier. The trace and extension theorems for multipliers in Sobolev spaces are proved in Chapter 5. The next chapter deals with multipliers in a domain. In particular, we discuss the change of variables in norms of Sobolev spaces and present some implicit function theorems for multipliers. Chapter 7 presents applications of these results to \(L_ p\) theory of elliptic boundary value problems in domains with non-smooth boundaries.

##### MSC:

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 |

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

35J25 | Boundary value problems for second-order elliptic equations |

35J40 | Boundary value problems for higher-order elliptic equations |

47Gxx | Integral, integro-differential, and pseudodifferential operators |

47F05 | General theory of partial differential operators |

45E10 | Integral equations of the convolution type (Abel, Picard, Toeplitz and Wiener-Hopf type) |

35B45 | A priori estimates in context of PDEs |

31B15 | Potentials and capacities, extremal length and related notions in higher dimensions |