##
**Hardy-type inequalities.**
*(English)*
Zbl 0698.26007

Pitman Research Notes in Mathematics, 219. Harlow: Longman Scientific & Technical; New York: John Wiley & Sons, Inc. 333 p. £26.00 (1990).

Various kinds of inequalities involving integrals of functions and their derivatives or partial derivatives play a vital role in the development of many branches of mathematics, especially in the theory of differential and integral equations, integro-differential equations, functional differential equations, and in the theory of approximation theory and numerical analysis.

The monograph under review is concerned with the important Hardy-type integral inequalities. An integral inequality of the following type and its modifications are called to be of the Hardy-type \[ (1)\quad [\int_{\Omega}| u(x)|^ qw(x)dx]^{1/q}\leq C\{\sum^{N}_{i=1}\int_{\Omega}| \partial u(x)/\partial x_ i|^ pv_ i(x)dx\}^{1/p}, \] where \(\Omega\) is a domain in the N- dimensional Euclidean space; C, q, and \(p\geq 1\) are positive constants; w, \(v_ 1,...,v_ N\) are weight functions, i.e. measurable and positive a.e. in \(\Omega\). Many important inequalities in the literature such as the Friedrichs inequality, the Poincaré inequality, the weighted Friedrichs-Poincaré inequality, and the weighted Sobolev inequality can all be collected under this common name Hardy-type inequalities.

The main question studied in the book is, under different choices of a function class K, what conditions on \(\Omega\), p, q, w, \(v_ 1,...,v_ N\) ensure the validity of inequality (1) with a constant C independent of the function u from K? The case \(N=1\) is treated in Chapter 1 which occupies over one half of pages of this book and is more or less closed. Three approaches via ordinary differential equations, formulas connecting the weight functions, and via imbedding theorems are extensively used. The Chapters 2 and 3 are devoted respectively to the N-dimensional Hardy- type inequalities and the applications of continuous and compact imbedding theorems for weighted Sobolev spaces to multidimensional Hardy- type inequalities. The book is written in a well-arranged way with many illustrating examples.

The monograph under review is concerned with the important Hardy-type integral inequalities. An integral inequality of the following type and its modifications are called to be of the Hardy-type \[ (1)\quad [\int_{\Omega}| u(x)|^ qw(x)dx]^{1/q}\leq C\{\sum^{N}_{i=1}\int_{\Omega}| \partial u(x)/\partial x_ i|^ pv_ i(x)dx\}^{1/p}, \] where \(\Omega\) is a domain in the N- dimensional Euclidean space; C, q, and \(p\geq 1\) are positive constants; w, \(v_ 1,...,v_ N\) are weight functions, i.e. measurable and positive a.e. in \(\Omega\). Many important inequalities in the literature such as the Friedrichs inequality, the Poincaré inequality, the weighted Friedrichs-Poincaré inequality, and the weighted Sobolev inequality can all be collected under this common name Hardy-type inequalities.

The main question studied in the book is, under different choices of a function class K, what conditions on \(\Omega\), p, q, w, \(v_ 1,...,v_ N\) ensure the validity of inequality (1) with a constant C independent of the function u from K? The case \(N=1\) is treated in Chapter 1 which occupies over one half of pages of this book and is more or less closed. Three approaches via ordinary differential equations, formulas connecting the weight functions, and via imbedding theorems are extensively used. The Chapters 2 and 3 are devoted respectively to the N-dimensional Hardy- type inequalities and the applications of continuous and compact imbedding theorems for weighted Sobolev spaces to multidimensional Hardy- type inequalities. The book is written in a well-arranged way with many illustrating examples.

Reviewer: Yang En-Hao

### MSC:

26D10 | Inequalities involving derivatives and differential and integral operators |

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

26-02 | Research exposition (monographs, survey articles) pertaining to real functions |