The blocking technique, weighted mean operators and Hardy's inequality
In recent years great interest has centered on the generalization of classical inequalities, e.g., on Hardy's inequality. The so-called blocking method or technique to prove such type of inequalities is also a very frequent accurrence in recent papers. The author, an expert in both research fields, unites the methods and results of these topics and presents in his lecture notes ``a comprehensive treatment of the so-called blocking technique, together with applications to the study of sequence and function spaces, to the study of operators between such spaces, and to classical inequalities''. The author also emphasizes that his research ``originated from a study of four papers by G. Bennett that revolve around the inequalities of Hardy and Copson''. It is also G. Bennett who has introduced the new concept of factorization of inequalities. This new concept enables us to replace the classical inequalities by identities. These notes deal with problems revolving around classical inequalities and factorizations. The blocking technique enables the author to replace norms in section form by norms in block form and vice versa, but meanwhile to solve the problem in question in block form, what is usually much simpler. The only disadvantage of blocking technique is loosing the control over constants, it does not convey the best-possible constants. The notes are divided into four chapters and an appendix. Its references contain 101 items. Chapter headings are: The Blocking Technique, The Sequence Spaces \(c(a,p,q)\) and \(d(a,p,q)\), Applications to Matrix Operators and Inequalities, Integral Analogues, Equivalence of Section Conditions with Block Conditions, References, Index. In my opinion this work is worth for reading. It gives a good survey of the relevant results including several new ones. It shows certainly the great practial value of the blocking technique. The exposition is clear, the results are lucidly arranged, the hints to the literature are practically complete, the proofs are elementary but not easy at all, the remarks and examples are very useful to compare the new and previous results. I warmly recommend these lecture notes to experts interested in inequalities and its reading is useful for every analysts.
scientific article; zbMATH DE number 1110776