Weighted Inequalities Involving Quasiconcave Operators

Some problems in mathematical analysis (e.g., in theory of function spaces, in approximation theory or in interpolation theory) lead to the investigation of weighted inequalities on certain classes of quasiconcave functions on the interval I=(a,b) ∊ R. In this book we analyse the class Qρ(I) of ρ-quasiconcave functions in a complete generality in order to establish results needed for a comprehensive study of weighted inequalities on the class Qρ(I). We illustrate our results on weighted inequalities of Hardy type, on weighted inequalities of Hardy type involving supremum, and on reverse forms of these inequalities.

Inequalities continue to play an essential role in mathematics. Perhaps, they form the last field comprehended and used by mathematicians in all areas of the discipline. Since the seminal work Inequalities (1934) by Hardy, Littlewood and Pólya, mathematicians have laboured to extend and sharpen their classical inequalities. New inequalities are discovered every year, some for their intrinsic interest whilst others flow from results obtained in various branches of mathematics. The study of inequalities reflects the many and various aspects of mathematics. On one hand, there is the systematic search for the basic principles and the study of inequalities for their own sake. On the other hand, the subject is the source of ingenious ideas and methods that give rise to seemingly elementary but nevertheless serious and challenging problems. There are numerous applications in a wide variety of fields, from mathematical physics to biology and economics. This volume contains the contributions of the participants of the Conference on Inequalities and Applications held in Noszvaj (Hungary) in September 2007. It is conceived in the spirit of the preceding volumes of the General Inequalities meetings held in Oberwolfach from 1976 to 1995 in the sense that it not only contains the latest results presented by the participants, but it is also a useful reference book for both lecturers and research workers. The contributions reflect the ramification of general inequalities into many areas of mathematics and also present a synthesis of results in both theory and practice.

This book is intended as a survey of latest results on weighted inequalities in Lorentz, Orlicz spaces and Zygmund classes. During the last few years they have become one of the mostdeveloped offshoots of the theory of the harmonic analysis operators. Up to now there has been no monograph devoted to these questions, the results are mostly scattered in various journals and a part of the book consists of results not published anywhere else. Many of theorems presented have only previously been published in Russian.

Inequalities play an important role in almost all branches of mathematics as well as in other areas of science and engineering. This book surveys the present state of the theory of weighted integral inequalities of Hardy type, including modifications concerning Hardy–Steklov operators, and some basic results about Hardy-type inequalities and their limit (Carleman–Knopp type) inequalities. It also describes some rather new areas such as higher order and fractional order Hardy-type inequalities and integral inequalities on the cone of monotone functions, together with some applications and open problems. In this second edition, all chapters in the first edition have been updated with new information. Moreover, a new chapter contains new and complementary information concerning: (a) a convexity approach to prove and explain Hardy-type inequalities; (b) sharp constants; (c) scales of inequalities to characterize Hardy-type inequalities; (d) Hardy-type inequalities in other function spaces; and (e) a number of new open questions. Request Inspection Copy

This book on Advance Elements of Laser circuits and systems Nonlinearity applications in engineering addresses two separate engineering and scientific areas, and presents advanced analysis methods for Laser circuits and systems that cover a broad range of engineering and scientific applications. The book analyzed Laser circuits and systems as linear and nonlinear dynamical systems and there limit cycles, bifurcation, and limit cycle stability by using nonlinear dynamic theory. Further, it discussed a broad range of bifurcations related to Laser systems and circuits, starting from laser system differential equations and their bifurcations, delay differential equations (DDEs) are a function of time delays, delay dependent parameters, followed by phase plane analysis, limit cycles and their bifurcations, chaos, iterated maps, period doubling. It combines graphical information with analytical analysis to effectively study the local stability of Laser systems models involving delay dependent parameters. Specifically, the stability of a given steady state is determined by the graphs of some functions of which can be expressed explicitly. The Laser circuits and systems are Laser diode circuits, MRI system Laser diode circuitry, Electron-photon exchanges into VCSEL, Ti: Sapphire laser systems, Ion channel and long-wavelength lasers, Solid state lasers, Solid state laser controlled by semiconductor devices, microchip solid-state laser, Q-switched diode-pumped solid-state laser, Nd:YAG, Mid-Infrared and Q-switched microchip lasers, Gas laser systems, copper vapor laser (CVL) circuitry, Dual-wavelength laser systems, Dual-wavelength operation of a Ti:sapphire laser, Diode-pumped Q-switched Nd:YVO4 yellow laser, Asymmetric dual quantum well lasers, Tm3+-doped silica fibre lasers, Terahertz dual-wavelength quantum cascade laser. The Book address also the additional areas, Laser X guiding system, Plasma diagnostics, Laser Beam shaping, Jitter and crosstalk, Plasma mirror systems, and High power Laser/Target diagnostic system optical elements. The book is unique in its emphasis on practical and innovative engineering and scientific applications. All conceptual Laser circuits are innovative and can be broadly implemented in many engineering applications. The dynamics of Laser circuits and systems provides several ways to use them in a variety of applications covering wide areas. This book is aimed at electrical and electronics engineers, students and researchers in physics as well. It is also aimed for research institutes in lasers and plasma physics and gives good comprehensive in laser and plasma systems. In each chapter, the concept is developed from basic assumptions up to the final engineering and scientific outcomes. The scientific background is explained at basic and advance levels and closely integrated with mathematical theory. Many examples are presented in this book and it is also ideal for intermediate level courses at graduate level studies. It is also ideal for engineer who has not had formal instruction in nonlinear dynamics, but who now desires to fill the gap between innovative Laser circuits/systems and advance mathematical analysis methods

This book is intended as a survey of latest results on weighted inequalities in Lorentz, Orlicz spaces and Zygmund classes. During the last few years they have become one of the mostdeveloped offshoots of the theory of the harmonic analysis operators. Up to now there has been no monograph devoted to these questions, the results are mostly scattered in various journals and a part of the book consists of results not published anywhere else. Many of theorems presented have only previously been published in Russian. Contents:Integral Operators in Nonweighted Orlicz ClassesMaximal Functions and Potentials in Weighted Orlicz ClassesSingular Integrals in Weighted Orlicz ClassesIntegral Operators in Weighted Zygmund ClassesFractional Maximal Function in Weighted Lorentz SpacesPotentials and Riesz Transforms in Weighted Lorentz Spaces Readership: Mathematicians, graduate students and researchers in real and complex analysis. keywords:Orlicz Space;Lorentz Space;Zygmund Space;Weighted Space;Ap Weight;Maximal Operator;Riesz Potential;Hilbert Transform;Singular Integral;Weighted Inequalities “The authors, together with various collaborators, have made important contributions to the field over the last decade … The exposition is clear with detailed proofs of all statements and the monograph will certainly be a good supplement to survey articles and books on the weighted inequalities.” Mathematical Reviews

The main objective of this work is to bring together two well known and, a priori, unrelated theories dealing with weighted inequalities for the Hardy-Littlewood maximal operator $M$. for this, the authors consider the boundedness of $M$ in the weighted Lorentz space $\Lambdap u(w)$. Two examples are historically relevant as a motivation: If $w=1$, this corresponds to the study of the boundedness of $M$ on $Lp(u)$, which was characterized by B. Muckenhoupt in 1972, and the solution is given by the so called $A p$ weights. The second case is when we take $u=1$. This is a more recent theory, and was completely solved by M.A. Arino and B. Muckenhoupt in 1991. It turns out that the boundedness of $M$ on $\Lambdap(w)$ can be seen to be equivalent to the boundedness of the Hardy operator $A$ restricted to decreasing functions of $Lp(w)$, since the nonincreasing rearrangement of $Mf$ is pointwise equivalent to $Af*$. The class of weights satisfying this boundedness is known as $B p$. Even though the $A p$ and $B p$ classes enjoy some similar features, they come from very different theories, and so are the techniques used on each case: Calderon-Zygmund decompositions and covering lemmas for $A p$, rearrangement invariant properties and positive integral operators for $B p$. This work aims to give a unified version of these two theories. Contrary to what one could expect, the solution is not given in terms of the limiting cases above considered (i.e., $u=1$ and $w=1$), but in a rather more complicated condition, which reflects the difficulty of estimating the distribution function of the Hardy-Littlewood maximal operator with respect to general measures.

This book presents the first comprehensive treatment of the blocking technique which consists in transforming norms in section form into norms in block form, and vice versa. Such norms appear throughout analysis. The blocking technique is a powerful, yet elementary, tool whose usefulnes is demonstrated in the book. In particular, it is shown to lead to the solution of three recent problems of Bennett concerning the inequalities of Hardy and Copson. The book is addressed to researchers and graduate students. An interesting feature is that it contains a dictionary of transformations between section and block norms and will thus be useful to researchers as a reference text. The book requires no knowledge beyond an introductory course in functional analysis.