Bounded and Compact Integral Operators

by Edmunds, D. E.; Kokilashvili, Vakhtang; Meskhi, Alexander

• ISBN13: 9781402006197
• ISBN10: 1402006195
• Format: Hardcover
• Copyright: 2002-05-01
• Publisher: Kluwer Academic Pub

Summary

The monograph presents some of the authors' recent and original results concerning boundedness and compactness problems in Banach function spaces both for classical operators and integral transforms defined, generally speaking, on nonhomogeneous spaces. It focuses on integral operators naturally arising in boundary value problems for PDE, the spectral theory of differential operators, continuum and quantum mechanics, stochastic processes, etc. The book may be considered as a systematic and detailed analysis of a large class of specific integral operators from the boundedness and compactness point of view. A characteristic feature of the monograph is that most of the statements proved here have the form of criteria. We provide a list of problems which were open at the time of completion of the book.Audience: The book is aimed at a rather wide audience, ranging from researchers in functional and harmonic analysis to experts in applied mathematics and prospective students.

Table of Contents

Preface	ix
Acknowledgments	xiii
Basic notation	xv
Hardy-Type Operators	1	(76)
Boundedness and compactness in BFS	1	(24)
An extension of the Hardy transform	25	(19)
Estimates for approximation numbers	44	(13)
Norms of positive operators	57	(18)
Notes and comments on Chapter 1	75	(2)
Fractional Integrals on the Line	77	(174)
Fractional integrals	77	(25)
Two-weight problems	102	(5)
Trace inequalities (``Diagonal case''). Examples	107	(28)
Weak-type inequalities Examples	135	(6)
Integral transforms with power- logarithmic kernels	141	(13)
Erdelyi-Kober operators	154	(6)
Integral transforms with positive kernels	160	(25)
Extended Erdelyi-Kober operators	185	(13)
Generalized one-sided potentials	198	(20)
One-sided potentials on the half-space	218	(12)
Weighted criteria in Lorentz spaces	230	(11)
Applications to Abel's integral equations	241	(3)
On some Volterra-type integral equations	244	(2)
Application to the existence of positive solutions of nonlinear integral equations	246	(2)
Notes and comments on Chapter 2	248	(3)
One-Sided Maximal Functions	251	(66)
One-sided maximal functions	251	(5)
Mapping properties of potentials on the line	256	(11)
Potentials Tγ on the line (the case 0 < γ < 1/p)	267	(8)
Weak-type inequalities	275	(4)
Potentials on bounded intervals	279	(2)
Two-weight criteria for fractional maximal functions	281	(2)
Generalized one-sided maximal functions	283	(5)
Potentials with power-logarithmic kernels	288	(8)
Multiple potentials	296	(14)
One-sided Hormander-type maximal functions	310	(5)
Notes and comments on Chapter 3	315	(2)
Ball Fractional Integrals	317	(26)
Boundedness criteria	317	(11)
Compactness criteria	328	(6)
The measure of non-compactness	334	(4)
Mapping properties in Lorentz spaces	338	(3)
Notes and comments on Chapter 4	341	(2)
Potentials on RN	343	(24)
Truncated potentials	343	(16)
Two-weight compactness conditions for Riesz potentials	359	(3)
Integral transforms with radial kernels	362	(3)
Notes and comments on Chapter 5	365	(2)
Fractional Integrals on Measure Spaces	367	(80)
Integral transforms on nonhomogeneous spaces	367	(9)
Theorems of Adams type	376	(4)
Truncated potentials on SHT	380	(9)
Riesz potentials in the half-space	389	(8)
Truncated potentials in the half-space	397	(15)
Two-weight (p, p) type inequalities	412	(16)
Theorems of Koosis type	428	(9)
Fractional maximal functions on SHT	437	(4)
Weighted estimates in Lorentz spaces	441	(5)
Notes and comments on Chapter 6	446	(1)
Singular Numbers	447	(54)
Volterra integral operators	447	(14)
Riemann-Liouville-type operators	461	(15)
Potential-type operators	476	(6)
Hardy-type operators	482	(6)
Asymptotic behaviour of singular and entropy numbers	488	(10)
Notes and comments on Chapter 7	498	(3)
Singular Integrals	501	(92)
Two-weight strong-type estimates	502	(20)
Weak-type estimates	522	(19)
Singular integrals on nonhomogeneous spaces	541	(11)
Two-weight inequaiities for the Hilbert transform	552	(5)
Estimates in Lorentz spaces	557	(21)
Cauchy-Szego projection	578	(6)
Singular integrals via Clifford analysis	584	(3)
Theorems of Koosis type	587	(5)
Notes and comments on Chapter 8	592	(1)
Multipliers of Fourier Transforms	593	(24)
Weighted Triebei-Lizorkin spaces	593	(5)
Two-weight multipliers in Triebel-Lizorkin spaces	598	(11)
(Fq,θ β,w, Fp,θ β,ν) multipliers. The case 1 < q ≤ p < ∞	609	(1)
Multipliers in weighted spaces with mixed norms	610	(3)
Examples	613	(2)
Notes and comments on Chapter 9	615	(2)
Problems	617	(5)
References	622	(18)
Index	640