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Function Spaces and Partial Differential Equations: Volume 2 - Contemporary Analysis [Kietas viršelis]

(Reader in Mathematics, University of Sussex)
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Function Spaces and Partial Differential Equations: Volume 2 - Contemporary AnalysisDidesnis paveikslėlis
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Pastovi nuoroda: https://www.kriso.lt/db/9780198733157.html
Raktažodžiai:
This is a book written primarily for graduate students and early researchers in the fields of Analysis and Partial Differential Equations (PDEs). Coverage of the material is essentially self-contained, extensive and novel with great attention to details and rigour. The strength of the book primarily lies in its clear and detailed explanations, scope and coverage, highlighting and presenting deep and profound inter-connections between different related and seemingly unrelated disciplines within classical and modern mathematics and above all the extensive collection of examples, worked-out and hinted exercises. There are well over 700 exercises of varying level leading the reader from the basics to the most advanced levels and frontiers of research. The book can be used either for independent study or for a year-long graduate level course. In fact it has its origin in a year-long graduate course taught by the author in Oxford in 2004-5 and various parts of it in other institutions later on. A good number of distinguished researchers and faculty in mathematics worldwide have started their research career from the course that formed the basis for this book.

Recenzijos

The book is clearly written and is a valuable addition to the literature. It is warmly recommended to the graduate student. For the instructor, it is an excellent resource for graduate courses and seminars in analysis and partial differential equations. * Valentin Keyantuo, MathSciNet *

Contents of Volume 2
13 Maximal Function; Bounding Averages and Pointwise Convergence
501
13.1 A Covering Lemma of Vitali Type
501(1)
13.2 The Hardy-Littlewood Maximal Function
504(3)
13.3 Applications to Differentiability
507(2)
13.4 Approximation to Identity: Pointwise Convergence and Bounds
509(7)
13.5 Local L1-Integrability of M[ f] and Stein's L log L Theorem
516(2)
13.6 Lp-Boundedness of Riesz Potentials via Maximal Function
518(6)
13.7 Young's Convolution Inequality: Lrw(W) * LP(Rn) ⊂ Lq(Rn)
524(2)
13.8 The Maximal Operator T*; Pointwise Convergence of Operator Families (Tεf)
526(4)
13.9 Exercises and Further Results
530(19)
14 Harmonic-Hardy Spaces hp(H)
549(40)
14.1 The Poisson Kernel P(ξζH)
549(4)
14.2 Poisson Integrals in Lp(Rn) (1 ≤ p ≤∞) and M(Rn)
553(2)
14.3 Characterisation of Harmonic-Hardy Spaces hp(H)
555(1)
14.4 Non-Tangential Convergence to Boundary Values
556(3)
14.5 The Hardy-Littlewood Maximal Function on Spheres
559(5)
14.6 Mobius Maps; The Kelvin Transform K[ u]
564(3)
14.7 Functions Harmonic at Infinity
567(7)
14.8 Positive Harmonic Functions in Rn+
574(3)
14.9 Exercises and Further Results
577(12)
15 Sobolev Spaces Wk,p(ω); A Resolution of the Dirichlet Principle
589(56)
15.1 Calculus of Weak Derivatives
589(5)
15.2 Wk,p -Approximation by Smooth Functions
594(4)
15.3 Trace Theorem for W1,P(ω); The Zero Trace Space W0k,p(ω)
598(6)
15.4 Poincare Inequality; Equivalent Norms on W0k,p
604(3)
15.5 Gagliardo-Nirenberg-Sobolev Inequality
607(8)
15.6 Embedding Theorems for W0k,p and Wk,p
615(5)
15.7 Rellich-Kondrachov Compactness Theorem
620(3)
15.8 The Spectrum of-Λ and the Perron-Frobenius Theorem
623(4)
15.9 Exercises and Further Results
627(18)
16 Singular Integral Operators and Vector-Valued Inequalities
645(56)
16.1 The Hilbert Transform on LP(R); Riesz's Theorem by Complex Methods
646(5)
16.2 The Maximal Hilbert Transforms; Riesz's Theorem by Real Methods
651(6)
16.3 Singular Integrals of Calderon-Zygmund Type
657(3)
16.4 The Riesz Transforms Rj (1 ≤ j ≤ n) on LP(Rn) and Beyond
660(3)
16.5 Homogeneous Kernels: L2-Boundedness
663(5)
16.6 Homogeneous Kernels: LP -Theory (1 ≤p < ∞)
668(2)
16.7 The Calderon-Zygmund Method of Rotations
670(5)
16.8 Vector-Valued Inequalities; Vector-Valued Singular Integrals
675(3)
16.9 More on the Newtonian Potential N[ f; ω2]
678(8)
16.10 Exercises and Further Results
686(15)
17 Littlewood-Paley Theory, Lp -Multipliers and Function Spaces
701(50)
17.1 Littlewood-Paley Theory on the Line
701(5)
17.2 Littlewood-Paley Theory on the Euclidean n-Space: Part I
706(6)
17.3 Littlewood-Paley Theory on the Euclidean n-Space: Part II
712(4)
17.4 The Hormander-Mihlin Multiplier Theorem
716(2)
17.5 A Littlewood-Paley Characterisation of Hs (Rn) and More
718(3)
17.6 Applications to Strichartz Estimates for the Wave Equation
721(6)
17.7 Slobodeckij Spaces WS,P(Rn) and Bessel Potential Spaces Hsp (Rn)
727(5)
17.8 Besov Spaces Bsp,q(Rn)and Triebel-Lizorkin Spaces Fp,q(Rn)
732(2)
17.9 Embeddings of Bsp,q(Rn)and Fsp,q(Rn)
734(3)
17.10 Exercises and Further Results
737(14)
18 Morrey and Campanato vs. Hardy and John-Nirenberg Spaces
751(48)
18.1 Morrey Spaces mp,λ
751(2)
18.2 Campanato Spaces £p,λ
753(1)
18.3 Relations Between mp,λ, £p,λ and c0,μ
754(6)
18.4 The John-Nirenberg Space BMO
760(5)
18.5 The Real Hardy Spaces Hp (Rn) (0 <p≤ ∞)
765(2)
18.6 H1(Rn) and the Div-Curl Lemma
767(3)
18.7 The L log L Integrability of det u on W-1,n
770(6)
18.8 Gehring's Higher L p-Integrability Lemma; Reverse Holder Inequalities
776(3)
18.9 Exercises and Further Results
779(20)
19 Layered Potentials, Jump Relations and Existence Theorems
799(42)
19.1 The Potential D = D[ φ δΩ] of a Double Layer
799(11)
19.2 The Inner and Outer Trace Operators yi, y0; The Jump Relations
810(1)
19.3 The Potential S = S[ φ δΩ] of a Single Layer
811(8)
19.4 The Inner and Outer Trace Operators yi, v0, y; The Jump Relations
819(2)
19.5 Existence Theorems Through the Method of Layered Potentials
821(1)
19.6 Spectral Analysis of T on L2(δω)
822(5)
19.7 An Eigen-Space Decomposition of L2(δω)
827(2)
19.8 A Resolution of the Dirichlet and Neumann Problems
829(2)
19.9 Exercises and Further Results
831(10)
20 Second Order Equations in Divergence Form: Continuous Coefficients
841(26)
20.1 Caccioppoli Inequality: The Classical Form
842(3)
20.2 Application to Higher Local Integrability of |u|2
845(3)
20.3 A-Harmonic Functions and the Decay Rate of their Integral Means
848(3)
20.4 Comparison with A-Harmonic Functions; Iteration Lemma
851(2)
20.5 L2,λ -Estimates for A-Harmonic Functions
853(2)
20.6 Continuous Coefficients: Gradient m2,λ-Estimates
855(2)
20.7 Gradient Holder Continuity: C1,μ-Estimates (0 < μ < 1)
857(4)
20.8 Exercises and Further Results
861(6)
21 Second Order Equations in Divergence Form: Measurable Coefficients
867(40)
21.1 Caccioppoli Inequality on Level Sets
867(4)
21.2 Local Boundedness of Weak Solutions; De Giorgi's Approach
871(3)
21.3 Holder Continuity of Weak Solutions; Oscillations on Balls
874(9)
21.4 Moser Iteration: Local Boundedness of Weak Solutions
883(7)
21.5 Moser Iteration: Holder Continuity of Weak Solutions
890(4)
21.6 Harnack Inequality and its Consequences
894(4)
21.7 Exercises and Further Results
898(9)
Appendices
A Partition of Unity
907(2)
B Total Boundedness and Compact Subsets of Lp
909(4)
C Gamma and Beta Functions
913(3)
D Volume of the Unit n-Ball: ω = |Bn|
916(2)
E Integrals Related to Abel and Gauss Kernels
918(4)
F The Hausdorff Measure H1 (0 ≤ s < ∞)
922(5)
G Evaluation of Some Integrals Over Sn-1
927(2)
H Sobolev Spaces W1,P(a,b)
929(10)
Bibliography 939(20)
Index 959
Dr Taheri is a Reader in Mathematics as the University of Sussex. His primary research area is in the field of years. During which time he has published research papers in prestigious journals, made valuable contributions to the field and has taught and conducted research in some of the leading institutions in the world including Oxford, Courant Institute, Max-Planck-Institute Leipzig and Warwick. He heads up the Analysis and PDEs research group in Sussex In June 2014 he was awarded the First University of Sussex Student Led Teaching Prize for "Outstanding and Innovative Postgraduate Teaching in Mathematics".