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Function Spaces and Partial Differential Equations: Volume 1 - Classical Analysis [Kietas viršelis]

(Reader in Mathematics, University of Sussex)
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Function Spaces and Partial Differential Equations: Volume 1 - Classical AnalysisDidesnis paveikslėlis
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Pastovi nuoroda: https://www.kriso.lt/db/9780198733133.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 *

1 Harmonic Functions and the Mean-Value Property
1(32)
1.1 Spherical Means
1(3)
1.2 Mean-Value Property and Smoothness
4(2)
1.3 Maximum Principles
6(1)
1.4 The Laplace-Beltrami Operator on Spheres
7(12)
1.5 Harnack's Monotone Convergence Theorem
19(1)
1.6 Interior Estimates and Uniform Gradient Bounds
20(3)
1.7 Weyl's Lemma on Weakly Harmonic Functions
23(1)
1.8 Exercises and Further Results
24(9)
2 Poisson Kernels and Green's Representation Formula
33(42)
2.1 The Fundamental Solution N of Δ
34(2)
2.2 Green's Identities and Representation Formulas
36(5)
2.3 The Green's Function G = G(x,y; Ω)
41(3)
2.4 The Poisson Kernel P = P(x,y; Ω)
44(1)
2.5 Explicit Constructions: Balls
45(7)
2.6 Explicit Constructions: Half-Spaces
52(1)
2.7 The Newtonian Potential N[ ƒ; Ω
53(6)
2.8 Decay of the Newtonian Potential
59(2)
2.9 Second Order Derivatives and ΔN[ ƒ; Ω]
61(5)
2.10 Exercises and Further Results
66(9)
3 Abel-Poisson and Fejer Means of Fourier Series
75(50)
3.1 Function Spaces on the Circle
76(3)
3.2 Conjugate Series; Magnitude of Fourier Coefficients
79(3)
3.3 Summability Methods; Tauberian Theorems
82(4)
3.4 Abel-Poisson vs. Fejer Means of Fourier Series
86(5)
3.5 L1(T) and M(T) as Convolution Banach Algebras
91(8)
3.6 Approximation to Identity: Strong Convergence in C and LP (p < ∞)
99(5)
3.7 Approximation to Identity: Weak* Convergence in M and L∞
104(4)
3.8 The Riemann-Lebesgue Lemma; An Isomorphism of L1 (T) into c0(Z)
108(3)
3.9 A Primer of Peter-Weyl Theory: Characters and Orthogonality in L2(T)
111(3)
3.10 Exercises and Further Results
114(11)
4 Convergence of Fourier Series: Dini vs. Dirichlet-Jordon
125(26)
4.1 The Wiener Algebra of the Circle A(T)
125(3)
4.2 Pointwise Convergence of Fourier Series
128(5)
4.3 Riemann's Localisation Principle
133(1)
4.4 Dini and Marcinkiewicz Convergence Criteria
133(2)
4.5 Dirichlet-Jordan Convergence Criterion
135(2)
4.6 The Frechet-Schwartz Space D(T)
137(3)
4.7 The Hilbert-Sobolev Spaces Hs(T) (-∞ < s < ∞)
140(4)
4.8 Exercises and Further Results
144(7)
5 Harmonic-Hardy Spaces Hp(D)
151(34)
5.1 The Poisson Kernel V(x,y; D)
151(4)
5.2 The Dirichlet Problem in a Jordan Domain
155(2)
5.3 Nodal Sets and the Rado-Kneser-Choquet Theorem
157(5)
5.4 Poisson Integrals in LP(T) (1 ≤ p ≤ ∞)
162(3)
5.5 Poisson Integrals in M(T)
165(3)
5.6 Non-Tangential Convergence
168(3)
5.7 Characterisation of Harmonic-Hardy Spaces hp(D)
171(3)
5.8 Harmonic Conjugation on hp(D) (1 ≤ p ≤ ∞
174(3)
5.9 Hadamard's Three Lines Theorem
177(1)
5.10 Exercises and Further Results
177(8)
6 Interpolation Theorems of Marcinkiewicz and Riesz-Thorin
185(62)
6.1 Interpolation of Integral Operators on LP(X, u, μ)
185(7)
6.2 Integration via the Distribution Function
192(3)
6.3 Marcinkiewicz Spaces Lw (X, u, μ)
195(4)
6.4 Real Interpolation Method of Marcinkiewicz: The Diagonal Case
199(7)
6.5 Complex Interpolation Method of Riesz-Thorin
206(6)
6.6 The Hausdorff-Young and Hardy-Littlewood-Paley Inequalities
212(2)
6.7 Real Interpolation Method of Marcinkiewicz: The General Case
214(5)
6.8 Decreasing Rearrangements; The Maximal Function Operator M [ ƒ*]
219(6)
6.9 The Lorentz Spaces LP,q(X, u, μ) and Interpolation
225(8)
6.10 Exercises and Further Results
233(14)
7 The Hilbert Transform on LP(T) and Riesz's Theorem
247(28)
7.1 Fourier Partial Sums and Riesz Projection on Lp(T) (1 ≤ p ≤ ∞)
247(4)
7.2 Higher Regularity of u = P[ /] Up to the Boundary
251(2)
7.3 The Hilbert Transform on L1(T); Existence a.e. and Finiteness
253(4)
7.4 The Hilbert Transform as an L2-Multiplier Operator
257(2)
7.5 Kolmogoroff's Theorem: The L1 - Weak Estimate on H
259(3)
7.6 Riesz's Theorem: The LP-Boundedness of H (1 < p < ∞)
262(3)
7.7 Zygmund's L log L Theorem and its Converse
265(1)
7.8 Riesz Projection and the Lp-Convergence of Fourier Series (1 < p < ∞)
266(1)
7.9 Exercises and Further Results
267(8)
8 Harmonic-Hardy Spaces hP(Bn)
275(34)
8.1 The Poisson Kernel V(x,y; Bn)
275(4)
8.2 Poisson Integrals in Lp (Sn-1) (1 ≤ p ≤ ∞) and M(Sn-1)
279(5)
8.3 Characterisation of Harmonic-Hardy Spaces hip(Bn)
284(1)
8.4 Herglotz's Theorem on Positive Harmonic Functions
285(1)
8.5 H.A. Schwarz's Reflection Principle; Removable Singularities
286(3)
8.6 Non-Tangential Maximal Function; Stoltz Domains Ωα (y) in Bn
289(3)
8.7 A Spectral Decomposition of L2(Sn-1) via Spherical Harmonics
292(1)
8.8 Orthogonal Projection of L2 (Sn-1) onto Hj; Zonal Harmonics
293(6)
8.9 Exercises and Further Results
299(10)
9 Convolution Semigroups; The Poisson and Heat Kernels on Rn
309(42)
9.1 Convolutions in Co(Rn), LP(Rn) and M(Rn)
309(4)
9.2 L1(Rn) and M(Rn) as Convolution Banach Algebras
313(3)
9.3 Approximation to Identity: Strong Convergence in Co and Lp (p < ∞)
316(3)
9.4 Approximation to Identity: Weak* Convergence in M and L∞
319(1)
9.5 Young's Convolution Inequality: Lr(Rn) * LP(Rn) C Lq(Rn)
320(1)
9.6 Friedrich Mollifiers and Approximation by Smooth Functions
321(4)
9.7 Continuity of Riesz Potentials by way of Young's Inequality
325(6)
9.8 LP Norm for Vector p; The Loomis-Whitney Inequality and Beyond
331(3)
9.9 Exercises and Further Results
334(17)
10 Perron's Method of Sub-Harmonic Functions
351(42)
10.1 Upper Semicontinuous Functions
351(2)
10.2 Sub-Harmonic Functions Revisited
353(4)
10.3 Perron's Existence Theorem
357(2)
10.4 Barriers and the Boundary Regularity of Perron's Solution
359(2)
10.5 Potentials; Capacity and Wiener's Criterion
361(13)
10.6 Harmonic Measure; Generalised Poisson Integrals
374(2)
10.7 The Riemann Mapping Theorem via Green's Functions
376(2)
10.8 Hardy's Theorem on the Convexity of log p [ ƒ;r]
378(2)
10.9 Solvability of the Poisson Equation; C2, α Estimates on N[ ƒ; Ω]
380(4)
10.10 Exercises and Further Results
384(9)
11 From Abel-Poisson to Bochner-Riesz Summability
393(44)
11.1 The L1 Theory of Fourier Transform
393(6)
11.2 Abel-Poisson vs. Gauss-Weierstrass Summability of Integrals
399(2)
11.3 Fourier Inversion Formula on L1(Rn)
401(3)
11.4 The Schwartz Space S(Rn) as a Frechet Space
404(7)
11.5 Fourier-Plancherel Transform and the L2 Theory
411(3)
11.6 The Calderon-Zygmund Decomposition Lemma
414(2)
11.7 Summability of Fourier Integrals; Fefferman's Ball Multiplier
416(4)
11.8 Bochner-Riesz Summability
420(2)
11.9 Exercises and Further Results
422(15)
12 Fourier Transform on 5'(Rn); The Hilbert-Sobolev Spaces Hs(Rn)
437(44)
12.1 S'(Rn) as a Dual Space
437(4)
12.2 Fourier Transform on S'(Rn)
441(8)
12.3 (Lp,Lq) Operators Commuting with Translations
449(4)
12.4 Fractional Integration and (-Δ)-α/2 (0 < α < n)
453(3)
12.5 LP - Estimates: Poisson, Heat and Schrodinger Semigroups
456(3)
12.6 The Wave Kernel Wt; The Light Cone and Huygens Principle
459(3)
12.7 The Hilbert-Sobolev Spaces Hs(Rn) (-∞ < s < ∞)
462(5)
12.8 Trace Theorems and Restrictions in Hs(Rn)
467(1)
12.9 Extensions and a Theorem of Slobodeckij
468(2)
12.10 Exercises and Further Results
470(11)
Bibliography 481
Index 1
Dr Taheri is a Reader in Mathematics as the University of Sussex. His primary research area is in the field of Analysis & PDEs where he has been working for over 15 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".