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E-raamat: Morrey Spaces

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In this set of lecture notes, the author includes some of the latest research on the theory of Morrey Spaces associated with Harmonic Analysis.  There are three main claims concerning these spaces that are covered: Determining the integrability classes of the trace of Riesz potentials of an arbitrary Morrey function; determining the dimensions of singular sets of weak solutions of PDE (e.g. The Meyers-Elcart System); determining whether there are any “full” interpolation results for linear operators between Morrey spaces.

This book will serve as a useful reference to graduate students and researchers interested in Potential Theory, Harmonic Analysis, PDE, and/or Morrey Spaces Theory.      

Arvustused

This book gives a comparatively systematic discussion on the theory of Morrey spaces. This book puts a lot of emphasis on the predual theory of the Morrey spaces as well as their applications. the book has an impressive level of generality on the modern theory of Morrey spaces . In addition, the related theories for Morrey spaces promise developments of this field in the near future. (Liguang Liu, Mathematical Reviews, May, 2017)



This book contains the latest results obtained by the author. It is a useful reference to mathematicians working in potential theory, harmonic analysis and partial differential equations, in particular, in Morrey spaces theory. (Sibei Yang, zbMATH 1339.42001, 2016)

1 Introduction
1(6)
2 Function Spaces
7(6)
2.1 LP, Cα, BMO, LP.λ, LPW.λ
7(1)
2.2 The Campanato scale Lp.λ, Lp.λ (Q0)
8(2)
2.3 Sobolev Spaces Wmp (Ω,), Gα(LP), Iα(LP)
10(1)
2.4 Morrey-Sobolev Spaces Iα(LPλ)
10(1)
2.5 Dense/non-dense subspaces, Zorko Spaces, VLpλ, VMO
10(2)
2.6 Note
12(1)
3 Hausdorff Capacity
13(8)
3.1 Set functions Λd, ΛHd, Ln, 0 < d ≤ n
13(1)
3.2 Dyadic versions: Λd, Λd0
14(1)
3.3 Frostman's Theorem
15(1)
3.4 Strong subadditivity of Λd0 and Λd ~ Λd0
16(1)
3.5 The operator Mα and Hausdorff capacity
16(1)
3.6 Notes
17(4)
3.6.1 Netrusov's capacity Λd;θ and a Netrusov-Frostman Theorem
17(1)
3.6.2 A strong type estimate for Mα
18(3)
4 Choquet Integrals
21(8)
4.1 Definition and basic properties: sublinear vs. strong subadditivity
21(3)
4.2 Adams-Orobitg-Verdera Theorem
24(2)
4.3 Notes
26(3)
4.3.1 Further estimates for Mαƒ
26(1)
4.3.2 Speculations on weighted Hausdorff Capacity
27(2)
5 Duality for Morrey Spaces
29(8)
5.1 Dual of L1 (Λd)
29(2)
5.2 Three equivalent predual spaces Xpλ, Kpλ, Zpλ
31(2)
5.3 The predual Hpp'λ
33(2)
5.4 The space Z0pλ and Zpλ
35(1)
5.5 Notes
36(1)
6 Maximal Operators and Morrey Spaces
37(6)
6.1 Mo on Lpλ - two proofs
37(2)
6.2 ||Iα μ||Lpλ ~ ||Mα μ|| Lpλ, ||Iα μ||Hpλ ~ ||Mα μ||Hpλ
39(1)
6.3 Proof of (6.6)
40(2)
6.4 Notes
42(1)
7 Potential Operators on Morrey Spaces
43(8)
7.1 Iα : Lpλ → Lpλ ∩ Lpλ--αp, p = λp/λ--αp, αp < λ Iα : Hpλ → Hpλ ∩ Hpλ+αp'
43(2)
7.2 Wolff potentials associated with ||Iα μ||Lp'
45(1)
7.3 Wolff potentials associated with ||Iα μ||Lp'λ, ||Iα μ||Hp'λ
46(2)
7.4 A "Morrey bridge" to Cα
48(1)
7.5 Notes
49(2)
7.5.1 Proof of Lemma 7.4
49(1)
7.5.2 Iα : L1λ → L1.λw, 1 = λ / (λ -- α), 0 < α < λ
50(1)
8 Singular Integrals on Morrey Spaces
51(2)
8.1 T : Lpλ → Lpλ and T : Hpλ → 1 < p < ∞, o < λ < n
51(2)
9 Morrey-Sobolev capacities
53(10)
9.1 Definitions and simple properties for Cαp(·)
53(1)
9.2 Definitions and simple properties for Cα(·; X), X = LPλ or Hpλ
54(2)
9.3 Cα(B(x, r); LPλ)
56(3)
9.4 Cα(B(x, r);Hpλ) and failure of CSI for Cα(·; Lpλ)
59(1)
9.5 Notes
60(3)
9.5.1 Weighted capacity vs. Choquet Integrals
60(1)
9.5.2 Relations between Cα,p and Cβ,q via Morrey Theory
60(1)
9.5.3 Speculations and parabolic capacities
61(2)
10 Traces of Morrey Potentials
63(8)
10.1 ||Iαƒ||Lq(μ) ≤ c ||ƒ||Lp
63(2)
10.2 ||Iαƒ||Lq(μ) ≤ c0 ||ƒ||Lpλ
65(2)
10.3 An Improved Trace Result
67(1)
10.4 Notes
68(3)
11 Interpolation of Morrey Spaces
71(6)
11.1 Stampacchia-Peetre interpolation; Interpolation via the new duality
71(4)
11.2 Counterexamples to interpolation with Morrey Spaces in the domain of the operator
75(1)
11.3 Integrability of Morrey Potentials
76(1)
12 Commutators of Morrey Potentials
77(8)
12.1 Some history for the operators [ b, T] and [ b, Iα]
77(1)
12.2 Commutators: b BMO
78(2)
12.3 Traces of Morrey commutators, |b| Ln
80(5)
13 Mock Morrey Spaces
85(4)
13.1 Marcinkiewicz Spaces
85(1)
13.2 Conti's Theorem
86(2)
13.3 Notes
88(1)
13.3.1 QPλ vs Lpλ
88(1)
14 Morrey-Besov Spaces and Besov Capacity
89(6)
14.1 Adams-Lewis inequality (Sobolev inequality for Morrey-Besov)
89(3)
14.2 Besov capacity and the Netrusov capacity
92(1)
14.3 Notes: CSI for Besov capacities
93(2)
15 Morrey Potentials and PDE I
95(8)
15.1 Δu = up, u ≥ 0
95(6)
15.2 Notes
101(2)
15.2.1 The Yamabe Case p = n+2/n-2
101(1)
15.2.2 Stationary Navier-Stokes (n = 5)
101(1)
15.2.3 A Comment
102(1)
16 Morrey Potentials and PDE II
103(8)
16.1 Examples of singular sets for elliptic systems
103(1)
16.2 Meyers-Elcrat system
104(4)
16.3 Notes
108(3)
16.3.1 Harmonic Maps
108(1)
16.3.2 Lane-Emden systems
109(2)
17 Morrey Spaces On Complete Riemannian Manifolds
111(4)
17.1 A counterexample
112(1)
17.2 A Morrey-Sobolev inequality on Mn with balls having maximal growth and Ric ≥ 0
112(2)
17.3 Further embedding and speculations
114(1)
Bibliography 115(6)
Index of Symbols 121(2)
Index 123
David R. Adams is a Professor in the Department of Mathematics at the University of Kentucky. He received his Ph.D. from the University of Minnesota in 1969. His research areas include analysis and partial differential equations.
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