Preface
Chapter 1 Preliminary
1.1 Galois field GF （p）
1.1.1 Galois field GF （p）， characteristic number p
1.1.2 Algebraic extension fields of Galois field GF （p）
1.2 Structures of local fields
1.2.1 Definitions of local fields
1.2.2 Valued structure of a local field Kq
1.2.3 Haar measure and Harr integral on a local field Kq
1.2.4 Important subsets in a local field Kq
1.2.5 Base for neighborhood system of a local field Kq
1.2.6 Expressions of elements in Kq， operations
1.2.7 Important properties of balls in a local field Kp
1.2.8 Order structures in a local field Kp
1.2.9 Relationship between local field Kq and Euclidean space R
Exercises
Chapter 2 Character Group Fp of Local Field Kp
2.1 Character groups of locally compact groups
2.1.1 Characters of groups
2.1.2 Characters and character groups of locally compact groups
2.1.3 Pontryagin dual theorem
2.1.4 Examples
2.2 Character group гp of Kp
2.2.1 Properties of X ∈гp and гp
2.2.2 Character group of p—series field Sp
2.2.3 Character group of p—adic field Ap
2.3 Some formulas in local fields
2.3.1 Haar measures of certain important sets in Kp
2.3.2 Integrals for characters in Kp
2.3.3 Integrals for some functions in Kp
Exercises
Chapter 3 Harmonic Analysis on Local Fields
3.1 Fourier analysis on a local field Kp
3.1.1 L1—theory
3.1.2 L2—theory
3.1.3 Lr—theory 1＜r＜2
3.1.4 Distribution theory on Kp
Exercises
3.2 Pseudo—differential operators on local fields
3.2.1 Symbol class Sαρδ（Kp）≡Sαρδ（Kp×гp）
3.2.2 Pseudo—differential operator Tα on local fields
3.3 p—type derivatives and p—type integrals on local fields
3.3.1 p—type calculus on local fields
3.3.2 Properties of p—type derivatives and p—type integrals of ψ∈S（Kp）
3.3.3 p—type derivatives and p—type integrals of T∈S*（Kp）
3.3.4 Background of establishing of p—type calculus
3.4 Operator and construction theory of function on local fields
3.4.1 Operators on a local field Kp
3.4.2 Construction theory of function on a local field Kp
Exercises
Chapter 4 Function Spaces on Local Fields
4.1 B—type spaces and F—type spaces on local fields
4.1.1 B—type spaces， F—type spaces
4.1.2 Special cases of B—type spaces and F—type spaces
4.1.3 Holder type spaces on local fields
4.1.4 Lebesgue type spaces and Sobolev type spaces
Exercises
4.2 Lipschitz classes on local fields
4.2.1 Lipschitz classes on local fields
4.2.2 Chains of function spaces on Euclidean spaces
4.2.3 The cases on local fields
4.2.4 Comparison of Euclidean space analysis with local field analysis
Exercises
4.3 Fractal spaces on local fields
4.3.1 Fractal spaces on Kp
4.3.2 Completeness of space K （（Kp），h） on Kp
4.3.3 Some useful transformations on Kp
Exercises
Chapter 5 Fractal Analysis on Local Fields
5.1 Fractal dimensions on local fields
5.1.1 Hausdorff measure and dimension
5.1.2 Box dimension
5.1.3 Packing measure and dimension
Exercises
5.2 Analytic expressions of dimensions of sets in local fields
5.2.1 Borel measure and Borel measurable sets
5.2.2 distribution dimension
5.2.3 Fourier dimension
Exercises
5.3 p—type calculus and fractal dimensions on local fields
5.3.1 Structures of Kp， 3—adic Cantor type set， 3—adic Cantor type function
5.3.2 p—type derivative and integral of r（x）on K3
5.3.3 p—type derivative and integral of Weierstrass type function on Kp
5.3.4 p—type derivative and integral of second Weierstrass type function on Kp
Exercises
Chapter 6 Fractal PDE on Local Fields
6.1 Special examples
6.1.1 Classical 2—dimension wave equation with fractal boundary
6.1.2 p—type 2—dimension wave equation with fractal boundary
6.2 Further study on fractal analysis over local fields
6.2.1 Pseudo—differential operator Tα
6.2.2 Further problems on fractal analysis over local fields
Exercises
Chapter 7 Applications to Medicine Science
7.1 Determine the malignancy of liver cancers
7.1.1 Terrible havocs of liver cancer， solving idea
7.1.2 The main methods in studying of liver cancers
7.2 Examples in clinical medicine
7.2.1 Take data from the materials ofliver cancers of patients
7.2.2 Mathematical treatment for data
7.2.3 Compute fractal dimensions
7.2.4 Induce to obtain mathematical models
7.2.5 Other problems in the research of liver cancers
References
Index
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