E-raamat: Stratified Lie Groups and Potential Theory for Their Sub-Laplacians

Part I Elements of Analysis of Stratified Groups
Stratified Groups and Sub-Laplacians		3	(84)
Vector Fields in RN: Exponential Maps and Lie Algebras		3	(10)
Vector Fields in RN		3	(3)
Integral Curves		6	(2)
Exponentials of Vector Fields		8	(2)
Lie Brackets of Vector Fields in RN		10	(3)
Lie Groups on Rw		13	(18)
The Lie Algebra of a Lie Group on RN		13	(6)
The Jacobian Basis		19	(3)
The (Jacobian) Total Gradient		22	(1)
The Exponential Map of a Lie Group on RN		23	(8)
Homogeneous Lie Groups on RN		31	(25)
homogeneous Functions and Differential Operators		32	(6)
The Composition Law of a Homogeneous Lie Group		38	(6)
The Lie Algebra of a Homogeneous Lie Group on RN		44	(4)
The Exponential Map of a Homogeneous Lie Group		48	(8)
Homogeneous Carnot Groups		56	(6)
The Sub-Laplacians on a Homogeneous Carnot Group		62	(11)
The Horizontal L-gradient		68	(5)
Exercises of Chapter 1		73	(14)
Abstract Lie Groups and Carnot Groups		87	(68)
Abstract Lie Groups		87	(34)
Differentiable Manifolds		87	(4)
Tangent Vectors		91	(4)
Differentials		95	(2)
Vector Fields		97	(5)
Commutators. φ-relatedness		102	(4)
Abstract Lie Groups		106	(1)
Left Invariant Vector Fields and the Lie Algebra		107	(5)
Homomorphisms		112	(4)
The Exponential Map		116	(5)
Carnot Groups		121	(26)
Some Properties of the Stratification of a Carnot Group		125	(3)
Some General Results on Nilpotent Lie Groups		128	(2)
Abstract and Homogeneous Carnot Groups		130	(8)
More Properties of the Lie Algebra		138	(6)
Sub-Laplacians of a Stratified Group		144	(3)
Exercises of Chapter 2		147	(8)
Carnot Groups of Step Two		155	(28)
The Heisenberg-Weyl Group		155	(3)
Homogeneous Carnot Groups of Step Two		158	(5)
Free Step-two Homogeneous Groups		163	(2)
Change of Basis		165	(1)
The Exponential Map of a Step-two Homogeneous Group		166	(3)
Prototype Groups of Heisenberg Type		169	(4)
H-groups (in the Sense of Metivier)		173	(4)
Exercises of Chapter 3		177	(6)
Examples of Carnot Groups		183	(44)
A Primer of Examples of Carnot Groups		183	(8)
Euclidean Group		183	(1)
Carnot Groups with Homogeneous Dimension Q ≤ 3		184	(1)
B-groups		184	(2)
K-type Groups		186	(4)
Sum of Carnot Groups		190	(1)
From a Set of Vector Fields to a Stratified Group		191	(7)
Further Examples		198	(12)
The Vector Fields ∂1,∂2+∂3		198	(2)
Classical and Kohn Laplacians		200	(2)
Bony-type Sub-Laplacians		202	(2)
Kolmogorov-type Sub-Laplacians		204	(1)
Sub-Laplacians Arising in Control Theory		205	(2)
Filiform Carnot Groups		207	(3)
Fields not Satisfying One of the Hypotheses (HO), (H1), (H2)		210	(5)
Fields not Satisfying Hypothesis (HO)		210	(2)
Fields not Satisfying Hypothesis (H1)		212	(3)
Fields not Satisfying Hypothesis (H2)		215	(1)
Exercises of Chapter 4		215	(12)
The Fundamental Solution for a Sub-Laplacian and Applications		227	(110)
Homogeneous Norms		229	(3)
Control Distances or Carnot-Caratheodory Distances		232	(4)
The Fundamental Solution		236	(10)
The Fundamental Solution in the Abstract Setting		244	(2)
L-gauges and L-radial Functions		246	(5)
Gauge Functions and Surface Mean Value Theorem		251	(6)
Superposition of Average Operators		257	(5)
Harnack Inequalities		262	(7)
Liouville-type Theorems		269	(7)
Asymptotic Liouville-type Theorems		274	(2)
Sobolev-Stein Embedding Inequality		276	(4)
Analytic Hypoellipticity of Sub-Laplacians		280	(7)
Harmonic Approximation		287	(4)
An Integral Representation Formula for Γ		291	(2)
Appendix A. Maximum Principles		293	(13)
A Decomposition Theorem for L-harmonic Functions		303	(3)
Appendix B. The Improved Pseudo-triangle Inequality		306	(3)
Appendix C. Existence of Geodesies		309	(10)
Exercises of Chapter 5		319	(18)
Part II Elements of Potential Theory for Sub-Laplacians
Abstract Harmonic Spaces		337	(44)
Preliminaries		338	(2)
Sheafs of Functions. Harmonic Sheafs		340	(5)
Harmonic Measures and Hyperharmonic Functions		341	(1)
Directed Families of Functions		342	(3)
Harmonic Spaces		345	(3)
Directed Families of Harmonic and Hyperharmonic Functions		347	(1)
β-hyperharmonic Functions. Minimum Principle		348	(5)
Subharmonic and Superharmonic Functions. Perron Families		353	(5)
Harmonic Majorants and Minorants		358	(1)
The Perron-Wiener-Brelot Operator		359	(4)
G-harmonic Spaces: Wiener Resolutivity Theorem		363	(4)
Appendix: The Stone-Weierstrass Theorem		366	(1)
H-harmonic Measures for Relatively Compact Open Sets		367	(3)
G-harmonic Spaces: Bouligand's Theorem		370	(5)
Reduced Functions and Balayage		375	(3)
Exercises of Chapter 6		378	(3)
The L-harmonic Space		381	(16)
The L-harmonic Space		381	(7)
Some Basic Definitions and Selecta of Properties		388	(4)
Exercises of Chapter 7		392	(5)
L-subharmonic Functions		397	(28)
Sub-mean Functions		397	(4)
Some Characterizations of L-subharmonic Functions		401	(10)
Continuous Convex Functions on G		411	(11)
Exercises of Chapter 8		422	(3)
Representation Theorems		425	(48)
L-Green Function for L-regular Domains		425	(2)
L-Green Function for General Domains		427	(5)
Potentials of Radon Measures		432	(9)
Potentials Related to the Average Operators		435	(6)
Riesz Representation Theorems for L-subharmonic Functions		441	(4)
The Poisson-Jensen Formula		445	(6)
Bounded-above L-subharmonic Functions in G		451	(4)
Smoothing of L-subharmonic Functions		455	(3)
Isolated Singularities---Bocher-type Theorems		458	(5)
An Application of Bocher's Theorem		462	(1)
Exercises of Chapter 9		463	(10)
Maximum Principle on Unbounded Domains		473	(16)
MP Sets and L-thinness at Infinity		473	(4)
q-sets and the Maximum Principle		477	(5)
The Maximum Principle on Unbounded Domains		482	(1)
The Proof of Lemma 10.2.3		483	(4)
Exercises of Chapter 10		487	(2)
L-capacity, L-polar Sets and Applications		489	(48)
The Continuity Principle for L-potentials		489	(2)
L-polar Sets		491	(3)
The Maria-Frostman Domination Principle		494	(3)
L-energy and L-equilibrium Potentials		497	(3)
L-balayage and L-capacity		500	(10)
The Fundamental Convergence Theorem		510	(4)
The Extended Poisson-Jensen Formula		514	(5)
Further Results. A Miscellanea		519	(8)
Further Reading and the Quasi-continuity Property		527	(6)
Exercises of Chapter 11		533	(4)
L-thinness and L-fine Topology		537	(20)
The L-fine Topology: A More Intrinsic Tool		537	(1)
L-thinness at a Point		538	(4)
L-thinness and L-regularity		542	(5)
Functions Peaking at a Point		542	(2)
L-thinness and L-regularity		544	(3)
Wiener's Criterion for Sub-Laplacians		547	(6)
A Technical Lemma		547	(3)
Wiener's Criterion for L		550	(3)
Exercises of Chapter 12		553	(4)
d-Hausdorff Measure and L-capacity		557	(20)
d-Hausdorff Measure and Dimension		557	(4)
d-Hausdorff Measure and L-capacity		561	(8)
New Phenomena Concerning the d-Hausdorff Dimension		569	(3)
Exercises of Chapter 13		572	(5)
Part III Further Topics on Carnot Groups
Some Remarks on Free Lie Algebras		577	(16)
Free Lie Algebras and Free Lie Groups		577	(7)
A Canonical Way to Construct Free Carnot Groups		584	(5)
The Campbell-Hausdorff Composition		584	(2)
A Canonical Way to Construct Free Carnot Groups		586	(3)
Exercises of Chapter 14		589	(4)
More on the Campbell-Hausdorff Formula		593	(28)
The Campbell-Hausdorff Formula for Stratified Fields		593	(6)
The Campbell-Hausdorff Formula for Formal Power Series-1		599	(6)
The Campbell-Hausdorff Formula for Formal Power Series-2		605	(5)
The Campbell-Hausdorff Formula for Smooth Vector Fields		610	(6)
Exercises of Chapter 15		616	(5)
Families of Diffeomorphic Sub-Laplacians		621	(28)
An Isomorphism Turning Σi, jai, j XiXi into Σi, jai, j XiXj into ΔG		622	(6)
Examples and Counter-examples		628	(9)
Canonical or Non-canonical?		637	(7)
An Example		641	(3)
Further Reading: An Application to PDE's		644	(1)
Exercises of Chapter 16		645	(4)
Lifting of Carnot Groups		649	(32)
Lifting to Free Carnot Groups		649	(10)
An Example of Lifting		659	(2)
An Example of Application to PDE's		661	(5)
Folland's Lifting of Homogeneous Vector Fields		666	(10)
The Hypotheses on the Vector Fields		669	(7)
Exercises of Chapter 17		676	(5)
Groups of Heisenberg Type		681	(34)
Heisenberg-type Groups		681	(5)
A Direct Characterization of H-type Groups		686	(9)
The Fundamental Solution on H-type Groups		695	(7)
H-type Groups of Iwasawa-type		702	(2)
The H-inversion and the H-Kelvin Transform		704	(5)
Exercises of Chapter18		709	(6)
The Caratheodory-Chow-Rashevsky Theorem		715	(18)
The Caratheodory-Chow-Rashevsky Theorem for Stratified Vector Fields		715	(12)
An Application of Caratheodory-Chow-Rashevsky Theorem		727	(3)
Exercises of Chapter 19		730	(3)
Taylor Formula on Carnot Groups		733	(40)
Polynomials and Derivatives on Homogeneous Carnot Groups		734	(7)
Polynomial Functions on G		734	(2)
Derivatives on G		736	(5)
Taylor Polynomials on Homogeneous Carnot Groups		741	(5)
Taylor Formula on Homogeneous Carnot Groups		746	(20)
Stratified Taylor Formula with Peano Remainder		751	(3)
Stratified Taylor Formula with Integral Remainder		754	(12)
Exercises of Chapter 20		766	(7)
References		773	(16)
Index of the Basic Notation		789	(6)
Index		795

1) ERMANNO LANCONELLI:



--Education and Undergraduate Studies: Dec. 1966, Universita' di Bologna (Mathematics).



Career/Employment:



1975-present: Full Professor of Mathematical Analysis at Dipartimento di Matematica, Universita' di Bologna (Italy); Member of the "Accademia dell'Istituto di Bologna" and of the "Accademia delle Scienze, Lettere ed Arti di Modena".



1968-1975: Theaching Assistant at Istituto di Matematica, Universita' di Bologna.



--Academic activity:



Director of the Istituto di Matematica di Bologna(1978/80),



Director of the Undergraduate Mathematics Program, University of Bologna (1990/1999, 2000-2002, 2006-present)



Director of PHD program, University of Bologna (1986/91, 1997/2000)



--INVITATIONS:



-University of Minnesota, Minneapolis (USA)



-University of Purdue, West La Fayette, Indiana (USA)



-Temple University, Philadelphia, Pennsylvania (USA)



-Rutgers University, New Brunswick, New Jersey (USA)



-University of Bern, Switzerland



-- Specialization main fields: Partial Differential Equations, Potential



Theory



--CURRENT RESEARCH INTEREST:



Second order linear and nonlinear partial differential equations with non- negative characteristic form and application to complex geometry and diffusion processes.



Potential Theory and Harmonic Analysis in sub-riemannian settings.



Real analysis and geometric methods.



--EDITORIAL BOARD: Nonlinear Differential Equations and Applications, Birkhauser.



--PUBLICATIONS: More than 70 papers in refereed journals.









2) UGUZZONI FRANCESCO:



--Education and Undergraduate Studies: Dec. 1994, Universita' di Bologna (Mathematics)



Career/Employment:



February 2000: Ph.D. in Mathematical Analysis at Dipartimento di Matematica, Universita' di Bologna (Italy).



October 1998: Assistant Professor at Dipartimento di Matematica, Universita' di Bologna.



--CURRENT RESEARCH INTEREST:



Second order linear and nonlinear partial differential equations with non- negative characteristic form and applications. Harmonic Analysis in sub- riemannian settings.



--PUBLICATIONS: About 30 papers in refereed journals.









3) ANDREA BONFIGLIOLI:



--Education and Undergraduate Studies: July 1998, Universita' di Bologna (Mathematics)



--Career/Employment:



March 2002: Ph.D. in Mathematical Analysis at Dipartimento di Matematica, Universita' di Bologna (Italy).



November 2006: Assistant Professor at Dipartimento di Matematica, Universita' di Bologna.



--CURRENT RESEARCH INTEREST:



Second order linear partial differential equations with non-negative characteristic form and applications. Potential Theory in stratified Lie groups.



--PUBLICATIONS: About 20 papers in refereed journals.