scholarly journals Homogeneous Triebel-Lizorkin Spaces on Stratified Lie Groups

2013 ◽  
Vol 2013 ◽  
pp. 1-16 ◽  
Author(s):  
Guorong Hu
Keyword(s):  
Lie Groups ◽  
Singular Integral ◽  
Full Range ◽  
Integral Operators ◽  
Function Spaces ◽  
Singular Integral Operators ◽  
Basic Properties ◽  
Stratified Lie Groups ◽  
Type Decomposition

Homogeneous Triebel-Lizorkin spaces with full range of parameters are introduced on stratified Lie groups in terms of Littlewood-Paley-type decomposition. It is shown that the scale of these spaces is independent of the choice of Littlewood-Paley-type decomposition and the sub-Laplacian used for the construction of the decomposition. Some basic properties of these spaces are given. As the main result of this paper, boundedness of a class of singular integral operators on these function spaces is obtained.

Multi-parameter Singular Integrals I: Examples

2017 ◽  
Author(s):  
Brian Street
Keyword(s):  
Singular Integral ◽  
Singular Integrals ◽  
Pseudodifferential Operators ◽  
Integral Operators ◽  
Singular Integral Operators ◽  
Special Cases ◽  
Stratified Lie Groups ◽  
Left And Right ◽  
Flag Kernels

This chapter discusses a few special cases where a theory of multi-parameter singular integral operators has already been developed. These include the product theory of singular integrals, convolution with flag kernels on graded groups, convolution with both the left and right invariant Calderón–Zygmund singular integral operators on stratified Lie groups, and composition of standard pseudodifferential operators with certain singular integrals corresponding to non-Euclidean geometries. The chapter outlines these examples and their applications and relates them to the trichotomy discussed in Chapter 1.

Multi-parameter Singular Integrals II: General Theory

2017 ◽  
Author(s):  
Brian Street
Keyword(s):  
General Theory ◽  
Singular Integral ◽  
Singular Integrals ◽  
Integral Operators ◽  
Singular Integral Operators ◽  
Main Issue

This chapter turns to a general theory which generalizes and unifies all of the examples in the preceding chapters. A main issue is that the first definition from the trichotomy does not generalize to the multi-parameter situation. To deal with this, strengthened cancellation conditions are introduced. This is done in two different ways, resulting in four total definitions for singular integral operators (the first two use the strengthened cancellation conditions, while the later two are generalizations of the later two parts of the trichotomy). Thus, we obtain four classes of singular integral operators, denoted by A1, A2, A3, and A4. The main theorem of the chapter is A1 = A2 = A3 = A4; i.e., all four of these definitions are equivalent. This leads to many nice properties of these singular integral operators.

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