eBook ISBN:  9781470441319 
Product Code:  MEMO/249/1182.E 
List Price:  $75.00 
MAA Member Price:  $67.50 
AMS Member Price:  $45.00 
eBook ISBN:  9781470441319 
Product Code:  MEMO/249/1182.E 
List Price:  $75.00 
MAA Member Price:  $67.50 
AMS Member Price:  $45.00 

Book DetailsMemoirs of the American Mathematical SocietyVolume: 249; 2017; 79 ppMSC: Primary 35
The authors consider operators of the form \(L=\sum_{i=1}^{n}X_{i}^{2}+X_{0}\) in a bounded domain of \(\mathbb{R}^{p}\) where \(X_{0},X_{1},\ldots,X_{n}\) are nonsmooth Hörmander's vector fields of step \(r\) such that the highest order commutators are only Hölder continuous. Applying Levi's parametrix method the authors construct a local fundamental solution \(\gamma\) for \(L\) and provide growth estimates for \(\gamma\) and its first derivatives with respect to the vector fields. Requiring the existence of one more derivative of the coefficients the authors prove that \(\gamma\) also possesses second derivatives, and they deduce the local solvability of \(L\), constructing, by means of \(\gamma\), a solution to \(Lu=f\) with Hölder continuous \(f\). The authors also prove \(C_{X,loc}^{2,\alpha}\) estimates on this solution.

Table of Contents

Chapters

1. Introduction

2. Some known results about nonsmooth Hörmander’s vector fields

3. Geometric estimates

4. The parametrix method

5. Further regularity of the fundamental solution and local solvability of $L$

6. Appendix. Examples of nonsmooth Hörmander’s operators satisfying assumptions A or B


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The authors consider operators of the form \(L=\sum_{i=1}^{n}X_{i}^{2}+X_{0}\) in a bounded domain of \(\mathbb{R}^{p}\) where \(X_{0},X_{1},\ldots,X_{n}\) are nonsmooth Hörmander's vector fields of step \(r\) such that the highest order commutators are only Hölder continuous. Applying Levi's parametrix method the authors construct a local fundamental solution \(\gamma\) for \(L\) and provide growth estimates for \(\gamma\) and its first derivatives with respect to the vector fields. Requiring the existence of one more derivative of the coefficients the authors prove that \(\gamma\) also possesses second derivatives, and they deduce the local solvability of \(L\), constructing, by means of \(\gamma\), a solution to \(Lu=f\) with Hölder continuous \(f\). The authors also prove \(C_{X,loc}^{2,\alpha}\) estimates on this solution.

Chapters

1. Introduction

2. Some known results about nonsmooth Hörmander’s vector fields

3. Geometric estimates

4. The parametrix method

5. Further regularity of the fundamental solution and local solvability of $L$

6. Appendix. Examples of nonsmooth Hörmander’s operators satisfying assumptions A or B