# $$C^{1,\alpha }$$ C 1 , α -subelliptic regularity on $${{\,\mathrm{SU}\,}}(3)$$ SU ( 3 ) and compact, semi-simple Lie groups

$$C^{1,\alpha }$$ C 1 , α -subelliptic regularity on $${{\,\mathrm{SU}\,}}(3)$$ SU... Let the vector fields $$X_1, \ldots , X_{6}$$ X 1 , … , X 6 form an orthonormal basis of $$\mathcal {H}$$ H , the orthogonal complement of a Cartan subalgebra (of dimension 2) in $${{\,\mathrm{SU}\,}}(3)$$ SU ( 3 ) . We prove that weak solutions u to the degenerate subelliptic p-Laplacian \begin{aligned} \Delta _{\mathcal {H},{p}} u(x)=\sum _{i=1}^{6} X_i^{*}\left( |\nabla _{\!{\mathcal {H}}}u|^{p-2}X_{i}u \right) =0, \end{aligned} Δ H , p u ( x ) = ∑ i = 1 6 X i ∗ | ∇ H u | p - 2 X i u = 0 , have Hölder continuous horizontal derivatives $$\nabla _{\!{\mathcal {H}}}u=(X_1u, \ldots , X_{6}u)$$ ∇ H u = ( X 1 u , … , X 6 u ) for $$p\ge 2$$ p ≥ 2 . We also prove that a similar result holds for all compact connected semisimple Lie groups. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Analysis and Mathematical Physics Springer Journals

# $$C^{1,\alpha }$$ C 1 , α -subelliptic regularity on $${{\,\mathrm{SU}\,}}(3)$$ SU ( 3 ) and compact, semi-simple Lie groups

, Volume 10 (1) – Dec 24, 2019
32 pages

/lp/springer-journals/c-1-alpha-c-1-subelliptic-regularity-on-mathrm-su-3-su-3-and-compact-Gw7jiZIrEk
Publisher
Springer Journals
Subject
Mathematics; Analysis; Mathematical Methods in Physics
ISSN
1664-2368
eISSN
1664-235X
DOI
10.1007/s13324-019-00350-6
Publisher site
See Article on Publisher Site

### Abstract

Let the vector fields $$X_1, \ldots , X_{6}$$ X 1 , … , X 6 form an orthonormal basis of $$\mathcal {H}$$ H , the orthogonal complement of a Cartan subalgebra (of dimension 2) in $${{\,\mathrm{SU}\,}}(3)$$ SU ( 3 ) . We prove that weak solutions u to the degenerate subelliptic p-Laplacian \begin{aligned} \Delta _{\mathcal {H},{p}} u(x)=\sum _{i=1}^{6} X_i^{*}\left( |\nabla _{\!{\mathcal {H}}}u|^{p-2}X_{i}u \right) =0, \end{aligned} Δ H , p u ( x ) = ∑ i = 1 6 X i ∗ | ∇ H u | p - 2 X i u = 0 , have Hölder continuous horizontal derivatives $$\nabla _{\!{\mathcal {H}}}u=(X_1u, \ldots , X_{6}u)$$ ∇ H u = ( X 1 u , … , X 6 u ) for $$p\ge 2$$ p ≥ 2 . We also prove that a similar result holds for all compact connected semisimple Lie groups.

### Journal

Analysis and Mathematical PhysicsSpringer Journals

Published: Dec 24, 2019