Riesz Transforms, Function Spaces, and Their Applications on Infinite Dimensional Compact Groups

Abstract On a compact connected group G , consider the infinitesimal generator $$-L$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>-</mml:mo> <mml:mi>L</mml:mi> </mml:mrow> </mml:math> of a central symmetric Gaussian convolution semigroup $$(\mu _t)_{t&gt;0}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>μ</mml:mi> <mml:mi>t</mml:mi> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> <mml:mrow> <mml:mi>t</mml:mi> <mml:mo>&gt;</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:msub> </mml:math> . We establish several regularity results of the solution to the Poisson equation $$LU=F$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>L</mml:mi> <mml:mi>U</mml:mi> <mml:mo>=</mml:mo> <mml:mi>F</mml:mi> </mml:mrow> </mml:math> , both in strong and weak senses. To this end, we introduce two classes of Lipschitz spaces for $$1\le p\le \infty $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>≤</mml:mo> <mml:mi>p</mml:mi> <mml:mo>≤</mml:mo> <mml:mi>∞</mml:mi> </mml:mrow> </mml:math> : $$\Lambda _{\theta }^p$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>Λ</mml:mi> <mml:mrow> <mml:mi>θ</mml:mi> </mml:mrow> <mml:mi>p</mml:mi> </mml:msubsup> </mml:math> , defined via the associated Markov semigroup, and $$\mathrm L_{\theta }^p$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>L</mml:mi> <mml:mrow> <mml:mi>θ</mml:mi> </mml:mrow> <mml:mi>p</mml:mi> </mml:msubsup> </mml:math> , defined via the intrinsic distance. In the strong sense, we prove a priori Sobolev regularity and Lipschitz regularity in the class of $$\Lambda _{\theta }^p$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>Λ</mml:mi> <mml:mrow> <mml:mi>θ</mml:mi> </mml:mrow> <mml:mi>p</mml:mi> </mml:msubsup> </mml:math> space. In the distributional sense, we further show local regularity in the class of $$\mathrm L_{\theta }^{\infty }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>L</mml:mi> <mml:mrow> <mml:mi>θ</mml:mi> </mml:mrow> <mml:mi>∞</mml:mi> </mml:msubsup> </mml:math> space. These results require some strong assumptions on $$-L$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>-</mml:mo> <mml:mi>L</mml:mi> </mml:mrow> </mml:math> . Our main techniques build on the differentiability of the associated semigroup, explicit dimension-free $$L^p$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>L</mml:mi> <mml:mi>p</mml:mi> </mml:msup> </mml:math> ( $$1&lt;p&lt;\infty $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>&lt;</mml:mo> <mml:mi>p</mml:mi> <mml:mo>&lt;</mml:mo> <mml:mi>∞</mml:mi> </mml:mrow> </mml:math> ) boundedness of first and second order Riesz transforms, and a comparison between the two Lipschitz norms.