<p>On a compact connected group <i>G</i>, consider the infinitesimal generator <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$-L$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>-</mo> <mi>L</mi> </mrow> </math></EquationSource> </InlineEquation> of a central symmetric Gaussian convolution semigroup <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$(\mu _t)_{t>0}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mrow> <mo stretchy="false">(</mo> <msub> <mi>μ</mi> <mi>t</mi> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>t</mi> <mo>></mo> <mn>0</mn> </mrow> </msub> </math></EquationSource> </InlineEquation>. We establish several regularity results of the solution to the Poisson equation <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$LU=F$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>L</mi> <mi>U</mi> <mo>=</mo> <mi>F</mi> </mrow> </math></EquationSource> </InlineEquation>, both in strong and weak senses. To this end, we introduce two classes of Lipschitz spaces for <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$1\le p\le \infty $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo>≤</mo> <mi>p</mi> <mo>≤</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation>: <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\Lambda _{\theta }^p$</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mi mathvariant="normal">Λ</mi> <mrow> <mi>θ</mi> </mrow> <mi>p</mi> </msubsup> </math></EquationSource> </InlineEquation>, defined via the associated Markov semigroup, and <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\mathrm L_{\theta }^p$</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mi mathvariant="normal">L</mi> <mrow> <mi>θ</mi> </mrow> <mi>p</mi> </msubsup> </math></EquationSource> </InlineEquation>, defined via the intrinsic distance. In the strong sense, we prove a priori Sobolev regularity and Lipschitz regularity in the class of <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\Lambda _{\theta }^p$</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mi mathvariant="normal">Λ</mi> <mrow> <mi>θ</mi> </mrow> <mi>p</mi> </msubsup> </math></EquationSource> </InlineEquation> space. In the distributional sense, we further show local regularity in the class of <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\mathrm L_{\theta }^{\infty }$</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mi mathvariant="normal">L</mi> <mrow> <mi>θ</mi> </mrow> <mi>∞</mi> </msubsup> </math></EquationSource> </InlineEquation> space. These results require some strong assumptions on <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$-L$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>-</mo> <mi>L</mi> </mrow> </math></EquationSource> </InlineEquation>. Our main techniques build on the differentiability of the associated semigroup, explicit dimension-free <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$L^p$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mi>p</mi> </msup> </math></EquationSource> </InlineEquation> (<InlineEquation ID="IEq11"> <EquationSource Format="TEX">$1<p<\infty $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo><</mo> <mi>p</mi> <mo><</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation>) boundedness of first and second order Riesz transforms, and a comparison between the two Lipschitz norms.</p>

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

期刊论文
出版模式：

开放获取
发表日期：

2026年3月27日

查看全文

Alexander Bendikov,
Li Chen,
Laurent Saloff-Coste

摘要

On a compact connected group G, consider the infinitesimal generator $-L$ $- L$ of a central symmetric Gaussian convolution semigroup $(\mu _t)_{t>0}$ ${(μ_{t})}_{t > 0}$ . We establish several regularity results of the solution to the Poisson equation $LU=F$ $L U = F$ , both in strong and weak senses. To this end, we introduce two classes of Lipschitz spaces for $1\le p\le \infty $ $1 \leq p \leq \infty$ : $\Lambda _{\theta }^p$ $Λ_{θ}^{p}$ , defined via the associated Markov semigroup, and $\mathrm L_{\theta }^p$ $L_{θ}^{p}$ , defined via the intrinsic distance. In the strong sense, we prove a priori Sobolev regularity and Lipschitz regularity in the class of $\Lambda _{\theta }^p$ $Λ_{θ}^{p}$ space. In the distributional sense, we further show local regularity in the class of $\mathrm L_{\theta }^{\infty }$ $L_{θ}^{\infty}$ space. These results require some strong assumptions on $-L$ $- L$ . Our main techniques build on the differentiability of the associated semigroup, explicit dimension-free $L^p$ $L^{p}$ ( $1<p<\infty $ $1 < p < \infty$ ) boundedness of first and second order Riesz transforms, and a comparison between the two Lipschitz norms.