We introduce a pointwise definition of Lipschitz (also called Hölder) spaces adapted to the parabolic Hermite operator \(\mathbb {H}= \partial _t- \Delta _x+|x|^2\) on \(\mathbb {{R}}^{n+1}\). Also for every \(\alpha >0\), we define the following spaces by means of the Poisson semigroup of \(\mathbb {H}\), \(\mathcal {P}_y^{\mathbb {H}}=e^{-y\sqrt{\mathbb {H}}}\):

with the obvious norm. We prove that both spaces do coincide and their norms are equivalent. For the harmonic oscillator, \(\mathcal {{H}}=-\Delta _x+|x|^2\), Stinga and Torrea introduced in 2011 adapted Hölder classes. Parallel to the parabolic case, we characterize these pointwise Hölder spaces via the \(L^\infty \) norm of the derivatives of the Poisson and heat semigroups, \(e^{-y\sqrt{\mathcal {{H}}}}\) and \(e^{-\tau \mathcal {{H}}}\), respectively. As important applications of these semigroups characterizations, we get regularity results regarding the boundedness in these adapted Lipschitz spaces of operators related to \(\mathbb {H}\) and \(\mathcal {{H}}\) as fractional (positive and negative) powers, Bessel potentials, Hermite Riesz transforms, and Laplace transform multipliers, in a more direct way. The proofs use in a fundamental way the semigroup definition of the operators considered along the paper. The non-convolution structure of the operators produces an extra difficulty on the arguments.

我们引入李普希茨（也称为保持体）的逐点定义的空间适合于抛物面埃尔米特操作者\（\ mathbb {H} = \局部_t- \德尔塔_x + | X | ^ 2 \）上\（\ mathbb {{R} } ^ {n + 1} \）。同样，对于每个\（\ alpha> 0 \），我们也通过\（\ mathbb {H} \），\（\ mathcal {P} _y ^ {\ mathbb {H}}的泊松半群来定义以下空间= e ^ {-y \ sqrt {\ mathbb {H}}} \）：

具有明显的规范。我们证明两个空间确实重合并且它们的范数是等价的。对于谐波振荡器，\（\ mathcal {{H}} =-\ Delta _x + | x | ^ 2 \），Stinga和Torrea于2011年引入，采用了Hölder类。与抛物线情形平行，我们通过泊松和热半群的导数\（e ^ {-y \ sqrt {\ mathcal {{H}} ）的\（L ^ \ infty \）范数来刻画这些点式Hölder空间}} \）和\（e ^ {-\ tau \ mathcal {{H}}} \）。作为这些半群特征的重要应用，我们获得了与\（\ mathbb {H} \）和\（\ mathcal {{H}} \）相关的算符在这些适应的Lipschitz空间中的有界性的正则性结果作为分数（正和负）幂，贝塞尔势，埃尔米·里兹变换和拉普拉斯变换乘数，以更直接的方式表示。证明从根本上使用了本文中所考虑的算子的半群定义。运算符的非卷积结构在参数上产生了额外的困难。