Riesz and Bessel potentials, the \(g^k\) functions and an area function for the Gaussian measure \(\gamma\) (Q2767706)

Let \(L=\frac 12\Delta_x-x\cdot\nabla_x\) be the Ornstein-Uhlenbeck operator on \(\mathbb R^n\), which is self-adjoint with respect to the Gaussian measure \(d\gamma=e^{-\left|x\right|^2}dx\). Denote by \(T_t=e^{tL}\) the Ornstein-Uhlenbeck semigroup and by \(P_t=e^{-t(-L)^{1/2}}\) the Poisson semigroup. NEWLINENEWLINENEWLINEThe authors study the Riesz potentials \(I_{\alpha}^{\gamma}=(-L)^{-\alpha}\) for \(\alpha>0\). They show that \(I_{\alpha}^{\gamma}\) is \(L^p(d\gamma)\)-bounded for \(1<p<+\infty\), which follows from the hypercontractivity of \((T_t)_{t>0}\). Note that \(I_{\alpha}^{\gamma}\) is not of weak type \((1,1)\) with respect to \(d\gamma\) (this is due to Garcia-Cuerva, Mauceri, Sjögren and Torrea). The authors also prove that, contrary to the classical case, \(I_{\alpha}^{\gamma}\) is not of strong type \((p,q)\) with \(1/q=1/p-\alpha/d\). However, an \(L^p\log L(d\gamma)\) inequality can be obtained. NEWLINENEWLINENEWLINELittlewood-Paley-Stein functionals are also considered. Define, for \(k\in \mathbb N\), \(k\geq 1\), NEWLINE\[NEWLINE g^k_Tf(x)=\left(\int_0^{+\infty} \left|t^k\frac{\partial^k}{\partial t^k}P_tf(x)\right|^2 \frac{dt}t\right)^{1/2} NEWLINE\]NEWLINE and NEWLINE\[NEWLINE g^k_Sf(x)=\left(\int_0^{+\infty} \left|t^k\nabla^kP_tf(x)\right|^2 \frac{dt}t\right)^{1/2}. NEWLINE\]NEWLINE It was proved by \textit{C. E. Gutierrez, C. Segovia} and \textit{J. L. Torrea} [J. Fourier Anal. Appl. 2, No. 6, 583-596 (1996; Zbl 0893.42007)] that these functionals are \(L^p(d\gamma)\) bounded for \(1<p<+\infty\). For \(k=1\) and \(k=2\), they are also of weak type \((1,1)\) (this is due to Scotto). For \(k\geq 3\), the authors prove that \(g^k_T\) is of weak type \((1,1)\), but \(g^k_S\) is not. NEWLINENEWLINENEWLINEFinally, a Lusin area functional for \(\gamma\) is defined and it is shown that this functional is \(L^p(d\gamma)\)-bounded for all \(1<p<+\infty\).