The local part and the strong type for operators related to the Gaussian measure.
The author gives results implying weak type \(1\) estimates for the local part of a linear operator associated with solutions of a differential equation given by the Ornstein-Uhlenbeck semigroup. Let \(L = {1 \over 2} \Delta_x - \langle x, \nabla_x\rangle\) be considered on the Euclidean space \(R^n\) where the solution of \({\partial u \over \partial t} = Lu, u(x, 0) = f(x)\) is given by the Ornstein-Uhlenbeck semigroup \[ u(x, t) = \pi^{-1/n} \int_{R^n} M_{e^{-t}}(x, y) f(y) \, dy, \] with \[ M_{e^{-t}}(x, y) = {\exp \left( -{ | y - e^{-t} x| ^2 \over 1 - e^{-2t} } \right) \over (1 - e^{-2t})^{n/2} }. \] The natural measure is the Gaussian measure \(\gamma(x) \, dx = e^{-| x| ^2} \, dx\), and in general, if \(T\) is a sublinear operator, and \(B_h(x) = \{ y: | y -x| \leq n(1 \wedge 1/| x| )\) is a hyperbolic ball about \(x\), the local part of \(T\) is \(T_0 f(x) = T(f \chi_{B_h(x)})(x)\), and its global part is \(T_1 f(x) = T(f \chi_{B_h^c(x)})(x)\). She shows under growth conditions on the kernel and its derivatives of order \(1\) on hyperbolic balls about the diagonal, that if the principal value integral maps \(L^p(\gamma)\) into itself, then its local part is of weak type \(1\) with respect to \(\gamma\). She verifies that the usual Riesz transform operators satisfy the growth conditions. She then verifies that for kernels of the form \[ K_F(x, y) = \int_0^1 r^{m - 1} \left( {-\log r \over 1 - r^2} \right)^{m-2 \over 2} F\left({y - rx \over \sqrt {1 - r^2}}\right) {e^{- | y - r x| ^2 \over 1 - r^2} \over (1 - r^2)^{n/2 + 1}} \, dr, \] where \(F\) is a \(C^1\) function of \(z\) orthogonal to \(\gamma\) that grows not faster than exponential of order \(2\), i.e., \[ \int_{R^n} F(z) e^{- | z| ^2} \, dz = 0, \] and for each \(\varepsilon > 0\), there is a \(C_{\varepsilon}, C_{1, \varepsilon} \) such that \( | F(z)| \leq C_{\varepsilon} e^{\varepsilon | z| ^2}, | \nabla F(z)| \leq C_{1,\varepsilon}e^{\varepsilon | z| ^2}, \) that the associated principal value operator maps \(L^p(\gamma)\) into \(L^p(\gamma)\) for \(1 < p < \infty\) and that its local part is of weak type \(1\) with respect to \(\gamma\). This incorporates the Riesz transforms of higher order as a special case. Finally, she gives growth conditions on an operator for which if \(T\) is sublinear and of weak type \(1\) with respect to Lebesgue measure, its local part \(T_0\) is of weak type \(p\) with respect to \(\gamma\). If it is of strong type \(p\) with respect to Lebesgue measure, then its local part \(T_0\) is also of strong type \(p\) with respect to \(\gamma\). The results apply to vector valued operators including the Littlewood-Paley \(g\) function; a later paper will give results for higher order Littlewood-Paley \(g\)-functions.
scientific article; zbMATH DE number 1777178
scientific article