Morrey-type spaces on Gauss measure spaces and boundedness of singular integrals
Let \(\mathcal{X}\) be a set endowed with a metric \(d\) and a non-negative regular Borel measure \(\mu\). A function \(m:\,\mathcal{X}\rightarrow(0,\infty)\) is called admissible, if for any given \(\tau\in(0,\infty)\), there exists a constant \(\theta_\tau\in[1,\infty)\) such that, for all \(x,\,y\in\mathcal{X}\) satisfying \(d(x,y)\leq \tau m(x)\), it holds that \[ \theta_\tau^{-1}m(y)\leq m(x)\leq\theta_\tau m(y). \] Moreover, for any \(a\in(0,\infty)\), denote by \(\mathcal{B}_a\) the set of all balls \(B\subset\mathcal{X}\) such that \(r_B\leq am(c_B)\), where \(r_B\) and \(c_B\) are, respectively, the center and the radius of \(B\). Balls in the set \(\mathcal{B}_a\) are referred to as admissible balls with scale \(a\). The triple \((\mathcal{X}, d,\mu)\) is called a locally doubling measure metric space if, for every \(a\in(0,\infty)\), there exist constants \(D_a,\,R_a\in(1,\infty)\) such that, for all \(B\in\mathcal{B}_a\), \[ \mu(2B)\leq D_a\mu(B) \] and \[ \mu(2B)\geq R_a\mu(B). \] It is worth pointing out that a typical but non-trivial example of the locally doubling measure metric space is the Gauss measure space \((\mathbb{R}^n,|\cdot|,\gamma)\), where \(|\cdot|\) denotes the usual Euclidean distance and the Gauss measure \(\gamma\) is given by \(d\gamma(x):=\pi^{-n/2}e^{-|x|^2}\,dx\) for all \(x\in\mathbb{R}^n\). In this case, the admissible function \(m\) is given by, , \[ m(x):=\min\left\{1, |x|^{-1}\right\} \;\;\text{for\;all}\;x\in\mathbb{R}^n. \] For \(1\leq q\leq p<\infty\) and \(a\in(0,\infty)\), the Morrey-type space \(\mathcal{M}^{p,q}_{\mathcal{B}_a}(\mathcal{X})\) is defined to be the collection of all locally integrable functions \(f\) such that \[ \|f\|_{\mathcal{M}^{p,q}_{\mathcal{B}_a}(\mathcal{X})}:=\sup_{B\in\mathcal{B}_a}\frac{1}{[\mu(B)]^{1/q-1/p}} \left\{\int_B|f(y)|^q\,d\mu(y)\right\}^{1/q}<\infty. \] For \(a\in(0,\infty)\), the local Hardy-Littlewood maximal operator \(M_{\mathcal{B}_a}\) is defined by setting, for all locally integrable functions \(f\) on \(\mathcal{X}\) and all \(x\in\mathcal{X}\), \[ M_{\mathcal{B}_a}(f)(x):=\sup_{B\ni x,\,B\in\mathcal{B}_a}\frac{1}{\mu(B)}\int_B|f(y)|\,d\mu(y). \] Moreover, denote by \(L^\infty_b(\mu)\) the set of all functions in \(L^\infty(\mu)\) with bounded support. For any \(a\in(0,\infty)\) and \(\beta\in(0,1)\), the local fractional integral operator \(I^{\beta}_a\) is defined by setting, for all \(f\in L^\infty_b(\mu)\) and \(x\in\mathcal{X}\), \[ I^{\beta}_a(f)(x):=\int_{B(x,am(x))}\frac{f(y)}{[\mu(B(x,d(x,y)))]^{1-\beta}}\,d\mu(y). \] In the article under review, the authors establish the boundedness of the local Hardy-Littlewood maximal operator \(M_{\mathcal{B}_a}\) and the local fractional integral operator \(I^{\beta}_a\) on such Morrey-type spaces \(\mathcal{M}^{p,q}_{\mathcal{B}_a}(\mathcal{X})\). Furthermore, the authors study some more properties of such Morrey-type spaces on the Gauss measure space, which are proved to be naturally adapted to singular integrals associated with the Ornstein-Uhlenbeck operator. More precisely, via skillfully applying the locally doubling property and the geometric properties of the Gauss measure, the authors of this article establish the equivalence between Morrey-type spaces and Campanato-type spaces on the Gauss measure space, and the boundedness for a class of singular integrals associated with the Ornstein-Uhlenbeck operator (including Riesz transforms of any order) on Morrey-type spaces over the Gauss measure space.
scientific article; zbMATH DE number 6370413
scientific article