Sharp estimates for the non-centered maximal operator associated to Gaussian and other radial measures.
Let \(\mu\) be a positive Borel measure on \(\mathbb R^n\). Associated with it, define the non-centered maximal function \[ M_\mu f(x)=\sup_{x\in B}\mu(B)^{-1} \int_B | f(y)| \,d \mu(y), \] where the supremum is taken over all balls \(B\) containing \(x\) with \(\mu(B)>0\). The authors give: For \(n\geq2\) and \(d\mu_\delta(x)=\exp (-| x| ^\delta)dx\) \((\delta>0)\), there exists \(C>0\) such that \[ \mu_\delta\{x\in\mathbb R^n; M_{\mu_\delta}f(x)>\lambda\}\leq C \int_{\mathbb R^n}| f| \lambda^{-1}(1+\log^+| f| /\lambda)^{(n+1)/2} d \mu_\delta,\quad \lambda>0. \] From this naturally follows the \(L^p (d\mu_\delta)\) boundedness \((p>1)\). They state further that their inequality holds for similar measures, i.e., for the measures \(d\mu(x)=\gamma(| x| )dx\) such that \(\gamma (t)\) is strictly decreasing and continuous, and \(\lim_{t\to\infty} \gamma (t)=0\), \(\lim_{t\to0}\gamma(t)<\infty\), and, furthermore, for \(\phi (t)\) defined by \(\gamma(t+\phi(t))=\phi(t)/2\) it holds \(\lim_{t\to \infty}\phi(t)/t=0\). For the Gaussian type measure \(d\mu_\delta (x)\), they also give the following: \[ \mu_\delta \{x\in\mathbb R^n; M_{\mu_\delta}f(x)>\lambda\}\leq C \lambda^{-1} (1+\log^+ \lambda)^{(n-1)/2}\int_{\mathbb R^n}| f| d\mu_\delta +C/ \lambda, \quad \lambda>0. \] They investigate the sharpness of the exponents \((n+1)/2\) and \((n-1)/2\) in the inequalities above.
scientific article; zbMATH DE number 2037178