The convergence in \(L^1\) of singular integrals in harmonic analysis and ergodic theory
The paper is devoted to the extension of results on the \(L^1\)-convergence of the truncated singular integrals of integrable functions to its limit in the case of Cesàro-bounded semiflows. Consider a Calderón-Zygmund kernel \(K(y)\) with support in the positive halfline \((0,\infty)\) and a nonsingular semiflow \(\{\tau_t, t\geq 0\}\). For \(f\in L^1(\mu)\), let \[ T_\varepsilon f(x)= \int^{1/\varepsilon}_\varepsilon K(t) f(\tau_t x) dt. \] The semiflow is assumed to be Cesàro-bounded. \(T_\varepsilon f\) converges in measure to the ergodic singular integral \(Tf\) for \(\varepsilon\to 0\). The author shows that convergence in \(L^1\)-norm holds if \(Tf\in L^1(\mu)\) and the kernel satisfies a weak condition. This generalizes joint results of the author with F. J. Martín-Reyes on ergodic Hilbert transforms.
scientific article; zbMATH DE number 1441223