Weighted maximal estimates along curve associated with dispersive equations

In the present paper, we give the local and global weighted [Formula: see text] maximal estimate for the operator [Formula: see text] which is defined by [Formula: see text] where [Formula: see text] satisfies some growth conditions, [Formula: see text] is a pseudo-differential operator with symbol [Formula: see text] and [Formula: see text] satisfies Hölder’s condition of order [Formula: see text] and bilipschitz conditions. As a corollary of the above conclusions, we show that if [Formula: see text] for [Formula: see text] and [Formula: see text], then [Formula: see text] In particular, if taking [Formula: see text], then this improves a result in [C. Cho, S. Lee and A. Vargas, Problems on pointwise convergence of solutions to the Schrödinger equation, J. Fourier Anal. Appl. 18 (2012) 972–994], where [Formula: see text] holds for [Formula: see text] only.

Almost Everywhere Convergence of Convolution Measures

Let (X, ℬ, m, τ) be a dynamical system with (X, ℬ, m) a probability space and τ an invertible, measure preserving transformation. This paper deals with the almost everywhere convergence in L1(X) of a sequence of operators of weighted averages. Almost everywhere convergence follows once we obtain an appropriate maximal estimate and once we provide a dense class where convergence holds almost everywhere. The weights are given by convolution products of members of a sequence of probability measures {vi} defined on ℤ. We then exhibit cases of such averages where convergence fails.