In this work we first introduce the concept of Poisson Stepanov-like almost automorphic (Poisson S2−almost automorphic) processes in distribution. We establish some interesting results on the functional space of such processes like an composition theorems. Next, under some suitable assumptions, we establish the existence, the uniqueness and the stability of the square-mean almost automorphic solutions in distribution to a class of abstract stochastic evolution equations driven by Lévy noise in case...

Let T be a d×d matrix with integer entries and with eigenvalues >1 in modulus. Let f be a lipschitzian function of positive order. We prove that the series ${\sum }_{n=1}^{\infty }{c}_{n}f\left(T^{n}x\right)$ converges almost everywhere with respect to Lebesgue measure provided that ${\sum }_{n=1}^{\infty }{\left|c_n\right|}^{2}lo{g}^{2}n<\infty$.

The maximal operator S⁎ for the spherical summation operator (or disc multiplier) $S_R$ associated with the Jacobi transform through the defining relation $\widehat{S_{R}f}\left(\lambda \right)=1_{\left|\lambda \right|\le R}f̂\left(t\right)$ for a function f on ℝ is shown to be bounded from $L^p(ℝ₊,d\mu )$ into $L^p(ℝ,d\mu )+L²(ℝ,d\mu )$ for (4α + 4)/(2α + 3) < p ≤ 2. Moreover S⁎ is bounded from $L^{p₀,1}(ℝ₊,d\mu )$ into $L^{p₀,\infty }(ℝ,d\mu )+L²(ℝ,d\mu )$. In particular ${S_{R}f\left(t\right)}_{R>0}$ converges almost everywhere towards f, for $f\in L^p(ℝ₊,d\mu )$, whenever (4α + 4)/(2α + 3) < p ≤ 2.

We apply a construction of generalized twisted convolution to investigate almost everywhere summability of expansions with respect to the orthonormal system of functions ${\ell }_n^a\left(x\right)={(n!/\Gamma (n+a+1))}^{1/2}{e}^{-x/2}{L}_n^a\left(x\right)$, n = 0,1,2,..., in $L^2({ℝ}_{+},x^{a}dx)$, a ≥ 0. We prove that the Cesàro means of order δ > a + 2/3 of any function $f\in L^p\left(x^{a}dx\right)$, 1 ≤ p ≤ ∞, converge to f a.e. The main tool we use is a Hardy-Littlewood type maximal operator associated with a generalized Euclidean convolution.

Using methods from [9] we prove the almost everywhere convergence of the Cesàro means of Laguerre series associated with the system of Laguerre functions $L_n^a\left(x\right)={(n!/\Gamma (n+a+1))}^{1/2}{e}^{-x/2}{x}^{a/2}{L}_n^a\left(x\right)$, n = 0,1,2,..., a ≥ 0. The novel ingredient we add to our previous technique is the $A_p$ weights theory. We also take the opportunity to comment and slightly improve on our results from [9].