Almost everywhere summability of Laguerre series. II

K. Stempak

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On a Summability Method Defined by Means of Hermite Polynomials Върху един метод на сумиране, дефиниран чрез полиномите на Ермит

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Here we prove results about Riesz summability of classical Laguerre series, locally uniformly or on the Lebesgue set of the function f such that (∫(1 + x)^(mp) |f(x)|^p dx )^(1/p) < ∞, for some p and m satisfying 1 ≤ p ≤ ∞, −∞ < m < ∞.

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Almost everywhere summability of Laguerre series

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We apply a construction of generalized twisted convolution to investigate almost everywhere summability of expansions with respect to the orthonormal system of functions $ℓ_{n}^{a} (x) = {(n! / Γ (n + a + 1))}^{1 / 2} e^{- x / 2} L_{n}^{a} (x)$ , n = 0,1,2,..., in $L^{2} (ℝ_{+}, x^{a} d x)$ , a ≥ 0. We prove that the Cesàro means of order δ > a + 2/3 of any function $f \in L^{p} (x^{a} d x)$ , 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.

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In this paper, we prove that integral $n$ -varifolds $μ$ in codimension 1 with $H_{μ} \in L_{loc}^{p} (μ)$ , $p > n$ , $p \geq 2$ have quadratic tilt-excess decay ${tiltex}_{μ} (x, ϱ, T_{x} μ) = O_{x} (ϱ^{2})$ for $μ$ -almost all $x$ , and a strong maximum principle which states that these varifolds cannot be touched by smooth manifolds whose mean curvature is given by the weak mean curvature $H_{μ}$ , unless the smooth manifold is locally contained in the support of $μ$ .