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Weighted weak type estimates are proved for some maximal operators on the weighted Hardy spaces$H$_{ω}^{$p$}(0 <$p$< 1, ω ∈$A$_{1}) (0<p<1, ω∞A_{1}); in particular, weighted weak type endpoint estimates are obtained for the maximal operators arising from the Bochner-Riesz means and the spherical means on$H$_{ω}^{$p$}.

In this note, we consider the regularity of solutions of the nonlinear elliptic systems of$n$-Laplacian type involving measures, and prove that the gradients of the solutions are in the weak Lebesgue space$L$^{$n$,∞}. We also obtain the$a priori$global and local estimates for the$L$^{$n$,∞}-norm of the gradients of the solutions without using BMO-estimates. The proofs are based on a new lemma on the higher integrability of functions.

Let each point of a homogeneous Poisson process in ℝ^{$d$}independently be equipped with a random number of stubs (half-edges) according to a given probability distribution$μ$on the positive integers. We consider translation-invariant schemes for perfectly matching the stubs to obtain a simple graph with degree distribution$μ$. Leaving aside degenerate cases, we prove that for any$μ$there exist schemes that give only finite components as well as schemes that give infinite components. For a particular matching scheme which is a natural extension of Gale–Shapley stable marriage, we give sufficient conditions on$μ$for the absence and presence of infinite components.

In this addendum note we fill in the gap left in the description of 2D homogeneous solutions to the stationary Euler system initiated in the previous publication. This completes the classification of all homogeneous stationary solutions. The note includes updated classification tables.