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## Online first articles

Articles are posted here individually soon after proof is returned from authors, before the corresponding journal issue is completed. All articles are in their final form, including issue number and pagination. For recently accepted articles, see Articles in press.

 $\mathfrak{D}^\perp$-invariant real hypersurfaces in complex Grassmannians of rank two. Ruenn-Huah Lee and Tee-How Loo Let $M$ be a real hypersurface in a complex Grassmannian of rank two. Denote by $\mathfrak{J}$ the quaternionic Kähler structure of the ambient space, $TM^\perp$ the normal bundle over $M$, and $\mathfrak{D}^\perp=\mathfrak{J}TM^\perp$. The real hypersurface $M$ is said to be $\mathfrak{D}^\perp$-invariant if $\mathfrak{D}^\perp$ is invariant under the shape operator of $M$. We show that if $M$ is $\mathfrak{D}^\perp$-invariant, then $M$ is Hopf. This improves the results of Berndt and Suh [Int. J. Math. 23 (2012) 1250103] and [Monatsh. Math. 127 (1999), 1–14]. We also classify $\mathfrak{D}^\perp$ real hypersurfaces in complex Grassmannians of rank two of noncompact type with constant principal curvatures. 197–207 Drazin inverse of singular adjacency matrices of directed weighted cycles. Andrés M. Encinas, Daniel A. Jaume, Cristian Panelo, and Adrián Pastine We present a necessary and sufficient condition for the singularity of circulant matrices associated with directed weighted cycles. This condition is simple and independent of the order of matrices from a complexity point of view. We give explicit and simple formulas for the Drazin inverse of these circulant matrices. We also provide a Bjerhammar-type condition for the Drazin inverse. 209–227 On the atomic and molecular decomposition of weighted Hardy spaces. Pablo Rocha The purpose of this article is to give another molecular decomposition for members of weighted Hardy spaces, different from that given by Lee and Lin [J. Funct. Anal. 188 (2002), no. 2, 442–460], and to review some overlooked details. As an application of this decomposition, we obtain the boundedness on $H^{p}_{w}(\mathbb{R}^{n})$ of every bounded linear operator on some $L^{p_0}(\mathbb{R}^n)$ with $1 < p_0 < +\infty$, for all weights $w \in \mathcal{A}_{\infty}$ and all $0 < p \leq 1$ if $1< \frac{r_w -1}{r_w} p_0$, or all $0 < p < \frac{r_w -1}{r_w} p_0$ if $\frac{r_w -1}{r_w} p_0 \leq 1$, where $r_w$ is the critical index of $w$ for the reverse Hölder condition. In particular, the well-known results about boundedness of singular integrals from $H^{p}_w(\mathbb{R}^{n})$ into $L^{p}_{w}(\mathbb{R}^{n})$ and on $H^{p}_{w}(\mathbb{R}^{n})$ for all $w \in \mathcal{A}_{\infty}$ and all $0 < p \leq 1$ are established. We also obtain the $H^{p}_{w^{p}}(\mathbb{R}^{n})$-$H^{q}_{w^{q}}(\mathbb{R}^{n})$ boundedness of the Riesz potential $I_{\alpha}$ for $0 < p \leq 1$, $\frac{1}{q}=\frac{1}{p} - \frac{\alpha}{n}$, and certain weights $w$. 229–247