On some classes of ω-plurisubharmonic functions on compact Kähler manifolds

Abstract

In this paper we establish a relation between classes of ω-plurisubharmonic functions $\mathcal{E}_{p}(X,\omega), p>0$ and DMA_loc(X,ω) introduced and investigated by Guedj and Zeriahi (J. Funct. Anal., 250:442–482, 2007) and Cegrell (Acta Math., 180:187–217, 1998; Ann. Inst. Fourier (Grenoble), 54:159–179, 2004), respectively, on a compact Kähler manifold X. We show that $\mathcal{E}_{n-1}(X,\omega)\subset\mathrm{DMA}_{\mathrm {loc}}(X,\omega)$ but $\bigcap_{0 < p < n-1}\mathcal{E}_{p}(\mathbb{CP}^{n},\omega)\not \subset\mathrm{DMA}_{\mathrm{loc}}(\mathbb{CP}^{n},\omega)$. At the same time we investigate a relation between the classes $\mathrm{DMA} (\mathbb{C}^{n})\cap\mathcal{L}(\mathbb{C}^{n})$ and $\mathrm {DMA}(\mathbb{CP}^{n},\omega)$, as well as $\widehat{\mathrm{DMA}}(\mathbb{C}^{n})\cap\mathcal {L}(\mathbb{C}^{n})$ and $\widehat{\mathrm{DMA}}$ $(\mathbb{CP}^{n},\omega)$.