Log truncated threshold and zero mass conjecture

For plurisubharmonic functions $\phi$ and $\psi$ lying in the Cegrell class of ${\mathbb{B}}^n$ and ${\mathbb{B}}^m$ respectively such that the Lelong number of $\phi$ at the origin vanishes, we show that the mass of the origin with respect to the measure $(d{d}^c\max \{\phi (z),\psi (Az)\}{)}^n$ on ${\mathbb{C}}^n$ is zero for $A\in Hom({\mathbb{C}}^n,{\mathbb{C}}^m)={\mathbb{C}}^{nm}$ outside a pluripolar set. For a plurisubharmonic function $\phi$ near the origin in ${\mathbb{C}}^n$, we introduce a new concept coined the log truncated threshold of $\phi$ at $0$ which reflects a singular property of $\phi$ via a log function near the origin (denoted by $lt(\phi ,0)$) and derive an optimal estimate of the residual Monge-Ampère mass of $\phi$ at $0$ in terms of its higher order Lelong numbers ${\nu }_j(\phi )$ at $0$ for $1\le j\le n-1$, in the case that $lt(\phi ,0)<\mathrm{\infty }$. These results provide a new approach to the zero mass conjecture of Guedj and Rashkovskii, and unify and strengthen well-known results about this conjecture.