Related papers: Disc functionals and Siciak-Zaharyuta extremal fun…

We prove two disc formulas for the Siciak-Zahariuta extremal function of an arbitrary open subset of complex affine space. We use these formulas to characterize the polynomial hull of an arbitrary compact subset of complex affine space in…

We establish disc formulas for the Siciak-Zahariuta extremal function of an arbitrary open subset of complex affine space, generalizing Lempert's formula for the convex case. This function is also known as the pluricomplex Green function…

We use our disc formula for the Siciak-Zahariuta extremal function to characterize the polynomial hull of a connected compact subset of complex affine space in terms of analytic discs.

We prove that for any given upper semicontinuous function $\varphi$ on an open subset $E$ of $\mathbb C^n\setminus\{0\}$, such that the complex cone generated by $E$ minus the origin is connected, the homogeneous Siciak-Zaharyuta function…

For each closed, positive (1,1)-current \omega on a complex manifold X and each \omega-upper semicontinuous function \phi on X we associate a disc functional and prove that its envelope is equal to the supremum of all…

We prove a disc formula for the weighted Siciak-Zahariuta extremal function $V_{X,q}$ for an upper semicontinuous function $q$ on an open connected subset $X$ in $\C^n$. This function is also known as the weighted Green function with…

Using a recent result of L\'arusson and Poletsky regarding plurisubharmonic subextensions we prove a disc formula for the quasiplurisubharmonic global extremal function for domains in complex projective space. As a corollary we get a…

We show that a disc functional on a complex manifold has a plurisubharmonic envelope if all its pullbacks by holomorphic submersions from domains of holomorphy in affine space do and it is locally bounded above and upper semicontinuous in a…

We generalize the Poletsky disc envelope formula for the function $\sup \{u \in \PSH(X,\omega) ; u\leq \phi\}$ on any complex manifold $X$ to the case where the real (1,1)-current $\omega=\omega_1-\omega_2$ is the difference of two positive…

A famous result of Siciak is how the Siciak-Zakharyuta functions, sometimes called global extremal functions or pluricomplex Green functions with a pole at infinity, of two sets relate to the Siciak-Zakharyuta function of their cartesian…

We establish plurisubharmonicity of the envelope of Poisson and Lelong functionals on almost complex manifolds. That is, we generalize the corresponding results for complex manifolds and almost complex manifolds of complex dimension two. We…

We establish the plurisubharmonicity of the envelope of the Poisson functional on almost complex manifolds. That is, we generalize the corresponding result for complex manifolds and almost complex manifolds of complex dimension two.

We prove that the upper envelope of a family of subharmonic functions defined on an open subset of $\mathbb{R}^{N}$, $(N\geq2)$, that is finite every where, is locally bounded above outside a closed nowhere dense set with no bounded…

For a compact subset in a compact Hermitian manifold, we prove that the continuity of the extremal function at a given point in the set is a local property and that the continuity of a weighted extremal function follows from the…

The classical Siciak-Zakharyuta theorem states that the Siciak-Zakharyuta function $V_{E}$ of a subset $E$ of $\mathbb C^n$, also called a pluricomplex Green function or global exremal function of $E$, equals the logarithm of the Siciak…

We study various regularization operators on plurisubharmonic functions that preserve Lelong classes with growth given by certain compact convex sets. The purpose is to show that the weighted Siciak-Zakharyuta functions associated with…

We establish plurisubharmonicity of the envelope of Lelong functional on almost complex manifolds of real dimension four, thereby we generalize the corresponding result for complex manifolds.

In this paper, we combine tools from pluripotential theory and commutative algebra to study singularity invariants of plurisubharmonic functions. We establish several relationships between the singularity invariants of plurisubharmonic…

We prove that the regularity of the extremal function of a compact subset of a compact K\"ahler manifold is a local property, and that the continuity and H\"older continuity are equivalent to classical notions of the local $L$-regularity…

We prove a sharp decay of capacity of sublevel sets of a $(\omega,m)$-subharmonic functions on a $n$-dimensional compact Hermitian manifold $(X,\omega)$ which generalizes the case $m=n$ as well as the case $1\leq m\leq n$ on a compact…