Partial Harmonicity of Continuous Maximal Plurisubharmonic Functions

If u is a sufficiently smooth maximal plurisubharmonic function such that the complex Hessian of u has constant rank, it is known that there exists a foliation by complex manifolds, such that u is harmonic along the leaves of the foliation. In this paper, we show a partial analogue of this result for maximal plurisubharmonic functions that are merely continuous, without the assumption on the complex Hessian. In this setting, we cannot expect a foliation by complex manifolds, but we prove the existence of positive currents of bidimension (1, 1) such that the function is harmonic along the currents.     

Holomorphic Minorants of Plurisubharmonic Functions

Let φ be a plurisubharmonic function on a pseudoconvex domain D in an n-dimensional complex space. We show that there exists a nonzero holomorphic function f on D such that some local mean value of φ with logarithmic additional terms majorizes log|f|. A similar problem is discussed for a locally integrable function on D in terms of balayage of positive measures.     

多重次调和函数的一个 Phragmen-Lindelof定理

文给出了 C^N中锥上关于多重次调和函数的一个 Phragme-Lindelof 定理, 从而推广了文献[8]的一个结论.     

Some Weighted Energy Classes of Plurisubharmonic Functions

In this paper we study relations between the weighted energy class E_cmathcal{E}_{chi} introduced by S. Benelkourchi, V. Guedj and A. Zeriahi recently with Cegrell’s classes Emathcal{E} and Nmathcal{N}. Next we establish a generalized comparison principle for the operator M _χ . As an application, we prove a version of existence of solutions of Monge–Ampère type equations in the class E_c(H,W)mathcal{E}_{chi}(H,Omega).     

Bounded Plurisubharmonic Exhaustion Functions and Levi-flat Hypersurfaces

In this paper, we survey some recent results on the existence of bounded plurisubharmonic functions on pseudoconvex domains, the Diederich–Forn?ss exponent and its relations with existence of domains with Levi-flat boundary in complex manifolds.     

A Bounded Plurisubharmonic Function Which is not an Infinite Sum of Continuous Plurisubharmonic Functions

Potential Analysis -     

Differential Tests for Plurisubharmonic Functions and Koch Curves

Potential Analysis - We study minimum sets of singular plurisubharmonic functions and their relation to upper contact sets. In particular we develop an algorithm checking when a naturally...     

Minimum Principle for Plurisubharmonic Functions and Related Topics

This is a survey about some recent developments of the minimum principle for plurisubharmonic functions and related topics.     

Maximal Hilbert Functions

The set of Hilbert functions of standard graded algebras is considered as a partially ordered set under numerical comparison. For the set of algebras H(d, e₀), of a given dimension d and multiplicity e₀, we describe the requirements its maximal elements must satisfy; under fairly general conditions, the extremal functions arise from Cohen-Macaulay algebras. We also examine the subset H(d, e₀, e₁), of those functions whose first two coefficients of their Hilbert polynomials are assigned. Finally, we show how these results and the use of certain extended multiplicities can be used to prove finiteness theorems for the number of corresponding functions.     

Plurisubharmonic domination

For a large class of separable Banach spaces we prove the following. Given a pseudoconvex open and that is locally bounded above, there is a plurisubharmonic such that . We also discuss applications of this result.

     

多重次调和函数的一个Phragmén-Lindel(o)f定理

该文给出了CN中锥上关于多重次调和函数的一个Phragmé-Lindel(o)f定理,从而推广了文献[8]的一个结论.     

一类具有极值亏量和的亚纯函数

本文结合亚纯函数的导函数，得到一个更一般的亏量和不等式，并对其极值情形进行了讨论，在更广泛的条件下，证明了相应的Ｆ．Ｎｅｖａｎｌｉｎｎａ猜测．     

Local Lipschitz-constant Functions and Maximal Subdifferentials

It is shown that if k(x) is an upper semicontinuous and quasi lower semicontinuous function on a Banach space X, then k(x)B _X* is the Clarke subdifferential of some locally Lipschitz function on X. Related results for approximate subdifferentials are also given. Moreover, on smooth Banach spaces, for every locally Lipschitz function with minimal Clarke subdifferential, one can obtain a maximal Clarke subdifferential map via its local Lipschitz-constant function. Finally, some results concerning the characterization and calculus of local Lipschitz-constant functions are developed.     

Spherical Maximal Operators on Radial Functions

Let A_tf(x) denote the mean of f over a sphere of radius t and center x. We prove sharp estimates for the maximal function M_E f(X) = sup_t∈_E |Atf(x)| where E is a fixed set in IR⁺ and f is a radial function ∈ L^p(IR^d). Let P_d = d/(d?1) (the critical exponent for Stein's maximal function). For the cases (i) p < p_d, d ? 2, and (ii) p = p_d, d ? 3, and for p ? q ? ∞ we prove necessary and sufficient conditions on E for M_E to map radial functions in L^p to the Lorentz space L^P,q.     

Best Constants for Uncentred Maximal Functions

We precisely evaluate the operator norm of the uncentred Hardy–Littlewoodmaximal function on L^p(R¹). Consequently, we compute the operatornorm of the ‘strong’ maximal function on L^p(Rⁿ),and we observe that the operator norm of the uncentred Hardy–Littlewoodmaximal function over balls on L^p(Rⁿ) grows exponentially asn. 1991 Mathematics Subject Classification 42B25.     

Pseudo-Simplicial Complexes from Maximal Locally Convex Functions

We introduce and discuss pseudo-simplicial complexes in ℝ^d as generalizations of pseudo-triangulations in ℝ². Our approach is based on the concept of maximal locally convex functions on polytopal domains.     

Maximal Functions Associated with The Jacobi Transform

In this paper, we define a Hardy–Littlewood maximal functionoperator M associated with the Jacobi transform, and prove thatM is of weak (1, 1) and strong (p, p) (p > 1) type. 1991Mathematics Subject Classification 42B25, 44A15.     

Plurisubharmonic Defining Functions, Good Vector Fields, and Exactness of a Certain One Form

 We show that the approaches to global regularity of the -Neumann problem via the methods listed in the title are equivalent when the conditions involved are suitably modified. These modified conditions are also equivalent to one that is relevant in the context of Stein neighborhood bases and Mergelyan type approximation.