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

Potential Analysis -     

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.     

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.     

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.     

Second structure relation for semiclassical orthogonal polynomials

Classical orthogonal polynomials are characterized from their orthogonality and by a first or second structure relation. For the semiclassical orthogonal polynomials (a generalization of the classical ones), we find only the first structure relation in the literature. In this paper, we establish a second structure relation. In particular, we deduce it by means of a general finite-type relation between a semiclassical polynomial sequence and the sequence of its monic derivatives.     

Plurisubharmonic Functions on the Octonionic Plane and <Emphasis Type="Italic">Spin</Emphasis>(9)-Invariant Valuations on Convex Sets

A new class of plurisubharmonic functions on the octonionic plane is introduced. An octonionic version of theorems of A.D. Aleksandrov (Vestnik Leningrad. Univ. Ser. Mat. Meh. Astr. 13(1):5–24, 1958) and Chern-Levine-Nirenberg (Global Analysis, pp. 119–139, 1969), and Błocki (Proc. Am. Math. Soc. 128(12):3595–3599, 2000) are proved. These results are used to construct new examples of continuous translation invariant valuations on convex subsets of . In particular, a new example of Spin(9)-invariant valuation on ℝ¹⁶ is given. Partially supported by ISF grant 1369/04.     

Plurisubharmonic and holomorphic functions relative to the plurifine topology

A weak and a strong concept of plurifinely plurisubharmonic and plurifinely holomorphic functions are introduced. Strong will imply weak. The weak concept is studied further. A function f is weakly plurifinely plurisubharmonic if and only if it is locally bounded from above in the plurifine topology and f°h is finely subharmonic for all complex affine-linear maps h. As a consequence, the regularization in the plurifine topology of a pointwise supremum of such functions is weakly plurifinely plurisubharmonic, and it differs from the pointwise supremum at most on a pluripolar set. Weak plurifine plurisubharmonicity and weak plurifine holomorphy are preserved under composition with weakly plurifinely holomorphic maps.     

A family of univalent polynomials in an angular domain

We study the properties of symmetric polynomials of a particular form and use them to find a maximal angular domain of univalence for some family of polynomials.     

Second structure relation for q-semiclassical polynomials of the Hahn Tableau

The q-classical orthogonal polynomials of the q-Hahn Tableau are characterized from their orthogonality condition and by a first and a second structure relation. Unfortunately, for the q-semiclassical orthogonal polynomials (a generalization of the classical ones) we find only in the literature the first structure relation. In this paper, a second structure relation is deduced. In particular, by means of a general finite-type relation between a q-semiclassical polynomial sequence and the sequence of its q-differences such a structure relation is obtained.     

Metrics and smoothing of translation-invariant Radon transforms along curves

In this paper new L_α^p→L_β^q estimates are proved for translation-invariant Radon transforms along curves for α?β and p<q. For a fixed α and β, if p is sufficiently close to 2 the best possible q is obtained, up to ε. The method is related to that of Greenblatt (Ph.D. Thesis, Princeton University, 1998).     

On the Cegrell Classes Associated to a Positive Closed Current

The aim of this paper is to study the operatoron■ on some classes of plurisubharmonic (psh) functions, which are not necessary bounded, where T is a positive closed current of bidimension (q, q) on an open set ? of C~n. The author introduces two classes F_p~T (?) and■ and shows first that they belong to the domain of definition of the operator■. Then the author proves that all functions that belong to these classes are C_T-quasi-continuous and that the comparison principle is valid for them.     

Plurisubharmonic functions in calibrated geometry and q-convexity

Let (M, ω) be a Kähler manifold. An integrable function ${\varphi}Let (M, ω) be a K?hler manifold. An integrable function j{\varphi} on M is called ω ^q -plurisubharmonic if the current dd^cjùw^q-1{dd^c\varphi\wedge \omega^{q-1}} is positive. We prove that j{\varphi} is ω ^q -plurisubharmonic if and only if j{\varphi} is subharmonic on all q-dimensional complex subvarieties. We prove that a ω ^q -plurisubharmonic function is q-convex, and admits a local approximation by smooth, ω ^q -plurisubharmonic functions. For any closed subvariety Z ì M{Z\subset M} , dim_\mathbbC Z ￡ q-1{\dim_\mathbb{C} Z\leq q-1} , there exists a strictly ω ^q -plurisubharmonic function in a neighbourhood of Z (this result is known for q-convex functions). This theorem is used to give a new proof of Sibony’s lemma on integrability of positive closed (p, p)-forms which are integrable outside of a complex subvariety of codimension ≥  p + 1.     

Inverse functions of polynomials and orthogonal polynomials as operator monotone functions

We study the operator monotonicity of the inverse of every polynomial with a positive leading coefficient. Let be a sequence of orthonormal polynomials and the restriction of to , where is the maximum zero of . Then and the composite are operator monotone on . Furthermore, for every polynomial with a positive leading coefficient there is a real number so that the inverse function of defined on is semi-operator monotone, that is, for matrices , implies

     

The inductive wedge product of positive currents

We define the wedge product of positive currents. Our main issue is about studying the convergence of dd^c_gj∧T$d{d}^c{g}_j\wedge T$ where _gj$g_j$ is a given sequence of plurisubharmonic functions and T is a positive current.     

Continuity of the complex Monge-Ampère operator

The complex Monge-Ampère operator is an important tool in complex analysis. It would be interesting to find the right notion of convergence such that in the weak topology. In this paper, using the -capacity, we give a sufficient condition of the weak convergence . We also show that our condition is quite sharp in some case.

     

Graphs that are not complete pluripolar

Let be domains in . Under very mild conditions on we show that there exist holomorphic functions , defined on with the property that is nowhere extendible across , while the graph of over is not complete pluripolar in . This refutes a conjecture of Levenberg, Martin and Poletsky (1992).

     

On roots of random polynomials

We study the distribution of the complex roots of random polynomials of degree with i.i.d. coefficients. Using techniques related to Rice's treatment of the real roots question, we derive, under appropriate moment and regularity conditions, an exact formula for the average density of this distribution, which yields appropriate limit average densities. Further, using a different technique, we prove limit distribution results for coefficients in the domain of attraction of the stable law.