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Plurisubharmonic functions with logarithmic singularities

E. Bedford, B. A. Taylor (1988)

Annales de l'institut Fourier

To a plurisubharmonic function u on C n with logarithmic growth at infinity, we may associate the Robin function ρ u ( z ) = lim sup λ → ∞ u ( λ z ) - log ( λ z ) defined on P n - 1 , the hyperplane at infinity. We study the classes L + , and (respectively) L p of plurisubharmonic functions which have the form u = log ( 1 + | z | ) + O ( 1 ) and (respectively) for which the function ρ u is not identically - ∞ . We obtain an integral formula which connects the Monge-Ampère measure on the space C n with the Robin function on P n - 1 . As an application we obtain a criterion on the convergence of the Monge-Ampère...

Plurisubharmonic saddles

Siegfried Momm (1996)

Annales Polonici Mathematici

A certain linear growth of the pluricomplex Green function of a bounded convex domain of ℂ N at a given boundary point is related to the existence of a certain plurisubharmonic function called a “plurisubharmonic saddle”. In view of classical results on the existence of angular derivatives of conformal mappings, for the case of a single complex variable, this allows us to deduce a criterion for the existence of subharmonic saddles.

Poincaré inequality and Hajłasz-Sobolev spaces on nested fractals

Katarzyna Pietruska-Pałuba, Andrzej Stós (2013)

Studia Mathematica

Given a nondegenerate harmonic structure, we prove a Poincaré-type inequality for functions in the domain of the Dirichlet form on nested fractals. We then study the Hajłasz-Sobolev spaces on nested fractals. In particular, we describe how the "weak"-type gradient on nested fractals relates to the upper gradient defined in the context of general metric spaces.

Pointwise and locally uniform convergence of holomorphic and harmonic functions

Libuše Štěpničková (1999)

Commentationes Mathematicae Universitatis Carolinae

We shall characterize the sets of locally uniform convergence of pointwise convergent sequences. Results obtained for sequences of holomorphic functions by Hartogs and Rosenthal in 1928 will be generalized for many other sheaves of functions. In particular, our Hartogs-Rosenthal type theorem holds for the sheaf of solutions to the second order elliptic PDE's as well as it has applications to the theory of harmonic spaces.

Pointwise inequalities and approximation in fractional Sobolev spaces

David Swanson (2002)

Studia Mathematica

We prove that a function belonging to a fractional Sobolev space L α , p ( ℝ ⁿ ) may be approximated in capacity and norm by smooth functions belonging to C m , λ ( ℝ ⁿ ) , 0 < m + λ < α. Our results generalize and extend those of [12], [4], [14], and [11].

Pointwise inequalities for Sobolev functions and some applications

Bogdan Bojarski, Piotr Hajłasz (1993)

Studia Mathematica

We get a class of pointwise inequalities for Sobolev functions. As a corollary we obtain a short proof of Michael-Ziemer’s theorem which states that Sobolev functions can be approximated by C m functions both in norm and capacity.

Positive Schatten class Toeplitz operators on the ball

Boo Rim Choe, Hyungwoon Koo, Young Joo Lee (2008)

Studia Mathematica

On the harmonic Bergman space of the ball, we give characterizations for an arbitrary positive Toeplitz operator to be a Schatten class operator in terms of averaging functions and Berezin transforms.

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