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Juhani Riihentaus

Publications and source records attributed to Juhani Riihentaus.

12 recordsLinked to original sources
arXiv2019

Removability results for subharmonic functions, for separately subharmonic functions, for harmonic functions, for separately harmonic functions and for holomorphic functions, a survey

We begin with an improvement to an extension result for subharmonic functions of Blanchet et al. With the aid of this improvement we then give extension results for subharmonic functions, for separately subharmonic functions, for harmonic functions, for separately harmonic functions and for holomorphic functions. Our results for holomorphic functions are related to Besicovitch's and Shiffman's well-known extension results, at least in some sense. Moreover, we recall another, slightly related and previous extension result for holomorphic functions.

math.AP↗
arXiv2015

First Baire class functions in the pluri-fine topology

Let $B_{1}(Ω, \mathbb R)$ be the first Baire class of real functions in the pluri-fine topology on an open set $Ω\subseteq \mathbb C^{n}$ and let $H_{1}^{*}(Ω, \mathbb R)$ be the first functional Lebesgue class of real functions in the same topology. We prove the equality $B_{1}(Ω, \mathbb R)=H_{1}^{*}(Ω, \mathbb R)$ and show that for every $f\in B_{1}(Ω, \mathbb R)$ there is a separately continuous function $g: Ω^{2} \to\mathbb R$ in the pluri-fine topology on $Ω^2$ such that $f$ is the diagonal of $g.$

math.GN↗
arXiv2015

An inequality type condition for quasinearly subharmonic functions and applications

Generalizing older works of Domar and Armitage and Gardiner, we give an inequality for quasinearly subharmonic functions. As an application of this inequality, we improve Domar's, Rippon's and our previous results concerning the existence of the largest subharmonic minorant of a given function. Moreover, and as an another application, we give a sufficient condition for a separately quasinearly subharmonic function to be quasinearly subharmonic. Our result contains the previous results of Lelong, of Avanissian, of Arsove, of Armitage and Gardiner, and of ours.

math.AP↗
arXiv2013

On separately subharmonic and harmonic functions

We improve our previous generalizations to Arsove's and Ko\lodziej's and Thorbiörnson's results concerning the subharmonicity of a function subharmonic with respect to the first variable and harmonic with respect to the second.

math.AP↗
arXiv2012

Mean value type inequalities for quasinearly subharmonic functions

The mean value inequality is characteristic for upper semicontinuous functions to be subharmonic. Quasinearly subharmonic functions generalize subharmonic functions. We find the necessary and sufficient conditions under which subsets of balls are big enough for the catheterization of nonnegative, quasinearly subharmonic functions by mean value inequalities. Similar result is obtained also for generalized mean value inequalities where, instead of balls, we consider arbitrary bounded sets which have nonvoid interiors and instead of the volume of ball some functions depending on the radius of this ball.

math.AP↗
arXiv2011

Domination conditions for families of quasinearly subharmonic functions

Domar has given a condition that ensures the existence of the largest subharmonic minorant of a given function. Later Rippon pointed out that a modification of Domar's argument gives in fact a better result. Using our previous, rather general and flexible, modification of Domar's original argument, we extend their results both to the subharmonic and quasinearly subharmonic settings.

math.CA↗
arXiv2010

On an inequality related to the radial growth of quasinearly subharmonic functions in locally uniformly homogeneous spaces

We begin by recalling the definition of nonnegative quasinearly subharmonic functions on locally uniformly homogeneous spaces. Recall that these spaces and this function class are rather general: among others subharmonic, quasisubharmonic and nearly subharmonic functions on domains of Euclidean spaces ${\mathbb{R}}^n$, $n\geq 2$, are included. The following result of Gehring and Hallenbeck is classical: Every subharmonic function, defined and ${\mathcal{L}}^p$-integrable for some $p$, $0<p<+\infty$, on the unit disk ${\mathbb{D}}$ of the complex plane ${\mathbb{C}}$ is for almost all $θ$ of the form $o((1-| z|)^{-1/p})$, uniformly as $z\to e^{iθ}$ in any Stolz domain. Recently both Pavlović and Riihentaus have given related and partly more general results on domains of ${\mathbb{R}}^n$, $n\geq 2$. Now we extend one of these results to quasinearly subharmonic functions on locally uniformly homogeneous spaces.

math.AP↗
arXiv2008

On an inequality related to the radial growth of subharmonic functions

It is a classical result that every subharmonic function, defined and ${\mathcal{L}}^p$-integrable for some $p$, $0<p<+\infty$, on the unit disk $\mathbb{D}$ of the complex plane ${\mathbb{C}}$ is for almost all $θ$ of the form $o((1-| z|)^{-1/p})$, uniformly as $z\to e^{iθ}$ in any Stolz domain. Recently Pavlović gave a related integral inequality for absolute values of harmonic functions, also defined on the unit disk in the complex plane. We generalize Pavlović's result to so called quasi-nearly subharmonic functions defined on rather general domains in ${\mathbb {R}}^n$, $n\geq 2$.

math.AP↗
arXiv2008

Quasi-nearly subharmonicity and separately quasi-nearly subharmonic functions

Wiegerinck has shown that a separately subharmonic function need not be subharmonic. Improving previous results of Lelong, of Avanissian, of Arsove and of us, Armitage and Gardiner gave an almost sharp integrability condition which ensures a separately subharmonic function to be subharmonic. Completing now our recent counterparts to the cited results of Lelong, Avanissian and Arsove for so called quasi-nearly subharmonic functions, we present a counterpart to the cited result of Armitage and Gardiner for separately quasi-nearly subharmonic functions. This counterpart enables us to slightly improve Armitage's and Gardiner's original result, too. The method we use is a rather straightforward and technical, but still by no means easy, modification of Armitage's and Gardiner's argument combined with an old argument of Domar.

math.AP↗
arXiv2008

Subharmonic functions, generalizations and separately subharmonic functions

First, we give the definition for quasi-nearly subharmonic functions, now for general, not necessarily nonnegative functions, unlike previously. We point out that our function class incudes, among others, quasisubharmonic functions, nearly subharmonic functions (in a slightly generalized sense) and almost subharmonic functions. We also give some basic properties of quasi-nearly subharmonic functions. Second, after recalling some of the existing subharmonicity results of separately subharmonic functions, we give the corresponding counterparts for separately quasi-nearly subharmonic functions, thus improving previous results of ours, of Lelong, of Avanissian and of Arsove. Third, we give two results concerning the subharmonicity of a function subharmonic with respect to the first variable and harmonic with respect to the second variable. The first result improves a result of Arsove, concerning the case when the function has, in addition, locally a negative integrable minorant. The second result improves a result of Kołodziej and Thorbiörnson concerning the subharmonicity of a function subharmonic and ${\mathcal{C}}^2$ in the first variable and harmonic in the second.

math.AP↗
arXiv2006

Subharmonic functions, mean value inequality, boundary behavior, nonintegrability and exceptional sets

We begin by shortly recalling a generalized mean value inequality for subharmonic functions, and two applications of it: first a weighted boundary behavior result (with some new references and remarks), and then a borderline case result to Suzuki's nonintegrability results for superharmonic and subharmonic functions. The main part of the talk consists, however, of partial improvements to Blanchet's removable singularity results for subharmonic, plurisubharmonic and convex functions.

math.AP↗