scholarly journals Estimates of Henstock-Kurzweil Poisson Integrals

2005 ◽  
Vol 48 (1) ◽  
pp. 133-146 ◽  
Author(s):  
Erik Talvila
Keyword(s):  
Dirichlet Problem ◽  
Boundary Data ◽  
Borel Measure ◽  
Poisson Integral ◽  
Uniqueness Theorems ◽  
Norm Estimates ◽  
Poisson Integrals ◽  
Finite Borel Measure ◽  
Harmonic Hardy Space ◽  
Similar Growth

AbstractIf f is a real-valued function on [−π, π] that is Henstock-Kurzweil integrable, let ur(θ) be its Poisson integral. It is shown that ∥ur∥p = o(1/(1 − r)) as r → 1 and this estimate is sharp for 1 ≤ p ≤ ∞. If μ is a finite Borel measure and ur(θ) is its Poisson integral then for each 1 ≤ p ≤ ∞ the estimate ∥ur∥p = O((1−r)1/p−1) as r → 1 is sharp. The Alexiewicz norm estimates ∥ur∥ ≤ ∥f ∥ (0 ≤ r < 1) and ∥ur − f∥ → 0 (r → 1) hold. These estimates lead to two uniqueness theorems for the Dirichlet problem in the unit disc with Henstock-Kurzweil integrable boundary data. There are similar growth estimates when u is in the harmonic Hardy space associated with the Alexiewicz norm and when f is of bounded variation.

scholarly journals Dirichlet Problem for the Schrödinger Operator in a Half Space

2012 ◽  
Vol 2012 ◽  
pp. 1-14 ◽  
Author(s):  
Baiyun Su
Keyword(s):  
Dirichlet Problem ◽  
Half Space ◽  
Schrödinger Operator ◽  
Boundary Data ◽  
Poisson Kernel ◽  
Boundary Function ◽  
Poisson Integral ◽  
Schrodinger Operator ◽  
Generalized Poisson ◽  
Continuous Boundary

For continuous boundary data, the modified Poisson integral is used to write solutions to the half space Dirichlet problem for the Schrödinger operator. Meanwhile, a solution of the Poisson integral for any continuous boundary function is also given explicitly by the Poisson integral with the generalized Poisson kernel depending on this boundary function.

scholarly journals Weighted Pluricomplex Energy II

2015 ◽  
Vol 2015 ◽  
pp. 1-8
Author(s):  
Slimane Benelkourchi
Keyword(s):  
Dirichlet Problem ◽  
Orlicz Space ◽  
Boundary Data ◽  
Borel Measure ◽  
A Priori Estimates ◽  
A Priori ◽  
Unique Function ◽  
A Priori Bounds ◽  
Priori Estimates ◽  
Monge Ampere

We continue our study of the complex Monge-Ampère operator on the weighted pluricomplex energy classes. We give more characterizations of the range of the classes Eχ by the complex Monge-Ampère operator. In particular, we prove that a nonnegative Borel measure μ is the Monge-Ampère of a unique function φ∈Eχ if and only if χ(Eχ)⊂L1(dμ). Then we show that if μ=(ddcφ)n for some φ∈Eχ then μ=(ddcu)n for some φ∈Eχ, where f is given boundary data. If moreover the nonnegative Borel measure μ is suitably dominated by the Monge-Ampère capacity, we establish a priori estimates on the capacity of sublevel sets of the solutions. As a consequence, we give a priori bounds of the solution of the Dirichlet problem in the case when the measure has a density in some Orlicz space.

On Dirichlet problem for Beltrami equations in finitely connected domains

2021 ◽  
Vol 34 ◽  
pp. 85-99
Author(s):  
Ihor Petkov ◽  
Vladimir Ryazanov
Keyword(s):  
Dirichlet Problem ◽  
Boundary Data ◽  
Boundary Behavior ◽  
Beltrami Equation ◽  
Elliptic Systems ◽  
Beltrami Equations ◽  
Prime Ends ◽  
Wide Circle ◽  
Continuous Boundary ◽  
Homeomorphic Solution

Boundary value problems for the Beltrami equations are due to the famous Riemann dissertation (1851) in the simplest case of analytic functions and to the known works of Hilbert (1904, 1924) and Poincare (1910) for the corresponding Cauchy--Riemann system. Of course, the Dirichlet problem was well studied for uniformly elliptic systems, see, e.g., \cite{Boj} and \cite{Vekua}. Moreover, the corresponding results on the Dirichlet problem for degenerate Beltrami equations in the unit disk can be found in the monograph \cite{GRSY}. In our article \cite{KPR1}, see also \cite{KPR3} and \cite{KPR5}, it was shown that each generalized homeomorphic solution of a Beltrami equation is the so-called lower $Q-$homeomorphism with its dilatation quotient as $Q$ and developed on this basis the theory of the boundary behavior of such solutions. In the next papers \cite{KPR2} and \cite{KPR4}, the latter made possible us to solve the Dirichlet problem with continuous boundary data for a wide circle of degenerate Beltrami equations in finitely connected Jordan domains, see also [\citen{KPR5}--\citen{KPR7}]. Similar problems were also investigated in the case of bounded finitely connected domains in terms of prime ends by Caratheodory in the papers [\citen{KPR9}--\citen{KPR10}] and [\citen{P1}--\citen{P2}]. Finally, in the present paper, we prove a series of effective criteria for the existence of pseudo\-re\-gu\-lar and multi-valued solutions of the Dirichlet problem for the degenerate Beltrami equations in arbitrary bounded finitely connected domains in terms of prime ends by Caratheodory.

The Dirichlet problem for the complex Hessian operator in the class $\mathcal{N}_m(\Omega,f)$

2021 ◽  
Vol 127 (2) ◽  
pp. 287-316
Author(s):  
Ayoub El Gasmi
Keyword(s):  
Weak Solution ◽  
Dirichlet Problem ◽  
Borel Measure ◽  
Positive Borel Measure ◽  
Hessian Equation ◽  
Hessian Operator ◽  
Hyperconvex Domain ◽  
Complex Hessian Operator

Let $\Omega\subset \mathbb{C}^{n}$ be a bounded $m$-hyperconvex domain, where $m$ is an integer such that $1\leq m\leq n$. Let $\mu$ be a positive Borel measure on $\Omega$. We show that if the complex Hessian equation $H_m (u) = \mu$ admits a (weak) subsolution in $\Omega$, then it admits a (weak) solution with a prescribed least maximal $m$-subharmonic majorant in $\Omega$.

scholarly journals Dirichlet problem for Poisson equations in Jordan domains

2018 ◽  
Vol 32 ◽  
pp. 30-41
Author(s):  
Vladimir Gutlyanskii ◽  
Vladimir Ryazanov ◽  
Eduard Yakubov
Keyword(s):  
Dirichlet Problem ◽  
Boundary Data ◽  
Continuous Functions ◽  
Logarithmic Capacity ◽  
Hölder Continuous ◽  
Arbitrary Boundary ◽  
Poisson Equations ◽  
Local Properties ◽  
Continuous Boundary ◽  
Angular Limits

First, we study the Dirichlet problem for the Poisson equations \(\triangle u(z) = g(z)\) with \(g\in L^p\), \(p>1\), and continuous boundary data \(\varphi :\partial D\to\mathbb{R}\) in arbitrary Jordan domains \(D\) in \(\mathbb{C}\) and prove the existence of continuous solutions \(u\) of the problem in the class \(W^{2,p}_{\rm loc}\). Moreover, \(u\in W^{1,q}_{\rm loc}\) for some \(q>2\) and \(u\) is locally Hölder continuous. Furthermore, \(u\in C^{1,\alpha}_{\rm loc}\) with \(\alpha = (p-2)/p\) if \(p>2\). Then, on this basis and applying the Leray-Schauder approach, we obtain the similar results for the Dirichlet problem with continuous data in arbitrary Jordan domains to the quasilinear Poisson equations of the form \(\triangle u(z) = h(z)\cdot f(u(z))\) with the same assumptions on \(h\) as for \(g\) above and continuous functions \(f:\mathbb{R}\to\mathbb{R}\), either bounded or with nondecreasing \(|f\,|\) of \( |t\,|\) such that \(f(t)/t \to 0\) as \(t\to\infty\). We also give here applications to mathematical physics that are relevant to problems of diffusion with absorbtion, plasma and combustion. In addition, we consider the Dirichlet problem for the Poisson equations in the unit disk \(\mathbb{D}\subset\mathbb{C}\) with arbitrary boundary data \(\varphi :\partial\mathbb{D}\to\mathbb{R}\) that are measurable with respect to logarithmic capacity. Here we establish the existence of continuous nonclassical solutions \(u\) of the problem in terms of the angular limits in \(\mathbb D\) a.e. on \(\partial\mathbb D\) with respect to logarithmic capacity with the same local properties as above. Finally, we extend these results to almost smooth Jordan domains with qusihyperbolic boundary condition by Gehring-Martio.

Convex cores of measures on R d

2001 ◽  
Vol 38 (1-4) ◽  
pp. 177-190 ◽  
Author(s):  
Imre Csiszár ◽  
F. Matúš
Keyword(s):  
Borel Measure ◽  
Convex Sets ◽  
Probability Measures ◽  
Countable Number ◽  
Borel Sets ◽  
Finite Borel Measure ◽  
Number Of Faces ◽  
Convex Core

We define the convex core of a finite Borel measure Q on R d as the intersection of all convex Borel sets C with Q(C) =Q(R d). It consists exactly of means of probability measures dominated by Q. Geometric and measure-theoretic properties of convex cores are studied, including behaviour under certain operations on measures. Convex cores are characterized as those convex sets that have at most countable number of faces.

scholarly journals Besicovitch Covering Property on graded groups and applications to measure differentiation

2019 ◽  
Vol 2019 (750) ◽  
pp. 241-297 ◽  
Author(s):  
Enrico Le Donne ◽  
Séverine Rigot
Keyword(s):  
Riemannian Manifold ◽  
Borel Measure ◽  
Blow Up ◽  
General Study ◽  
Covering Property ◽  
Locally Finite ◽  
Homogeneous Groups ◽  
Finite Borel Measure ◽  
Stratified Group ◽  
Differentiation Theorem

Abstract We give a complete answer to which homogeneous groups admit homogeneous distances for which the Besicovitch Covering Property (BCP) holds. In particular, we prove that a stratified group admits homogeneous distances for which BCP holds if and only if the group has step 1 or 2. These results are obtained as consequences of a more general study of homogeneous quasi-distances on graded groups. Namely, we prove that a positively graded group admits continuous homogeneous quasi-distances satisfying BCP if and only if any two different layers of the associated positive grading of its Lie algebra commute. The validity of BCP has several consequences. Its connections with the theory of differentiation of measures is one of the main motivations of the present paper. As a consequence of our results, we get for instance that a stratified group can be equipped with some homogeneous distance so that the differentiation theorem holds for each locally finite Borel measure if and only if the group has step 1 or 2. The techniques developed in this paper allow also us to prove that sub-Riemannian distances on stratified groups of step 2 or higher never satisfy BCP. Using blow-up techniques this is shown to imply that on a sub-Riemannian manifold the differentiation theorem does not hold for some locally finite Borel measure.

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