Tangential Limits Research Articles

Boundary limits of fractional derivatives of univalent functions

ABSTRACTEach function which is analytic and univalent in the open unit disk in the complex plane has a tangential boundary limit almost everywhere with an approach having exponential contact. In this paper, we obtain corresponding results for the fractional derivatives of univalent functions, for .

Tangential Limits for Harmonic Functions with Respect to ϕ(Δ): Stable and Beyond

In this paper, we discuss tangential limits for regular harmonic functions with respect to ϕ(Δ):= −ϕ(−Δ) in the C 1,1 open set D in ℝ d , where ϕ is the complete Bernstein function and d ≥ 2. When the exterior function f is local L p -Holder continuous of order β on D c with p ∈ (1, ∞] and β > 1/p, for a large class of Bernstein function ϕ, we show that the regular harmonic function u f with respect to ϕ(Δ), whose value is f on D c , converges a.e. through a certain parabola that depends on ϕ and ϕ ′. Our result includes the case ϕ(λ) = log(1 + λ α/2). Our proofs use both the probabilistic and analytic methods. In particular, the Poisson kernel estimates recently obtained in Kang and Kim (J. Korean Math. Soc. 50(5), 1009–1031, 2013) are essential to our approach.

Boundary behavior of superharmonic functions satisfying nonlinear inequalities in uniform domains

In a uniform domain Ω, we investigate the boundary behavior of positive superharmonic functions u satisfying the nonlinear inequality ―Δu(x) 0, α ∈ ℝ and p > 0, where Δ is the Laplacian and δ Ω (x) is the distance from a point x to the boundary of Ω. In particular, we present a Fatou type theorem concerning the existence of nontangential limits and a Littlewood type theorem concerning the nonexistence of tangential limits.

On first order linear PDE systems all of whose solutions are harmonic functions

We study the first order linear system $u_{\tilde{z}} + \bar{v}_{w} = 0, u_{\bar{w}} - \bar{v}_{z} = 0$ in a domain $\Omega \subset \mathbb{C}^{2}$ (first considered by G. Cimmino, [3]). We prove a Morera type theorem, emphasizing the analogy to the Cauchy-Riemann system, and a representation formula yielding a result on removable singularities of solutions to (2). We derive (by a Hilbert space technique outlined in [5]) compatibility relations among the free terms and boundary data in the boundary value problem $u_{\tilde{z}}+ \bar{v}_{w} = f, u_{\bar{w}} - \bar{v}_{z} = g$ in $\Omega$, and $u = \varphi,v = \psi$ on $\partial \Omega$. If $F = (u, v) : \Omega \to \mathbb{C}^{2}$ is a solution to (2) such that $\sup_{\varepsilon \gt O}$\int_{\partial \Omega_{\varepsilon}}|F(z, w)|^{p} d\sigma_{\varepsilon}(z, w} \lt \infty$ for some $p \geq 2$, then we show that $F$ admits non tangential limits at almost every $(\xi, \omega) \in \partial\Omega$.

Existence of Tangential Limits for α-Harmonic Functions on Half Spaces

Our aim in this paper is to prove the existence of tangential limits for Poisson integrals of the fractional order of functions in the L p Hölder space on half spaces.

Boundary behaviour of analytic functions in spaces of Dirichlet type

For and, we let be the space of all analytic functions in such that belongs to the weighted Bergman space. We obtain a number of sharp results concerning the existence of tangential limits for functions in the spaces. We also study the size of the exceptional set, where denotes the radial variation of along the radius, for functions.

STRONG APPROXIMATE CONTINUITY PROPERTIES OF CERTAIN CONJUGATE FUNCTIONS

The conjugate function $F(x) = - \lim _{n \to 0} \frac{1}{{2\pi }}\int_n^\pi {[F(x + t) - F(x - t)]\cot \frac{1}{2}dt} $ of an integrable and 2π-periodic function F exists finitely for almost all x. As our principal result we prove here that if F̃(0) is finite and F is monotonic in some neighbourhood of 0, then F̃ satisfies strong continuity-type properties at 0. Using an auxiliary result concerning the existence of tangential limits for the Poisson integral, we derive some consequences of our main theorem for the boundary behaviour of analytic and univalent functions.

Fatou and littlewood theorems for poisson integrals with respect to non-integrable kernels*

Sjögren and others studied the boundary behavior of fractional Poisson integrals with respect to the fractional power of the Poisson kernel. We extend the fractional power of the Poisson kernel to a non-integrable kernel and investigate the boundary behavior of associated Poisson integrals. The existence of certain tangential limit (Fatou type theorem) as well as its sharpness (Littlewood type theorem) are given. The admissible tangency varies according to the integrability of the boundary function. Our Littlewood type theorem is new even for the fractional power of the Poisson kernel.

Tangential limits and removable sets for weighted Sobolev spaces

The aim in the present paper is to give a weighted version of Koskela [Ark. Mat. 37 (1999), 291–304] concerning removable sets for Sobolev functions.

Tangential Limits of Potentials on Homogeneous Trees

Let T be a homogeneous tree of homogeneity q+1. Let Δ denote the boundary of T, consisting of all infinite geodesics b=[b0,b1,b2,] beginning at the root, 0. For each beΔ, τ≥1, and a≥0 we define the approach region Ωτ,a(b) to be the set of all vertices t such that, for some j, t is a descendant of bj and the geodesic distance of t to bj is at most (τ−1)j+a. If τ>1, we view these as tangential approach regions to b with degree of tangency τ. We consider potentials Gf on T for which the Riesz mass f satisfies the growth condition ∑Tfp(t)q−γ|t| 1 and 0 0inf ∑iq−τγ|ti|:E a subset of the boundary points passing through ti for some i,|ti|>log q(1/δ).

Radial variation of functions in Dirichlet‐type spaces

Let W denote a positive, increasing and continuous function on [1, ∞]. We write to denote the Dirichlettype space of functions f that are holomorphic in the unit disc and for which Where If W ( x ) = x for all x , then is the classicial Dirichlet space for which Note also that for every so, by Fatu's theoreum, every function in . ha finite radial(and angular) limits a.e. on the boundary of U . The question of the existence a.e. on ∂ U of certain tangential limits for functions in has been considered in [6,11], but we shall be concerned here with the radial variation i.e., the length of the image of the ray from 0 to e i θ under the mapping w = f ( z ), and, in particular, with the size of the set of values of θ for which L f (θ) can be infinite when

Tangential limits of cauchy — stieltjes transforms

This paper studies the tangential limit of functions defined by Where μ is a complex valued Borel measure the results depend on local properties of the measure μ Various tangential limits associated with f are shown to hold off exceptional sets.

Boundary limits of pluri-green potentials in the unit ball of

In this paper we investigate radial and tangential limits of pluri-Green potentials Vμ on the unit ball B in , which for suitable μ are defined by where for z ∊ B,φ z denotes the holomorphic automorphism of B satisfying φz(0)=z, φz(z) = 0, and φz(φz(w))=w. The main result of this paper is as follows: Let λ be the M-invariant measure on B and let f be a non-negative measurable function on B satisfying , and for some p<1, then the pluri-Green potential V f has tangential limit at almost all points of the boundary S of B. It is shown that the result is best possible.

On the existence of tangential limits of monotone BLD functions

On the existence of tangential limits of monotone BLD functions

Non-isotropic hausdorff capacity of exceptional sets of invariant potentials

In the paper we investigate tangential boundary limits of invariant Green potentials on the unit ballB in ℂ n ,n≥1. LetG(z, w) denote the Green function for the Laplace-Beltrami operator onB, and let λ denote the invariant measure onB. If μ is a non-negative measure, orf is a non-negative measurable function onB,G μ andG f denote the Green potential of μ andf respectively. For ξ∈S=δB, τ≥1, andc>0, let $$\mathcal{T}_{\tau ,c} (\zeta ) = \{ z \in B:\left| {1 - \left\langle {z,\xi } \right\rangle } \right|^\tau< c(1 - \left| z \right|^2 )\} $$ . The main result of the paper is as follows: Letf be a non-negative measurable function onB satisfying $$\int_B {(1 - \left| w \right|^2 )^\beta f^p (w)d\lambda (w)< \infty } $$ for some β, 0 n. Then for each τ, 1≤τ<n/β, there exists a setE t ⊂S withH βτ (E t )=0, such that $$\mathop {\lim }\limits_{\mathop {z \to \zeta }\limits_{z \in \mathcal{T}_{\tau ,c} (\zeta )} } G_f (z) = 0,forall\zeta \in S \sim E_\tau $$ In the above, for 0<α≤n,H α denotes the non-isotropic α-dimensional Hausdorff capacity onS. We also prove that if {a k } is a sequence inB satisfying Σ(1−|a k |2) β <∞ for some β, 0 <β<n, and μ=Σδ ak , where δ a denotes point mass measure ata, then the same conclusion holds for the potentialG μ .

Boundary limits of polyharmonic functions in sobolev-orlicz spaces

This paper deals with weighted boundary limits of polyharmonic functions on a bounded Lipschitz domain G with finite Dirichlet-type integral. There are many known results about the existence of angular limits for harmonic functions. Recently, tangential behaviours are studied mainly for harmonic functions. Our aim in this paper is to study the existence os weighted limit: limk(x)u(x) as x tends to the bounded, the limit is replaced by the usual tangential limit.

Möbius-invariant Hilbert spaces in polydiscs

We define the Dirichlet space 31 on the unit polydisc of . 2 is a semi-Hilbert space of of holomorphic functions, contains the holomorphic polynomials densely, is invariant under compositions with the biholomorphic automorphisms of , and its semi-norm is preserved under such compositions. We show that 2 is unique with these properties. We also prove 3ί is unique if we assume that the semi-norm of a function in 3f composed with an automorphism is only equivalent in the metric sense to the semi-norm of the original function. Members of a subclass of 3f given by a norm can be written as potentials of «5 5>2-functions on the «-torus . We prove that the functions in this subclass satisfy strong-type inequalities and have tangential limits almost everywhere on dVn . We also make capacitory estimates on the size of the exceptional sets on f)Vn . 1. Introduction. Mόbius-invariant spaces. Let U be the open unit disc in C and T be the unit circle bounding it. The open unit polydisc Vn and the torus Tn in Cn are the cartesian products of n unit discs and n unit circles, respectively. Ίn is the distinguished boundary of Uw and forms only a small part of the topological boundary dVn of Un . We denote by Jΐ the group of all biholomorphic automorphisms of Vn (the Mδbius group). The subgroup of linear automorphisms in Jf is denoted by %. The space of holomorphic functions with domain Un will be called ^(ϋn) and will carry the topology of uniform convergence on compact subsets of Vn. A semi-inner product on a complex vector space β? is a sesquilinear functional on β(? x %? with all the properties of an inner product except that it is possible to have {(a, a)) = 0 when a Φ 0. IWI = \/(( α> a)) *s the associated semi-norm. We assume ((• , •)) is

Tangential limits and exceptional sets for holomorphic Besov functions in the unit ball of $\mathbf{C}^n$

Tangential limits and exceptional sets for holomorphic Besov functions in the unit ball of $\mathbf{C}^n$

Harmonic Functions and Green Potentials Having No Tangential Limits

Harmonic Functions and Green Potentials Having No Tangential Limits

Solutions positives et mesure harmonique pour des opérateurs paraboliques dans des ouverts «lipschitziens»

Let L be a parabolic operator on R n+1 written in divergence form and with Lipschitz coefficients relatively to an adapted metric. We compare, near the boundary, the relative behavior of positive L-solutions on a Lipschitz domain. We first establish a so-called weak boundary Harnack principle. We then establish a uniform Harnack principle for certain particular positive L-solutions. This principle then allows us to prove another strong boundary Harnack principle for certain pairs of positive L-solutions. Then, we can generalize to L-operators some of J. T. Kemper results: we characterize the Martin boundary for “Lipschitz” domains and we show that the positive L-solutions on such domains admit non tangential limits except for a negligible set with respect to harmonic measure. Finally, in the last part, and for slightly more regular domains, we establish the equivalence between harmonic measure, adjoint harmonic measure and surface measure thus developing some results of J. M. Wu and R. Kaufman.