# Showing papers in "Complex Variables and Elliptic Equations in 2010"

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TL;DR: In this paper, a Bohr's inequality for analytic functions subordinated to univalent functions is established for harmonic functions mapping D into D. Theorem 5.2.1.

Abstract: We first establish Bohr's inequality for the class of analytic functions subordinate to univalent functions. Then, we establish a Bohr's inequality for harmonic functions mapping D into D.

106 citations

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TL;DR: In this article, the problem of boundedness of the supremal operator in weighted L p -spaces on the cone of non-negative nondecreasing functions is reduced to the boundedness problem of the fractional maximal operator M α, 0 ≤ α < n.

Abstract: The problem of boundedness of the fractional maximal operator M α, 0 ≤ α < n, in general local Morrey-type spaces is reduced to the problem of boundedness of the supremal operator in weighted L p -spaces on the cone of non-negative non-decreasing functions. This allows obtaining sharp sufficient conditions for boundedness for all admissible values of the parameters, which, for a certain range of the parameters wider than known before, coincide with the necessary ones.

103 citations

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TL;DR: In this article, the authors proposed a quasicrystals-based irregular sampling strategy to reduce the number of measures needed to recover a signal or an image whose Fourier transform is supported by a compact set with a given measure.

Abstract: This contribution is addressing an issue named in signal processing. Let be a lattice and be the dual lattice. Then the standard Shannon–Nyquist theorem says that any signal f whose Fourier transform is supported by a compact subset can be recovered from the samples if and only if the translated sets are pairwise disjoint. This sufficient condition on K is also necessary. When it is not satisfied may occur. Olevskii and Ulanovskii designed irregular sampling strategies which remedy . Then one can optimally reduce the number of measures needed to recover a signal or an image whose Fourier transform is supported by a compact set K with a given measure. The present contribution is aimed at bridging the gap between this advance on irregular sampling and the theory of quasicrystals.

68 citations

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TL;DR: In this paper, the authors considered open, discrete mappings between domains from R n satisfying condition (N), having local ACL n inverses on D∖B f, so that μ n (B f ) = 0, H*(·, f) < ∞ on B f and.

Abstract: We consider open, discrete mappings between domains from R n satisfying condition (N), having local ACL n inverses on D∖B f , so that μ n (B f ) = 0, H*(·, f) < ∞ on B f and . For this class of mappings (or even for larger classes of open, discrete mappings) we generalize the important modular inequality of Poleckii, also using the modular estimates of the spherical rings from Cristea (Local homeomorphism having local ACL n inverses, Complex Var. Elliptic Equ. 53(1) (2008), 77–99). We continue the work from the same paper by generalizing some basic facts from the theory of quasiregular mappings. We give equicontinuity results, Picard, Montel and Liouville type theorems, estimates of the modulus of continuity, analogues of Schwarz's lemma, eliminability results and boundary extensions theorems. Together with the multiple extensions of Zoric's theorem from the paper, we establish strong generalizations of one of the most important theorems from the theory of quasiregular mappings. We also extend similar res...

64 citations

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TL;DR: In this article, the boundedness of modified maximal operators and potentials with variable parameter in variable exponent Morrey spaces with non-doubling measure is established, and the Holder continuity properties for fractional integrals of functions in a Morrey space with variable exponent defined on non-homogeneous spaces are investigated.

Abstract: The boundedness of modified maximal operators and potentials with variable parameter in variable exponent Morrey spaces with non-doubling measure is established. Moreover, Holder continuity properties for fractional integrals of functions in Morrey spaces with variable exponent defined on non-homogeneous spaces are investigated.

40 citations

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TL;DR: In this paper, the authors formulate general principles on the existence of homeomorphic absolutely continuous on lines (ACL) solutions for the Beltrami equations with degeneration and derive from them a series of criteria and, in particular, a generalization and strengthening of the well-known Lehto existence theorem.

Abstract: We formulate general principles on the existence of homeomorphic absolutely continuous on lines (ACL) solutions for the Beltrami equations with degeneration and derive from them a series of criteria and, in particular, a generalization and strengthening of the well-known Lehto existence theorem. Furthermore, we prove that in all these cases there exist the so-called strong ring solutions satisfying additional moduli conditions which play a great role in the research of various properties of such solutions.

40 citations

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TL;DR: In this article, it was shown that under suitable assumptions the above problem has a family of solutions {u(e, ·)}e∈]0,e′[ for e′ sufficiently small.

Abstract: Let Ω i and Ω o be two bounded open subsets of ℝ n containing 0. Let G i be a (nonlinear) map of ∂Ω i × ℝ n to ℝ n . Let a o be a map of ∂Ω o to the set M n (ℝ) of n × n matrices with real entries. Let g be a function of ∂Ω o to ℝ n . Let γ be a positive valued function defined on a right neighbourhood of 0 on the real line. Let T be a map of] 1 − (2/n), +∞[×M n (ℝ) to M n (ℝ). Then we consider the problem where νeΩ i and ν o denote the outward unit normal to e∂Ω i and ∂Ω o , respectively, and where e > 0 is a small parameter. Here (ω − 1) plays the role of ratio between the first and second Lame constants and T(ω, ·) plays the role of (a constant multiple of) the linearized Piola Kirchhoff stress tensor. Under the condition that lime→0 γ−1(e)e(log e)δ2,n exists in ℝ, we prove that under suitable assumptions the above problem has a family of solutions {u(e, ·)}e∈]0,e′[ for e′ sufficiently small and we analyse the behaviour of such a family as e approaches 0 by an approach which is alternative to those of ...

35 citations

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TL;DR: In this paper, the authors studied the analytical properties of quasiconformal homeomorphisms with bounded (p, q)-distortion in the limit case q = n − 1.

Abstract: We study generalizations of the quasiconformal homeomorphisms (the so-called homeomorphisms with bounded (p, q)-distortion) that induce bounded composition operators on the Sobolev spaces with the first weak derivatives. If a homeomorphism between domains of the Euclidean space ℝ n has bounded (p, q)-distortion and q > n − 1 then its inverse mapping has bounded (q/(q − n + 1), p/(p − n + 1))-distortion. In this article, we study in detail the analytical properties of these homeomorphisms in the limit case q = n − 1. The study of these classes is important because of applications of mappings with bounded (p, q)-distortion to the Sobolev type embedding theorems and nonlinear elasticity problems.

34 citations

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TL;DR: In this paper, the right invertibility of the divergence on Ω in weighted Lebesgue and Sobolev spaces was studied, and it was shown that Ω is a John domain.

Abstract: Let Ω be an arbitrary bounded domain of ℝ n . We study the right invertibility of the divergence on Ω in weighted Lebesgue and Sobolev spaces on Ω, and rely this invertibility to a geometric characterization of Ω and to weighted Poincare inequalities on Ω. We recover, in particular, well-known results on the right invertibility of the divergence in Sobolev spaces when Ω is Lipschitz or, more generally, when Ω is a John domain, and focus on the case of s-John domains.

31 citations

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TL;DR: In this article, conditions on the parameters a, b, c were derived so that the function zF(a, b; c; z) is starlike in 𝔻, where F denotes the classical hypergeometric function.

Abstract: In this article, we derive conditions on the parameters a, b, c so that the function zF(a, b; c; z) is starlike in 𝔻, where F(a, b; c; z) denotes the classical hypergeometric function. We give some consequences of our results including some mapping properties of convolution operator.

30 citations

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TL;DR: The constructive and numerical solution of minimizing the energy for the Gauss variational problem involving the Newtonian potential is studied and the fast multipole method provides an efficient solution algorithm.

Abstract: We study the constructive and numerical solution of minimizing the energy for the Gauss variational problem involving the Newtonian potential. As a special case, we also treat the corresponding condenser problem. These problems are considered for two two-dimensional compact, disjoint Lipschitz manifolds Γ j ⊂ ℝ3, j = 1, 2, charged with measures of opposite sign. Since this minimizing problem over an affine cone of Borel measures with finite Newtonian energy can also be formulated as the minimum problem over an affine cone of surface distributions belonging to the Sobolev–Slobodetski space , which allows the application of simple layer boundary integral operators on Γ, a penalty approximation for the Gauss variational problem can be used. The numerical approximation is based on a Galerkin–Bubnov discretization with piecewise constant boundary elements. To the discretized problem, the projection-iteration is applied where the matrix times vector operations are executed with the fast multipole method. For th...

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TL;DR: In this paper, a special Muckenhoupt weight of type w(x) ∼ |x|α was used to determine the trace spaces with respect to the hyperplane ℝ n−1.

Abstract: We study Besov and Triebel–Lizorkin spaces with a special Muckenhoupt weight of type w(x) ∼ |x|α and determine their trace spaces with respect to the hyperplane ℝ n−1. The approach is based on atomic decomposition results.

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TL;DR: In this article, it was proved that open discrete ring Q-mappings are differentiable and belong to the class ACL in ℝ n, n ≥ 2, provided that.

Abstract: We study the so-called ring Q-mappings which are the natural generalization of quasiregular mappings. It is proved that open discrete ring Q-mappings are differentiable a.e. and belong to the class ACL in ℝ n , n ≥ 2; furthermore, provided that .

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TL;DR: A series of new criteria have been established for the existence of regular solutions of the Dirichlet problem for the Beltrami equations and, in particular, interims of majorants for dilatations of finite mean oscillation.

Abstract: A series of new criteria have been established for the existence of regular solutions of the Dirichlet problem for the Beltrami equations and, in particular, interims of majorants for dilatations of finite mean oscillation. Moreover, integral criteria of the type of Lehto–Miklyukov–Suvorov–Zorich are given.

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TL;DR: In this paper, a Levi non-degenerate real analytic hypersurface M of ℂ2 represented in local coordinates by a complex defining equation of the form, which satisfies an appropriate reality condition, is spherical if and only if its complex graphing function Θ satisfies an explicitly written sixth-order polynomial complex partial differential equation.

Abstract: A Levi nondegenerate real analytic hypersurface M of ℂ2 represented in local coordinates (z, w) ∈ ℂ2 by a complex defining equation of the form , which satisfies an appropriate reality condition, is spherical if and only if its complex graphing function Θ satisfies an explicitly written sixth-order polynomial complex partial differential equation. In the rigid case (known before), this system simplifies considerably, but in the general nonrigid case, its combinatorial complexity shows well why the two fundamental curvature tensors constructed by Cartan [Sur la geometrie pseudo-conforme des hypersurfaces de l'espace de deux variables complexes, II, Ann. Scuola Norm. Sup. Pisa 1 (1932), pp. 333–354] in his classification of hypersurfaces have, since then, never been reached in parametric representation.

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TL;DR: In this article, a Serrin-like condition is found for removability of isolated singularity for high-order elliptic equations in divergent form whose coefficients satisfy a polynomial growth condition and an ellipticity condition stronger than one that usually is considered for highorder EDEs.

Abstract: A Serrin-like condition is found for removability of the isolated singularity for high-order elliptic equations in divergent form whose coefficients satisfy a polynomial growth conditions and an ellipticity condition stronger than one that usually is considered for high-order elliptic equations.

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TL;DR: In this paper, the area formula for contact C 1-mappings of Carnot manifolds was proved and a suitable notion of the sub-Riemannian Jacobian was introduced.

Abstract: We prove the area formula for contact C 1-mappings of Carnot manifolds. In particular, we introduce a suitable notion of the sub-Riemannian Jacobian and investigate the relation of Riemannian and sub-Riemannian measures on image.

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TL;DR: In 2008, Cabiria Andreian Cazacu reached the beautiful age of 80 as discussed by the authors and it is a joyful occasion for the mathematical Romanian community (colleagues, ex-students and teachers) to mark her path a...

Abstract: In 2008, Professor Cabiria Andreian Cazacu reached the beautiful age of 80. It is a joyful occasion for the mathematical Romanian community (colleagues, ex-students and teachers) to mark her path a...

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TL;DR: In this article, necessary and sufficient conditions for Nemytskij operators are discussed in terms of functions g : [0, 1] × Ω → ℝ respectively, under which the corresponding operators map a certain function space into themselves.

Abstract: We discuss necessary and sufficient conditions, in terms of functions g : ℝ → ℝ and g : [0, 1] × ℝ → ℝ respectively, under which the corresponding Nemytskij operators T g (f)(x) = g(f (x)) and T g (f)(x) = g(x, f (x)) respectively map a certain function space into themselves.

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TL;DR: The main goal of as discussed by the authors is to study special polynomial power series expansions for axially monogenic functions, and the results obtained contain previous ones of complex setting as special cases.

Abstract: The main goal of this article is to study special polynomial power series expansions for entire axially monogenic functions. In relation to the difference and sum bases of monogenic polynomials, which have link with Bernoulli polynomials, the best possible bounds for the orders of such bases are determined. Moreover the T ρ-property of such differenced base and the sum base is discussed. The results obtained contain previous ones of complex setting as special cases.

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TL;DR: The well-known Vaisala inequality for quasiregular mappings is extended to open discrete mappings with finite length distortion in this article, where the authors show that the inequality holds even for finite length distortions.

Abstract: The well-known Vaisala inequality for quasiregular mappings is extended to open discrete mappings with finite length distortion.

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TL;DR: In this article, Hardy-type operators on the cones of monotone functions with general positive σ-finite Borel measure are considered. And two-sided Hardy-types are proved for the parameter −∞ < p < ∞.

Abstract: We consider Hardy-type operators on the cones of monotone functions with general positive σ-finite Borel measure. Some two-sided Hardy-type inequalities are proved for the parameter −∞ < p < ∞. It is pointed out that such equivalences, in particular, imply a new characterization of the discrete Hardy inequality for the (most difficult) case 0 < q < p ≤ 1.

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TL;DR: In this paper, the authors investigated the Gromov hyperbolicity of Denjoy domains equipped with the Hyperbolic or the quasihyperbolic metric.

Abstract: In this article, we investigate the Gromov hyperbolicity of Denjoy domains equipped with the hyperbolic or the quasihyperbolic metric. The focus are on comparative or decomposition results, which allow us to reduce the question of whether a given domain is Gromov hyperbolic to a series of questions concerning simpler domains. We also give several concrete examples of applications of the results.

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TL;DR: In this article, a visualization of Blaschke product mappings can be obtained by treating them as canonical projections of covering Riemann surfaces and finding fundamental domains and covering transformations corresponding to these surfaces.

Abstract: A visualization of Blaschke product mappings can be obtained by treating them as canonical projections of covering Riemann surfaces and finding fundamental domains and covering transformations corresponding to these surfaces. A working tool is the technique of simultaneous continuation we introduced in previous papers. Here, we are refining this technique for some particular types of Blaschke products for which colouring pre-images of annuli centred at the origin allow us to describe the mappings with a high degree of fidelity. Additional graphics and animations are provided on the website of the project (http://math.holycross.edu/~cballant/complex/complex-functions.html).

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TL;DR: In this article, an anti-maximum principle for second-order elliptic operators was extended to the degenerate case, which includes the Dirichlet and Robin boundary conditions as special cases.

Abstract: This article deals with an anti-maximum principle to the degenerate case and to elliptic boundary value problems with indefinite weight function which include the Dirichlet and Robin boundary conditions as special cases. The result extends an earlier theorem by Clement and Peletier [An anti-maximum principle for second-order elliptic operators, J. Diff. Eqns. 34 (1979), pp. 218–229] to the degenerate case.

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TL;DR: The local behavior of ACL-homeomorphisms in the plane is studied in this article, where extremal length techniques, modules of ring domains, quadrilaterals and families of curves that are arcs of spirals are used to provide geometric sufficient and necessary conditions for conformality at a point.

Abstract: We study the local behaviour of ACL-homeomorphisms in the plane, defined in a neighbourhood of a point and satisfying the Beltrami equation with coefficient We use extremal length techniques, modules of ring domains, of quadrilaterals and of families of curves that are arcs of spirals to provide geometric sufficient and necessary conditions for conformality at a point.

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TL;DR: In this paper, the Salagean differential operator and sharp subordination results are used to define analytic p-valent functions defined by using the salagean operator and a sharp subordinator.

Abstract: In this article, we introduce and investigate various properties of a certain class of analytic p-valent functions defined by using the Salagean differential operator and sharp subordination results

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TL;DR: In this paper, the authors give an application of Ricceri's multiplicity result to the Neumann problem associated with a quasilinear elliptic equation and give three critical points theorem.

Abstract: In this article, we give an application of our recent multiplicity result obtained in Ricceri [B. Ricceri, A further three critical points theorem, Nonlinear Anal. 71 (2009), pp. 4151–4157] to the Neumann problem associated with a quasilinear elliptic equation.

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TL;DR: In this article, six new generalized convolutions of the integral transforms of Fourier type were given, and a class of integral equations of convolution type by using the constructed convolutions was investigated.

Abstract: This article gives six new generalized convolutions of the integral transforms of Fourier type, and investigates a class of integral equations of convolution type by using the constructed convolutions Namely, the explicit solutions in L 1(ℝ d ) of a class of integral equations of convolution type are obtained

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TL;DR: In this article, a Cauchy integral representation formula is obtained explicitly for the equation for, and different forms of boundary conditions originating from the well-known Schwarz, Dirichlet and Neumann problems from complex analysis are studied in the upper half plane.

Abstract: The equation for , is investigated in the upper half plane. A Cauchy integral representation formula is obtained explicitly. Different forms of boundary conditions originating from the well-known Schwarz, Dirichlet and Neumann problems from complex analysis are studied. These boundary value problems are solved in the upper half plane.