Authors:Aymen Ammar; Aref Jeribi; Bilel Saadaoui Pages: 325 - 350 Abstract: The main goal of this paper is to give a characterization of the essential pseudospectra of \(2\times 2\) matrix of linear relations on a Banach space. We start by giving the definition and we investigate the characterization and some properties of the essential pseudospectra. Furthermore, we apply the obtained result to determine the essential pseudospectra of two-group transport equation with general boundary conditions in the Banach space. PubDate: 2018-09-01 DOI: 10.1007/s13324-017-0170-z Issue No:Vol. 8, No. 3 (2018)

Authors:Palle Jorgensen; Erin Pearse; Feng Tian Pages: 351 - 382 Abstract: Our main theorem is in the generality of the axioms of Hilbert space, and the theory of unbounded operators. Consider two Hilbert spaces whose intersection contains a fixed vector space \(\mathscr {D}\) . In the case when \(\mathscr {D}\) is dense in one of the Hilbert spaces (but not necessarily in the other), we make precise an operator-theoretic linking between the two Hilbert spaces. No relative boundedness is assumed. Nonetheless, under natural assumptions (motivated by potential theory), we prove a theorem where a comparison between the two Hilbert spaces is made via a specific selfadjoint semibounded operator. Applications include physical Hamiltonians, both continuous and discrete (infinite network models), and the operator theory of reflection positivity. PubDate: 2018-09-01 DOI: 10.1007/s13324-017-0173-9 Issue No:Vol. 8, No. 3 (2018)

Authors:Diganta Borah; Pranav Haridas; Kaushal Verma Pages: 383 - 414 Abstract: We study several quantities associated to the Green’s function of a multiply connected domain in the complex plane. Among them are some intrinsic properties such as geodesics, curvature, and \(L^2\) -cohomology of the capacity metric and critical points of the Green’s function. The principal idea used is an affine scaling of the domain that furnishes quantitative boundary behaviour of the Green’s function and related objects. PubDate: 2018-09-01 DOI: 10.1007/s13324-017-0177-5 Issue No:Vol. 8, No. 3 (2018)

Authors:Hui Nie; Junyi Zhu; Xianguo Geng Pages: 415 - 426 Abstract: The Gerdjikov–Ivanov equation is investigated by the Riemann–Hilbert approach and the technique of regularization. The trace formula and new form of N-soliton solution are given. The dynamics of the stationary solitons and non-stationary solitons are discussed. PubDate: 2018-09-01 DOI: 10.1007/s13324-017-0179-3 Issue No:Vol. 8, No. 3 (2018)

Authors:Jin-Yun Yang; Wen-Xiu Ma; Zhenyun Qin Pages: 427 - 436 Abstract: Based on the Hirota bilinear form of the \((2+1)\) -dimensional Ito equation, one class of lump solutions and two classes of interaction solutions between lumps and line solitons are generated through analysis and symbolic computations with Maple. Analyticity is naturally guaranteed for the presented lump and interaction solutions, and the interaction solutions reduce to lumps (or line solitons) while the hyperbolic-cosine (or the quadratic function) disappears. Three-dimensional plots and contour plots are made for two specific examples of the resulting interaction solutions. PubDate: 2018-09-01 DOI: 10.1007/s13324-017-0181-9 Issue No:Vol. 8, No. 3 (2018)

Authors:Yehonatan Salman Pages: 437 - 463 Abstract: The aim of the article is to recover a certain type of finite parametric distributions and functions using their spherical mean transform which is given on a certain family of spheres whose centers belong to a finite set \(\Gamma \) . For this, we show how the problem of reconstruction can be converted to a Prony’s type system of equations whose regularity is guaranteed by the assumption that the points in the set \(\Gamma \) are in general position. By solving the corresponding Prony’s system we can extract the set of parameters which define the corresponding function or distribution. PubDate: 2018-09-01 DOI: 10.1007/s13324-017-0171-y Issue No:Vol. 8, No. 3 (2018)

Authors:Nguyen Xuan Hong; Tran Van Thuy Pages: 465 - 484 Abstract: In this paper, we prove the Hölder continuity for solutions to the complex Monge–Ampère equations on non-smooth pseudoconvex domains of plurisubharmonic type m. PubDate: 2018-09-01 DOI: 10.1007/s13324-017-0175-7 Issue No:Vol. 8, No. 3 (2018)

Authors:Stephen J. Gardiner; Hermann Render Pages: 213 - 220 Abstract: The Schwarz reflection principle applies to a harmonic function which continuously vanishes on a relatively open subset of a planar or spherical boundary surface. It yields a harmonic extension to a predefined larger domain and provides a simple formula for this extension. Although such a point-to-point reflection law is unavailable for other types of surface in higher dimensions, it is natural to investigate whether similar harmonic extension results still hold. This article describes recent progress on such results for the particular case of cylindrical surfaces, and concludes with several open questions. PubDate: 2018-06-01 DOI: 10.1007/s13324-018-0213-0 Issue No:Vol. 8, No. 2 (2018)

Authors:Steven R. Bell Pages: 221 - 236 Abstract: The adjoint of the classic composition operator on the Hardy space of the unit disc determined by a holomorphic self map of the unit disc is well known to send the Szegő kernel function associated to a point in the unit disc to the Szegő kernel associated to the image of that point under the self map. The purpose of this paper is to show that a constructive proof that holomorphic functions that extend past the boundary can be well approximated by complex linear combinations of the Szegő kernel function gives an explicit formula for the adjoint of a composition operator that yields a new way of looking at these objects and provides inspiration for new ways of thinking about operators that act on linear spans of the Szegő kernel. Composition operators associated to multivalued self mappings will arise naturally, and out of necessity. A parallel set of ideas will be applied to composition operators on the Bergman space. PubDate: 2018-06-01 DOI: 10.1007/s13324-018-0215-y Issue No:Vol. 8, No. 2 (2018)

Authors:Pu Zhang Abstract: We give some characterizations of the boundedness of the maximal or nonlinear commutators of the Hardy–Littlewood maximal function and sharp maximal function in variable exponent Lebesgue spaces when the symbols b belong to the Lipschitz spaces, by which some new characterizations of Lipschitz spaces and nonnegative Lipschitz functions are obtained. Some equivalent relations between the Lipschitz norm and the variable exponent Lebesgue norm are also given. PubDate: 2018-08-11 DOI: 10.1007/s13324-018-0245-5

Authors:Natalia P. Bondarenko Abstract: In this paper, a partial inverse problem for the quadratic Sturm–Liouville pencil on a geometrical graph of arbitrary structure is studied. We suppose that the coefficients of differential expressions are known a priori on all the graph edges except one, and recover the coefficients on the remaining edge, using a part of the spectrum. The results of the paper are uniqueness theorems and a constructive algorithm for solving the partial inverse problem. PubDate: 2018-08-03 DOI: 10.1007/s13324-018-0244-6

Authors:Jihoon Lee Abstract: Let \(\Omega \) be a bounded domain in \(\mathbb {R}^N=\mathbb {R}^{N_1} \times \mathbb {R}^{N_2}\) with \(N_1, N_2 \ge 1\) , and \(N(s) = N_1 + (1+s)N_2\) be the homogeneous dimension of \(\mathbb {R}^N\) for \(s \ge 0\) . In this paper, we prove the existence and uniqueness of boundary blow-up solutions to the following semilinear degenerate elliptic equation $$\begin{aligned} {\left\{ \begin{array}{ll} G_s u = { x ^{2s}} u^p_+ \; &{}\text { in } \Omega ,\\ u(z)\rightarrow +\infty \; &{}\text { as } {d}(z) \rightarrow 0, \end{array}\right. } \end{aligned}$$ where \(u_+ = \max \{u,0\}\) , \(1<p<{{N(s)+2s} \over {N(s)-2}}\) , and d(z) denotes the Grushin distance from z to the boundary of \(\Omega \) . Here \(G_s\) is the Grushin operator of the form $$\begin{aligned} G_s u= \Delta _x u + x ^{2s}\Delta _y u, \; s\ge 0. \end{aligned}$$ It is worth noticing that our results do not require any assumption on the smoothness of the domain \(\Omega \) , and when \(s=0\) , we cover the previous results for the Laplace operator \(\Delta \) . PubDate: 2018-07-23 DOI: 10.1007/s13324-018-0241-9

Authors:Deepak Bansal; Manoj Kumar Soni; Amit Soni Abstract: In the present investigation we first introduce modified Dini function \(R^k_{\nu }(z)\) and then find sufficient conditions so that the modified Dini function \(R^k_{\nu }(z)\) have certain geometric properties like close-to-convexity, starlikeness and strongly starlikeness in the open unit disk. Relevance with some known results are also pointed out. PubDate: 2018-07-16 DOI: 10.1007/s13324-018-0243-7

Authors:O. Sh. Mukhtarov; K. Aydemir Abstract: In this paper a Sturm–Liouville equation together with eigenparameter-dependent boundary-transmission conditions are considered on two disjoint intervals. We construct the resolvent operator and Green’s function and obtain asymptotic approximate formulas for eigenvalues and corresponding eigenfunctions. The obtained results are implemented to the investigation of the basis properties of the system of eigenfunctions in the Lebesgue space \(L_2\) with new measures. In particular, we show that the eigenfunction expansion series regarding the convergence behaves in the same way as an ordinary Fourier series. PubDate: 2018-07-14 DOI: 10.1007/s13324-018-0242-8

Authors:Sahanous Mallick; Uday Chand De Abstract: In this paper, we investigate pseudo Q-symmetric spacetimes \((PQS)_{4}\) . At first, we prove that a \((PQS)_{4}\) spacetime is a quasi-Einstein spacetime. Then we investigate perfect fluid \((PQS)_{4}\) spacetimes and interesting properties are pointed out. From a result of Mantica and Suh (Int J Geom Methods Mod Phys 10:1350013, 2013) we have shown that \((PQS)_{4}\) spacetime is the Robertson-Walker spacetime. Further, it is shown that a \((PQS)_{4}\) spacetime with cyclic parallel Ricci tensor is an Einstein spacetime. Finally, we construct an example of a \((PQS)_{4}\) spacetime. PubDate: 2018-06-27 DOI: 10.1007/s13324-018-0240-x

Authors:Der-Chen Chang; M. Hedayat Mahmoudi; Bert-Wolfgang Schulze Abstract: We study the Volterra property of a class of anisotropic pseudo-differential operators on \(\mathbb {R}\times B\) for a manifold B with edge Y and time-variable t. This exposition belongs to a program for studying parabolicity in such a situation. In the present consideration we establish non-smoothing elements in a subalgebra with anisotropic operator-valued symbols of Mellin type with holomorphic symbols in the complex Mellin covariable from the cone theory, where the covariable \(\tau \) of t extends to symbols with respect to \(\tau \) to the lower complex v half-plane. The resulting space of Volterra operators enlarges an approach of Buchholz (Parabolische Pseudodifferentialoperatoren mit operatorwertigen Symbolen. Ph.D. thesis, Universitat Potsdam, 1996) by necessary elements to a new operator algebra containing Volterra parametrices under an appropriate condition of anisotropic ellipticity. Our approach avoids some difficulty in choosing Volterra quantizations in the edge case by generalizing specific achievements from the isotropic edge-calculus, obtained by Seiler (Pseudodifferential calculus on manifolds with non-compact edges, Ph.D. thesis, University of Potsdam, 1997), see also Gil et al. (in: Demuth et al (eds) Mathematical research, vol 100. Akademic Verlag, Berlin, pp 113–137, 1997; Osaka J Math 37:221–260, 2000). PubDate: 2018-06-19 DOI: 10.1007/s13324-018-0238-4

Authors:Björn Gustafsson Abstract: For conformal maps defined in the unit disk one can define a certain Poisson bracket that involves the harmonic moments of the image domain. When this bracket is applied to the conformal map itself together with its conformally reflected map the result is identically one. This is called the string equation, and it is closely connected to the governing equation, the Polubarinova–Galin equation, for the evolution of a Hele-Shaw blob of a viscous fluid (or, by another name, Laplacian growth). In the present paper we show that the string equation makes sense and holds for general polynomials. PubDate: 2018-06-15 DOI: 10.1007/s13324-018-0239-3

Authors:Thomas Bieske; Robert D. Freeman Abstract: We prove a \(\texttt {p}(\cdot )\) -Poincaré-type inequality for variable exponent Sobolev spaces with zero boundary values in Carnot groups. We then establish the existence and uniqueness (up to a set of zero \(\texttt {p}(\cdot )\) -capacity) of a minimizer to the Dirichlet energy integral for the variable exponent case. PubDate: 2018-05-22 DOI: 10.1007/s13324-018-0235-7

Authors:Andrew Thomack; Zachariah Tyree Abstract: Li and Wei (Proc Am Math Soc 137:195–204, 2009) studied the density of zeros of Gaussian harmonic polynomials with independent Gaussian coefficients. They derived a formula for the expected number of zeros of random harmonic polynomials as well as asymptotics for the case that the polynomials are drawn from the Kostlan ensemble. In this paper we extend their work to cover the case that the polynomials are drawn from the Weyl ensemble by deriving asymptotics for this class of harmonic polynomials. PubDate: 2018-03-16 DOI: 10.1007/s13324-018-0220-1