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Previously, the parabolic equation method was used to study the high-frequency acoustic diffraction by a strongly elongated spheroid. This paper represents a continuation of that study. We justify some formal steps of the parabolic equation method at the level typical for the general PDE theory. In particular, we prove that a formal solution of the parabolic equation is actually the classical solution. We prove its uniqueness. We use various asymptotic properties of the higher functions. Some of these properties are new. We study location of zeros of the Whittaker functions. [ABSTRACT FROM AUTHOR]

Abstract: Harmonic analysis has been an efficient tool in control theory for a long time, see, e.g., Russell [D.L. Russell, Controllability and stabilizability theory for linear partial differential equations. Recent progress and open questions, SIAM Rev. 20 (1978) 639–739] and its numerous references. Here we establish discrete Ingham type and Haraux type inequalities for exponential sums satisfying a weakened gap condition. They enable us to obtain discrete simultaneous observability theorems for systems of vibrating strings or beams. [Copyright &y& Elsevier]

Abstract: We extend a well-known result of Bonami and Clerc on the almost everywhere (a.e.) convergence of Cesàro means of spherical harmonic expansions. For smooth functions measured in terms of φ-derivatives on the unit sphere, we obtained the sharp a.e. convergence rate of Cesàro means of their spherical harmonic expansions. [Copyright &y& Elsevier]

Abstract: For , let denote a weighted radial measure on normalized so that . If f is harmonic and integrable with respect to over the open unit disc D, then for every . Equivalently f is invariant under the weighted Berezin transform; . Conversely, does the invariance under the weighted Berezin transform imply the harmonicity of a function? In this paper, we prove that for any and , a function which is invariant under the weighted Berezin transform; needs not be 2-harmonic. [Copyright &y& Elsevier]

Abstract: In this paper it is shown that irregular boundary points for p-harmonic functions as well as for quasiminimizers can be divided into semiregular and strongly irregular points with vastly different boundary behaviour. This division is emphasized by a large number of characterizations of semiregular points. The results hold in complete metric spaces equipped with a doubling measure supporting a Poincaré inequality. They also apply to Cheeger p-harmonic functions and in the Euclidean setting to -harmonic functions, with the usual assumptions on . [Copyright &y& Elsevier]

Abstract: We prove versions of the Ahlfors–Schwarz lemma for quasiconformal euclidean harmonic functions and harmonic mappings with respect to the Poincaré metric. [Copyright &y& Elsevier]

Abstract: We give an explicit derivative computation for the restriction of a harmonic function on SG to segments at specific points of the segments: The derivative is zero at points dividing the segment in ratio 1:3. This shows that the restriction of a harmonic function to a segment of SG has the following curious property: The restriction has infinite derivatives on a dense subset of the segment (at junction points) and vanishing derivatives on another dense subset. [Copyright &y& Elsevier]

Abstract: We characterize the Radon–Nikodým property of a Banach space in terms of the existence of non-tangential limits of -valued harmonic functions u defined in a domain , , with Lipschitz boundary and belonging to maximal Hardy spaces. This extends the same result previously known for the unit disk of . We also prove an atomic decomposition of the Borel -valued measures in ∂D that arise as boundary limits of -valued harmonic functions whose non-tangential maximal function is integrable with respect to harmonic measure of ∂D. [Copyright &y& Elsevier]

Abstract: For any complete manifold with nonnegative Bakry–Emery''s Ricci curvature, we prove the gradient estimate of L-harmonic function. As application, we use this gradient estimate to deduce the localized version of the Harnack inequality for L-harmonic operator and some Liouville properties of positive or bounded L-harmonic function. [Copyright &y& Elsevier]

Abstract: It is proved that if u is the solution of PDE , that maps two annuli on the space , then the annulus in co-domain cannot be with arbitrary small modulus, providing that the annulus of domain is fixed. Also it is improved the inequality obtained in [D. Kalaj, On the Nitsche conjecture for harmonic mappings in and , Israel J. Math. 150 (2005) 241–253] for harmonic functions in . Finally it is given the new conjecture for harmonic mappings in the space similar to the conjecture of J.C.C. Nitsche for harmonic mapping in the plane related to the modulus of annuli. [Copyright &y& Elsevier]

Abstract: We provide a simple method for obtaining boundary asymptotics of the Poisson kernel on a domain in . [Copyright &y& Elsevier]

Abstract: We generalize a Hardy–Littlewood inequality and a Privalov inequality for conjugate harmonic functions in the plane to components of Clifford-valued monogenic functions. [Copyright &y& Elsevier]