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Abstract: In this paper we study the asymptotic growth behavior of solutions to the Dirac–Hodge equation on upper half-space of . By means of the Fourier transform we introduce lower and upper growth orders and generalizations of the maximum term and central index for this function class. Together with a Cauchy estimate we obtain an explicit lower and upper bound estimate of the maximum modulus in terms of these notions. [Copyright &y& Elsevier]

Abstract: In this note we will present a stability property of the reverse isoperimetric inequality newly obtained in [S.L. Pan, H. Zhang, A reverse isoperimetric inequality for convex plane curves, Beiträge Algebra Geom. 48 (2007) 303–308], which states that if K is a convex domain in the plane with perimeter and area , then one gets , where denotes the oriented area of the domain enclosed by the locus of curvature centers of the boundary curve ∂K, and the equality holds if and only if K is a circular disc. [Copyright &y& Elsevier]

Abstract: We consider high stresses in stiff-fiber reinforced materials, which increase rapidly as fibers approximate to one another. This paper presents the optimal blow-up rate of the stresses with respect to the distance between a pair of stiff fibers in . The blow-up result plays an important role in our understanding of low strengths of fiber-reinforced composites. Referring to a problem of anti-plane shear, the stresses can be interpreted as the electric fields outside closely spaced perfect conductors in , under the action of applied electric field ∇H. It has been shown by Ammari, Kang et al. that in the particular case of circular inclusions, the electric field blows up at the optimal rate as , where ϵ is the distance between conductors. Recently, Yun has extended the blow-up result to pairs of conductors associated with a large class of shapes whose complements can be transformed conformally to the outside of a circle with mapping. However, it presented a suboptimal result that only for a special uniform field , the electric fields blow up at the exact rate . In this paper, an upper bound with the rate of electric field for any harmonic function H is established. This yields the optimal blow-up rate for the inclusions in the same class of shapes as Yun. [Copyright &y& Elsevier]

Abstract: We construct an example of a non-convex star-shaped origin-symmetric body such that its section function is decreasing in for every fixed direction . [Copyright &y& Elsevier]

Abstract: The notions of column and row operator space were extended by A. Lambert from Hilbert spaces to general Banach spaces. In this paper, we use column and row spaces over quotients of subspaces of general -spaces to equip several Banach algebras occurring naturally in abstract harmonic analysis with canonical, yet not obvious operator space structures that turn them into completely bounded Banach algebras. We use these operator space structures to gain new insights on those algebras. [Copyright &y& Elsevier]

Abstract: In this paper, the existence of positive doubly periodic solutions for nonlinear telegraph system is discussed using the method of upper and lower solutions. [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: We study a plane problem with mixed boundary conditions for a harmonic function in an unbounded Lipschitz domain contained in a strip. The problem is obtained by linearizing the hydrodynamic equations which describe the steady flow of a heavy ideal fluid over an obstacle lying on the flat bottom of a channel. In the case of obstacles of rectangular shape we prove unique solvability for all velocities of the (unperturbed) flow above a critical value depending on the obstacle depth. We also discuss regularity and asymptotic properties of the solutions. [Copyright &y& Elsevier]

Abstract: We study the smoothness property of a function f with absolutely convergent Fourier series, and give best possible sufficient conditions in terms of its Fourier coefficients to ensure that f belongs to one of the Zygmund classes and for some . This paper is a natural supplement to our earlier one [F. Móricz, Absolutely convergent Fourier series and function classes, J. Math. Anal. Appl. 324 (2) (2006) 1168–1177] under the same title, and we keep its notations. [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 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 obtain square function estimates and bounds for maximal singular integral operators associated with bilinear multipliers given by characteristic functions of dyadic dilations of certain planar sets. As a consequence, we deduce pointwise almost everywhere convergence for lacunary partial sums of bilinear Fourier series with respect to methods of summation determined by the corresponding planar sets. [Copyright &y& Elsevier]

Abstract: We find the condition on a bounded function f under which the Walsh–Fourier series of f converge in the norm of Orlicz space . [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 study the smoothness property of a function f with absolutely convergent Fourier series, and give best possible sufficient conditions in terms of its Fourier coefficients to ensure that f belongs either to one of the Lipschitz classes and for some , or to one of the Zygmund classes and . Our theorems generalize some of those by Boas [R.P. Boas Jr., Fourier series with positive coefficients, J. Math. Anal. Appl. 17 (1967) 463–483] and one by Németh [J. Németh, Fourier series with positive coefficients and generalized Lipschitz classes, Acta Sci. Math. (Szeged) 54 (1990) 291–304]. We also prove a localized version of a theorem by Paley [R.E.A.C. Paley, On Fourier series with positive coefficients, J. London Math. Soc. 7 (1932) 205–208] on the existence and continuity of the derivative of f. [Copyright &y& Elsevier]

Abstract: In this paper, we examine a class of averaging operators which exhibit local smoothing. That is, viewed as a function of space and time variables, the operators yield more smoothing than the fixed-time estimates. Sogge showed in a more general setting that if these operators satisfy a cinematic curvature condition, they will exhibit some local smoothing [C.D. Sogge, Propagation of singularities and maximal functions in the plane, Invent. Math. 104 (1991) 231–251]. Here we translate this condition into the setting of averaging operators in the plane. We prove that cinematic curvature is not necessary for local smoothing to occur, exhibiting a class of operators which fail the cinematic curvature condition but still satisfy a local smoothing estimate. Furthermore, the amount of local smoothing exhibited by these operators is strictly less than that conjectured for operators satisfying the cinematic curvature condition. [Copyright &y& Elsevier]

Abstract: We give an example of a permutation of integers which preserves all convergent Fourier series and makes some divergent Fourier series converge. [Copyright &y& Elsevier]

Abstract: The main aim of this paper is to prove that the logarithmic means of quadratical partial sums of the double Walsh–Fourier series does not improve the convergence in measure. In other words, we prove that for any Orlicz space, which is not a subspace of , the set of the functions having logarithmic means of quadratical partial sums of the double Walsh–Fourier series convergent in measure is of first Baire category. [Copyright &y& Elsevier]

Abstract: For define the symmetric q-linear operator acting on a suitable function by . The q-linear initial value problem , , has two entire functions and as linearly independent solutions, which are orthogonal on a discrete set. Sufficient conditions for pointwise convergence and for uniform convergence of the corresponding Fourier expansion are given. [Copyright &y& Elsevier]

Abstract: A Henstock type integral is defined on compact subsets of a locally compact zero-dimensional abelian group. This integral is applied to obtain an inversion formula for the multiplicative integral transform. [Copyright &y& Elsevier]

Abstract: In this note, a sufficient condition for summability of Fourier series has been obtained which in conjunction with the author''s Tauberian theorem [M.L. Mittal, A Tauberian theorem on strong Nörlund summability, J. Indian Math. Soc. 44 (1980) 369–377] on strong Nörlund summability gives a sufficient condition for summability of a Fourier series. This generalizes results due to Prasad [G. Prasad, On strong Nörlund summability of Fourier series, Univ. Roorkee Res. J. 9 (1966–1967) 1–10] and Varshney [O.P. Varshney, Note on summability of Fourier series, Boll. Un. Mat. Ital. 16 (1961) 383–385]. [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]

Abstract: In this article, we study the effect of the chosen representation of a point value (and point evaluation) on the class of periodic signals realizable using a certain type of infinite-dimensional linear system. By suitably representing the point evaluation at the origin in a Hilbert space, we are able to give a complete characterization of its extensions. These extensions involve a new concept called δ-sequence, the use of which as an observation operator of an infinite-dimensional linear system is studied in this article. In particular, we consider their use in the realization of periodic signals. We also investigate how the use of δ-sequences affects the convergence properties of such realizations; we consider the rate and character of convergence and the removal of the Gibbs phenomenon. As still a further demonstration of the significance of the chosen concept of a point value, we discuss the use of distributional point values in the realization of periodic distributions. The possible applications of this work lie in regulator problems of infinite-dimensional control theory, as is indicated by the well-known internal model principle. [Copyright &y& Elsevier]