Volume 33, Issue 3

We give some sufficient and necessary conditions for the L 2 boundedness of Littlewood-Paley operator g ψ under some conditions on the kernel function ψ, which includes T. Walsh’s result, and recent results by Al-Salman, and Ding, Xue and Yabuta.

We prove Hölder continuity for solutions to the n-dimensional H-System assuming logarithmic higher integrability of the solution.

In the paper, the convergence, monotonicity, and inequalities of sequences involving continued powers [xxx] are investigated and established.

We obtain the equations that allow us to compute the position of the weighted Fermat-Torricelli point on the two dimensional sphere S K 2 of constant Gaussian curvature K and on the two dimensional hyperbolic plane H K 2 of constant Gaussian curvature K for K <0; by introducing a method of symmetrical differentiation of a length of a geodesic arc with respect to two variable lengths of geodesic arcs, respectively. The method of differentiating a length of a geodesic arc with respect to two variable lengths of geodesic arcs is a generalization of the first variation formula of the length of a geodesic arc with respect to one variable length of geodesic arc on the K-plane.

In this article we introduce the sequence spaces Z I (M),Z I 0 (M) and Z I ∞ (M) using the Orlicz function M . We study the algebraic properties and inclusion relations on these spaces.

We consider the Dirichlet problem for so-called G-minimal graphs in two dimensions. These are immersions of minimal surface type which can be presented as graphs over a planar domain. With the aid of a weight matrix G we derive a quasilinear elliptic and homogeneous differential equation for this height function. Then we solve the Dirichlet problem over convex domains Ω without differentiability assumptions and continuous boundary data with a constructive continuity and approximation method. We firstly establish an a priori C 1+α -estimate up to the boundary of the solution as we take the dense problem class of strictly convex C 2+α -domains and C 2+α -boundary data. By proving theorems on stability and compactness of graphs we solve this boundary value problem with a nonlinear continuity method. Then we introduce weighted conformal parameters in the graph and consider the parametric problem on the unit disc. Finally, we solve the original Dirichlet problem by using an approximation argument and the important parametric compactness theorem.

In this paper, some algorithms for constructing tight wavelet frames on local fields using the unitary extension principles are suggested. We present a sufficient condition for finite number of functions to form a tight wavelet frame and establish general principles for constructing tight wavelet frames on local fields