On the convergence of double Fourier integrals of functions of bounded variation on ℝ2

We investigate the pointwise and uniform convergence of the symmetric rectangular partial (also called Dirichlet) integrals of the double Fourier integral of a function that is Lebesgue integrable and of bounded variation over ℝ2. Our theorem is a two-dimensional extension of a theorem of Móricz (see Theorem 3 in [10]) concerning the single Fourier integrals, which is more general than the two-dimensional extension given by Móricz himself (see Theorem 3 in [11]).

On conformal, harmonic mappings and Dirichlet’s integral

This paper has an expository character, however we present as well some new results and new proofs. We prove a complex version of Dirichlet?s principle in the plane and give some applications of it as well as estimates of Dirichlet?s integral from below. Some of the results in the plane are generalized to higher dimensions. Roughly speaking, under the appropriate conditions we estimate the n-Dirichlet integral of a mapping u defined on a domain ? ? Rn , n ? 2, by the measure of u(?) and show that equality holds if and only if it is injective conformal. Also some sharp inequalities related to the L2 norms of the radial derivatives of vector harmonic mappings from the unit ball in Rn, n ? 2, are given. As an application, we estimate the 2-Dirichlet integrals of mappings in the Sobolev space Wi2.