The BMO norm of / is equivalent to sup (f\f(y)-u(x,t)\"Pt(x-y)dy\ , where Pt is the Poisson kernel. In this note, we show that Pt can be replaced by a nonnegative radial function h, which is positive in a neighbourhood of 0, with J h(x) dx = 1 and J rTM" (Inr)h(r) dr < oo, where h is the least decreasing radial majorant of h. 1980 Mathematics subject classification (Amer. Math. Soc): primary 42...

R.P. Boas has found necessary and sufficient conditions of belonging of function to Lipschitz class. From his findings it turned out, that the conditions on sine and cosine coefficients for belonging of function to Lip α (0 < α < 1) are the same, but for Lip 1 are different. Later his results were generalized by many authors in the viewpoint of generalization of condition on the majorant of mod...

Based on empirical Levy-type concentration functions a new graphical representation of the ML-density estimator under order restrictions is given. This representation generalizes the well-known representation of the Grenander estimator of a monotone density as the slope of the least concave majorant of the empirical distribution function. From the given representation it follows that a density ...

Recently a new class of schemes, called Time Relaxed Monte Carlo (TRMC) has been introduced for the numerical solution of the Boltzmann equation of gas dynamics. The motivation is to propose a systematic framework to derive Monte Carlo methods effective near the fluid dynamic regime. Before the methods can be accepted as alternative tools to other methods, they have to show that they are able t...

fails to possess a solution in general if X is equal to any of the characteristic values X¡, ¿ = 1,2,3, , of the kernel __(x, y), it is not surprising that all treatments of (1) have been limited to the cases in which equation (2) is in some sense (to be made more precise later) a majorant for (1) when X =Xi, the smallest characteristic value of F"(x, y). Thus, if FT(x, y) is assumed to be posi...

We consider a free interpolation problem in Nevanlinna and Smirnov classes and find a characterization of the corresponding interpolating sequences in terms of the existence of harmonic majorants of certain functions. We also consider the related problem of characterizing positive functions in the disc having a harmonic majorant. An answer is given in terms of a dual relation which involves pos...

In this paper a 2-dimensional mapping is investigated in the complex field C for the existence of analytic invariant curves. Employing the method of majorant series, we need to discuss the eigenvalue α of the mapping at a fixed point. Besides the hyperbolic case |α| = 1, we focus on those α on the unit circle S1, i.e., |α| = 1. We discuss not only those α at resonance, i.e., at a root of the un...

We study ground state solutions for linear and nonlinear elliptic PDEs in $\mathbb{R}^n$ with (pseudo-)differential operators of arbitrary order. prove a general symmetry result the case as well uniqueness states case. In particular, we can deal problems (e.g., higher order PDEs) that cannot be tackled by usual methods such maximum principles, moving planes, or Polya--Szegö inequalities. Instea...

The linear Beltrami equation on the Riemann sphere is studied under assumption that its measurable complex-valued coefficient μ(z) has a compact support in ℂ and ‖μ‖∞ = 1 Sufficient conditions for existence of regular homeomorphic $$ {W}_{\mathrm{loc}}^{1,1} solutions to with hydrodynamic normalization at infinity are given, particular, provided either dilatation Kμ boundedmean- oscillation maj...

fails to possess a solution in general if X is equal to any of the characteristic values X¡, ¿ = 1,2,3, , of the kernel __(x, y), it is not surprising that all treatments of (1) have been limited to the cases in which equation (2) is in some sense (to be made more precise later) a majorant for (1) when X =Xi, the smallest characteristic value of F"(x, y). Thus, if FT(x, y) is assumed to be posi...