Let $ℒ$ be a right-invariant sub-Laplacian on a connected Lie group $G,$ and let $S_{R}f:={\int }_0^{R}d{E}_{\lambda }f,R\ge 0,$ denote the associated “spherical partial sums,” where $ℒ={\int }_0^{\infty }\lambda d{E}_{\lambda }$ is the spectral resolution of $ℒ.$ We prove that $S_{R}f\left(x\right)$ converges a.e. to $f\left(x\right)$ as $R\to \infty$ under the assumption $log(2+ℒ)f\in L^2\left(G\right).$

We compute upper and lower bounds for the approximation of hyperbolic functions at points $1/s$ $(s=1,2,\cdots )$ by rationals $x/y$, such that $x,y$ satisfy a quadratic equation. For instance, all positive integers $x,y$ with $y\equiv 0(mod2)$ solving the Pythagorean equation $x^2+y^2=z^2$ satisfy $$|ysinh(1/s)-x|\gg \frac{loglogy}{logy}.$$ Conversely, for every $s=1,2,\cdots$ there are infinitely many coprime integers $x,y$, such that $$|ysinh(1/s)-x|\ll \frac{loglogy}{logy}$$ and $x^2+y^2=z^2$ hold simultaneously for some integer $z$. A generalization to the approximation of $h\left(e^{1/s}\right)$ for rational...

Let $G$ be a locally compact group and $\rho$ the left Haar measure on $G$. Given a non-negative Radon measure $\mu$, we establish a necessary condition on the pairs $\left(q,p\right)$ for which $\mu$ is a multiplier from $L^q\left(G,\rho \right)$ to $L^p\left(G,\rho \right)$. Applied to ${ℝ}^n$, our result is stronger than the necessary condition established by Oberlin in [14] and is closely related to a class of measures defined by Fofana in [7]. When $G$...

The note develops results from [5] where an invariance under the Möbius transform mapping the upper halfplane onto itself of the Weinstein operator $W_k:=\Delta +\frac{k}{x_n}\frac{\partial }{\partial x_n}$ on ${ℝ}^n$ is proved. In this note there is shown that in the cases $k\ne 0$, $k\ne 2$ no other transforms of this kind exist and for case $k=2$, all such transforms are described.

We prove that minimizers $u\in W^{1,n}$ of the functional $E_{}\left(u\right)=1/n{\int }_{|}{\nabla u|}^{n}dx+1/\left(4^n\right){\int }_{(}1-{\left|u\right|}^2{)}^{2}dx$, ⊂ ${ℝ}^n$, n ≥ 3, which satisfy the Dirichlet boundary condition $u_{}=g$ on for g: → $S^{n-1}$ with zero topological degree, converge in $W^{1,n}$ and $C_{loc}^{\alpha }$ for any α<1 - upon passing to a subsequence ${}_k\to 0$ - to some minimizing n-harmonic map. This is a generalization of an earlier result obtained for n=2 by Bethuel, Brezis, and Hélein. An example of nonunique asymptotic behaviour (which cannot occur in two dimensions if deg g = 0) is presented.