Gradient estimates of q-harmonic functions of fractional Schrodinger operator

We study gradient estimates of $q$-harmonic functions $u$ of the fractional Schr{\"o}dinger operator $\Delta^{\alpha/2} + q$, $\alpha \in (0,1]$ in bounded domains $D \subset \R^d$. For nonnegative $u$ we show that if $q$ is H{\"o}lder continuous of order $\eta > 1 - \alpha$ then $\nabla u(x)$ exists for any $x \in D$ and |\nabla u(x)| \le c u(x)/ (\dist(x,\partial D) \wedge 1). The exponent $1 - \alpha$ is critical i.e. when $q$ is only $1 - \alpha$ H{\"o}lder continuous $\nabla u(x)$ may not exist. The above gradient estimates are well known for $\alpha \in (1,2]$ under the assumption that $q$ belongs to the Kato class \calJ^{\alpha - 1}. The case $\alpha \in (0,1]$ is different. To obtain results for $\alpha \in (0,1]$ we use probabilistic methods. As a corollary, we obtain for $\alpha \in (0,1)$ that a weak solution of $\Delta^{\alpha/2}u + q u = 0$ is in fact a strong solution

A class of continua that are not attractors of any IFS

This paper presents a sufficient condition for a continuum in $R^n$ to be embeddable in $R^n$ in such a way that its image is not an attractor of any iterated function system. An example of a continuum in $R^2$ that is not an attractor of any weakly contracting iterated function system is also given

Axial gravitational waves in FLRW cosmology and memory effects

We show initial data for gravitational axial waves, that are twice differentiable but which are not $C^{2}$. They generate wave pulses that interact with matter in the radiation cosmological era. This forces the radiation matter to rotate. This rotation is permanent - it persists after the passage of the gravitational pulse. The observed inhomogeneities of the cosmic microwave background radiation put a bound onto discontinuities of superhorizon metric perturbations. We explicitly show that a class of smooth initial metrics that are at least $C^{2}$ gives rise to gravitational wave pulses that do not interact with the background during the radiation epoch.Comment: This version matches the published article (Phys. Rev. D 96, 063523 (2017)). A note is added on interaction of axial gravitational waves with stars' interiors, with 3 new reference