Domestic and International Firearm Laws: Can Implementation Be Used to Nationally Decrease Firearm Violence and Mass Shootings

The issue of firearm violence in the United States is highly controversial, as there are sound arguments on both sides of the discussion. Advocates of stricter gun laws often refer to both international and domestic examples that highlight the effectiveness of more restrictive firearm policies. Japan and Australia are two such countries that are continually referred to when a tragedy, such as a mass shooting, occurs in the United States and initial reactions often emphasis a need for fewer guns in the general public. Opposition to the proposed reforms of firearm policies cite the importance of the Second Amendment which grants their right to bear arms. To better understand both sides of the argument, this paper examines the effectiveness of firearm policies on an international and domestic level, and seeks to address whether or not such policies would aid in addressing firearm crime

Probabilistic Approach to Fractional Integrals and the Hardy-Littlewood-Sobolev Inequality

We give a short summary of Varopoulos' generalised Hardy-Littlewood-Sobolev inequality for self-adjoint $C_{0}$ semigroups and give a new probabilistic representation of the classical fractional integral operators on $\R^n$ as projections of martingale transforms. Using this formula we derive a new proof of the classical Hardy-Littlewood-Sobolev inequality based on Burkholder-Gundy and Doob's inequalities for martingales

On Astala's theorem for martingales and Fourier multipliers

We exhibit a large class of symbols $m$ on $\R^d$, $d\geq 2$, for which the corresponding Fourier multipliers $T_m$ satisfy the following inequality. If $D$, $E$ are measurable subsets of $\R^d$ with $E\subseteq D$ and $|D|<\infty$, then \int_{D\setminus E} |T_{m}\chi_E(x)|\mbox{d}x\leq \begin{cases} |E|+|E|\ln\left(\frac{|D|}{2|E|}\right), & \mbox{if}|E|<|D|/2, |D\setminus E|+\frac{1}{2}|D \setminus E|\ln \left(\frac{|E|}{|D\setminus E|}\right), & \mbox{if}|E|\geq |D|/2. \end{cases}. Here $|\cdot|$ denotes the Lebesgue measure on \bR^d. When $d=2$, these multipliers include the real and imaginary parts of the Beurling-Ahlfors operator $B$ and hence the inequality is also valid for $B$ with the right-hand side multiplied by $\sqrt{2}$. The inequality is sharp for the real and imaginary parts of $B$. This work is motivated by K. Astala's celebrated results on the Gehring-Reich conjecture concerning the distortion of area by quasiconformal maps. The proof rests on probabilistic methods and exploits a family of appropriate novel sharp inequalities for differentially subordinate martingales. These martingale bounds are of interest on their own right

Sharp Integrability for Brownian Motion in Parabola-shaped Regions

We study the sharp order of integrability of the exit position of Brownian motion from the planar domains {\cal P}_\alpha = \{(x,y)\in \bR\times \bR\colon x> 0, |y| < Ax^{\alpha}\}, $0<\alpha<1$. Together with some simple good-$\lambda$ type arguments, this implies the order of integrability for the exit time of these domains; a result first proved for $\alpha =1/2$ by Ba\~nuelos, DeBlassie and Smits \cite{ba} and for general $\alpha$ by Li \cite{li}. A sharp version of this result is also proved in higher dimensions