Hardy-Stein identities and square functions for semigroups

We prove a Hardy-Stein type identity for the semigroups of symmetric, pure-jump L\'evy processes. Combined with the Burkholder-Gundy inequalities, it gives the $L^p$ two-way boundedness, for $1<p<\infty$, of the corresponding Littlewood-Paley square function. The square function yields a direct proof of the $L^p$ boundedness of Fourier multipliers obtained by transforms of martingales of L\'evy processes.Comment: 17 page

Martin representation and Relative Fatou Theorem for fractional Laplacian with a gradient perturbation

Let $L=\Delta^{\alpha/2}+ b\cdot\nabla$ with $\alpha\in(1,2)$. We prove the Martin representation and the Relative Fatou Theorem for non-negative singular $L$-harmonic functions on ${\mathcal C}^{1,1}$ bounded open sets.Comment: 28 pages, editorial change

Law of corresponding state for the transport properties of molten salts

Corresponding state theory for transport properties of ionic fused alkali salt