Narrow Results

We study the problem of extending a complex structure to a given Lie algebra 𝔤, which is firstly defined on an ideal 𝔥 ⊂ 𝔤. We consider the next situations: 𝔥 is either complex or it is totally real. The next question is to equip 𝔤 with an additional structure, such as a (non)-definite metric or a symplectic structure and to ask either 𝔥 is non-degenerate, isotropic, etc. with respect to this structure, by imposing a compatibility assumption. We show that this implies certain constraints on the algebraic structure of 𝔤. Constructive examples illustrating this situation are shown, in particular computations in dimension six are given.

We study the fractional Laplacian ${(-\mathrm{\Delta })}^{\sigma /2}$ on the $n$-dimensional torus ${𝕋}^n$, $n\ge 1$. First, we present a general extension problem that describes any fractional power $L^{\gamma }$, $\gamma >0$, where $L$ is a general nonnegative self-adjoint operator defined in an $L^2$-space. This generalizes to all $\gamma >0$ and to a large class of operators the previous known results by Caffarelli and Silvestre. In particular, it applies to the fractional Laplacian on the torus. The extension problem is used to prove interior and boundary Harnack’s inequalities for ${(-\mathrm{\Delta })}^{\sigma /2}$, when $0<\sigma <2$. We deduce regularity estimates on Hölder, Lipschitz and Zygmund spaces. Finally, we obtain the pointwise integro-differential formula for the operator. Our method is based on the semigroup language approach.

In this paper, we establish a scale invariant Harnack inequality for the fractional powers of parabolic operators ${({\partial }_t-\mathcal{ℒ})}^s$, $0<s<1$, where $\mathcal{ℒ}$ is the infinitesimal generator of a class of symmetric semigroups. As a by-product, we also obtain a similar result for the nonlocal operators ${\left(-\mathcal{ℒ}\right)}^s$. Our focus is on non-Euclidean situations.