English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

This article discusses several definitions of the fractional Laplace operator  L  = — (—Δ)  α /2  in  R  d  , also known as the Riesz fractional derivative operator; here  α  ∈ (0,2) and  d  ≥ 1. This is a core example of a nonlocal pseudo-differential operator, appearing in various areas of theoretical and applied mathematics. As an operator on Lebesgue spaces ℒ  p   (with  p  ∈ [1,∞)), on the space  0  of continuous functions vanishing at infinity and on the space   bu   of bounded uniformly continuous functions,  L  can be defined, among others, as a singular integral operator, as the generator of an appropriate semigroup of operators, by Bochner’s subordination, or using harmonic extensions. It is relatively easy to see that all these definitions agree on the space of appropriately smooth functions. We collect and extend known results in order to prove that in fact all these definitions are completely equivalent: on each of the above function spaces, the corresponding operators have a common domain and they coincide on that common domain.

Ten equivalent definitions of the fractional laplace operator