Abstract
In this paper we construct some nonlocal operators in the Heisenberg group. Specifically, starting from the Grünwald-Letnikov derivative and Marchaud derivative in the Euclidean setting, we revisit those definitions with respect to the one of the fractional Laplace operator. Then, we define some nonlocal operators in the non-commutative structure of the first Heisenberg group adapting the approach applied in the Euclidean case to the new framework.
Highlights
In 1927, in his PhD thesis André Marchaud (see [1], p. 47, Section 27, (23), or the published paper [2], p. 383, (23)), defined the following fractional differentiation for sufficiently regular real functions f : R → R for every α ∈ (0, 1) : Dα f ( x ) = c Z +∞f ( x ) − f ( x − t)t 1+ α dt, where c is a suitable normalizing constant depending on α only.There exist two Marchaud derivatives: one from the right and the other from the left
They are respectively defined for functions defined for R and α ∈ (0, 1) in such a way that
We prove that that fixing cH1 (s) = E 22, where c E ( 2s, 2) denotes the analogous normalizing constant for the fractional Laplace operator in R2 we obtain that (−LH1 ) 2 f ( P) → −∆H1 f ( P), s s → 2− and (−LH1 ) 2 f ( P) → f ( P), as s → 0+
Summary
We prove that that fixing cH1 (s) = E 22 , where c E ( 2s , 2) denotes the analogous normalizing constant for the fractional Laplace operator in R2 (see Section 3 where we recall some topics about s the fractional Laplace operator in the Euclidean setting) we obtain that (−LH1 ) 2 f ( P) → −∆H1 f ( P), s s → 2− and (−LH1 ) 2 f ( P) → f ( P), as s → 0+. While for denoting Marchaud fractional α) differentiation in several variables, along the vector ξ, we use the symbol D±,ξ when α 6∈ N, otherwise we write Dlξ if l ∈ N
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