TY - JOUR

T1 - Flett potentials associated with differential-difference Laplace operators

AU - Ben Saïd, Salem

AU - Negzaoui, Selma

N1 - Funding Information:
S.B.S. would like to acknowledge the financial support awarded by UAEU through the UPAR Grant No. 12S002.
Publisher Copyright:
© 2022 Author(s).

PY - 2022/3/1

Y1 - 2022/3/1

N2 - A large family of Flett potentials is investigated. Formally, these potentials are negative powers of the operators id + |x|1-1/mΔk, where Δk is the Dunkl Laplace (differential and difference) operator on ℝ. Here, k ≥ 0 and m ϵ ℕ\{0}. In the (k = 0, m = 1) case, our family of potentials reduces to the classical one studied by Flett [Proc. London Math. Soc. s3-22, 385-451 (1971)]. An explicit inversion formula of the Flett potentials is obtained for functions belonging to C0(ℝ) and weighted Lp spaces, 1 ≤ p < ∞. As a tool, we use a wavelet-like transforms generated by a Poisson type semigroup and signed Borel measures. In this context, a fundamental theorem proving an almost everywhere convergence of a convolution operator for an approximate identity was given. The k = 0 case is already new.

AB - A large family of Flett potentials is investigated. Formally, these potentials are negative powers of the operators id + |x|1-1/mΔk, where Δk is the Dunkl Laplace (differential and difference) operator on ℝ. Here, k ≥ 0 and m ϵ ℕ\{0}. In the (k = 0, m = 1) case, our family of potentials reduces to the classical one studied by Flett [Proc. London Math. Soc. s3-22, 385-451 (1971)]. An explicit inversion formula of the Flett potentials is obtained for functions belonging to C0(ℝ) and weighted Lp spaces, 1 ≤ p < ∞. As a tool, we use a wavelet-like transforms generated by a Poisson type semigroup and signed Borel measures. In this context, a fundamental theorem proving an almost everywhere convergence of a convolution operator for an approximate identity was given. The k = 0 case is already new.

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U2 - 10.1063/5.0063053

DO - 10.1063/5.0063053

M3 - Article

AN - SCOPUS:85126136643

SN - 0022-2488

VL - 63

JO - Journal of Mathematical Physics

JF - Journal of Mathematical Physics

IS - 3

M1 - 033504

ER -