Fractional calculus for the one-dimensional Laplacian with Robin boundary condition

Fethi Bouzeffour; Fethi Bouzeffour

doi:10.3934/math.2026861

AIMS Mathematics

2026, Volume 11, Issue 7: 21190-21234. doi: 10.3934/math.2026861

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Fractional calculus for the one-dimensional Laplacian with Robin boundary condition

Fethi Bouzeffour ^,

Department of Mathematics, College of Sciences, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia

Received: 10 May 2026 Revised: 10 June 2026 Accepted: 30 June 2026 Published: 16 July 2026
MSC : Primary 47A60, 35R11, 26A33; Secondary 44A15, 47B25, 35K05, 47G30

We study fractional powers of the one-dimensional Robin Laplacian $ A_h = -d^2/dx^2 $ on the half-line, subject to the boundary condition $ u'(0) = hu(0) $, in the non-negative regime $ h\geq0 $. The analysis is based on a Robin-deformed cosine transform with kernel
$ \cos_h(x, \lambda) = \cos(\lambda x)+\frac{h}{\lambda}\sin(\lambda x), $
which diagonalizes $ A_h $. We prove a product formula for these eigenfunctions and use it to define positive generalized translations and a commutative Robin convolution. We then derive an explicit Robin heat kernel and Gaussian bounds. Finally, we define the fractional powers $ A_h^{s/2} $, $ 0 < s < 2 $, by spectral calculus and obtain their Bochner representation and singular-integral formula in terms of the Robin translations and an associated Lévy kernel. The Caffarelli–Silvestre extension is formulated for an extension exponent $ 0 < \sigma < 1 $; when it is applied to the operator $ A_h^{s/2} $, one takes $ \sigma = s/2 $. The case $ h = 0 $ recovers the fractional Neumann Laplacian on the half-line.
- Robin Laplacian,
- fractional powers of operators,
- Robin-deformed cosine transform,
- Bochner subordination,
- Caffarelli–Silvestre extension,
- maximum principle,
- spectral positivity estimate,
- Lévy kernel,
- heat semigroup
Citation: Fethi Bouzeffour. Fractional calculus for the one-dimensional Laplacian with Robin boundary condition[J]. AIMS Mathematics, 2026, 11(7): 21190-21234. doi: 10.3934/math.2026861

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Abstract

We study fractional powers of the one-dimensional Robin Laplacian $ A_h = -d^2/dx^2 $ on the half-line, subject to the boundary condition $ u'(0) = hu(0) $, in the non-negative regime $ h\geq0 $. The analysis is based on a Robin-deformed cosine transform with kernel

$ \cos_h(x, \lambda) = \cos(\lambda x)+\frac{h}{\lambda}\sin(\lambda x), $

which diagonalizes $ A_h $. We prove a product formula for these eigenfunctions and use it to define positive generalized translations and a commutative Robin convolution. We then derive an explicit Robin heat kernel and Gaussian bounds. Finally, we define the fractional powers $ A_h^{s/2} $, $ 0 < s < 2 $, by spectral calculus and obtain their Bochner representation and singular-integral formula in terms of the Robin translations and an associated Lévy kernel. The Caffarelli–Silvestre extension is formulated for an extension exponent $ 0 < \sigma < 1 $; when it is applied to the operator $ A_h^{s/2} $, one takes $ \sigma = s/2 $. The case $ h = 0 $ recovers the fractional Neumann Laplacian on the half-line.

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