@article{Bouzeffour2026, 
author = {Fethi Bouzeffour},
title = {Fractional calculus for the one-dimensional Laplacian with Robin boundary condition},
year = {2026},
journal = {AIMS Mathematics},
volume = {11},
number = {7},
pages = {21190-21234},
keywords = {Robin Laplacian, fractional powers of operators, Robin-deformed cosine transform, Bochner subordination, Caffarelli–Silvestre extension, maximum principle, spectral positivity estimate, Lévy kernel, heat semigroup},
url = {https://www.sciopen.com/article/10.3934/math.2026861},
doi = {10.3934/math.2026861},
abstract = {We study fractional powers of the one-dimensional Robin Laplacian        A    h    =  −      d    2        /    d      x    2   on the half-line, subject to the boundary condition        u    ′    (  0  )  =  h  u  (  0  ), in the non-negative regime    h  ≥  0. The analysis is based on a Robin-deformed cosine transform with kernel         cos    h    ⁡  (  x  ,  λ  )  =  cos  ⁡  (  λ  x  )  +      h    λ    sin  ⁡  (  λ  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  <  σ  <  1; when it is applied to the operator        A    h          s              /            2      , one takes    σ  =  s      /    2. The case    h  =  0 recovers the fractional Neumann Laplacian on the half-line.}
}