2022/08/18 by Roland Duduchava, Duduchava, Roland
Computer Science · Mathematics · #42A45 #43A25 #45A05 #45E10 #Advanced Mathematical Modeling in Engineering #Differential Equations and Boundary Problems #FOS: Physical sciences #Mathematical Physics (math-ph) #advanced mathematical theories
paper · pdf · doi:10.48550/arxiv.2208.08765
openalex publication_date 2022/08/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The interval j=[-1,1] turns into an Abelian group \cA(\cJ) under the group operation x+_\cJ y:=(x+y)(1+xy)-1, x,y∈\cJ. This enables definition of the invariant measure d_\cJ x=(1-x2)-1dx and the Fourier transform \cF_\cJ on the interval \cJ and, as a consequence, we can consider Fourier convolution operators W0\cJ,\cA:=\cF_\cJ-1\cA\cF_\cJ on \cJ. This class of convolutions includes celebrated Prandtl, Tricomi and Lavrentjev-Bitsadze equations and, also, differential equations of arbitrary order with the natural weighted derivative \fD_\cJ u(x)=-(1-x2)u'(x), t∈\cJ. Equations are solved in the scale of Bessel potential \bHsp(\cJ,d_\cJ x), 1\leqslant p\leqslant∞, and Hölder-Zygmound \bZν(\cJ,(1-x2)μ), 0<μ,ν<∞ spaces, adapted to the group \cA(\cJ). Boundedness of convolution operators (the problem of multipliers) is discussed. The symbol \cA(ξ), ξ∈\bR, of a convolution equation W0\cJ,\cAu=f defines solvability: the equation is uniquely solvable if and only if the symbol \cA is elliptic. The solution is written explicitely with the help of the inverse symbol. We touch shortly the multidimensional analogue-the Abelian group \cA(\cJn).