The interval G=(−1 ,1) turns into a Lie group under the group operation x ◦ y : = ( x + y ) ( 1 + x y ) − 1 , x , y ∈ G . This enables definition of the invariant measure d G ( x ) : = ( 1 − x 2 ) − 1 d x and the Fourier transformation F G on the interval G and, as a consequence, we can consider Fourier convolution operators W G , a 0 : = F G − 1 a F G on G. This class of convolutions includes celebrated Prandtl, Tricomi and Lavrentjev-Bitsadze equations and, also, differential equations of arbitrary order with the natural weighted derivative G u ( x ) = ( 1 − x 2 ) u ′ ( x ) , x∈ G. Equations are solved in the scale of Bessel potential H p s ( G , d G ( x ) ) , 1⩽ p⩽∞, and Hölder-Zygmound Z ν ( G ) , 0 ∞ spaces, adapted to the group G. Boundedness of convolution operators (the problem of multipliers) is discussed. The symbol a( ξ), ξ∈R, of a convolution equation W G , a 0 u = f defines solvability: the equation is uniquely solvable if and only if the symbol a is elliptic. The solution is written explicitly with the help of the inverse symbol. We touch shortly the multidimensional analogue-the Lie group G n .

The interval $j=[-1,1]$ turns into an Abelian group $\cA(\cJ)$ under the group operation $x+_\cJ y:=(x+y)(1+xy)^{-1},\qquad x,y\in\cJ$. This enables definition of the invariant measure $d_\cJ x=(1-x^2)^{-1}dx$ and the Fourier transform $\cF_\cJ$ on the interval $\cJ$ and, as a consequence, we can consider Fourier convolution operators $W^0_{\cJ,\cA}:=\cF_\cJ^{-1}\cA\cF_\cJ$ 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 $\fD_\cJ u(x)=-(1-x^2)u’(x)$, $t\in\cJ$. Equations are solved in the scale of Bessel potential $\bH^s_p(\cJ,d_\cJ x)$, $1\leqslant p\leqslant\infty$, and H\”older-Zygmound $\bZ^\nu(\cJ,(1-x^2)^\mu)$, $0<\mu,\nu<\infty$ spaces, adapted to the group $\cA(\cJ)$. Boundedness of convolution operators (the problem of multipliers) is discussed. The symbol $\cA(\xi)$, $\xi\in\bR$, of a convolution equation $W^0_{\cJ,\cA}u=f$ 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 $\cA(\cJ^n)$.

A mixed boundary value problem for the L\’ame equation in a thin layer $\Omega^h:\cC\times[-h,h]$ around a surface $\cC$ with the Lipshitz boundary is investigated. The main goal is to find out what happens when the thickness of the layer tends to zero $h\to0$. To this end we reformulate BVP into an equivalent variational problem and prove that the energy functional has the $\Gamma$-limit being the energy functional on the mid-surface $\cC$. The corresponding BVP on $\cC$, considered as the $\Gamma$-limit of the initial BVP, is written in terms of G\”unter’s tangential derivatives on $\cC$ and represents a new form of the shell equation. It is shown that the Neumann boundary condition from the initial BVP on the upper and lower surfaces transforms into a right-hand side term of the basic equation of the limit BVP.