论文标题
谎言组的卷积方程（-1,1）
Convolution equations on the Lie group (-1,1)
论文作者
论文摘要
间隔$ j = [ - 1,1] $变成group操作$ x+_ \ cj y：=（x+y）（x+y）（1+xy）^{ - 1}，\ qquad x，y \ in \ cj $。此功能可以定义不变度度量$ d_ \ cj x =（1-x^2）^{ - 1} dx $和founier transform $ \ cf_ \ cj $ in Interval $ \ cj $，因此，我们可以考虑傅立叶卷积运算符$ w^0 _ {\ cj，\ ca}：= \ cf_ \ cj^{ - 1} \ ca \ cf_ \ cf_ \ cj $ on $ \ cj $。这类卷积包括著名的Prandtl，Tricomi和Lavrentjev-Bitsadze方程式，以及与自然加权派生$ \ fd_ \ fd_ \ cj U（x）= - （1-x^2）U'（1-x^2）u'（x）u'（x）$，$ t \ in \ cj $。方程以贝塞尔潜在$ \ bh^s_p（\ cj，d_ \ cj x）$，$ 1 \ leqslant p \ leqslant \ infty $和hölder-zygmound $ \ bz^n（\ bz^n（\ cj，\ cj，（1- x^2）$ 0 <适用于组$ \ ca（\ cj）$。讨论了卷积操作员的界限（乘数问题）。卷积方程的符号$ \ ca（ξ）$，$ξ\ in \ br $，$ w^0 _ {\ cj，\ ca} u = f $定义了解决性：该方程在符号$ \ ca $的情况下才能唯一解决。该解决方案是在逆符号的帮助下明确编写的。 我们很快就会触摸多维模拟 - Abelian Group $ \ ca（\ cj^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 Hölder-Zygmound $\bZ^ν(\cJ,(1-x^2)^μ)$, $0<μ,ν<\infty$ spaces, adapted to the group $\cA(\cJ)$. Boundedness of convolution operators (the problem of multipliers) is discussed. The symbol $\cA(ξ)$, $ξ\in\bR$, 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 $\cA(\cJ^n)$.
