A family of integral inequalities on the circle \(S^{1}\)
The purpose of this paper is to present a family of integral inequalities on the unit circle \(S^1\) which provide interpolation between the Sobolev, the logarithmic Sobolev (of C.\,E.\thinspace Mueller and F.\,B.\thinspace Weissler) and Poincaré inequalities. These types of integral inequalities are deeply related to the aspects of the large-time behavior of parabolic PDE's [cf.\ \textit{A.\,Pazy}, ``Semigroups of linear operators and applications to partial differential equations'' (Applied Mathematical Sciences 44; New York etc.:\ Springer-Verlag) (1983; Zbl 0516.47023)]). One of the applications of the paper's main result is a hypercontractive estimation for the Chebyshev semigroup of operators \((P_t)_{t>0}\) generated by the Chebyshev operator \(\mathfrak{L}\) defined by \[ \mathfrak{L}:=(1-x^2)\frac{d^2}{dx^2}-x\frac{d}{dx},\quad x\in I=[-1,1] \] and acting on \(L^2(I,\mu)\) with respect to the probability measure \[ \mu(dx):=\frac{1}{\pi\sqrt{1-x^2}}dx. \]
scientific article; zbMATH DE number 5773630