A family of integral inequalities on the circle \(S^{1}\). (English) Zbl 1204.47046

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. Mueller and F. B. 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. 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
\[ \mu(dx):=\frac{1}{\pi\sqrt{1-x^2}}dx. \]


47D06 One-parameter semigroups and linear evolution equations
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
26D15 Inequalities for sums, series and integrals
42A99 Harmonic analysis in one variable


Zbl 0516.47023
Full Text: DOI


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