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

MSC:

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

Citations:

Zbl 0516.47023
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References:

[1] A. Arnold, J.-P. Bartier and J. Dolbeault, Interpolation between logarithmic Sobolev and Poincaré inequalities, Commun. Math. Sci. 5 (2007), no. 4, 971-979. · Zbl 1146.60063
[2] W. Beckner, A generalized Poincaré inequality for Gaussian measures, Proc. Amer. Math. Soc. 105 (1989), no. 2, 397-400. · Zbl 0677.42020
[3] M. Émery and J. E. Yukich, A simple proof of the logarithmic Sobolev inequality on the circle, in Séminaire de Probabilités, XXI , 173-175, Lecture Notes in Math., 1247, Springer, Berlin. · Zbl 0616.46023
[4] L. Gross, Logarithmic Sobolev inequalities, Amer. J. Math. 97 (1975), no. 4, 1061-1083. · Zbl 0318.46049
[5] R. Latala and K. Oleszkiewicz, Between Sobolev and Poincaré, in Geometric aspects of functional analysis , 147-168, Lecture Notes in Math., 1745, Springer, Berlin. · Zbl 0986.60017
[6] C. E. Mueller and F. B. Weissler, Hypercontractivity for the heat semigroup for ultraspherical polynomials and on the n -sphere, J. Funct. Anal. 48 (1982), no. 2, 252-283. · Zbl 0506.46022
[7] A. Pazy, Semigroups of linear operators and applications to partial differential equations , Springer, New York, 1983. · Zbl 0516.47023
[8] G. Szego, Orthogonal polynomials , Third edition, Amer. Math. Soc., Providence, R.I., 1967.
[9] F.-Y. Wang, A generalization of Poincaré and log-Sobolev inequalities, Potential Anal. 22 (2005), no. 1, 1-15. · Zbl 1068.47051
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