Log-Sobolev Inequality

My 2009 Williams College NSF “SMALL” undergraduate research Geometry Group has the following inequality for any C^1 function on the unit interval and for any p ≥ 1:

\left(\int_0^1{f^{\frac{p+1}{p}}}\right)^\frac{p}{p+1}\le\int_0^1{\left(f^2+f'^2/\pi^2\right)^{1/2}}

with equality for constant functions and if p>1 only for constant functions. They conjecture that these results still hold if \pi^2 on the right-hand side is replaced by p\pi^2 (sharp).

The case p=1 is standard and follows from Wirtinger’s Inequality.

Are any inequalities like this known?

5 Comments

  1. LJS:

    Isn’t the LHS decreasing in p, so p=1 is worst case?
    This does not of course help with the conjecture.

    Right you are, thanks very much. The undergraduates came upon it in a completely different way, as a generalization of the isoperimetric inequality (p=1) to sectors of planes with density r^{p-1}, using the four vertex theorem. Your comment inspired a simpler geometric comparison argument. Do you believe the conjecture?—FM

  2. Frank Morgan:

    Comment from David Thompson, SMALL undergraduate research Geometry Group ’09

    The conjectured inequality says that for all 1\leq q \leq 2,

    \left[\int_{0}^{1}f^{q}\right]^{1/q}\leq\int_{0}^{1}\sqrt{f^2+(q-1)f'^2/\pi^2}.

    The q=1 case holds trivially; q=2 is the standard isoperimetric inequality. Both sides are concave as functions of q. Would some kind of interpolation argument prove the result for all q?

  3. Frank Morgan:

    Francesco Maggi at Firenze notes that differentiating with respect to q at q=1 implies the following log-Sobolev inequality for uniform measure on the interval for a nonnegative function f of mean 1:

    Corollary of Conjecture: 2\pi^2\int_{0}^{1}f log f\leq\int_{0}^{1}f'^2/f.

    Similarly the corollary, on small perturbations of f = 1, implies the sharp, known Wirtinger inequality:
    [\int_0^1 u^2\le(1/\pi^2)\int_0^1 (u')^2]
    for (nonperiodic) u of mean 0. (For u periodic the sharp constant is 1/4\pi^2.)

  4. Frank Morgan:

    My 2012 undergraduate research Geometry Group has observed that the sharp log-Sobolev inequality on the interval follows immediately from a similar inequality on the circle.

  5. Frank Morgan » Blog Archive » Manifolds with Density: Fuller References:

    […] [2012] Alexander Díaz, Nate Harman, Sean Howe, David Thompson, Isoperimetric problems in sectors with density, Adv. Geom. 12 (2012), 589–619; arXiv.org (2010); see blog posts 1 and 2. […]

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