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I'm trying to prove or disprove the inequality $$\int_{\mathbb{R}}f(x)^{p-2}f'(x)^4dx\leq \frac{3}{p(p-1)}\int_{\mathbb{R}}f(x)^pf''(x)^2dx$$ for any real-valued function $f$ (provided the RHS is finite) where $p\geq 6$ is an even integer.

I tried to disprove the inequality above by plugging in some nice functions (such as $e^{-x^2}, \frac{1}{1+x^2}$ or multiplying them by trig functions), but it failed. Proving the inequality seems to be very difficult since using Hölder, Galiardo-Nirenberg and so on splits the integral into two or more pieces. Is there any similar inequalities with a single integral on the right hand side like this, proved to be true? Otherwise, is there any counterexample to disprove such inequality? (Since this inequality is scaling-invariant, so scaling argument does not work.)

bellcircle
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    Some context for where this comes from would probably be of help to anyone who can help you in order to actually help you. – Mariano Suárez-Álvarez Apr 03 '23 at 08:14
  • I'm a Ph.D student and searching for a monotonicity formula applicable for higher-order KdV equations, and proving the inequality above is necessary to complete the proof of the monotonicity formula I've found. – bellcircle Apr 03 '23 at 08:21
  • (Always add information to the question itself, not in comments!) Do you have any evidence that the inequality is true or false? If your monotonicity formula (I have no idea what that is ）is true, does it follow that this inequality is true? and in that case, you clearly have hopes that your monotonicity formula is true, because of what exactly? – Mariano Suárez-Álvarez Apr 03 '23 at 08:26
  • It becomes more and more complicated to explain, so I give you some references about the matter: (1) Tao, Terence. "Two remarks on the generalised Korteweg de-Vries equation." Discrete and Continuous Dynamical Systems 18, no. 1 (2007): 1-14. // (2) Kim, Kihyun, and Soonsik Kwon. "Scattering for defocusing generalized Benjamin-Ono equation in the energy space ^{\frac {1} 2}(ℝ)." Transactions of the American Mathematical Society 372, no. 7 (2019): 5011-5067. – bellcircle Apr 03 '23 at 09:32
  • I do not know anything about Korteweg de-Vries equations and that stuff. But it seems to me that integration by parts (twice) gets you close to what you need. – Martin R Apr 03 '23 at 09:48
  • I tried it already, but it did not work at all. – bellcircle Apr 03 '23 at 10:31

1 Answers1

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No, that inequality does not hold in general. A counterexample is $f(x) = e^{-x^4}$ with $p = 6$. The following calculations were done with the Maxima compute algebra system: $$ \int_{-\infty}^\infty f(x)^{p-2} f'(x)^4 \, dx = \frac{90}{8^{13/4}} \Gamma\left( \frac 14 \right) \\ > \frac{381}{5 \cdot 8^{13/4}} \Gamma\left( \frac 14 \right) = \frac{3}{p(p-1)}\int_{-\infty}^\infty f(x)^p f''(x)^2 \, dx \, . $$

Remark: Generally, if $f$ and all its derivatives vanish at infinity, repeated integration by parts gives $$ \int_{-\infty}^\infty f(x)^{p-2} f'(x)^4 \, dx = \int_{-\infty}^\infty f(x)^{p-2} f'(x) \cdot f'(x)^3 \, dx \\ = - \frac{3}{p-1}\int_{-\infty}^\infty f(x)^{p-1} \cdot f'(x)^2 f''(x) \, dx \\ = - \frac{3}{p-1}\int_{-\infty}^\infty f(x)^{p-1} f'(x) \cdot f'(x) f''(x) \, dx \\ = \frac{3}{p(p-1)} \int_{-\infty}^\infty f(x)^{p} \left(f''(x)^2 + f'(x) f'''(x)\right) \, dx \\ = \frac{3}{p(p-1)} \int_{-\infty}^\infty f(x)^{p} f''(x)^2 \, dx - \frac{3}{(p+1)p(p-1)} \int_{-\infty}^\infty f(x)^{p+1} f^{(4)}(x) \, dx $$ so that your conjectured inequality holds if and only if $$ \int_{-\infty}^\infty f(x)^{p+1} f^{(4)}(x) \, dx \ge 0 \, . $$

Martin R
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