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The following result can be found in this article

(Jensen’s inequality) Let $v = v(x, t)$ be any nonnegative function. Then it holds that, for all $t > 0$, $$[S(t)v(s)]^q \leq S(t)v^q(s)$$ if $q \geq 1,$ and $$[S(t)v(s)]^q \geq S(t)v^q(s)$$ if $q \leq 1.$

The demonstration is omitted and I do not know how these inequalities are obtained. I was wondering if they remain true if $ S (t) $ is Schrödinger semigroup? Because the author uses them assuming that $S(t)$ is the heat semigroup. If true, can you give me a reference for consultation?

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$$S(t)v(x)=\int_E k(t,x,y)v(y) dy=\int_E k(t,x,y)^{\frac1q+\frac{1}{q'}}v(y) dy \le\left (\int_E k(t,x,y)v(y)^q dy\right )^{\frac1q} \left (\int_E k(t,x,y) dy \right )^{\frac{1}{q'}}.$$ What you need is $\int_E k(t,x,y) dy \le 1$.

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  • $\begingroup$ It really is very simple, I thought it would be complicated. Thank you for your help. $\endgroup$
    – Ilovemath
    Commented Dec 4, 2020 at 16:38
  • $\begingroup$ Yes it is! You need only a control for $S(t)1$. $\endgroup$
    – Giorgio Metafune
    Commented Dec 4, 2020 at 16:45

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