Here is a problem from my homework, which asks me to show a Gronwall-type inequality.

Let $1\leq\beta<\gamma\leq\infty, 0<T\leq\infty$ and let $f\in L^\rho(0,T)$, where $1\leq\rho<\infty$ is defined by $\frac1\rho=\frac1\beta-\frac1\gamma$. If $\eta\geq0$ and $\varphi\in L_{\text{loc}}^\gamma([0,T))$ satisfy $$\|\varphi\|_{L^\gamma(0,t)}\leq\eta+\|f\varphi\|_{L^\beta(0,t)},$$ for all $0<t<T$. Prove that $$\|\varphi\|_{L^\gamma(0,t)}\leq\eta\Phi(\|f\|_{L^\rho(0,t)}),$$ for all $0<t<T$, where $\Phi(s)=2\Gamma(3+2s)$ and $\Gamma$ is the Gamma function.

I cannot see why the Gamma function appears here. Applying Hölder's inequality to the assumption gives that $$\|\varphi\|_{L^\gamma(0,t)}\leq\eta+\|f\|_{L^\rho(0,t)}\|\varphi\|_{L^\gamma(0,t)}.$$ Now we can do the iteration, but this process fails to give the desired result.

Any help would be appreciated.

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  • $\begingroup$ there's no further context to the question? $\endgroup$ – Calvin Khor Apr 5 at 5:37
  • $\begingroup$ @CalvinKhor No. I’m afraid the above is all that I can provide. $\endgroup$ – Feng Shao Apr 6 at 8:43

I’m grateful for all efforts people made to solve my problem. With the help of my teacher, I finally find a solution. And it turns out that there is some thing wrong in the original problem.

Let $0=x_0<x_1<\cdots<x_n=T$ be such that $\|f\|_{L^\rho(x_{k-1},x_k)}=1/2$ for each $1\leq k\leq n-1$ and $\|f\|_{L^\rho(x_{n-1},x_n)}\leq1/2$. A simple calculation gives that $\|f\|_{L^\rho(0,x_k)}=\frac{k^{1/\rho}}2$ for $1\leq k\leq n-1$.

Next we do the iteration. By Minkowski's inequality and Hölder's inequality \begin{align*} \|\varphi\|_{L^\gamma(0,x_{k+1})}&\leq\eta+\|f\varphi\|_{L^\beta(0,x_{k+1})}\\ &\leq\eta+\|f\varphi\|_{L^\beta(0,x_{k})}+\|f\varphi\|_{L^\beta(x_k,x_{k+1})}\\ &\leq\eta+\|f\|_{L^\rho(0,x_k)}\|\varphi\|_{L^\gamma(0,x_k)}+\|f\|_{L^\rho(x_k,x_{k+1})}\|\varphi\|_{L^\gamma(x_k,x_{k+1})}\\ &\leq\eta+\frac{k^{1/\rho}}2\|\varphi\|_{L^\gamma(0,x_k)}+\frac12\|\varphi\|_{L^\gamma(x_k,x_{k+1})}\\ &\leq\eta+\frac{k^{1/\rho}}2\|\varphi\|_{L^\gamma(0,x_k)}+\frac12\|\varphi\|_{L^\gamma(0,x_{k+1})}. \end{align*} Note that $1\leq \rho<\infty$, we thus have $$\|\varphi\|_{L^\gamma(0,x_{k+1})}\leq 2\eta+k^{1/\rho}\|\varphi\|_{L^\gamma(0,x_k)}\leq 2\eta+k\|\varphi\|_{L^\gamma(0,x_k)}.$$ By induction we can easily deduce that $$\|\varphi\|_{L^\gamma(0,x_k)}\leq 2\eta (k+1)!$$

Fix $t\in(0,T)$, then there is some $1\leq k\leq n$ such that $t\in [x_{k-1},x_k)$, so \begin{align*} \|\varphi\|_{L^\gamma(0,t)}&\leq 2\eta (k+1)!\\ &=2\eta \Gamma(k+2)\\ &=2\eta \Gamma\left(3+\frac{k-1}2\cdot2\right). \end{align*} Since $\|f\|_{L^\rho(0,t)}\geq \|f\|_{L^\rho(0,x_{k-1})}=\frac{(k-1)^{1/\rho}}2$, we conclude that \begin{align*} \|\varphi\|_{L^\gamma(0,t)}&\leq 2\eta\Gamma\left(3+2(2\|f\|_{L^\rho(0,t)})^\rho\right)\\ &\leq\eta\Phi\left(2^\rho\|f\|_{L^\rho(0,t)}^\rho\right). \end{align*}

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