# A Gronwall-type inequality for $L^p$ norms

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

Let $$1\leq\beta<\gamma\leq\infty, 0 and let $$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. Prove that $$\|\varphi\|_{L^\gamma(0,t)}\leq\eta\Phi(\|f\|_{L^\rho(0,t)}),$$ for all $$0, where $$\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.

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

Let $$0=x_0 be such that $$\|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*}