Inequality in Sobolev Space ($L^p$ norm)

Question

I want to find a constant $C$ that depends on the parameters $a$ and $p$ that satisfies the inequality

$$\|f\|_p \leq a\|f'\|_1+C\|f\|_1$$

for all $f \in W^{1,1}(0,1)$. This is for arbitrary $p \in [1,\infty)$.

I know the definition of the Sobolev space, so

$$\|f\|_{W^{1,1}}=\int_{0}^{1}|f|d\mu+\int_{0}^{1}|f'|d\mu.$$

I know about inequalities in $L^p$ spaces, namely

$$\|f\|_p \leq \|f\|_q$$ when $p \leq q$.

Answer 1

Note that a function $f \in W^{1,1}(0,1)$ has a continuous representative of the form

$$f(x)=f(0)+\int_0^x f'(y)dy$$

for all $x \in [0,1]$. Then we are imitating the proof of the continuous embedding $W^{1,1}(0,1) \hookrightarrow C([0,1])$ to get

$$\begin{aligned} \|f\|_{L^p} \leq \max_{x \in [0,1]} |f(x)| &\leq |f(0)| + \int_0^1 |f'(y)|dy \leq \int_0^1 |f(0)|dx + \| f' \|_{L^1} \end{aligned} $$

and

$$\int_0^1 |f(0)| dx = \int_0^1 \bigg|f(x)-\int_0^x f'(y)dy\bigg|dx\leq \|f\|_{L^1} + \int_0^1 \int_0^x |f'(y)|dy dx \leq \|f\|_{L^1} +\|f'\|_{L^1}.$$

Therefore for all $p \in [1,\infty]$

$$\|f\|_{L^p} \leq \|f\|_{L^1} + 2 \|f'\|_{L^1}.$$