Prove that there exists $g\in L^1(K)$ such that $a_1 +a_2 |y_n|^{p-1}\le g(x)$ a.e. in $\mathbb R$, for all $n\in\mathbb N$

Question

Let $q\in [1, 5)$ and $K$ denote a compact subset of $\mathbb R$. Let $\{y_n\}_n$ be a sequence of functions such that:

$$\mbox{there exists } h\in L^q(K) \mbox{ such that } |y_n(x)|\le h(x) \quad\mbox{ a.e. in }\mathbb R, \forall n\in\mathbb N.$$

The question is: given two positive constants $a_1, a_2$, is it true that there exists $g\in L^1(K)$ such that $$a_1 +a_2 |y_n|^{q-1}\le g(x) \quad\mbox{ a.e. in }\mathbb R, \forall n\in\mathbb N?$$

The lecturer wrote this few days ago while he was proving a Lemma, but I am not able to detect all the steps which lead to this. Moreover, there is any relation between $h$ and $g$? Anyone could please help me with that?

Answer 1

Assume $q > 1$ (for $q=1$ just take $a_1 + a_2$). By Young's inequality $ab \le \frac{a^p}{p} + \frac{b^q}{q}$ (where $\frac{1}{p} + \frac{1}{q}=1$),we have $$ \begin{align*} a_1 + a_2|y_n|^{q-1} &\le a_1 + \frac{a_2^q}{q} + \frac{|y_n|^q}{p} \\ &\le a_1 + \frac{a_2^q}{q} + \frac{|h|^q}{p} =: g \in L^1(K) \end{align*} $$