Let $\|f\|_{L^p(\omega) }$ denote the standard $L^p$ norm of the function $f*\omega$ where $\omega$ is non-negative and measurable. I am trying to show that certain norms are bounded while changing the weights. Given two weights $\omega_1,\omega_2$ let $m=\min(\omega_1,\omega_2)$. I am trying to show that $$\|f\|_{L^p(m)}\le \|f\|_{L^p(\omega_1)+L^p(\omega_2)}\le 2^{1/q}\|f\|_{L^p(m)}$$ where $1/p+1/q=1$.
here $$\|f_1\|_{L^p(\omega_1)+L^p(\omega_2)}=\inf\{\|f_1\|_{L^p(\omega_1)}+\|f\|_{L^p(\omega_2)}:f_1+f_2=g\}$$ and q is the conjugate of p. The first inequality isn't too challenging and I am able to figure that out. However I am struggling to figure out how to solve the second inequality and how to introduce the $2^{1/q}$.
First note that the concavity of $(\cdot)^{1/p}$ implies that \begin{align*} \dfrac{1}{2}a^{1/p}+\dfrac{1}{2}b^{1/p}\leq\left(\dfrac{a+b}{2}\right)^{1/p}, \end{align*} and hence \begin{align*} a^{1/p}+b^{1/p}\leq 2^{1/q}(a+b)^{1/p}. \end{align*} Now $|f|=|f|\chi_{\omega_{1}\leq\omega_{2}}+|f|\chi_{\omega_{1}>\omega_{2}}:=f_{1}+f_{2}$ and hence \begin{align*} &\|f\|_{L^{p}(\omega_{1})+L^{p}(\omega_{2})}\\ &\leq\left(\int f_{1}^{p}\omega_{1}\right)^{1/p}+\left(\int f_{2}^{p}\omega_{2}\right)^{1/p}\\ &\leq 2^{1/q}\left(\int f_{1}^{p}\omega_{1}+f_{2}^{p}\omega_{2}\right)^{1/p}\\ &=2^{1/q}\left(\int |f|^{p}\chi_{\omega_{1}\leq\omega_{2}}\omega_{1}+|f|^{p}\chi_{\omega_{1}>\omega_{2}}\omega_{2}\right)^{1/p}\\ &= 2^{1/q}\left(\int|f|^{p}\chi_{\omega_{1}\leq\omega_{2}}m+|f|^{p}\chi_{\omega_{1}>\omega_{2}}m\right)^{1/p}\\ &=2^{1/q}\left(\int|f|^{p}m\right)^{1/p}\\ &=2^{1/q}\|f\|_{L^{p}(m)}. \end{align*}