I want to check if a function $f$ defined on $[0,T]$ is a $L_2$ function.
What I know is $f$ is a $L_1$ function. (but $f$ could be not bounded)
So I want to use an inequality like
$$ ||\cdot||_{L^{2}} \leq C ||\cdot||_{L^{1}} $$ where some constant $C$ or function.
Is there the above inequality?
Or how can I prove $f$ is an $L_2$ function and what analytic conditions guarantee $f$ is $L_2$ when $f$ is $L_1$?
Relations between functions from \begin{equation} L_p, L_q \end{equation} called embedding. In general, for 0 ≤ p < q ≤ ∞. \begin{equation} L_q(S, μ) ⊂ L_p(S, μ) \end{equation} when S does not contain sets of arbitrarily large measure, i.e. we have bounded interval and \begin{equation} L_p(S, μ) ⊂ L_q(S, μ) \end{equation} when S does not contain arbitrarily small sets of non-zero measure, i.e. function does not have non-integrable spikes. Following relation holds: \begin{equation} \lVert f \lVert _p \le \lVert f \lVert _q \mu(S)^{\frac{1}{p}-\frac{1}{q}}\end{equation} There are other similar relations between different functional spaces. For example, search for Sobolev inequality and its generalizations, Nash inequality, etc.; In general, answer to your question is "yes, but" it requires additional conditions.