Let ($\Omega, M, \mu$) be a complete measure space. Is it true that $$L^1(\Omega, M, \mu) \cap L^\infty(\Omega, M, \mu) ↪ L^p(\Omega, M, \mu)$$ for all $p \in [1,\infty]$?
May you help me answering this question please? I have no clue
2026-05-05 18:41:09.1778006469
Question on continuous injection between $L^p$ spaces
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In fact something more general is true. If $p < r < q$ then one has the inequality $$||f||_r \leq ||f||_p^{1/r - 1/q) /(1/p - 1/q)} ||f||_q^{(1/r-1/q)/(1/p-1/q)}.$$ See for instance here. This implies that $L^p \cap L^q \subseteq L^r$ always. See also a related discussion here.