Continuous embedding of $L^p$ into $L^q + L^\infty$

Question

Let $L^p$ denote the Lebesgue-space over a $\sigma$-finite measure space $(\Omega,\mu)$.

It is known that $L^{p_0} \cap L^{p_1} \hookrightarrow L^p \hookrightarrow L^{p_0} + L^{p_1}$ continuously where $1 \le p_0 \le p \le p_1 < \infty$. Does the assertion stay true if $p_1 = \infty$, i.e. does it hold that $$L^{p_0} \cap L^\infty \hookrightarrow L^p \hookrightarrow L^{p_0} + L^\infty$$ whenever $1 \le p_0 \le p < \infty$?

The first inclusion is true by Lyapunov's inequality. But $L^p \hookrightarrow L^{p_0} + L^\infty$ continuously?

Answer 1

Let $q < p < \infty$. The norm of $L^q + L^\infty$ is $$||f||_{L^q + L^\infty} = \inf \left\{ \|g\|_{L^q} + \|h\|_{L^\infty} : f = g + h \right\}.$$ One way to decompose $f \in L^p$ into the sum of an $L^q$ function and an $L^\infty$ function is $$f = f \chi_{\{f > \|f\|_{L^p}\}} + f \chi_{\{f \le \|f\|_{L^p}\}} = g + h.$$ Chebyshev's inequality gives you $$\mu(\{ f > \|f\|_{L^p}\}) \le \frac{1}{\|f\|_{L^p}^p} \int_\Omega |f|^p \, d\mu = 1$$ so that $$\|g\|_{L^q} = \left( \int_{\{f > \|f\|_{L^p}\}} |f|^q \, d\mu \right)^\frac{1}{q} \le \left( \int_{\{f > \|f\|_{L^p}\}} |f|^p \, d\mu \right)^\frac{1}{p} \le \|f\|_{L^p}$$ by Holder's inequality. On the other hand $\|h\|_{L^\infty} \le \|f\|_{L^p}$ by definition. Thus $$\|f\|_{L^q + L^\infty} \le \|g\|_{L^q} + \|h\|_{L^\infty} \le 2 \|f\|_{L^p}.$$

A more judicious choice of $g$ and $h$ may yield a better constant.