Several inequalities concerning interpolation in classical Fourier analysis

In this note, we establish several interpolation inequalities in $\mathbb R^n$ in the Lebesgue spaces and Morrey spaces. By using the classical Calderon--Zygmund decomposition, we will reprove that $L^{p}(\mathbb R^n)\cap\mathrm{BMO}(\mathbb R^n)\subset L^{q}(\mathbb R^n)$ for all $q$ with $p<q<\infty$, where $1\leq p<\infty$. We also reprove that there exists a constant $C(p,q,n)$ depending on $p,q,n$ such that the following inequality \begin{equation*} \|f\|_{L^q}\leq C(p,q,n)\cdot\big(\|f\|_{L^p}\big)^{p/q}\cdot\big(\|f\|_{\mathrm{BMO}}\big)^{1-p/q} \end{equation*} holds for all $f\in L^{p}(\mathbb R^n)\cap\mathrm{BMO}(\mathbb R^n)$ with $1\leq p<\infty$. Moreover, this embedding constant has the optimal growth order $q$ as $q\to\infty$, which was given by Chen--Zhu, and Kozono--Wadade. We will show that $L^{p,\kappa}(\mathbb R^n)\cap\mathrm{BMO}(\mathbb R^n)\subset L^{q,\kappa}(\mathbb R^n)$ for all $q$ with $p<q<\infty$, where $1\leq p<\infty$ and $0<\kappa<1$. Moreover, there exists a constant $\widetilde{C}(p,q,n)$ depending on $p,q,n$ such that \begin{equation*} \|f\|_{L^{q,\kappa}}\leq \widetilde{C}(p,q,n)\cdot\big(\|f\|_{L^{p,\kappa}}\big)^{p/q}\cdot\big(\|f\|_{\mathrm{BMO}}\big)^{1-p/q} \end{equation*} holds for all $f\in L^{p,\kappa}(\mathbb R^n)\cap\mathrm{BMO}(\mathbb R^n)$ with $1\leq p<\infty$ and $0<\kappa<1$. This embedding constant is shown to have the linear growth order as $q\to\infty$, that is, $\widetilde{C}(p,q,n)\leq C_n\cdot q$ with the constant $C_n$ depending only on the dimension $n$, when $q$ is large. As an application of the above results, some new bilinear estimates are also established, which can be used in the study of the global existence and regularity of weak solutions to elliptic and parabolic partial differential equations of the second order.