In this paper, we establish several interpolation inequalities in the weighted Lebesgue spaces and Morrey spaces. By using the classical Calderón–Zygmund decomposition with respect to ω (a weight function), we prove that $L^{p}(\omega )\cap \mathrm{BMO}(\mathbb{R}^{n})\subset L^{q}(\omega )$ for all q with $p< q<\infty $, where $1\leq p<\infty $ and $\omega \in A_{p}$. We also prove that there exists a constant $C(p,q,n,[\omega ]_{A_{p}})$ depending on $p,q,n$ and $[\omega ]_{A_{p}}$ such that the following inequality \( \|f\|_{L^{q}(\omega )}\leq C\big(p,q,n,[\omega ]_{A_{p}}\big)\cdot \big(\|f\|_{L^{p}(\omega )}\big)^{p/q}\cdot \big(\|f\|_{\mathrm{BMO}} \big)^{1-p/q} \) holds for all $f\in L^{p}(\omega )\cap \mathrm{BMO}(\mathbb{R}^{n})$ with $1\leq p<\infty $ and $\omega \in A_{p}$. Moreover, this embedding constant is shown to have the optimal growth order q as $q\to \infty $, which was given by Chen–Zhu, Kozono–Wadade and Milman in the unweighted case. Furthermore, we show that $L^{p,\kappa }(\omega )\cap \mathrm{BMO}(\mathbb{R}^{n})\subset L^{q, \kappa }(\omega )$ for all q with $p< q<\infty $, where $1\leq p<\infty $, $\omega \in A_{p}$ and $0<\kappa <1$. Moreover, there exists a constant $\widetilde{C}(p,q,n,[\omega ]_{A_{p}})$ depending on $p,q,n$ and $[\omega ]_{A_{p}}$ such that the following inequality \( \|f\|_{L^{q,\kappa }(\omega )}\leq \widetilde{C}\big(p,q,n,[\omega ]_{A_{p}} \big) \cdot \big(\|f\|_{L^{p,\kappa }(\omega )}\big)^{p/q}\cdot \big( \|f\|_{\mathrm{BMO}}\big)^{1-p/q} \) holds for all $f\in L^{p,\kappa }(\omega )\cap \mathrm{BMO}(\mathbb{R}^{n})$ with $1\leq p<\infty $, $\omega \in A_{p}$ and $0<\kappa <1$. This embedding constant is shown to have the linear growth order as $q\to \infty $, that is, $\widetilde{C}(p,q,n,[\omega ]_{A_{p}})\leq C_{n}\cdot q$ with the constant $C_{n}$ depending only on the dimension n, when q is sufficiently 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. In addition, we investigate the inclusion relation between weighted weak Morrey spaces and Morrey spaces, and the size of the embedding constant from weighted weak Morrey spaces into weighted Morrey spaces is specified.
