For \(0<p< \infty\) and \(-1<\alpha <+\infty,\) let \(L_a^p(\omega_\alpha)\) be the weighted Bergman space which contains analytic functions on the unit disk \(\mathbb{D}\) such that \[ \|f\|_{L_a^p(\omega_\alpha)}^p =\int_\mathbb{D} |f(z)|^p \omega_\alpha (z)dA(z)<\infty. \] For a finite positive Borel measure \(\mu\) on \(\mathbb{D}\), the Toeplitz operator \(T_{\mu,\alpha}\) is defined by \[ T_{\mu,\alpha} f(z)= \int_\mathbb{D} \frac{f(w)}{(1-z\overline{w})^{2+\alpha}}d\mu (w), z\in \mathbb{D}. \] In operator theory, the boundedness and compactness of the Toeplitz operator \(T_{\mu,\alpha}\) has obtained increasing interest in recent years. The case of \(1<p<\infty\) was attacked by Luecking and Zhu, followed by many experts. Very recently, Duan, Guo, Wang and et al. extended Peláez, Rättyä and Sierra’s work on bounded and compact Toeplitz operator. Using some characterizations on Carleson measures for weighted Bergman spaces, Pau and Zhao presented full characterizations of boundedness and compactness of the Toeplitz operator \(T_{\mu,\beta}:L_a^p(\omega_\alpha) \to L_a^q(\omega_\gamma) \), where the parameters satisfy special conditions.
By using technical tricks and methods in harmonic analysis, the authors generalized in a great deal the above results to provide descriptions of the boundedness and compactness of the Toeplitz operator \(T_{\mu,\beta}\) between distinct weighted Bergman spaces \(L_a^p(\omega_\alpha)\) and \( L_a^p(\omega_\beta)\) for \(0<p\leq 1, q=1,-1<\alpha,\beta<\infty\) and \(0<p\leq 1 <q <\infty,\) \(-1<\beta \leq \alpha <\infty\), respectively. It is worthy to mention that partial characterizations depend on the action on \(1\) by the Toeplitz operators. Precisely, for \(\alpha >0,\) let \(\mathcal{LB}^\alpha\) be the Banach space of analytic functions \(f\) on \(\mathbb{D}\) such that \[ \|f\|_{\mathcal{LB}^\alpha}:= |f(0)| +\sup_{z\in \mathbb{D}} (1-|z|^2)^\alpha \log (\frac{2}{1-|z|^2})|f'(z)|<\infty, \] and \(\mathcal{LB}^\alpha_0\) be the set of members \(f\) in \(\mathcal{LB}^\alpha\) such that \[ \lim_{|z|\to 1^-} (1-|z|^2)^\alpha \log (\frac{2}{1-|z|^2})|f'(z)| =0. \] Assume \(\mu\) is a positive Borel measure, \(0<p\leq 1\) and \(-1<\alpha, \beta <\infty\). The authors show that \(T_{\mu,\beta}:L_a^p(\omega_\alpha) \to L_a^1(\omega_\beta) \) is bounded if and only if \(\mu\) is a \(1/p \)-Carleson measure for \( L_a^1(\omega_\beta)\) and \(T_{\mu, \frac{2+\alpha}{p}-2}(1)\in \mathcal{LB}^1\); and that \(T_{\mu,\beta}:L_a^p(\omega_\alpha) \to L_a^1(\omega_\beta) \) is compact if and only if \(\mu\) is a vanishing \(1/p \)-Carleson measure for \( L_a^1(\omega_\beta)\) and \(T_{\mu, \frac{2+\alpha}{p}-2}(1)\in \mathcal{LB}_0^1\).