In this paper, we obtain sharp remainder terms for the Hardy-Poincaré inequalities with general non-radial weights in the setting of Baouendi-Grushin vector fields (see Theorem 2. 5). It is worth emphasizing that all of our results are new both in the Baouendi-Grushin and standard Euclidean settings. The method employed allows us to not only unify, but also improve the results of Kombe and Yener KY18 for any 1<p< while holding true for complex-valued functions and providing explicit constants (Corollary 2. 7). As a result, we are able to obtain sharp remainder terms to many known weighted Hardy-type inequalities (see Section 3. 1). Aside from weighted Hardy-type inequalities, we also recover a sharp remainder formula for the L^p-Poincaré inequality (Corollary 3. 5). In the special case of radial weights, we are naturally able to introduce the notion of Baouendi-Grushin p-Bessel pairs (see Definition 2. 9). Finally, we apply the technique to establish the sharp remainder term of the Heisenberg-Pauli-Weyl inequality in L^p for 1<p< (Corollary 3. 13), which includes the sharp constant. This makes it possible to obtain the L^p-analogue for 2 p < n (Theorem 3. 17) of a stability result by Cazacu, Flynn, Lam and Lu CFLL24.
Shaimerdenov et al. (Fri,) studied this question.