arXiv subjects

Yerkin Shaimerdenov

Publications and source records attributed to Yerkin Shaimerdenov.

3 recordsLinked to original sources

arXiv2026

Sharp remainder formulae for general weighted Hardy and Rellich type inequalities for $1<p<\infty$

Inspired by the work of Cossetti and D'Arca [CD25], we show that the general weighted $L^{p}$-Hardy type inequalities [CD25, Theorems 1.1 and 1.2] and the corresponding identities hold for all $1<p<\infty$, thus extending their results beyond the case $p\geq 2$. In addition, we present a general weighted $L^{p}$-Rellich type inequality with a sharp remainder term for quasilinear second order degenerate elliptic differential operators. In particular, even for the classical Laplacian, these identities appear to be new.

math.AP↗

arXiv2026

Sharp remainder terms and stability of weighted Hardy-Poincaré and Heisenberg-Pauli-Weyl inequalities related to the Baouendi-Grushin operator

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 $1<p<\infty$ 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). Furthermore, we apply the technique to establish the sharp remainder term of the Heisenberg-Pauli-Weyl inequality in $L^{p}$ for $1<p<\infty$ (Corollary 3.13), which includes the sharp constant. This makes it possible to obtain the $L^{p}$-analogue for $2\leq p < n$ (Theorem 3.17) of a stability result by Cazacu, Flynn, Lam and Lu [CFLL24]. Lastly, as another application, the non-existence of positive solutions to nonlinear parabolic partial differential equations is investigated (Theorem 3.22).

math.AP↗

arXiv2025

Cylindrical extensions of critical Sobolev type inequalities and identities

In this paper, we investigate cylindrical extensions of critical Sobolev type (improved Hardy) inequalities and identities in the style of Badiale-Tarantello [BT02], which in a special case give a critical Hardy inequality and its stability results. We also obtain higher-order identities, which interestingly include well-known numbers like double factorial, Oblong numbers, and Stirling numbers of the second kind. All functional identities are obtained in $L^{p}$ for $p\in (1,\infty)$ without the real-valued function assumption, which gives a simple and direct understanding of the corresponding inequalities as well as the nonexistence of nontrivial extremizers. As applications, we obtain Caffarelli-Kohn-Nirenberg type inequalities with logarithmic weights, which in a particular case give the critical case of the Heisenberg-Pauli-Weyl type uncertainty principle. We also discuss these results in the setting of Folland and Stein's homogeneous Lie groups. A special focus is devoted to stratified Lie groups, where Sobolev type inequalities become intricately intertwined with the properties of sub-Laplacians and more general subelliptic partial differential equations. The obtained results are already new even in the classical Euclidean setting with respect to the range of parameters and the arbitrariness of the choice of any homogeneous quasi-norm. Most inequalities are obtained with sharp constants.

math.AP↗