arXiv subjects

Nurgissa Yessirkegenov

Publications and source records attributed to Nurgissa Yessirkegenov.

At least 19 recordsLinked to original sources

arXiv2026

Explicit sharp gradient stability of the $L^{p}$-Poincaré inequality and fundamental gaps for the Dirichlet $p$-Laplacian operator

In this paper, we prove the gradient stability of the $L^{p}$-Poincaré inequality with explicit constants on bounded convex domains for any $1<p<\infty$. These estimates control the distance to the optimizers in the strongest possible norm with the optimal stability exponent $\max\{2,p\}$, exhibiting the same phenomenon initially observed by Figalli and Zhang [FZ22a] for the $L^p$-Sobolev inequality. As a result, we also confirm the conjecture posed by Fleckinger-Pellé and Takáč [FPT02, Section 8]. We then apply the inequality to obtain positive fundamental gap estimates in terms of the inradius for the Dirichlet $p$-Laplacian.

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arXiv2026

Stability of the $L^{p}$-Poincaré inequality for the Lebesgue measure and Gaussian probability measure with explicit geometric dependence and applications to spectral gaps

In this paper, we obtain stability results for the $L^{p}$-Poincaré inequality for both Lebesgue measure and Gaussian probability measure (Theorem 3.3 and Theorem 3.13) that involve explicit dependence on the geometry of the domain. As a byproduct, the explicit constant allows us to recover important results of Yu, Zhong [YZ86] and Smits [Smi96] (Corollary 3.9), related to the fundamental gap conjecture of the Laplacian (resolved by Andrews and Clutterbuck [AC11]), thereby providing an alternative proof. Moreover, we extend this spectral gap result to the $p$-Laplacian (Corollary 3.6). Such gap estimates for the Dirichlet $p$-Laplacian appear to be unavailable, as also observed in [DSW18]. Our approach relies on properties of the first eigenfunction of the (Gaussian) $p$-Laplacian operator and weighted Poincaré inequalities for log-concave measures on convex domains.

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arXiv2026

Sharp forms and quantitative stability for general weighted discrete $p$-Hardy inequalities

In this paper, we provide a sharp remainder term for the general weighted discrete $p$-Hardy inequality. By choosing appropriate weights and specifying $1<p<\infty$, we are able to recover the identity by Krej{č}i{ř}{\'ı}k-Štampach [KS22, Theorem 1], obtain the sharp form of the $p$-Hardy inequality by Fischer-Keller-Pogorzelski [FKP23, Theorem 1] and generalize the power weighted inequality by Gupta [Gup22, Theorem 2.1] with a sharp remainder. In addition, we prove a quantitative stability type result, thereby showing that the deficit of the discrete $p$-Hardy inequality controls the weighted distance to the family of non-trivial minimizers.

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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.

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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).

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arXiv2025

Refined general weighted $L^{p}$-Hardy and Caffarelli-Kohn-Nirenberg type inequalities and identities related to the Baouendi-Grushin operator

In this paper, we present a sufficient condition on a pair of nonnegative weights $v$ and $w$ such that we have a general weighted $L^{p}$-Hardy type identity. The result, for a certain choice of weights, gives weighted $L^{p}$-Hardy type inequalities and identities with explicit remainder terms, thereby improving previously known results. Furthermore, we obtain the corresponding general weighted Caffarelli-Kohn-Nirenberg type inequality with remainder terms, which, as a result, imply Heisenberg-Pauli-Weyl type inequalities.

math.FA↗

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.

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arXiv2025

Sharp remainder of the $L^{p}$-Poincaré inequality for Baouendi-Grushin vector fields

In this paper, we establish a sharp remainder formula for the Poincaré inequality for Baouendi-Grushin vector fields in the setting of $L^{p}$ for complex-valued functions. In special cases, we recover previously known results. Consequently, we also derive the $L^{p}$-Poincaré inequality with an explicit optimal constant under a certain assumption. Additionally, we provide estimates of the remainder term for $p\geq2$ and $1<p<2\leq n<\infty$. As an application, we obtain a blow-up in finite time and global existence of the positive solutions to the initial-boundary value problem of the doubly nonlinear porous medium equation involving a degenerate nonlinear operator $Δ_{γ,p}$.

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arXiv2025

Critical, stability and higher-order analysis for Hardy type inequalities on Cartan-Hadamard manifolds

In this paper, we focus on three main objectives related to Hardy-type inequalities on Cartan-Hadamard manifolds. Firstly, we explore critical Hardy-type inequalities that contain logarithmic terms, highlighting their significance. Secondly, we examine the stability of both critical and subcritical cases of the Hardy inequality. Lastly, we establish two weighted Hardy-type inequalities where singularities appear at the origin as well as boundary and we discuss their implications for higher-order operators. Our results improve upon previous findings and also present new higher-order versions of these inequalities as additional outcomes.

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arXiv2025

A comparison principle for nonlinear parabolic equations with nonlocal source and gradient absorption

This paper investigates the initial-boundary value problem for a nonlinear parabolic equation involving the $p$-Laplacian operator, nonlocal source terms, gradient absorption, and various nonlinearities: \[ \frac{\partial u}{\partial t} - \text{div}(|\nabla u|^{p-2} \nabla u ) = α|u|^{k-1}u \int_Ω|u|^s \, dx - β|u|^{l-1}u |\nabla u|^q + γu^m + μ|\nabla u|^r - ν|u|^{σ-1}u, \] where $ Ω$ is a bounded domain in $\mathbb{R}^N$, $N \geq 1$, with a smooth boundary $\partial Ω$. The parameters satisfy $ α, l, σ> 0 $, $ β, ν\geq 0 $, $ k, m, s \geq 1 $, $ r \geq p - 1 \geq \frac{p}{2}$, and $γ, μ\in \mathbb{R}$. We establish a comparison principle for this problem. Using this principle, we derive blow-up results as well as global-in-time boundedness of solutions. Our results extend and unify previous studies in the literature.

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arXiv2024

Cylindrical Hardy, Sobolev type and Caffarelli-Kohn-Nirenberg type inequalities and identities

In this paper we discuss cylindrical extensions of improved Hardy, Sobolev type and Caffarelli-Kohn-Nirenberg type inequalities with sharp constants and identities in the spirit of Badiale-Tarantello [2]. All identities are obtained in the setting of $L^p$ for all $p\in (1,\infty)$ without the real-valued function assumption. The obtained identities provide a simple and direct understanding of these inequalities as well as the nonexistence of nontrivial extremizers. As a byproduct, we show extended Caffarelli-Kohn-Nirenberg type inequalities with remainder terms that imply a cylindrical extension of the classical Heisenberg-Pauli-Weyl uncertainty principle. Furthermore, we prove $L^p$-Hardy type identities with logarithmic weights that imply the critical Hardy inequality in the special case. Lastly, we also discuss extensions of these results on homogeneous Lie groups. Particular attention is paid to stratified Lie groups, where the functional inequalities become intricately intertwined with the properties of sub-Laplacians and related hypoelliptic partial differential equations. The obtained results already imply new insights even in the classical Euclidean setting with respect to the range of parameters and the arbitrariness of the choice of any homogeneous quasi-norm.

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arXiv2024

Hardy-Sobolev-Rellich, Hardy-Littlewood-Sobolev and Caffarelli-Kohn-Nirenberg inequalities on general Lie groups

In this paper we establish a number of geometrical inequalities such as Hardy, Sobolev, Rellich, Hardy-Littlewood-Sobolev, Caffarelli-Kohn-Nirenberg, Gagliardo-Nirenberg inequalities and their critical versions for an ample class of sub-elliptic differential operators on general connected Lie groups, which include both unimodular and non-unimodular cases in compact and noncompact settings. We also obtain the corresponding uncertainty type principles.

math.FA↗

arXiv2023

Unified weighted Hardy-type inequalities

We present a unified approach to obtain Hardy-type inequalities in the context of nilpotent Lie groups with sharp constants. The unified methodology employed herein allows for exploration of the sharp Hardy inequalities on various Lie group structures and improves previously known inequalities even in the classical Euclidean setting. By leveraging this framework, as an application, we derive new Caffarelli-Kohn-Nirenberg type inequalities both in the classical Euclidean setting and on general homogeneous Lie groups.

math.FA↗

arXiv2021

Existence and non-existence of global solutions for semilinear heat equations and inequalities on sub-Riemannian manifolds, and Fujita exponent on unimodular Lie groups

In this paper we study the global well-posedness of the following Cauchy problem on a sub-Riemannian manifold $M$: \begin{equation*} \begin{cases} u_{t}-\mathfrak{L}_{M} u=f(u), \;x\in M, \;t>0, \\u(0,x)=u_{0}(x), \;x\in M, \end{cases} \end{equation*} for $u_{0}\geq 0$, where $\mathfrak{L}_{M}$ is a sub-Laplacian of $M$. In the case when $M$ is a connected unimodular Lie group $\mathbb G$, which has polynomial volume growth, we obtain a critical Fujita exponent, namely, we prove that all solutions of the Cauchy problem with $u_{0}\not\equiv 0$, blow up in finite time if and only if $1 0$, we also show that the differential inequality $$ u_{t}-\mathfrak{L}_{M} u\geq f(u) $$ does not admit any nontrivial distributional (a function $u\in L^{p}_{loc}(Q)$ which satisfies the differential inequality in $\mathcal{D}^{\prime}(Q)$) solution $u\geq 0$ in $Q:=(0,\infty)\times\mathbb G$. Furthermore, in the case when $\mathbb G$ has exponential volume growth and $f:[0,\infty)\to[0,\infty)$ is a continuous increasing function such that $f(u)\leq K_{1}u^{p}$ for some $K_{1}>0$, we prove that the Cauchy problem has a global, classical solution for $1<p<\infty$ and some positive $u_{0}\in L^{q}(\mathbb G)$ with $1\leq q<\infty$. Moreover, we also discuss all these results in more general settings of sub-Riemannian manifolds $M$.

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arXiv2021

Critical Gagliardo-Nirenberg, Trudinger, Brezis-Gallouet-Wainger inequalities on graded groups and ground states

In this paper we investigate critical Gagliardo-Nirenberg, Trudinger-type and Brezis-Gallouet-Wainger inequalities associated with the positive Rockland operators on graded Lie groups, which includes the cases of $\mathbb R^n$, Heisenberg, and general stratified Lie groups. As an application, using the critical Gagliardo-Nirenberg inequality, the existence of least energy solutions of nonlinear Schrödinger type equations is obtained. We also express the best constant in the critical Gagliardo-Nirenberg and Trudinger inequalities in the variational form as well as in terms of the ground state solutions of the corresponding nonlinear subelliptic equations. The obtained results are already new in the setting of general stratified Lie groups (homogeneous Carnot groups). Among new technical methods, we also extend Folland's analysis of Hölder spaces from stratified Lie groups to general homogeneous Lie groups.

math.FA↗

arXiv2020

A comparison principle for higher order nonlinear hypoelliptic heat operators on graded Lie groups

In this paper we present a comparison principle for higher order nonlinear hypoelliptic heat operators on graded Lie groups. Moreover, using the comparison principle we obtain blow-up type results and global in $t$-boundedness of solutions of nonlinear equations for the heat $p$-sub-Laplacian on stratified Lie groups. In particular, this paper generalises and extends previous results obtained by the first author and Suragan in [RS18].

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arXiv2019

Rellich, Gagliardo-Nirenberg, Trudinger and Caffarelli-Kohn-Nirenberg inequalities for Dunkl operators and applications

In this paper we obtain weighted higher order Rellich, weighted Gagliardo-Nirenberg, Trudinger, Caffarelli-Kohn-Nirenberg inequalities and the uncertainty principle for Dunkl operators. Moreover, we introduce an extension of the classical Caffarelli-Kohn-Nirenberg inequalities. Furthermore, we give an application of Gagliardo-Nirenberg inequality to the Cauchy problem for the nonlinear damped wave equations for the Dunkl Laplacian.

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arXiv2018

Very weak solutions to hypoelliptic wave equations

In this paper we study the Cauchy problem for the wave equations for hypoelliptic homogeneous left-invariant operators on graded Lie groups when the time-dependent non-negative propagation speed is regular, Hölder, and distributional. For Hölder coefficients we derive the well-posedness in the spaces of ultradistributions associated to Rockland operators on graded groups. In the case when the propagation speed is a distribution, we employ the notion of "very weak solutions" to the Cauchy problem, that was already successfully used in similar contexts in [GR15] and [RT17b]. We show that the Cauchy problem for the wave equation with the distributional coefficient has a unique "very weak solution" in an appropriate sense, which coincides with classical or distributional solutions when the latter exist. Examples include the time dependent wave equation for the sub-Laplacian on the Heisenberg group or on general stratified Lie groups, or $p$-evolution equations for higher order operators on $\mathbb{R}^{n}$ or on groups, the results already being new in all these cases.

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