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Vishvesh Kumar

Publications and source records attributed to Vishvesh Kumar.

At least 37 records · Page 2Linked to original sources
arXiv2025

Critical Equations Involving Nonlocal Subelliptic Operators on Stratified Lie Groups: Spectrum, Bifurcation and Multiplicity

In this paper, we explore the bifurcation phenomena and establish the existence of multiple solutions for the nonlocal subelliptic Brezis-Nirenberg problem: \begin{equation*} \begin{cases} (-Δ_{\mathbb{G}})^s u= |u|^{2_s^*-2}u+λu \quad &\text{in}\quad Ω, \\ u=0\quad & \text{in}\quad \mathbb{G}\backslash Ω, \end{cases} \end{equation*} where $(-Δ_{\mathbb{G}})^s$ is the fractional sub-Laplacian on the stratified Lie group $\mathbb{G}$ with homogeneous dimension $Q,$ $Ω$ is a open bounded subset of $\mathbb{G},$ $s \in (0,1)$, $Q> 2s,$ $2_s^*:=\frac{2Q}{Q-2s}$ is subelliptic fractional Sobolev critical exponent, $λ>0$ is a real parameter. This work extends the seminal contributions of Cerami, Fortunato, and Struwe to nonlocal subelliptic operators on stratified Lie groups. A key component of our study involves analyzing the subelliptic $(s, p)$-eigenvalue problem for the (nonlinear) fractional $p$-sub-Laplacian $(-Δ_{p,{\mathbb{G}}})^s$ \begin{align*} (-Δ_{p,{\mathbb{G}}})^s u&=λ|u|^{p-2}u,~\text{in}~Ω,\nonumber u&=0~\text{ in }~{\mathbb{G}}\setminusΩ, \end{align*} with $0 ps$, over the fractional Folland-Stein-Sobolev spaces on stratified Lie groups applying variational methods. Particularly, we prove that the $(s, p)$-spectrum of $(-Δ_{p,{\mathbb{G}}})^s$ is closed and the second eigenvalue $λ_2(Ω)$ with $λ_2(Ω)>λ_1(Ω)$ is well-defined and provides a variational characterization of $λ_2(Ω)$. We emphasize that the results obtained here are also novel for $\mathbb{G}$ being the Heisenberg group.

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arXiv2024

Classical inequalities for all Fourier matrix coefficients of $\mathrm{SL}(2,\mathbb{R})$ and their applications

In this article, we establish three fundamental Fourier inequalities: the Hausdorff-Young inequality, the Paley inequality, and the Hausdorff-Young-Paley inequality for $(l, n)$-type functions on $\mathrm{SL}(2,\mathbb{R})$. Utilizing these inequalities, we demonstrate the $L^p$-$L^q$ boundedness of $(l, n)$-type Fourier multipliers on $\mathrm{SL}(2,\mathbb{R})$. Furthermore, we explore applications related to the $L^p$-$L^q$ estimates of the heat kernel of the Casimir element on $\mathrm{SL}(2,\mathbb{R})$ and address the global well-posedness of certain parabolic and hyperbolic nonlinear equations.

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arXiv2024

Higher order hypoelliptic damped wave equations on graded Lie groups with data from negative order Sobolev spaces: the critical case

Let $\mathbb G$ be a graded Lie group with homogeneous dimension $Q$. In this paper, we study the Cauchy problem for a semilinear hypoelliptic damped wave equation involving a positive Rockland operator $\mathcal{R}$ of homogeneous degree $ν\geq 2$ on $\mathbb G$ with power type nonlinearity $|u|^p$ and initial data taken from negative order homogeneous Sobolev space $\dot H^{-γ}(\mathbb G), γ>0,$ for the critical exponent case $p=1+\frac{2ν}{Q+2γ}.$ We also explore the diffusion phenomenon of the higher-order hypoelliptic damped wave equations on graded Lie groups with initial data belonging to Sobolev spaces of negative order. We emphasize that our results are also new, even in the setting of higher-order differential operators on $\mathbb{R}^n$, and more generally, on stratified Lie groups.

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arXiv2024

Subelliptic Nonlocal Brezis-Nirenberg Problems on Stratified Lie Groups

In this paper, we investigate the subelliptic nonlocal Brezis-Nirenberg problem on stratified Lie groups involving critical nonlinearities, namely, \begin{align*} (-Δ_{\mathbb{G}, p})^s u&= μ|u|^{p_s^*-2}u+λh(x, u) \quad \text{in}\quad Ω, \\ u&=0\quad \text{in}\quad \mathbb{G}\backslash Ω, \end{align*} where $(-Δ_{\mathbb{G}, p})^s$ is the fractional $p$-sub-Laplacian on a stratified Lie group $\mathbb{G}$ with homogeneous dimension $Q,$ $Ω$ is an open bounded subset of $\mathbb{G},$ $s \in (0,1)$, $\frac{Q}{s}>p\geq2,$ $p_s^*:=\frac{pQ}{Q-ps}$ is subelliptic fractional Sobolev critical exponent, $μ, λ>0$ are real parameters and $h$ is a lower order perturbation of the critical power $|u|^{p_s^*-2}u$. Utilising direct methods of the calculus of variation, we establish the existence of at least one weak solution for the above problem under the condition that the real parameter $λ$ is sufficiently small. Additionally, we examine the problem for $μ= 0$, representing subelliptic nonlocal equations on stratified Lie groups depending on one real positive parameter and involving a subcritical nonlinearity. We demonstrate the existence of at least one solution in this scenario as well. We emphasize that the results obtained here are also novel for $p=2$ even for the Heisenberg group.

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arXiv2024

Higher order hypoelliptic damped wave equations on graded Lie groups with data from negative order Sobolev spaces

Let $\mathbb G$ be a graded Lie group with homogeneous dimension $Q$. In this paper, we study the Cauchy problem for a semilinear hypoelliptic damped wave equation involving a positive Rockland operator $\mathcal{R}$ of homogeneous degree $ν\geq 2$ on $\mathbb G$ with power type nonlinearity $|u|^p$ and initial data taken from negative order homogeneous Sobolev space $\dot H^{-γ}(\mathbb G), γ>0$. In the framework of Sobolev spaces of negative order, we prove that $p_{\text{Crit}}(Q, γ, ν) :=1+\frac{2ν}{Q+2γ}$ is the new critical exponent for $γ\in (0, \frac{Q}{2})$. More precisely, we show the global-in-time existence of small data Sobolev solutions of lower regularity for $p>p_{\text{Crit}}(Q, γ, ν) $ in the energy evolution space $ \mathcal{C}\left([0, T], H^{s}(\mathbb{G})\right), s\in (0, 1]$. Under certain conditions on the initial data, we also prove a finite-time blow-up of weak solutions for $1<p<p_{\text{Crit}}(Q, γ, ν)$. Furthermore, to precisely characterize the blow-up time, we derive sharp upper bound and lower bound estimates for the lifespan in the subcritical cases. We emphasize that our results are also new, even in the setting of higher-order differential operators on $\mathbb{R}^n$, and more generally, on stratified Lie groups.

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arXiv2024

Semilinear damped wave equations on the Heisenberg group with initial data from Sobolev spaces of negative order

In this paper, we focus on studying the Cauchy problem for semilinear damped wave equations involving the sub-Laplacian $\mathcal{L}$ on the Heisenberg group $\mathbb{H}^n$ with power type nonlinearity $|u|^p$ and initial data taken from Sobolev spaces of negative order homogeneous Sobolev space $\dot H^{-γ}_{\mathcal{L}}(\mathbb{H}^n), γ>0$, on $\mathbb{H}^n$. In particular, in the framework of Sobolev spaces of negative order, we prove that the critical exponent is the exponent $p_{\text{crit}}(Q, γ)=1+\frac{4}{Q+2γ},$ for some $γ\in (0, \frac{Q}{2})$, where $Q:=2n+2$ is the homogeneous dimension of $\mathbb{H}^n$. More precisely, we establish a global-in-time existence of small data Sobolev solutions of lower regularity for $p>p_{\text{crit}}(Q, γ)$ in the energy evolution space; a finite time blow-up of weak solutions for $1<p<p_{\text{crit}}(Q, γ)$ under certain conditions on the initial data by using the test function method. Furthermore, to precisely characterize the blow-up time, we derive sharp upper bound and lower bound estimates for the lifespan in the subcritical case.

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arXiv2024

Functional inequalities on symmetric spaces of noncompact type and applications

The aim of this paper is to begin a systematic study of functional inequalities on symmetric spaces of noncompact type of higher rank. Our first main goal of this study is to establish the Stein-Weiss inequality, also known as a weighted Hardy-Littlewood-Sobolev inequality, for the Riesz potential on symmetric spaces of noncompact type. This is achieved by performing delicate estimates of ground spherical function with the use of polyhedral distance on symmetric spaces and by combining the integral Hardy inequality developed by Ruzhansky and Verma with the sharp Bessel-Green-Riesz kernel estimates on symmetric spaces of noncompact type obtained by Anker and Ji. As a consequence of the Stein-Weiss inequality, we deduce Hardy-Sobolev, Hardy-Littlewood-Sobolev, Gagliardo-Nirenberg and Caffarelli-Kohn-Nirenberg inequalities on symmetric spaces of noncompact type. The second main purpose of this paper is to show the applications of aforementioned inequalities for studying nonlinear PDEs on symmetric spaces. Specifically, we show that the Gagliardo-Nirenberg inequality can be used to establish small data global existence results for the semilinear wave equations with damping and mass terms for the Laplace-Beltrami operator on symmetric spaces.

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arXiv2024

A Direct Method of Moving Planes for Logarithmic Schrödinger Operator

In this paper, we study the radial symmetry and monotonicity of nonnegative solutions to nonlinear equations involving the logarithmic Schr$\ddot{\text{o}}$dinger operator $(\mathcal{I}-Δ)^{\log}$ corresponding to the logarithmic symbol $\log(1 + |ξ|^2)$, which is a singular integral operator given by $$(\mathcal{I}-Δ)^{\log}u(x) =c_{N}P.V.\int_{\mathbb{R}^{N}}\frac{u(x)-u(y)}{|x-y|^{N}}κ(|x-y|)dy,$$ where $c_{N}=π^{-\frac{N}{2}}$, $κ(r)=2^{1-\frac{N}{2}}r^{\frac{N}{2}}\mathcal{K}_{\frac{N}{2}}(r)$ and $\mathcal{K}_ν$ is the modified Bessel function of second kind with index $ν$. The proof hinges on a direct method of moving planes for the logarithmic Schr$\ddot{\text{o}}$dinger operator.

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arXiv2024

Pseudo-differential operators on Homogeneous vector bundles over compact homogeneous manifolds

In this work, we introduce a global theory of subelliptic pseudo-differential operators on arbitrary homogeneous vector bundles over orientable compact homogeneous manifolds. We will show that a global pseudo-differential calculus can be associated to the operators acting on any pair of homogeneous vector-bundles with base space $M,$ if the compact Lie group $G$ that acts on $M=G/K$ is endowed with a (Riemannian or) sub-Riemannian structure. This is always possible if we choose on $G$ a sub-Laplacian associated to a Hörmander system of vector-fields or we fix the Laplace-Beltrami operator on $G$. We begin with developing a global subelliptic symbolic calculus for vector-valued pseudo-differential operators on $G$ and then, we show that this vector-valued calculus induces a pseudo-differential calculus on homogeneous vector bundles, which, among other things, is stable under the action of the complex functional calculus. We prove global versions of the Calderón-Vaillancourt theorem, Fefferman theorem and also, of the Gårding inequality. We present applications of the obtained Gårding inequality to the wellposedness of evolution problems. We characterise the classes of pseudo-differential operators on homogeneous vector bundles in the sense of Hörmander (which are defined by using local coordinate systems) in terms of their global symbols. Finally, using this formalism, we compute the global symbol of the exterior derivative, its adjoint, and the symbol of the Dirac operator on the vector bundle of differential forms. We hope that this work will provide a solid foundation for further research using the global quantisation of operators on (vector-bundles over) compact homogeneous manifolds.

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arXiv2024

Anharmonic semigroups and applications to global well-posedness of nonlinear heat equations

In this work we consider the semigroup $e^{-t\mathcal{A}_{k,\,\ell}^γ}$ for $γ>0$ associated to an anharmonic oscillator of the form $ \mathcal{A}_{k,\,\ell}=(-Δ)^{\ell}+|x|^{2k}$ where $k,\ell$ are integers $\geq 1$. By introducing a suitable Hörmander metric on the phase-space we analyse the semigroup $e^{-t\mathcal{A}_{k,\,\ell}^γ}$ within the framework of Hörmander $S(M,g)$ classes and obtain mapping properties in the scale of modulation spaces $M^{p,q},\, 0<p,q\leq \infty,$ with respect to an anharmonic modulation weight. As an application, we apply the obtained bounds to establish the well-posedness for the nonlinear heat equation associated with $\mathcal{A}_{k,\,\ell}^γ$. It is worth noting that the results presented in this paper are novel, even in the case where $γ=1.$

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arXiv2023

Stein-Weiss inequality on non-compact symmetric spaces

Let $Δ$ be the Laplace-Beltrami operator on a non-compact symmetric space of any rank, and denote the bottom of its $L^2$-spectrum as $-|ρ|^{2}$. In this paper, we provide a comprehensive characterization of both the sufficient and necessary conditions ensuring the validity of the Stein-Weiss inequality for the entire family of operators $\lbrace{(-Δ+b)^{-\fracσ{2}}}\rbrace_{σ\ge0,\,b\ge-|ρ|^{2}}$. As an application, some weighted functional inequalities, such as Heisenberg's uncertainty principle, Gagliardo-Nirenberg's interpolation inequality, Pitt's inequality, etc., become available in this context. In particular, their sets of admissible indices are larger than those in the Euclidean setting.

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arXiv2023

$L^p$-$L^q$ estimates for subelliptic pseudo-differential operators on compact Lie groups

We establish the $L^p$-$L^q$-boundedness of subelliptic pseudo-differential operators on a compact Lie group $G$. Effectively, we deal with the $L^p$-$L^q$-bounds for operators in the sub-Riemmanian setting because the subelliptic classes are associated to a Hörmander sub-Laplacian. The Riemannian case associated with the Laplacian is also included as a special case. Then, applications to the $L^p$-$L^q$-boundedness of pseudo-differential operators in the Hörmander classes on $G$ are given in the complete range $0\leq δ\leq ρ\leq 1,$ $δ<1.$ This also gives the $L^p$-$L^q$-bounds in the Riemannian setting, because the later classes are associated with the Laplacian on $G$. In both cases, in the Riemannian and the sub-Riemannian settings, necessary and sufficient conditions for the $L^p$-$L^q$-boundedness of operators are also anaysed.

math.AP↗
arXiv2023

$L^p$-$L^q$ boundedness of pseudo-differential operators on graded Lie groups

In this paper we establish the $L^p$-$L^q$ estimates for global pseudo-differential operators on graded Lie groups. We provide both necessary and sufficient conditions for the $L^p$-$L^q$ boundedness of pseudo-differential operators associated with the global Hörmander symbol classes on graded Lie groups, within the range $1<p\leq 2 \leq q<\infty$. Additionally, we present a sufficient condition for the $L^p$-$L^q$ estimates of pseudo-differential operators within the range $1<p\leq q\leq 2$ or $2\leq p\leq q<\infty$. The proofs rely on estimates of the Riesz and Bessel potentials associated with Rockland operators, along with previously established results on $L^p$-boundedness of global pseudo-differential operators on graded Lie groups. Notably, as a byproduct, we also establish the sharpness of the Sobolev embedding theorem for the inhomogeneous Sobolev spaces on graded Lie groups.

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arXiv2023

Non-harmonic $M$-elliptic pseudo differential operators on manifolds

In this article, we introduce and study $M$-elliptic pseudo-differential operators in the framework of non-harmonic analysis of boundary value problems on a manifold $Ω$ with boundary $\partial Ω$, introduced by Ruzhansky and Tokmagambetov ( Int. Math. Res. Not. IMRN, (12), 3548-3615, 2016) in terms of a model operator $\mathfrak{L}$. More precisely, we consider a weighted $\mathfrak{L}$-symbol class $M_{ρ, 0, Λ}^{m}, m\in \mathbb{R},$ associated to a suitable weight function $Λ$ on a countable set $\mathcal{I} $ and study elements of the symbolic calculus for pseudo-differential operators associated with $\mathfrak{L}$-symbol class $M_{ρ, 0, Λ}^{m},$ by deriving formulae for the composition, adjoint, and transpose. Using the notion of $M$-ellipticity for symbols belonging to $\mathfrak{L}$-symbol class $M_{ρ, 0, Λ}^{m}$, we construct the parametrix of $M$-elliptic pseudo-differential operators. Further, we investigate the minimal and maximal extensions for $M$-elliptic pseudo-differential operators and show that they coincide when the symbol $σ\in M_{ρ, 0, Λ}^{m}, $ is $M$-elliptic. We provide a necessary and sufficient condition to ensure that the pseudo-differential operators $T_σ$ with symbol in the $\mathfrak{L}$-symbol class $M_{ρ, 0,Λ}^{0} $ is a compact operator in $L^{2}(Ω)$ or a Riesz operator in $L^{p}(Ω).$ Finally, we prove Gärding's inequality for pseudo-differential operators associated with symbol from $M_{ρ, 0,Λ}^{0} $ in the setting of non-harmonic analysis.

math.FA↗
arXiv2023

Asymptotic estimates for the growth of deformed Hankel transform by modulus of continuity

We derive asymptotic estimates for the growth of the norm of the deformed Hankel transform on the deformed Hankel--Lipschitz space defined via a generalised modulus of continuity. The established results are similar in nature to the well-known Titchmarsh theorem, which provide a characterization of the square integrable functions satisfying certain Cauchy--Lipschitz condition in terms of an asymptotic estimate for the growth of the norm of their Fourier transform. We also give some necessary conditions in terms of the generalised modulus of continuity for the boundedness of the Dunkl transform of functions in Dunkl-Lipschitz spaces, improving the Hausdorff-Young inequality for the Dunkl transform in this special scenario.

math.FA↗
arXiv2023

Best constants in subelliptic fractional Sobolev and Gagliardo-Nirenberg inequalities and ground states on stratified Lie groups

In this paper, we establish the sharp fractional subelliptic Sobolev inequalities and Gagliardo-Nirenberg inequalities on stratified Lie groups. The best constants are given in terms of a ground state solution of a fractional subelliptic equation involving the fractional $p$-sublaplacian ($1<p<\infty$) on stratified Lie groups. We also prove the existence of ground state (least energy) solutions to nonlinear subelliptic fractional Schrödinger equation on stratified Lie groups. Different from the proofs of analogous results in the setting of classical Sobolev spaces on Euclidean spaces given by Weinstein (Comm. Math. Phys. 87(4):576-676 (1982/1983)) using the rearrangement inequality which is not available in stratified Lie groups, we apply a subelliptic version of vanishing lemma due to Lions extended in the setting of stratified Lie groups combining it with the compact embedding theorem for subelliptic fractional Sobolev spaces obtained in our previous paper (Math. Ann. (2023)). We also present subelliptic fractional logarithmic Sobolev inequalities with explicit constants on stratified Lie groups. The main results are new for $p=2$ even in the context of the Heisenberg group.

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arXiv2023

Compact Embeddings, Eigenvalue Problems, and subelliptic Brezis-Nirenberg equations involving singularity on stratified Lie groups

The purpose of this paper is twofold: first we study an eigenvalue problem for the fractional $p$-sub-Laplacian over the fractional Folland-Stein-Sobolev spaces on stratified Lie groups. We apply variational methods to investigate the eigenvalue problems. We conclude the positivity of the first eigenfunction via the strong minimum principle for the fractional $p$-sub-Laplacian. Moreover, we deduce that the first eigenvalue is simple and isolated. Secondly, utilising established properties, we prove the existence of at least two weak solutions via the Nehari manifold technique to a class of subelliptic singular problems associated with the fractional $p$-sub-Laplacian on stratified Lie groups. We also investigate the boundedness of positive weak solutions to the considered problem via the Moser iteration technique. The results obtained here are also new even for the case $p=2$ with $\mathbb{G}$ being the Heisenberg group.

math.AP↗
arXiv2023

Liouville type theorems for subelliptic systems on the Heisenberg group with general nonlinearity

In this paper, we establish Liouville type results for semilinear subelliptic systems associated with the sub-Laplacian on the Heisenberg group $\mathbb{H}^{n}$ involving two different kinds of general nonlinearities. The main technique of the proof is the method of moving planes combined with some integral inequalities replacing the role of maximum principles. As a special case, we obtain the Liouville theorem for the Lane-Emden system on the Heisenberg group $\mathbb{H}^{n}$, which also appears to be a new result in the literature.

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