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Tapendu Rana

Publications and source records attributed to Tapendu Rana.

12 recordsLinked to original sources
arXiv2026

Topological Effects on Bubbling in the Critical Dirichlet Problem on Hyperbolic Domains: $3\leq N \leq5$

Let \(\Omega\Subset\mathbb H^N\), \(N\in\{3,4,5\}\), be a bounded connected \(C^2\) domain. We prove that the pure critical Dirichlet problem \[ -\Delta_{\mathbb H}u=u^{\frac{N+2}{N-2}} \quad\text{in }\Omega, \qquad u=0 \quad\text{on }\partial\Omega \] admits a positive solution whenever \(H_d(\Omega;\mathbb F_2)\neq0\) for some \(1\le d\le N-1\). This gives a hyperbolic Bahri-Coron theorem for \(3\le N\le5\) under \(C^2\) boundary regularity. Under conformal reduction, the hyperbolic geometry produces a positive potential and leads to dimension-dependent bubbling mechanisms. In dimension three, the required energy drop follows from the balance between diagonal corrections and pair interactions at fixed large multiplicity. In dimensions four and five, it is obtained at the matched scale through a normalized defect estimate and an all-pairs source-transfer bound. A unified Thom-barycenter construction converts these analytic estimates into the topological contradiction. Thus nontrivial domain topology forces existence despite critical loss of compactness and the additional geometric potential.

math.AP↗
arXiv2026

Profile decomposition and multiple positive solutions for the perturbed CR Yamabe equation on the Heisenberg group

In this article, we study an inhomogeneous critical nonlinear equation involving the sub-Laplacian on the Heisenberg group $\mathbb H^n$. We prove the multiplicity of positive solutions for the critical problem \begin{align*} \mathcal{L}_{\mathbb H^n} u=|u|^{2^\star-2}u+f(\xi) \quad \text{in } \mathbb H^n, \qquad u>0,\quad u\in S^{1,2}(\mathbb H^n), \end{align*} where $ \mathcal{L}_{\mathbb H^n} $ is the sub-Laplacian on $\mathbb H^n$, $2^\star=\frac{2Q}{Q-2}$, $Q=2n+2$, $n\geq 1$, $S^{1,2}(\mathbb H^n)$ is the homogeneous Sobolev space on $\mathbb H^n$, and $f$ is a nontrivial nonnegative functional in the dual space $(S^{1,2}(\mathbb H^n))'$ satisfying a suitable smallness condition. The above mentioned equation appeared as a perturbation of the CR Yamabe equation on the Heisenberg group. A major difficulty comes from the lack of compactness of the critical Folland-Stein embedding into critical Lebesgue space. To overcome this, we establish a Palais-Smale profile decomposition for the associated energy functional. The obtained Palais-Smale profile decomposition identifies the precise energy levels at which lack of compactness may occur via energy quantization, and shows that every noncompact Palais-Smale sequence decomposes into a finite superposition of weakly interacting bubbles. As a key analytic ingredient, we establish an improved Folland-Stein-Sobolev inequality involving the Morrey norm, which serves as a fundamental interpolation inequality and plays a crucial role in detecting the concentration of noncompact Palais-Smale sequences.

math.AP↗
arXiv2026

Multiplicity of Solutions to the Brezis-Nirenberg Problem on Hyperbolic Spaces

This article investigates the multiplicity of solutions to the Brezis-Nirenberg problem on smooth bounded domains in the hyperbolic space $\mathbb{B}^N$ for $N \ge 4$. Specifically, we study the critical semilinear equation $-Δ_{\mathbb{B}^N} u = λu + |u|^{2^*-2}u$ under Dirichlet boundary conditions for $λ> \frac{N(N-2)}{4}$. Overcoming the analytic challenges induced by the hyperbolic geometry and the intricate concentration profiles of Palais-Smale sequences, we establish the existence of multiple pairs of nontrivial solutions. Using the equivariant Ljusternik-Schnirelmann category, we obtain lower bounds on the number of solutions depending on the position of the parameter $λ$ relative to the Dirichlet spectrum of the Laplace-Beltrami operator.

math.AP↗
arXiv2025

Shifted Pitt and uncertainty inequalities on Riemannian symmetric spaces of noncompact type

Our primary objective is to study Pitt-type inequalities on Riemannian symmetric spaces $\mathbb{X}$ of noncompact type, as well as within the framework of Jacobi analysis. Inspired by the spectral gap of the Laplacian on $\mathbb{X}$, we introduce the notion of a \textit{shifted} Pitt's inequality as a natural and intrinsic analogue tailored to symmetric spaces, capturing key aspects of the underlying non-Euclidean geometry. In the rank one case (in particular, for hyperbolic spaces), we show that the sufficient condition for the \textit{shifted} Pitt's inequality matches the necessary condition in the range $p \leq q \leq p'$, yielding a sharp characterization of admissible polynomial weights with non-negative exponents. In the Jacobi setting, we modify the transform so that the associated measure exhibits polynomial volume growth. This modification enables us to fully characterize the class of polynomial weights with non-negative exponents for which Pitt-type inequalities hold for the modified Jacobi transforms. As applications of the \textit{shifted} Pitt's inequalities, we derive $L^2$-type Heisenberg-Pauli-Weyl uncertainty inequalities and further establish generalized $L^p$ versions. Moreover, the geometric structure of symmetric spaces allows us to formulate a broader version of the uncertainty inequalities previously obtained by Ciatti-Cowling-Ricci in the setting of stratified Lie groups.

math.FA↗
arXiv2024

$L^p$-$L^q$ Fourier multipliers and Hausdorff-Young-Paley inequalities on Riemannian symmetric spaces of noncompact type

Our primary objective in this article is to establish Hörmander type $L^p \rightarrow L^q$ Fourier multiplier theorems in the context of noncompact type Riemannian symmetric spaces $\mathbb{X}$ of arbitrary rank for the range $1 < p \leq 2 \leq q < \infty$. As a consequence of the Fourier multiplier theorem, we also derive a spectral multiplier theorem on $\mathbb{X}$. We then apply this theorem to prove $L^p \rightarrow L^q$ boundedness for functions of the Laplace-Beltrami operator and to obtain embedding theorems and operator estimates for the potentials and heat semigroups. Additionally, we provide mixed-norm versions of the Hausdorff-Young and Paley inequalities. In this context, where the Fourier transform is holomorphic, and its domain consists of various strips, we present two versions of these inequalities and explore their interrelation. Furthermore, our findings and methods are also applicable to harmonic $NA$ groups, also known as Damek-Ricci spaces.

math.FA↗
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.

math.FA↗
arXiv2024

Weighted Fourier inequalities and application of restriction theorems on rank one Riemannian symmetric spaces of noncompact type

This article explores weighted $(L^p, L^q)$ inequalities for the Fourier transform in rank one Riemannian symmetric spaces of noncompact type. We establish both necessary and sufficient conditions for these inequalities to hold. To prove the weighted Fourier inequalities, we apply restriction theorems on symmetric spaces and utilize Calder{ó}n's estimate for sublinear operators. While establishing the necessary conditions, we demonstrate that Harish-Chandra's elementary spherical functions play a crucial role in this setting. Furthermore, we apply our findings to derive Fourier inequalities with polynomial and exponential weights.

math.CA↗
arXiv2023

Pseudo-differential operators on radial sections of line bundles over the Poincaré upper half plane

In this article, we explore the boundedness properties of pseudo-differential operators on radial sections of line bundles over the Poincaré upper half plane, even when dealing with symbols of limited regularity. We first prove the boundedness of these operators when the symbol is smooth. To achieve this, we establish a connection between the operator norm of the local part of our pseudo-differential operators and the corresponding Euclidean pseudo-differential operators. Additionally, we introduce a class of rough symbols that lack any regularity conditions in the space variable and investigate the boundedness properties of the associated pseudo-differential operators. As a crucial part of our proof, we provide asymptotic estimates and functional identities for certain matrix coefficients of the principal and discrete series representations of the group $\mathrm{SL(2,\mathbb{R})}$.

math.CA↗
arXiv2023

Weighted estimates for Hardy-Littlewood maximal functions on Harmonic $NA$ groups

Our aim in this article is to study the weighted boundedness of the centered Hardy-Littlewood maximal operator in Harmonic $NA$ groups. Following Ombrosi et al. \cite{ORR}, we define a suitable notion of $A_p$ weights, and for such weights, we prove the weighted $L^p$-boundedness of the maximal operator. Furthermore, as an endpoint case, we prove a variant of the Fefferman-Stein inequality, from which vector-valued maximal inequality has been established. We also provide various examples of weights to substantiate many aspects of our results. In particular, we have shown certain spherical functions of the Harmonic $NA$ group constitute examples of $A_p$ weights. The purely exponential volume growth property of the Harmonic $NA$ group has played a crucial role in our proofs.

math.CA↗
arXiv2022

$L^p$-boundedness of pseudo-differential operators on homogeneous trees

The aim of this article is to study the $L^{p}$-boundedness of pseudo-differential operators on a homogeneous tree $ \mathfrak{X} $. For $p\in (1,2)$, we establish a connection between the $L^{p}$-boundedness of the pseudo-differential operators on $ \mathfrak{X} $ and that on the group of integers $\mathbb{Z}$. We also prove an analogue of the Calderon-Vaillancourt theorem in the setting of homogeneous trees, for $p\in(1,\infty)\setminus\{2\}$.

math.CA↗