Searcharxiv⌕ Search

arXiv subjects

Utsav Dewan

Publications and source records attributed to Utsav Dewan.

17 recordsLinked to original sources
arXiv2026

Boundary behaviour of eigenfunctions and superharmonic functions on harmonic manifolds of purely exponential volume growth

On $\mathbb{X}$, a non-positively curved harmonic manifold of purely exponential volume growth, of dimension $n \ge 3$, we study certain quantitative aspects of the boundary behaviour of eigenfunctions and superharmonic functions. We first focus on complex-valued eigenfunctions lying outside the $L^2$-spectrum of $\Delta$ and obtain the almost everywhere existence of weighted non-tangential limits, sharp Hausdorff dimension and Hausdorff measure estimates of the boundary exceptional sets for radial limits. Then in the second part, we shift our attention to non-tangential and tangential boundary behaviour of positive superharmonic functions. Most of our results are new even for the homogeneous setting of rank one Riemannian symmetric spaces of non-compact type and Damek-Ricci spaces. Our arguments are based on potential theory adapted to the intrinsic Gromov hyperbolic geometry of $\mathbb{X}$.

math.FA↗
arXiv2026

Ces\`aro summability of H\"older functions and Talbot effect on rank one Riemannian symmetric spaces of compact type

On rank one Riemannian symmetric spaces of compact type (of dimension $\ge 2$), we first obtain a quantitative characterization of H\"older continuity in terms of Ces\`aro means. In addition to some approximation theoretic applications, we also apply it to study the celebrated physical phenomenon known as `Talbot effect' arising from diffraction theory. More precisely, for almost every fixed time instance, we study the H\"older continuity and the fractal profile of the Schr\"odinger propagation in terms of the decay of the Littlewood-Paley projections of the initial data. In the process, we also obtain oscillatory expansions of zonal spherical functions uniformly near the origin and near the cut locus respectively, which may be of independent interest.

math.CA↗
arXiv2025

Regularity and pointwise convergence for dispersive equations on Riemannian symmetric spaces of compact type

In this article, we first prove that for general dispersive equations on Riemannian symmetric spaces of compact type $\mathbb{X}=U/K$, of rank $1$ and $2$, the Sobolev regularity thresholds for the initial data, $\alpha >1/2$ and $\alpha >1$ respectively, are sufficient to obtain pointwise convergence of the solution a.e. on $\mathbb{X}$. We next focus on $K$-biinvariant initial data for rank $1$ and prove that the sufficiency of the regularity threshold can be improved down to $\alpha>1/3$, whereas the phenomenon fails for $\alpha<1/4$ for the Schr\"odinger equation. We also obtain the same results for other dispersive equations: the Boussinesq equation and the Beam equation, also known as the fourth order Wave equation, by a novel transference principle, which seems to be new even for the circle $\mathbb{T} \cong SO(2)$ and may be of independent interest. Our arguments involve harmonic analysis arising from the representation theory of compact semi-simple Lie groups and also number theory.

math.AP↗
arXiv2025

Regularity and pointwise convergence for dispersive equations on $\mathbb{H}^2$

In the prototypical setting of non-Euclidean geometry, the 2-dimensional Real Hyperbolic space $\mathbb{H}^2$, we consider the Carleson's problem for the Schr\"odinger equation and improve the best known result until now by proving that the Sobolev regularity threshold $\beta \ge 1/2$ for the initial data, is sufficient to obtain pointwise convergence of the solution a.e. on $\mathbb{H}^2$. In fact, we prove the same bound for a wide class of dispersive equations that include the fractional Schr\"odinger equations with convex phase, the Boussinesq equation and the Beam equation, also known as the fourth order Wave equation. For the Schr\"odinger equation, we improve the result of Wang-Zhang (Canad J Math 71(4), 983-995, 2019) and for the fractional Schr\"odinger equations with convex phase, we improve the result of Cowling (Lecture Notes Math 992, 83-90, 1983).

math.CA↗
arXiv2025

Regularity and pointwise convergence for dispersive equations with asymptotically concave phase on Damek-Ricci spaces

We study the Carleson's problem on Damek-Ricci spaces $S$ for dispersive equations: \begin{equation*} \begin{cases} i\frac{\partial u}{\partial t} +\Psi(\sqrt{-\mathcal{L}} )u=0\:,\: (x,t) \in S \times \mathbb{R} \:, \\ u(0,\cdot)=f\:,\: \text{ on } S \:, \end{cases} \end{equation*} where $\mathcal{L}= \Delta$, the Laplace-Beltrami operator or $\tilde{\Delta}$, the shifted Laplace-Beltrami operator, so that the corresponding phase function $\psi$ satisfies for some $a \in (0,1)$, the large frequency asymptotic: \begin{equation*} \psi(\lambda)=\lambda^a + \mathcal{O}(1)\:,\:\: \lambda \gg 1\:. \end{equation*} For almost everywhere pointwise convergence of the solution $u$ to its radial initial data $f$, we obtain the almost sharp regularity threshold $\beta>a/4$. This result is new even for $\mathbb{R}^n$ and in the special case of the fractional Schr\"odinger equations, generalizes classical Euclidean results of Walther.

math.AP↗
arXiv2025

Iterated convolution inequalities on $\mathbb{R}^d$ and Riemannian Symmetric Spaces of non-compact type

In a recent work (Int Math Res Not 24:18604-18612, 2021), Carlen-Jauslin-Lieb-Loss studied the convolution inequality $f \ge f*f$ on $\mathbb{R}^d$ and proved that the real integrable solutions of the above inequality must be non-negative and satisfy the non-trivial bound $\int_{\mathbb{R}^d} f \le \frac{1}{2}$. Nakamura-Sawano then generalized their result to $m$-fold convolution (J Geom Anal 35:68, 2025). In this article, we replace the monomials by genuine polynomials and study the real-valued solutions $f \in L^1(\mathbb{R}^d)$ of the iterated convolution inequality \begin{equation*} f \ge \displaystyle\sum_{n=2}^N a_n \left(*^n f\right) \:, \end{equation*} where $N \ge 2$ is an integer and for $2 \le n \le N$, $a_n$ are non-negative integers with at least one of them positive. We prove that $f$ must be non-negative and satisfy the non-trivial bound $\int_{\mathbb{R}^d} f \le t_{\mathcal{Q}}\:$ where $\mathcal{Q}(t):=t-\displaystyle\sum_{n=2}^N a_n\:t^n$ and $t_{\mathcal{Q}}$ is the unique zero of $\mathcal{Q}'$ in $(0,\infty)$. We also have an analogue of our result for Riemannian Symmetric Spaces of non-compact type. Our arguments involve Fourier Analysis and Complex analysis. We then apply our result to obtain an a priori estimate for solutions of an integro-differential equation which is related to the physical problem of the ground state energy of the Bose gas in the classical Euclidean setting.

math.FA↗
arXiv2025

Pointwise convergence of solutions of the Schrödinger equation along general curves on Damek-Ricci spaces

One of the most celebrated problems in Euclidean Harmonic analysis is the Carleson's problem: determining the optimal regularity of the initial condition $f$ of the Schrödinger equation given by \begin{equation*} \begin{cases} i\frac{\partial u}{\partial t} =Δu\:,\: (x,t) \in \mathbb{R}^n \times \mathbb{R} \newline u(0,\cdot)=f\:, \text{ on } \mathbb{R}^n \:, \end{cases} \end{equation*} in terms of the index $β$ such that $f$ belongs to the inhomogeneous Sobolev space $H^β(\mathbb{R}^n)$, so that the solution of the Schrödinger operator $u$ converges pointwise to $f$, \begin{equation*} \displaystyle\lim_{t \to 0+} u(x,t)=f(x), \text{ almost everywhere}. \end{equation*} Recently, the author considered the Carleson's problem for the Schrödinger equation with radial initial data on Damek-Ricci spaces and obtained the sharp bound up to the endpoint $β\ge 1/4$. Interpreting the above as convergence along vertical lines, in this article, we consider the problem of pointwise convergence via more general approach paths. By constructing a counter-example on the $3$-dimensional Real Hyperbolic space, we show that the solutions of the Schrödinger equation, unlike Harmonic functions or solutions of the Heat equation, do not admit any natural wide approach region. We then study their pointwise convergence properties on Damek-Ricci spaces along general curves that satisfy certain Hölder conditions and bilipschitz conditions in the distance from the identity and again obtain the sharp bound up to the endpoint $β\ge 1/4$. Certain Euclidean analogues are also obtained.

math.FA↗
arXiv2025

Maximal estimates and pointwise convergence for solutions of certain dispersive equations with radial initial data on Damek-Ricci spaces

One of the most celebrated problems in Euclidean Harmonic analysis is the Carleson's problem: determining the optimal regularity of the initial condition $f$ of the Schrödinger equation given by \begin{equation*} \begin{cases} i\frac{\partial u}{\partial t} -Δ_{\mathbb{R}^n} u=0\:,\:\:\: (x,t) \in \mathbb{R}^n \times \mathbb{R}\:, \newline u(0,\cdot)=f\:, \text{ on } \mathbb{R}^n \:, \end{cases} \end{equation*} in terms of the index $β$ such that $f$ belongs to the inhomogeneous Sobolev space $H^β(\mathbb{R}^n)$ , so that the solution of the Schrödinger operator $u$ converges pointwise to $f$, $\displaystyle\lim_{t \to 0+} u(x,t)=f(x)$, almost everywhere. In this article, we address the Carleson's problem for the fractional Schrödinger equation, the Boussinesq equation and the Beam equation corresponding to both the Laplace-Beltrami operator $Δ$ and the shifted Laplace-Beltrami operator $\tildeΔ$, with radial initial data on Damek-Ricci spaces, by obtaining a complete description of the local (in space) mapping properties for the corresponding local (in time) maximal functions. Consequently, we obtain the sharp bound up to the endpoint $β\ge 1/4$, for (almost everywhere) pointwise convergence. We also establish an abstract transference principle for dispersive equations whose corresponding multipliers have comparable oscillation and also apply it in the proof of our main result.

math.AP↗
arXiv2024

Estimates of the Poisson kernel on negatively curved Hadamard manifolds

Let $M$ be an $n$-dimensional Hadamard manifold of pinched negative curvature $-b^2 \leq K_M \leq -a^2$. The solution of the Dirichlet problem at infinity for $M$ leads to the construction of a family of mutually absolutely continuous probability measures $\{μ_x\}_{x \in M}$ called the harmonic measures. Fixing a basepoint $o \in M$, the Poisson kernel of $M$ is the function $P : M \times \partial M \to (0, \infty)$ defined by \begin{equation*} P(x, ξ) = \frac{dμ_x}{dμ_o}(ξ) \ , \ x \in M, ξ\in \partial M. \end{equation*} We prove the following global upper and lower bounds for the Poisson kernel: \begin{equation*} \frac{1}{C}\: e^{-2K{(o|ξ)}_x}\: e^{a d(x, o)} \le P(x,ξ) \le C\: e^{2K{(x|ξ)}_o}\: e^{-a d(x,o)} \:, \end{equation*} for some positive constants $C \geq 1, K > 0$ depending solely on $a, b$ and $n$. The above estimates may be viewed as a generalization of the well-known formula for the Poisson kernel in terms of Busemann functions for the special case of Gromov hyperbolic harmonic manifolds. These estimates do not follow directly from known estimates on Green's functions or harmonic measures. Instead we use techniques due to Anderson-Schoen for estimating positive harmonic functions in cones. As applications, we obtain quantitative estimates for the convergence $μ_x \to δ_ξ$ as $x \in M \to ξ\in \partial M$, and for the convergence of harmonic measures on finite spheres to the harmonic measures on the boundary at infinity as the radius of the spheres tends to infinity.

math.DG↗
arXiv2024

Regularity and pointwise convergence of solutions of the Schrödinger operator with radial initial data on Damek-Ricci spaces

One of the most celebrated problems in Euclidean Harmonic analysis is the Carleson's problem: determining the optimal regularity of the initial condition $f$ of the Schrödinger equation given by \begin{equation*}\begin{cases} i\frac{\partial u}{\partial t} =Δu\:,\: (x,t) \in \mathbb{R}^n \times \mathbb{R} \\ u(0,\cdot)=f\:, \text{ on } \mathbb{R}^n \:, \end{cases}\end{equation*} in terms of the index $α$ such that $f$ belongs to the inhomogeneous Sobolev space $H^α(\mathbb{R}^n)$ , so that the solution of the Schrödinger operator $u$ converges pointwise to $f$, $\lim_{t \to 0+} u(x,t)=f(x)$, almost everywhere. In this article, we consider the Carleson's problem for the Schrödinger equation with radial initial data on Damek-Ricci spaces and obtain the sharp bound up to the endpoint $α\ge 1/4$, which agrees with the classical Euclidean case.

math.CA↗
arXiv2024

Restricted Mean Value Property with non-tangential boundary behavior on Riemannian manifolds

A well studied classical problem is the harmonicity of functions satisfying the restricted mean-value property (RMVP) for domains in $\mathbb{R}^n$. Recently, the author along with Biswas investigated the problem in the general setting of Riemannian manifolds and obtained results in terms of unrestricted boundary limits of the function on a full measure subset of the boundary. However in the context of classical Fatou-Littlewood type theorems for the boundary behavior of harmonic functions, a genuine query is to replace the condition on unrestricted boundary limits with the more natural notion of non-tangential boundary limits. The aim of this article is to answer this question in the local setup for pre-compact domains with smooth boundary in Riemannian manifolds and in the global setup for non-positively curved Harmonic manifolds of purely exponential volume growth. This extends a classical result of Fenton for the unit disk in $\mathbb{R}^2$.

math.CA↗
arXiv2023

Initial singularities of positive solutions of the Heat equation on Stratified Lie groups

Let $(\mathbb{G},\circ)$ be a stratified Lie group. We estimate the Hausdorff dimension (with respect to the Carnot-Carath\'eodory metric) of the singular sets in $\mathbb{G}$, where a positive solution of the Heat equation corresponding to a sub-Laplacian, blows up faster than a prescribed rate along normal limits, in terms of the homogeneous dimension of $\mathbb{G}$ and the rate of the blowup parameter. This generalizes a classical result of Watson for the Euclidean Heat. We also obtain the corresponding sharpness result, which is new even for $\mathbb{R}^n$.

math.CA↗
arXiv2023

Boundary exceptional sets for radial limits of superharmonic functions on non-positively curved Harmonic manifolds of purely exponential volume growth

By classical Fatou type theorems in various setups, it is well-known that positive harmonic functions have non-tangential limit at almost every point on the boundary. In this paper, in the setting of non-positively curved Harmonic manifolds of purely exponential volume growth, we are interested in the size of the exceptional sets of points on the boundary at infinity, where a suitable function blows up faster than a prescribed growth rate, along radial geodesic rays. For Poisson integrals of complex measures, we obtain a sharp bound on the Hausdorff dimension of the exceptional sets, in terms of the mean curvature of horospheres and the parameter of the growth rate. In the case of the Green potentials, we obtain similar upper bounds and also construct Green potentials that blow up faster than a prescribed rate on lower Hausdorff dimensional realizable sets. So we get a gap in the corresponding Hausdorff dimensions due to the assumption of variable pinched non-positive sectional curvature. We also obtain a Riesz decomposition theorem for subharmonic functions. Combining the above results we get our main result concerning Hausdorff dimensions of the exceptional sets of positive superharmonic functions.

math.CA↗
arXiv2023

Restricted mean value property on Riemannian manifolds

A well studied classical problem is the harmonicity of functions satisfying the restricted mean-value property (RMVP). While this has so far been studied mainly for domains in $\mathbb{R}^n$, we consider this problem in the general setting of domains in Riemannian manifolds, and obtain results generalizing classical results of Fenton. We also obtain a result for complete, simply connected Riemannian manifolds of pinched negative curvature where there is no restriction on the radius function in the RMVP.

math.DG↗
arXiv2023

Spectral projections and resolvent estimates on Damek-Ricci spaces and their applications

We prove $L^p-L^{p^\prime}$ boundedness of spectral projections and the resolvent of the Laplace-Beltrami operator on Damek-Ricci spaces with the explicit norms in terms of the spectral parameter. To prove these results we established pointwise sharp bounds on the spherical functions and their derivatives. As an application, we study the eigenvalue bounds of Schrödinger operators with complex valued potential.

math.FA↗
arXiv2022

Admissible and sectorial convergence of generalized Poisson integrals on Harmonic $NA$ groups

We prove a converse of Fatou type result for certain eigenfunctions of the Lalplace-Beltrami operator on Harmonic NA groups relating sectorial convergence and admissible convergence of Poisson type integrals of complex (signed) measures. This result extends several results of this kind proved eariler in the context of the classical upper half space $\mathbb{R}_+^{n+1}$. Similar results are also obtained in the degenerate case of the real hyperbolic spaces.

math.CA↗