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Durvudkhan Suragan

Publications and source records attributed to Durvudkhan Suragan.

At least 19 recordsLinked to original sources
arXiv2026

An Improved Bound for the Ovals Problem

Let $γ\subset\mathbb R^{m},\,m\geq2,$ be a closed curve of length $2π$ with its curvature $κ$, parametrized by arc length, and let $λ_γ$ be the first eigenvalue of the periodic curvature Schrödinger operator $-d^2/d s^2+κ(s)^2$. We obtain \[ λ_γ\geq \frac{\sqrtπ}{2} \left(\frac{Γ(7/6)}{Γ(5/3)}\right)^3. \] This is a near-sharp lower bound for the Ovals problem. Our proof introduces a new geometric approach. We derive a convolution identity from the closure condition and combine it with projection averaging over tangent directions and sharp Poincaré inequalities on antipodal arcs. As applications, we provide an improved two-state kinetic Lieb-Thirring inequality and the corresponding two-eigenvalue constant.

math.SP↗
arXiv2026

The nonlinear Hausdorff-Young inequality

We prove the constant-one discrete nonlinear Hausdorff-Young inequality. As a consequence, by a discrete-to-continuous limiting argument, we obtain $$ \|(\log|a_f|^{2})^{1/2}\|_{L^{p'}(\mathbb{R})} \le \|f\|_{L^{p}(\mathbb{R})},\quad 1\le p<2, $$ for all $f\in L^{p}(\mathbb{R})$, where $a_f$ denotes the transmission coefficient of the nonlinear Fourier transform. In particular, this resolves the Muscalu-Tao-Thiele uniformity problem. The proof uncovers a hidden Hilbert-space structure that reduces the nonlinear inequality to classical interpolation.

math.FA↗
arXiv2026

Explicit constants in $L^p$-Hardy inequalities for Aharonov-Bohm potentials

For the two-dimensional Aharonov-Bohm potential $A_β$ with flux $β\notin\mathbb{Z}$ and $1<p<2$, Cazacu, Krejčiř\'ık, Lam and Laptev proved by a compactness argument that their constant $λ_β(p)$ in the $L^p$-Hardy inequality strictly exceeds the free constant $\big(\tfrac{2-p}{p}\big)^p$, and asked for a constructive proof with explicit estimates and for comparability of $λ_β(p)$ with a quantity depending on $\text{dist}(β,\mathbb{Z})$. We answer both questions by using a compactness-free two-sided bound for the twisted angular constant. Our explicit Hardy constant is $$\big[\big(\tfrac{2-p}{p}\big)^{2}+\big(\tfrac{\sin(π\text{dist}(β,\mathbb{Z}))}π\big)^{2}\big]^{p/2},\quad 1<p<2.$$ As a byproduct we observe that when $p\ge 2$ the Aharonov--Bohm field produces an $L^p$-Hardy inequality with the usual homogeneous weight $|x|^{-p}$. Our approach also provides new $L^p$-Hardy inequalities with explicit constants for the complex AB potentials.

math.AP↗
arXiv2026

Inhomogeneous parabolic equations with Hardy potential and memory on the Heisenberg group

We study a class of inhomogeneous parabolic equations on the Heisenberg group $\mathbbm{H}^N$ with Hardy-type singular potentials, nonlocal memory terms, and a space-time forcing term: \begin{align} \partial_tu-Δ_{H}u=λ\frac{ψu}{\|\cdot\|^{2}_{H}}+\frac{1}{Γ(γ)}\int_0^t(t-τ)^{γ-1}|u(τ)|^{p}dτ+t^αf \text{ in } \,\mathbbm{H}^N\times (0,T). \end{align} Here, $γ\in [0,1),$ $α\in (-1,\infty),$ $p>1,$ $λ>0,$ and $ψ(\cdot)=|\nabla_{H}\|\cdot\|_{H}|^2,$ where $\nabla_H$ is the horizontal gradient associated to $Δ_H.$ Also, $\|\cdot\|_{H}$ and $Δ_{H}$ denote the Korányi norm and sub-Laplacian associated with the sub-Riemannian geometry of $\mathbbm{H}^N,$ respectively. The combination of a singular Hardy potential and a memory kernel introduces significant analytical challenges. Using a Harnack-type inequality adapted to the Heisenberg group setting, we obtain quantitative positivity estimates that enable a detailed blow-up analysis. We identify parameter regimes depending on $p,γ,α$ leading to finite-time blow-up or instantaneous blow-up, and establish local well-posedness in the absence of the Hardy potential. These results reveal an interplay between the spatial singularity, temporal nonlocality and a time-dependent forcing term. Finally, under a suitable lower bound on the forcing term $f,$ we derive an explicit lifespan estimate for local-in-time solutions.

math.AP↗
arXiv2026

The Stein-Weiss inequality in variable exponent Morrey spaces

In this paper we prove the Stein-Weiss inequality in variable exponent Morrey spaces over a bounded domain. Our work extends earlier results in the variable exponent Lebesgue and Morrey settings, and utilizes new proof techniques applicable to Morrey spaces. We build on the foundational paper by Almeida, Hasanov, and Samko, which introduced Morrey spaces of variable exponents. As an application of our main result, we prove Poincaré-type inequalities using the approach of a recent paper by the first and third authors.

math.CA↗
arXiv2025

Eigenvalue lower bounds through a generalized inradius

Lieb has shown a lower bound on the smallest Dirichlet eigenvalue of the Laplace operator in terms of a generalized inradius. We derive similar bounds for Robin eigenvalues, for eigenvalues of the polyharmonic operator and the sub-Laplacian on the Heisenberg group. We propose a method based on Hardy inequalities that is different from Lieb's approach.

math.SP↗
arXiv2025

Subelliptic $p$-Laplacian spectral problem for Hörmander vector fields

Based on variational methods, we study the spectral problem for the subelliptic $p$-Laplacian arising from smooth Hörmander vector fields. We derive the smallest eigenvalue, prove its simplicity and isolatedness, establish the positivity of the first eigenfunction and show Hölder regularity of eigenfunctions. Moreover, we determine the best constant for the $L^{p}$-Poincaré-Friedrichs inequality for Hörmander vector fields as a byproduct.

math.AP↗
arXiv2025

On Herz-Bochkarev limiting problem

This paper studies Hausdorff-Young-type inequalities within the framework of Lorentz spaces $L_{p,q}$. Focusing on the dependence of the associated constants on the integrability parameter $p$, we derive optimal bounds in the limiting case $p\rightarrow 2$, addressing the Herz-Bochkarev problem. The results obtained refine the pioneering estimates in [3] and are comparable to recent advances in [16]. The main ingredients of our approach are new grand Lorentz space techniques.

math.FA↗
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.

math.AP↗
arXiv2025

One-dimensional integral Rellich type inequalities

The motive of this note is twofold. Inspired by the recent development of a new kind of Hardy inequality, here we discuss the corresponding Hardy-Rellich and Rellich inequality versions in the integral form. The obtained sharp Hardy-Rellich type inequality improves the previously known result. Meanwhile, the established sharp Rellich type integral inequality seems new.

math.FA↗
arXiv2025

Fujita exponent for the fractional sub-Laplace semilinear heat equation with forcing term on the Heisenberg group

In this paper, we study the semilinear heat equation with a forcing term, driven by the fractional sub-Laplacian (-Δ_{\mathbbm{H}^N})^s of order $s\in (0,1),$ on the Heisenberg group $\mathbbm{H}^N$. We establish that the Fujita exponent, a critical threshold that delimits different dynamical regimes of this equation, is $$p_F\coloneqq\frac{Q}{Q-2s},$$ where $Q\coloneqq 2N+2$ is the homogeneous dimension of $\mathbbm{H}^N$. We prove the existence of global-in-time solutions for the supercritical case $(p>p_F),$ and the non-existence of global-in-time solutions for the subcritical case $(1<p<p_F).$ For the critical case $p=p_F,$ we provide a class of functions for which the solution blows up in finite time. These results extend the classical Fujita phenomenon to a sub-Riemannian setting with the nonlocal effects of the fractional sub-Laplacian. Our proof methods intertwine analytic techniques with the geometric structure of the Heisenberg group.

math.AP↗
arXiv2025

Grand Net Spaces and Applications to Integral Operators

This paper introduces the concept of grand net spaces, a new framework that provides a unified setting for studying various function spaces. Building on the seminal works of [8] and [15], we define grand net spaces and establish their key properties, including embedding results, norm equivalences, and interpolation theorems. We prove that these spaces coincide with grand Lorentz spaces under certain conditions and derive boundedness criteria for integral operators acting on grand net spaces. The latter extends the Nursultanov-Tikhonov theorem established in [16].

math.FA↗
arXiv2025

Convolution-type operators in grand Lorentz spaces

We introduce and study a novel grand Lorentz space-that we believe is appropriate for critical cases-that lies "between" the Lorentz-Karamata space and the recently defined grand Lorentz space from [1]. We prove both Young's and O'Neil's inequalities in the newly introduced grand Lorentz spaces, which allows us to derive a Hardy-Littlewood-Sobolev-type inequality. We also discuss Köthe duality for grand Lorentz spaces, from which we obtain a new Köthe dual space theorem in grand Lebesgue spaces.

math.FA↗
arXiv2024

Nonexistence of solutions of certain semilinear heat equations

We consider a semilinear heat equation involving a forcing term which depends only on the space variable. To start with, the existence of a local mild solution is proved through an application of the Banach fixed-point theorem. With the help of carefully defined test functions, we then prove the nonexistence of global weak solutions. The most crucial step is to find the function $d(x)$ used in our proofs, which seems to depends only upon the considered vector fields. This leads to lower bounds for a possible critical Fujita-type exponent. The same function $d(x)$ could lead to a potential norm function which would be most suitable while working with these vector fields. Section 4 is the attraction of this paper in which we apply our approach to all of the vector fields discussed by Biagi, Bonfiglioli and Bramanti, giving rise to Grushin-type and Engel-type PDOs, and more. An upper bound for the blow-up time of local solutions is also provided in each of these cases.

math.AP↗
arXiv2024

Improved Stein inequalities for the Fourier transform

In this paper, we present a refined version of the (classical) Stein inequality for the Fourier transform, elevating it to a new level of accuracy. Furthermore, we establish extended analogues of a more precise version of the Stein inequality for the Fourier transform, broadening its applicability from the range $1<p<2$ to $2\leq p<\infty$.

math.FA↗
arXiv2024

Local and Global Analysis of Semilinear Heat Equations with Hardy Potential on Stratified Lie Groups

On stratified Lie groups we study a semilinear heat equation with the Hardy potential, a power non-linearity and a forcing term which depends only upon the spacial variable. The analysis of an equivalent formulation to the problem and an application of a decade old result of Avelin et al. facilitates the management of the singularity in the Hardy potential, thereby yielding results pertaining to both local and global nonexistence. In addition, local existence is verified when the gradient term appearing in the Hardy potential is unimodular almost everywhere. The global existence is also proved under an additional assumption that the forcing term depends on the time variable as well. Through these results this paper sheds light on the possible pivotal exponents for the existence of both local and global solutions to the equation, offering a deeper understanding of the interplay between the model's parameters and the underlying stratified Lie group structure.

math.AP↗
arXiv2024

Improvement of the discrete Hardy inequality

We establish a novel improvement of the classical discrete Hardy inequality, which gives the discrete version of a recent (continuous) inequality of Frank, Laptev, and Weidl. Our arguments build on certain weighted inequalities based on discrete analogues of symmetric decreasing rearrangement techniques.

math.FA↗