My research interests lie broadly in Analysis, Partial Differential Equations, Geometric Analysis, and Harmonic Analysis, with particular emphasis on the following directions:
Functional Inequalities: Functional inequalities appear in the study of several PDEs; for instance, they can often be shown to be equivalent to smoothing properties of suitably associated flows. There are many natural questions to ask, ranging from establishing the inequality itself to finding the best constant and proving stability. As a doctoral student, I began my research on this topic, including Hardy, Poincaré, and Rellich-type inequalities, particularly on negatively curved manifolds such as hyperbolic space. This study relies on criticality theory for second-order elliptic operators and the careful application of Bessel pair concepts. Later, I developed an interest in examining these inequalities in Euclidean space as well as in discrete settings. Recently, I have also become interested in studying these inequalities in the context of sub-elliptic operators generated by Grushin vector fields. These operators serve as a crucial link between elliptic and non-elliptic theory and represent a fundamental example in sub-Riemannian geometry. More recently, my interests have expanded toward the study of stability phenomena for functional inequalities, an increasingly active and celebrated area of research. In this direction, I have studied the stability of Hardy-type inequalities on Cartan–Hadamard manifolds satisfying centered isoperimetric inequalities. Broadly speaking, quantitative stability aims to understand how far a function is from achieving equality in an inequality, thereby providing refined information beyond the inequality itself.
Geometric Analysis: I have investigated the Gelfand-type equation on Cartan–Hadamard manifolds, which involves classifying solutions based on their asymptotic behavior, stability, and intersection properties. The qualitative characteristics of these solutions can be significantly influenced by the global geometry of the underlying manifold, particularly through stochastic completeness. In continuation of this line of research, I also explore Dirichlet p-parabolicity, positivity-preserving properties, and the stochastic completeness of second-order operators on complete Riemannian manifolds. These questions naturally lie at the interface of geometry, analysis, and potential theory, where the geometry of the ambient space plays a fundamental role in determining the behavior of solutions. While much of the existing potential theory has been developed in the linear setting, I am particularly interested in understanding how these phenomena extend to nonlinear operators, where several fundamental analytical and geometric questions remain widely open. More recently, I have also developed an interest in geometric and analytical problems arising in sub-Riemannian settings, particularly on the Heisenberg group. In this direction, I study improved Sobolev-type interpolation inequalities and their applications to perturbed CR Yamabe-type problems, where the interplay between geometric structures and functional inequalities plays a central role. Since many analytical and geometric aspects of nonlinear problems in such non-Euclidean settings remain far from fully understood, I am particularly interested in exploring these directions further.
Analysis & PDEs: My research focuses on Hamilton–Jacobi equations arising in specific ergodic control problems involving diffusions in a switching environment. In this direction, I have studied the existence and uniqueness of non-negative solutions related to the critical eigenvalue problem associated with these equations. More broadly, my interests in partial differential equations have expanded toward the study of spectral and eigenvalue properties of nonlinear operators in Euclidean spaces. Recently, I have become particularly interested in the quantitative spectral stability of eigenvalues for certain classes of domains under prescribed boundary conditions. In these problems, the sharp asymptotic behavior of eigenvalues with respect to perturbation parameters is expected to depend strongly on the vanishing order of the limiting eigenfunction. The precise rate at which eigenvalues of the perturbed problem converge to those of the limit problem can often be analyzed through blow-up techniques. The study of spectral stability in singularly perturbed problems has important applications across several fields, including quantum mechanics, materials science, heat conduction, climate modeling, and acoustics.
Harmonic Analysis: Over time, I have developed an interest in harmonic analysis through the study of fine properties of function spaces and critical embedding phenomena. In the critical regime of Sobolev embeddings, one encounters remarkable exponential integrability results, most notably the Moser–Trudinger inequality. My research interests extend to developments in this direction, particularly trace exponential integrability, where the dimension of integration plays a fundamental role. These critical embeddings are also closely related to bounded oscillation spaces introduced by Brezis and Nirenberg and can be further refined to capture delicate function behavior through capacitary BMO-type spaces. Such spaces enjoy analogs of the John–Nirenberg inequality, a fundamental result in the theory of functions of bounded mean oscillation. From this perspective, I study several tools from harmonic analysis, including maximal function estimates, bounded mean oscillation (BMO), vanishing mean oscillation (VMO), and John–Nirenberg-type inequalities associated with dyadic Hausdorff contents adapted to cubes, Choquet integrals defined through these contents, and the function spaces generated by them, inspired by the pioneering work of Adams. More recently, I have also developed an interest in harmonic analysis on non-Euclidean spaces, particularly hyperbolic spaces, where the exponential volume growth gives rise to analytical phenomena that differ significantly from the Euclidean setting and leave many interesting directions yet to be explored. I am particularly interested in exploring further connections between harmonic analysis and PDEs, especially in geometric settings where many analytical phenomena remain to be understood.
The title of my thesis is "Study of the Poincaré-Hardy type Inequalities and eigenvalue problems for second-order elliptic PDEs." You can find my doctoral thesis here.