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In this paper we introduce the new notion of complex isoparametric functions on Riemannian manifolds. These are then employed to devise a general method for constructing proper $p$-harmonic functions. We then apply this to construct the…

Using a four points inequality for the boundary of CAT(-1)-spaces, we study the relation between Gromov hyperbolic spaces and CAT(-1)-spaces.

We derive the sharp constants for the inequalities on the Heisenberg group H^n whose analogues on Euclidean space R^n are the well known Hardy-Littlewood-Sobolev inequalities. Only one special case had been known previously, due to…

In the setting of higher-dimensional anisotropic Heisenberg group, we compute the fundamental solution for the sub-Laplacian, and we prove Poincar\'e and $\beta-$Logarithmic Sobolev inequalities for measures as a function of this…

The hypercontractive inequality is a fundamental result in analysis, with many applications throughout discrete mathematics, theoretical computer science, combinatorics and more. So far, variants of this inequality have been proved mainly…

In this review paper, we survey Hardy type inequalities from the point of view of Folland and Stein's homogeneous groups. Particular attention is paid to Hardy type inequalities on stratified groups which give a special class of homogeneous…

In this note, we prove a theorem \`a la Fatou for the square root of Poisson Kernel in the context of quasi-convex cocompact discrete groups of isometries of $\delta$-hyperbolic spaces. As a corollary we show that some matrix coefficients…

Complex hyperbolic triangle groups were first considered by Mostow in building the first nonarithmetic lattices in PU(2, 1). They are a natural generalization of the classical triangle groups acting on the hyperbolic plane. A well-known…

Let $\mathbb{H}^{n}=\mathbb{C}^{n}\times\mathbb{R}$ be the $n$-dimensional Heisenberg group, $Q=2n+2$ be the homogeneous dimension of $\mathbb{H}^{n}$. We extend the well-known concentration-compactness principle on finite domains in the…

We prove an improved version of Poincar\'e-Hardy inequality in suitable subspaces of the Sobolev space on the hyperbolic space via Bessel pairs. As a consequence, we obtain a new Hardy type inequality with an improved constant (than the…

A sharp isoperimetric inequality for the Hamming cube is proved at the critical exponent $\beta=\frac12$. This follows up on previous work, where such bounds were established for $\beta$ near $\frac12$. As a consequence, this result settles…

We shall discuss a higher-rank Khovanskii-Teissier inequality, generalizing a theorem of Li. In the course of the proof, we develop new Hodge-Riemann bilinear relations in certain mixed settings, which in themselves slightly extend the…

Existence and uniqueness of the solution of the Neumann problem for the Kohn-Laplacian on the Kor\'anyi ball of the Heisenberg group $\mathbb{H}_n$ are discussed. Explicit representations of Green's type function (Neumann function) for the…

In this paper, we consider Carleman estimates and inverse problems for the coupled quantitative thermoacoustic equations. In Part I, we establish Carleman estimates for the coupled quantitative thermoacoustic equations by assuming that the…

We prove Hardy inequalities for the conformally invariant fractional powers of the sublaplacian on the Heisenberg group $\mathbb{H}^n$. We prove two versions of such inequalities depending on whether the weights involved are non-homogeneous…

We present two analytic applications of the fact that a hyperbolic group can be endowed with a strongly hyperbolic metric. The first application concerns the crossed-product C*-algebra defined by the action of a hyperbolic group on its…

We give a combinatorial criterion that implies both the non-strong relative hyperbolicity and the one-endedness of a finitely generated group. We use this to show that many important classes of groups do not admit a strong relatively…

Using methods from algebraic topology and group cohomology, I pursue Grothendieck's question on equality of geometric and cohomological Brauer groups in the context of complex-analytic spaces. The main result is that equality holds under…

The Riemann hypothesis is proved by quantum-extending the zeta Riemann function to a quantum mapping between quantum $1$-spheres with quantum algebra $A=\mathbb{C}$, in the sense of A. Pr\'astaro \cite{PRAS01, PRAS02}. Algebraic topologic…

In this paper, we prove sharp isoperimetric inequalities for lower order eigenvalues of Neumann Laplacian on bounded domains in both compact and noncompact rank-1 symmetric spaces. Our results generalize the work of Wang and Xia for bounded…