We consider perturbations of Dirac type operators on complete, connected metric spaces equipped with a doubling measure. Under a suitable set of assumptions, we prove quadratic estimates for such operators and hence deduce that these operators have a bounded functional calculus. In particular, we deduce a Kato square root type estimate.

Let $(X, \mu)$ be a non-homogeneous space in the sense that $X$ is a metric space equipped with an upper doubling measure $\mu$. The aim of this paper is to study the endpoint estimate of the maximal operator associated to a Calderón-Zygmund operator $T$ and the $L^p$ boundedness of the maximal commutator with RBMO functions

Let $M$ be a manifold with ends constructed in $[2]$ and $\Delta$ be the Laplace-Beltrami operator on $M$. In this note, we show the weak type $(1,1)$ and $L^p$ boundedness of the Hardy-Littlewood maximal function and of the maximal function associated with the heat semigroup $\mathcal{M}_{\Delta}f(x) = sup_{t \gt o} | exp(-t \Delta)f(x)|$ on $L^p(M)$ for $1 \lt p \leq \infty$. The significance of these results comes from the fact that $M$ does not satisfies the doubling condition.

This paper is concerned with quaternion-valued functions on the plane and operators which act on such functions. In particular, we investigate the space $L^2(\mathbb{R}^2, \mathbb{H})$ of square-integrable quaternion-valued functions on the plane and apply the recently developed Clifford-Fourier transform and associated convolution theorem to characterise the closed translation-invariant submodules of $L^2(\mathbb{R}^2, \mathbb{H})$ and its bounded linear translation-invariant operators. The Clifford-Fourier characterisation of Hardy-type spaces on $\mathbb{R}^d$ is also explored.

Isoperimetric estimates stretch back for thousands of years in geometry, and for more than a hundred years in harmonic analysis and mathematical physics. We will touch on some of these highlights before describing recent progress that uses rotational symmetry to prove sharp upper bounds on sums of eigenvalues of the Laplacian. For example, we prove in 2 dimensions that the scale-normalized eigenvalue sum \[ (\lambda_1 + \cdots + \lambda_n) \frac{A^3}{I} \] (where $A$ denotes area and $I$ is moment of inertia about the centroid) is maximized among triangles by the equilateral triangle, for each $n \geq 1$. This theorem, which is due to the author and B. A. Siudeja, generalizes a result of Pólya for the fundamental tone.

Numerous related problems will be discussed, such as the inverse spectral and spectral gap problems for triangular domains.

This is a short survey paper based on the topics of the au- thor's research along with his collaborators. Main topics presented in the paper are on the regularity and rigidity theorems for the solutions of the elliptic differential equations. In particular, the author poses several open problems in these topics for further study. The paper contains six sections. Regularity theorems for the elliptic equations of the non- divergence form with rough coefficients are presented in Section 1. In Section 2, we introduce and summarize some recent developments on the rigidity problems and theorems for the solution of some linear de- generate elliptic equations. A typical example is the rigidity problem for the solution of the Dirichlet problem of the Laplace-Beltrami op- erator on the unit ball in $C^n$ with Bergman metric. In Section 3, the rigidity theorems and problems for harmonic maps between two com- plete non-compact Kahler manifolds are discussed. In Section 4, we summarize the approximation formula for the potential function of the Kahler-Einstein metric. In Section 5, we summarize some rigidity theo- rems for the degenerate Monge-Ampère equations as well as some characterization theorems for some strictly pseudoconvex pseudo-Hermitian manifolds. Finally, in Section 6, we summarize some recent results on the bottom of the spectrum of the Laplace-Beltrami operators on Kahler manifolds.

We study boundedness on $L^p([0,T]) \times \mathbb{R}^N$ of Riesz transforms $\nabla(\mathcal{A})^{-1/2}$ for class of parabolic operators such as $A = \frac{\partial}{\partial t} - \Delta + V(t,x)$. Here $V(t,x)$ is a non-negative potential depending on time ${t}$ and space variable ${x}$. As a consequence, we obtain $W_{x}^{1,p}$-solutions for the non-homogeneous problem \[ \partial_{t}u - \Delta u + V(t,.)u = f(t,i), u(0) = 0 \] for initial data $f \in L^p([0,T] \times \mathbb{R}^N)$.

We review recent results on the uniqueness of solutions of the diffusion equation \[ \partial \psi_{t} / \partial t + H \psi_{t} = 0 \] where $H$ is a strictly elliptic, symmetric, second-order operator on an open subset $\Omega$ of $\mathbf{R}^d$. In particular we discuss $L_1$-uniqueness, the existence of a unique continuous solution on $L_1(\Omega)$, and Markov uniqueness, the existence of a unique submarkovian solution on the spaces $L_p(\Omega)$. We give various criteria for uniqueness in terms of capacity estimates and the Riemannian geometry associated with $H$.

A family of equivalent submultiplicative weights on the to- tally disconnected, locally compact group $G$ is defined in terms of the conjugation action of $G$ on itself. These weights therefore reflect the structure of $G$, and the corresponding weighted convolution algebra is intrinsic to $G$ in the same way that $L^1(G) is.