In this paper, we begin with clarifying spaces obtained as limits of sequences of finite networks from an analytic point of view, and we discuss convergence of finite networks with respect to the topology of both the Gromov-Hausdorff distance and variational convergence called $\Gamma$-convergence. Relevantly to convergence of finite networks to infinite ones, we investigate the space of harmonic functions of finite Dirichlet sums on infinite networks and their Kuramochi compactifications.

Clifford analysis offers a higher dimensional function theory studying the null solutions of the rotation invariant, vector valued, first order Dirac operator $\partial$. In the more recent branch Hermitean Clifford analysis, this rotational invariance has been broken by introducing a complex structure $J$ on Euclidean space and a corresponding second Dirac operator $\partial_J$, leading to the system of equations $\partial f = 0 = \partial_J f$ expressing so-called Hermitean monogenicity. The invariance of this system is reduced to the unitary group. In this paper we show that this choice of equations is fully justified. Indeed, constructing the Howe dual for the action of the unitary group on the space of all spinor valued polynomials, the generators of the resulting Lie superalgebra reveal the natural set of equations to be considered in this context, which exactly coincide with the chosen ones.

In this paper we prove the Strong $L^p$-stability of the heat semigroup generated by the Hodge Laplacian on complete Riemannian manifolds with non-negative Weitzenböck curvature. Based on a probabilistic representation formula, we obtain an explicit upper bound of the $L^p$-norm of the Riesz transforms on forms on complete Riemannian manifolds with suitable curvature conditions. Moreover, we establish the Weak $L^p$-Hodge decomposition theorem on complete Riemannian manifolds with non-negative Weitzenböck curvature.

We show that for any Cayley graph, the probability (at any $p$) that the cluster of the origin has size $n$ decays at a well-defined exponential rate (possibly 0). For general graphs, we relate this rate being positive in the supercritical regime with the amenability/nonamenability of the underlying graph.

In this paper, we consider the following type of non-local (pseudo-differential) operators $\mathcal{L}$ on $\mathbb{R}^d$: \begin{align*} \mathcal{L} u(x) = \frac{1}{2} \sum_{i, j=1}^d \frac{\partial}{\partial x_i} \Big(a_{ij}(x) \frac{\partial u(x)}{\partial x_j}\Big) \\+ \lim_{\varepsilon \downarrow 0} \int_{\{y\in \mathbb{R}^d: |y-x|>\varepsilon\}} (u(y)-u(x)) J(x, y) dy, \end{align*} where $A(x)=(a_{ij}(x))_{1\leq i,j\leq d}$ is a measurable $d\times d$ matrix-valued function on $\mathbb{R}^d$ that is uniformly elliptic and bounded and $J$ is a symmetric measurable non-trivial non-negative kernel on $\mathbb{R}^d \times \mathbb{R}^d$ satisfying certain conditions. Corresponding to $\mathcal{L}$ is a symmetric strong Markov process $X$ on $\mathbb{R}^d$ that has both the diffusion component and pure jump component. We establish a priori Hölder estimate for bounded parabolic functions of $\mathcal{L}$ and parabolic Harnack principle for positive parabolic functions of $\mathcal{L}$. Moreover, two-sided sharp heat kernel estimates are derived for such operator $\mathcal{L}$ and jump-diffusion $X$. In particular, our results apply to the mixture of symmetric diffusion of uniformly elliptic divergence form operator and mixed stable-like processes on $\mathbb{R}^d$. To establish these results, we employ methods from both probability theory and analysis.

Analogues of multi-parameter multiplier operators on $\mathbb{R}^d$ are defined on the torus $\mathbb{T}^d$. It is shown that these operators satisfy the classical Coifman-Meyer theorem. In addition, $L(\log L)^n$ end-point estimates are proved.

The main aim of the paper is to give a socle theory for Leavitt path algebras of arbitrary graphs. We use both the desingularization process and combinatorial methods to study Morita invariant properties concerning the socle and to characterize it, respectively. Leavitt path algebras with nonzero socle are described as those which have line points, and it is shown that the line points generate the socle of a Leavitt path algebra. A concrete description of the socle of a Leavitt path algebra is obtained: it is a direct sum of matrix rings (of finite or infinite size) over the base field. New proofs of the Graded Uniqueness and of the Cuntz-Krieger Uniqueness Theorems are given, by using very different means.

A Kleinian group containing a Schottky group as a finite index subgroup is called a Schottky extension group. If $\Omega$ is the region of discontinuity of a Schottky extension group $K$, then the quotient $\Omega/K$ is a closed Riemann orbifold; called a Schottky orbifold. Closed Riemann surfaces are examples of Schottky orbifolds as a consequence of the Retrosection Theorem. Necessary and sufficient conditions for a Riemann orbifold to be a Schottky orbifold are due to M. Reni and B. Zimmermann in terms of the signature of the orbifold. It is well known that the lowest uniformizations of a closed Riemann surface are exactly those for which the Deck group is a Schottky group. In this paper we extend such a result to the class of Schottky orbifolds, that is, we prove that the lowest uniformizations of a Schottky orbifold are exactly those for which the Deck group is a Schottky extension group.

Given a calibrated Riemannian manifold $\overline{M}$ with parallel calibration $\Omega$ of rank $m$ and $M$ an orientable m-submanifold with parallel mean curvature $H$, we prove that if $\cos\theta$ is bounded away from zero, where $\theta$ is the $\Omega$-angle of $M$, and if $M$ has zero Cheeger constant, then $M$ is minimal. In the particular case $M$ is complete with $Ricci^M\geq 0$ we may replace the boundedness condition on $\cos\theta$ by $\cos\theta\geq Cr^{-\beta}$, when $r\rightarrow+\infty$, where $0 < \beta < 1$ and $C > 0$ are constants and $r$ is the distance function to a point in $M$. Our proof is surprisingly simple and extends to a very large class of submanifolds in calibrated manifolds, in a unified way, the problem started by Heinz and Chern of estimating the mean curvature of graphic hypersurfaces in Euclidean spaces. It is based on an estimation of $\|H\|$ in terms of $\cos\theta$ and an isoperimetric inequality. In a similar way, we also give some conditions to conclude $M$ is totally geodesic. We study some particular cases.

We study the boundedness of Toeplitz operators $T_a$ with locally integrable symbols on Bergman spaces $A^p(\mathbb{D})$, $1 < p < \infty$. Our main result gives a sufficient condition for the boundedness of $T_a$ in terms of some ``averages'' (related to hyperbolic rectangles) of its symbol. If the averages satisfy an ${o}$-type condition on the boundary of $\mathbb{D}$, we show that the corresponding Toeplitz operator is compact on $A^p$. Both conditions coincide with the known necessary conditions in the case of nonnegative symbols and $p=2$. We also show that Toeplitz operators with symbols of vanishing mean oscillation are Fredholm on $A^p$ provided that the averages are bounded away from zero, and derive an index formula for these operators.

Given three transversal and sufficiently regular hypersurfaces in $\mathbb{R}^3$ it follows from work of Bennett-Carbery-Wright that the convolution of two $L^2$ functions supported of the first and second hypersurface, respectively, can be restricted to an $L^2$ function on the third hypersurface, which can be considered as a nonlinear version of the Loomis-Whitney inequality. We generalize this result to a class of $C^{1,\beta}$ hypersurfaces in $\mathbb{R}^3$, under scaleable assumptions. The resulting uniform $L^2$ estimate has applications to nonlinear dispersive equations.