Po-Lam Yung

Calculus on the Sierpinski gasket I: polynomials, exponentials and power series

Abstract We study the analog of power series expansions on the Sierpinski gasket, for analysis based on the Kigami Laplacian. The analog of polynomials are multiharmonic functions, which have previously been studied in connection with Taylor approximations and splines. Here the main technical result is an estimate of the size of the monomials analogous to x n / n !. We propose a definition of entire analytic functions as functions represented by power series whose coefficients satisfy exponential growth conditions that are stronger than what is required to guarantee uniform convergence. We present a characterization of these functions in terms of exponential growth conditions on powers of the Laplacian of the function. These entire analytic functions enjoy properties, such as rearrangement and unique determination by infinite jets, that one would expect. However, not all exponential functions (eigenfunctions of the Laplacian) are entire analytic, and also many other natural candidates, such as the heat kernel, do not belong to this class. Nevertheless, we are able to use spectral decimation to study exponentials, and in particular to create exponentially decaying functions for negative eigenvalues.

An Improved Strichartz Estimate for Systems with Divergence Free Data

Using the div-curl inequalities of Bourgain and Brezis [1] and van Schaftingen [9], we prove an improved Strichartz estimate for systems of inhomogeneous wave and Schrodinger equations, for which the inhomogeneity is a divergence-free vector field at each given time. The novelty of the result is that one can allow norms of the inhomogeneity in the right hand side of the estimate.

A subelliptic Bourgain–Brezis inequality

We prove an approximation lemma on (stratified) homogeneous groups that allows one to approximate a function in the non-isotropic Sobolev space

Variations on a proof of a borderline Bourgain-Brezis Sobolev embedding theorem

\dot{NL}^{1,Q}

Applications of Bourgain–Brézis inequalities to fluid mechanics and magnetism☆

by

Bourgain–Brezis inequalities on symmetric spaces of non-compact type

L^{\infty}

Polynomial Carleson Operators Along Monomial Curves in the Plane

functions, generalizing a result of Bourgain-Brezis \cite{MR2293957}. We then use this to obtain a Gagliardo-Nirenberg inequality for

Approximation in higher-order Sobolev spaces and Hodge systems

\bar{\partial}_b

The Incompressible Navier Stokes Flow in Two Dimensions with Prescribed Vorticity

on the Heisenberg group

Doubling Properties of Self-similar Measures

\mathbb{H}^n