We establish the equivalence between the Sobolev seminorm ‖∇u‖Lp and a quantity obtained when replacing strong Lp by weak Lp in the Gagliardo seminorm |u|Ws,p computed at s=1. As corollaries we derive alternative estimates in some exceptional cases (involving W1,1) where the “anticipated” fractional Sobolev and Gagliardo–Nirenberg inequalities fail.

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

We prove an approximation lemma on (stratified) homogeneous groups that allows one to approximate a function in the non-isotropic Sobolev spaceṄ L 1,Q by L ∞ functions, generalizing a result of Bourgain-Brezis [BB2].We then use this to obtain a Gagliardo-Nirenberg inequality for ∂ b on the Heisenberg group H n .