Permanence of metric fractals.
Our understanding of the interplay between Poincaré inequalities, Sobolev inequalities and the geometry of the underlying space has changed considerably in recent years. These changes have simultaneously provided new insights into the classical theory and allowed much of that theory to be extended to a wide variety of different settings. This paper reviews some of these new results and techniques and concludes with an example on the preservation of Sobolev spaces by the maximal function.[Proceedings...
Given a nondegenerate harmonic structure, we prove a Poincaré-type inequality for functions in the domain of the Dirichlet form on nested fractals. We then study the Hajłasz-Sobolev spaces on nested fractals. In particular, we describe how the "weak"-type gradient on nested fractals relates to the upper gradient defined in the context of general metric spaces.
We prove that a function belonging to a fractional Sobolev space may be approximated in capacity and norm by smooth functions belonging to , 0 < m + λ < α. Our results generalize and extend those of [12], [4], [14], and [11].
We get a class of pointwise inequalities for Sobolev functions. As a corollary we obtain a short proof of Michael-Ziemer’s theorem which states that Sobolev functions can be approximated by functions both in norm and capacity.
Let E and F be spaces of real- or complex-valued functions defined on a set X. A real- or complex-valued function g defined on X is called a pointwise multiplier from E to F if the pointwise product fg belongs to F for each f ∈ E. We denote by PWM(E,F) the set of all pointwise multipliers from E to F. Let X be a space of homogeneous type in the sense of Coifman-Weiss. For 1 ≤ p < ∞ and for , we denote by the set of all functions such that , where B(a,r) is the ball centered at a and of...
We prove several results concerning density of , behaviour at infinity and integral representations for elements of the space .
We present definitions of Banach spaces predual to Campanato spaces and Sobolev-Campanato spaces, respectively, and we announce some results on embeddings and isomorphisms between these spaces. Detailed proofs will appear in our paper in Math. Nachr.