Multiplication in Sobolev spaces, revisited

Multiplication in Sobolev Spaces, Revisited

In this article, we re-examine some of the classical pointwise multiplication theorems in Sobolev-Slobodeckij spaces, and along the way we cite a simple counterexample that illustrates how certain multiplication theorems fail in Sobolev-Slobodeckij spaces when a bounded domain is replaced by R. We identify the source of the failure, and examine why the same failure is not encountered in Bessel ...

New results on multiplication in Sobolev spaces

We consider the Sobolev (Bessel potential) spaces Hl(R,C), and their standard norms ‖ ‖l (with l integer or noninteger). We are interested in the unknown sharp constant Klmnd in the inequality ‖fg‖l 6 Klmnd‖f‖m‖g‖n (f ∈ Hm(R,C), g ∈ Hn(R,C); 0 6 l 6 m 6 n, m + n − l > d/2); we derive upper and lower bounds K lmnd for this constant. As examples, we give a table of these bounds for d = 1, d = 3 a...

The Multiplication Operator in Sobolev Spaces with Respect to Measures

We consider the multiplication operator, M, in Sobolev spaces with respect to general measures and give a characterization for M to be bounded, in terms of sequentially dominated measures. This has important consequences for the asymptotic behaviour of Sobolev orthogonal polynomials. Also, we study properties of Sobolev spaces with respect to measures. 2001 Academic Press

On the constants for multiplication in Sobolev spaces

For n > d/2, the Sobolev (Bessel potential) space Hn(R,C) is known to be a Banach algebra with its standard norm ‖ ‖n and the pointwise product; so, there is a best constant Knd such that ‖fg‖n 6 Knd‖f‖n‖g‖n for all f, g in this space. In this paper we derive upper and lower bounds for these constants, for any dimension d and any (possibly noninteger) n ∈ (d/2,+∞). Our analysis also includes th...

Sobolev Spaces