New results on multiplication in Sobolev spaces

TL;DR

This paper investigates the sharp constant in a Sobolev space multiplication inequality, providing bounds, tables for specific dimensions, and asymptotic behavior analysis, extending previous work on the special case where all indices are equal.

Contribution

It derives new upper and lower bounds for the sharp constant in Sobolev space multiplication inequalities and analyzes their asymptotic behavior for large indices.

Findings

Bounds for the sharp constant are close, with ratios often near 0.90 or higher.

Tables of bounds are provided for dimensions 1 and 3 across various parameters.

Asymptotic analysis reveals the behavior of bounds as indices grow large.

Abstract

We consider the Sobolev (Bessel potential) spaces H^ell(R^d, C), and their standard norms || ||_ell (with ell integer or noninteger). We are interested in the unknown sharp constant K_{ell m n d} in the inequality || f g ||_{ell} \leqs K_{ell m n d} || f ||_{m} || g ||_n (f in H^m(R^d, C), g in H^n(R^d, C); 0 <= ell <= m <= n, m + n - ell > d/2); we derive upper and lower bounds K^{+}_{ell m n d}, K^{-}_{ell m n d} for this constant. As examples, we give a table of these bounds for d=1, d=3 and many values of (ell, m, n); here the ratio K^{-}_{ell m n d}/K^{+}_{ell m n d} ranges between 0.75 and 1 (being often near 0.90, or larger), a fact indicating that the bounds are close to the sharp constant. Finally, we discuss the asymptotic behavior of the upper and lower bounds for K_{ell, b ell, c ell, d} when 1 <= b <= c and ell -> + Infinity. As an example, from this analysis we obtain the ell -> + Infinity limiting behavior of the sharp constant K_{ell, 2 ell, 2 ell, d}; a second example concerns the ell -> + Infinity limit for K_{ell, 2 ell, 3 ell, d}. The present work generalizes our previous paper [16], entirely devoted to the constant K_{ell m n d} in the special case ell = m = n; many results given therein can be recovered here for this special case.