On the constants for multiplication in Sobolev spaces

TL;DR

This work derives quantitative bounds for the best constant K_{nd} in the Sobolev algebra inequality for H^{n}(R^{d}, C), valid for any d and n>d/2. It provides a rigorous upper bound via a Fourier-convolution construction leading to a hypergeometric-type majorant, and two complementary families of lower bounds based on Bessel and Fourier-type trial functions, each with explicit formulas and asymptotic behavior. The authors analyze limiting regimes as n approaches d/2 and infinity, obtaining precise leading-order constants M_d and T_d and showing consistent exponential-type growth in n with a (2/√3) factor, while maintaining practical accuracy demonstrated by numerical data for d=1–4. Together, the results give 75–88% sandwich accuracy between upper and lower bounds across tested cases and supply a framework for sharp constants in Sobolev multiplication inequalities with potential PDE applications.

Abstract

For n > d/2, the Sobolev (Bessel potential) space H^n(R^d, C) is known to be a Banach algebra with its standard norm || ||_n and the pointwise product; so, there is a best constant K_{n d} such that || f g ||_{n} <= K_{n d} || 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 the limit cases n -> (d/2) and n -> + Infinity, for which asymptotic formulas are presented. Both in these limit cases and for intermediate values of n, the lower bounds are fairly close to the upper bounds. Numerical tables are given for d=1,2,3,4, where the lower bounds are always between 75% and 88% of the upper bounds.