Reproducing Kernels of H^m (a, b) (m = 1, 2, 3) and Least Constants in Sobolev’s Inequalities

Abstract

In [J.T. Marti (1983). Evaluation of the least constant in Sobolev’s inequality for H ^l(0,s). SIAM J. Numer. Anal., 20(6), 1239–1242.], Marti proves that the least constant c _l in Sobolev’s inequality for the embedding of the Sobolev space H ^l(a, b) into the space C[a, b] of bounded continuous functions on the interval [a, b] is . In this article, we compute the least constants in Sobolev’s inequalities for the spaces H^m (a,b) (m = 1, 2, 3) and H^m (ℝ), by the construction of the reproducing kernels of the Sobolev spaces.

Keywords:

• Least constant
• Sobolev’s inequality
• Reproducing kernel
• Green’s function

AMS Subject Classifications:

• 46E35
• 34B27