A note on Sobolev type inequalities in two independent variables
The inequality \[ \iint^{\infty}_{-\infty}u^ 4dx dy\leq (\alpha /2)(\iint^{\infty}_{-\infty}u^ 2dx dy)(\iint^{\infty}_{- \infty}| \text{grad} u|^ 2dx dy), \] where u(x,y) is a smooth function of compact support and \(\alpha\) is a constant, is known as Sobolev's inequality. The author establishes some new inequalities of this type. The proofs are quite elementary and the results yield in the special cases some known inequalities of Sobolev type.
scientific article; zbMATH DE number 3991819