Improved Sobolev inequalities

For a function $f$ defined on ${{\mathbf{R}}^n}$, Sobolev's inequality $\parallel f{\parallel_q} \leqslant c(\parallel \,f\;{\parallel_{p}} + \parallel \nabla f{\parallel_{p}})$, where $1 < p < q < \infty$ and $1/p - 1/q = 1/n$, can be improved if the Fourier transform $\hat f$ is assumed to have support in a set $A$ which satisfies an estimate $\vert\{ \xi \in A:\vert\vert\xi \vert\vert \leqslant s\} \vert \leqslant c{s^d}$ for some $d < n$ the improvement being that we can take $1/p - 1/q = 1/d$, provided we also assume $p \leqslant 2 \leqslant q$. Analogous results are proved for other Sobolev inequalities, for embeddings into Lipschitz-Zygmund spaces, and for functions on symmetric spaces whose Fourier expansions are suitably limited. Improved Sobolev inequalities are established locally for solutions of the wave equation. An application to the Radon transform on spheres is given.