Hardy-Sobolev inequalities for vector fields and canceling linear differential operators
(2014) Indiana University Mathematics Journal — Vol. 63, n° 5, p. 1419-1445 (2014)
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- Abstract
- Given a homogeneous $k$-th order differential operator A (D) on \R^n between two finite dimensional spaces, we establish the Hardy inequality $$ \int_{\R^n} \frac{\abs{D^{k-1}u}}{\abs{x}} \dif x \leq C \int_{\R^n} \abs{A(D)u} $$ and the Sobolev inequality $$ \norm{D^{k-n} u}_{L^{\infty}(\R^n)}\leq C \int_{\R^n} \abs{A(D)u} $$ when $A(D)$ is elliptic and satisfies a recently introduced cancellation property. We also study the necessity of these two conditions.
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Citations
Bousquet, P., & Van Schaftingen, J. (2014). Hardy-Sobolev inequalities for vector fields and canceling linear differential operators. Indiana University Mathematics Journal, 63(5), 1419-1445. https://doi.org/10.1512/iumj.2014.63.5395 (Original work published 2014)