Language:   Search:   Contact
Zentralblatt MATH has released its new interface!
For an improved author identification, see the new author database of ZBMATH.

Query:
Fill in the form and click »Search«...
Format:
Display: entries per page entries
Zbl 0563.46024
Caffarelli, L.; Kohn, R.; Nirenberg, Louis
First order interpolation inequalities with weights.
(English)
[J] Compos. Math. 53, 259-275 (1984). ISSN 0010-437X; ISSN 1570-5846/e

The authors prove a necessary and sufficient condition for there to exist a constant C such that for each $u\in C\sb 0\sp{\infty}$ $(R\sp n)$, $$\Vert \vert x\vert\sp{\gamma}u\Vert\sb{L\sp r}\le C\Vert \vert x\vert\sp{\alpha}\vert Du\vert \Vert\sp a\sb{L\sp p}\Vert \vert x\vert\sp{\beta}u\Vert\sp{1-a}\sb{L\sp q},$$ where $\alpha$, $\beta$, $\gamma$, a, r, p, q, and n are fixed real numbers satisfying a number of specified relationships. Special cases of this inequality have appeared in a number of papers, including a previous paper of the authors [Comm. Pure Appl. Math. 35, 771-831 (1982; Zbl 0509.35067)] and a paper of {\it B. Muckenhoupt} and {\it R. Wheeden} [Trans. Am. Math. Soc. 192, 261-274 (1974; Zbl 0289.26010)]. The proof is lengthy but elementary, and consists of verifying a large number of cases.
[P.Lappan]
MSC 2000:
*46E35 Sobolev spaces and generalizations
26D10 Inequalities involving derivatives, diff. and integral operators
46M35 Abstract interpolation of topological linear spaces
26D20 Analytical inequalities involving real functions

Citations: Zbl 0509.35067; Zbl 0289.26010

Highlights
Master Server