Isoperimetric functional inequalities via the maximum principle: the exterior differential systems approach

Square functions and the Hamming cube: duality

“…The degeneracy of (33) means that the determinant of the matrix in (33) is zero. This is a general Monge-Ampère type equation and, after an application of the exterior differential systems of Bryant-Griffiths (see [12]), we obtain that the solutions can be locally characterized as follows:…”

“…In [12] we used u(p,t) = −U(p, √ 2t) instead of U(p, q), in which case (35) becomes just the backward heat equation for u(p,t). We will not formulate a formal statement but we do make a remark that such a reasoning allows us to guess the dual of M, i.e., to find U given M. The way this guess works will be illustrated in Section 3.4.…”

“…The function M(x, y) = x ln x − y 2 2x satisfies (33) and, therefore, it gives the log-Sobolev inequality [12]. Its dual in the sense of (34) is U(p, q) = e p−q 2 /2 (see Section 3.1.1 in [12] where t = q 2 /2).…”

“…The function M(x, y) = x ln x − y 2 2x satisfies (33) and, therefore, it gives the log-Sobolev inequality [12]. Its dual in the sense of (34) is U(p, q) = e p−q 2 /2 (see Section 3.1.1 in [12] where t = q 2 /2). Notice that for this U, inequality (25) simplifies to 2e a 2 /2 ≥ e a + e −a , which is true since (2k)!…”

“…Proof. M(x, y) is a solution of the homogeneous Monge-Ampère equation ( 33), and therefore it has a representation of the form (34) (see Section 3.1.4 in [12]):…”