Paneitz-type operators and applications

Abstract

Following standard notations, · p in the above expression stands for the L p -norm (with respect to the Riemannian measure dv g ). As is well known and easy to see, for all u ∈ C ∞ (M), ( g u) 2 ≤ n|∇ 2 u| 2 .Conversely, by the Bochner-Lichnerowicz-Weitzenböck formula we get that DJADLI, HEBEY, AND LEDOUX

Q‐Curvature on a Class of Manifolds with Dimension at Least 5

“…(1) Y 4 .g/ Ä Y 4 .S n /; here S n has the standard metric. We start with the following standard fact (see [6,20]…”

“…The passage from (2.20) to (2.21) is standard, and we refer the readers to [6,20] for further details. The above almost-sharp Sobolev inequality can be used to prove the following concentration compactness lemma.…”

“…Then on B 2ı .p/ n B ı .p/, Note that this equation is critical in the sense that if we start with f 2 L 2n=.nC4/ and use the equation, the usual bootstrap method simply ends with f 2 L 2n=.nC4/ again. Approaches in deriving further regularity for such equations have been well understood (see, for example, [6,9,30,34] and so on). Here is a regularity result particularly tailored for our purpose; we refer the readers to [6,9,30,34]…”

“…It follows from Lemma 3.2 that f 2 C 1 .M / and f > 0. Moreover, the compactness of M under the C 1 .M /-topology follows from its compactness in L 2n=.nC4/ .M / and the proofs of Lemma 3.2 in [6,9,30,34].…”

Concentration of solutions for the fractional Nirenberg problem

“…N is the Q-curvature and Ric g is the Ricci curvature of g, A N , B N are suitable constants depending on N . See [16,31] for more details. Graham, Jenne, Mason and Sparling [20] constructed a sequence of conformally covariant elliptic operators {P g k } on Riemannian manifolds for all positive integers k if N is odd, and for k ∈ {1, 2, • • • , N/2} if N is even.…”