Uniform Sobolev inequalities for second order non-elliptic differential operators

Abstract

Abstract. We study uniform Sobolev inequalities for the second order differential operators P pDq of non-elliptic type. For d ě 3 we prove that the Sobolev type estimate }u} L q pR d q ď C}P pDqu} L p pR d q holds with C independent of the first order and the constant terms of P pDq if and only if 1{p´1{q " 2{d and. We also obtain restricted weak type endpoint estimates for the critical pp, qq " pd´2 q. As a consequence, the result extends the class of functions for which the unique continuation for the inequality |P pDqu| ď |V u| holds.

Sharp Resolvent Estimates Outside of the Uniform Boundedness Range

“…We prove (4.6) and (4.7) by making use of Theorem 2.14 as in [30,32]. Both arguments to show (4.6) and (4.7) are not much different from each other except for using different estimates in Theorem 2.14.…”

An endpoint version of uniform Sobolev inequalities

“…The estimates cannot be uniform, but depend on z, because the exponent pairs are not on the line 1 p´1 q " 2 n . Finally, for non-elliptic P pDq, uniform restricted weak type estimates have been established in a recent work of Jeong-Kwon-Lee [4], which completely characterizes the range of pp, qq for which (1) holds in the non-elliptic case. Theorem 2 follows from Theorem 1, a restricted weak type Stein-Tomas inequality (see Section 3) and the localization argument in [8, p.335-337], after adapting several classical results to Lorentz spaces.…”

“…Theorem 2 follows from Theorem 1, a restricted weak type Stein-Tomas inequality (see Section 3) and the localization argument in [8, p.335-337], after adapting several classical results to Lorentz spaces. To our knowledge, (4) is still open when n " 3. The difficulty here is the failure of Littlewood-Paley inequality when an exponent becomes 8 (see Proposition 1).…”

Existence uniqueness and regularity theory for elliptic equations with complex-valued potentials

“…Let c : R n → C be a given measurable complex valued potential function. Motivated by the study of L p -diffusion phenomena for dissipative wave equations [30] and by the study of other related topics such as [1,2,3,7,5,9,11,13,14,15,18,19,20,22,23,21,34], we are interested in studying the unique solvability and regularity estimates in the Sobolev space W 2,p (R n , C) for a strong solution u of the equations…”

“…Different from the known work regarding the W 2,p and W 1,p -regularity estimates of solutions such as [6,10,9,22,23,21,24,25] and also motivated by [2,13,14,15,18,20,30], this paper investigates the case that the potential c is a measurable complex valued function. Throughout the paper, we write c(x) = c 1 (x) + ic 2 (x) where c 1 , c 2 : R n → R are measurable functions.…”