Parabolic BMO estimates for pseudo-differential operators of arbitrary order

• Published in: Journal of mathematical analysis and applications Vol. 427; no. 2; pp. 557 - 580
• Main Authors: Kim, Ildoo, Kim, Kyeong-Hun, Lim, Sungbin
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 15-07-2015
• Subjects: [formula omitted]-estimate; Non-local operator; Parabolic BMO estimate; Pseudo-differential operator; Parabolic BMO estimate; Non-local operator; Pseudo-differential operator; Lp-estimate
• ISSN: 0022-247X; 1096-0813
• DOI: 10.1016/j.jmaa.2015.02.065

In this article we prove the BMO-L∞ estimate‖(−Δ)γ/2u‖BMO(Rd+1)≤N‖∂∂tu−A(t)u‖L∞(Rd+1),∀u∈Cc∞(Rd+1) for a wide class of pseudo-differential operators A(t) of order γ∈(0,∞). The coefficients of A(t) are assumed to be merely measurable in time variable. As an application to the equation∂∂tu=A(t)u+f,t∈R we prove that for any u∈Cc∞(Rd+1)‖ut‖Lp(Rd+1)+‖(−Δ)γ/2u‖Lp(Rd+1)≤N‖ut−A(t)u‖Lp(Rd+1), where p∈(1,∞) and the constant N is independent of u.