Parabolic BMO estimates for pseudo-differential operators of arbitrary order

Ildoo Kim; Kyeong Hun Kim; Sungbin Lim

doi:10.1016/j.jmaa.2015.02.065

Parabolic BMO estimates for pseudo-differential operators of arbitrary order

Ildoo Kim, Kyeong Hun Kim, Sungbin Lim

Research output: Contribution to journal › Article › peer-review

5 Citations (Scopus)

Abstract

In this article we prove the BMO-L_∞ estimate(-δ)γ/2u BMO(Rd+1)≤N∂∂tu-A(t)uL∞(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/2uLp(Rd+1)≤N ut - A(t)u Lp(Rd+1), where p∈(1, ∞) and the constant N is independent of u.

Original language	English
Pages (from-to)	557-580
Number of pages	24
Journal	Journal of Mathematical Analysis and Applications
Volume	427
Issue number	2
DOIs	https://doi.org/10.1016/j.jmaa.2015.02.065
Publication status	Published - 2015 Jul 15

Keywords

L-estimate
Non-local operator
Parabolic BMO estimate
Pseudo-differential operator

ASJC Scopus subject areas

Analysis
Applied Mathematics

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@article{5891f815b9a74e27b5269637df438d92,

title = "Parabolic BMO estimates for pseudo-differential operators of arbitrary order",

abstract = "In this article we prove the BMO-L∞ estimate(-δ)γ/2u BMO(Rd+1)≤N∂∂tu-A(t)uL∞(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/2uLp(Rd+1)≤N ut - A(t)u Lp(Rd+1), where p∈(1, ∞) and the constant N is independent of u.",

keywords = "L-estimate, Non-local operator, Parabolic BMO estimate, Pseudo-differential operator",

author = "Ildoo Kim and Kim, {Kyeong Hun} and Sungbin Lim",

year = "2015",

month = jul,

day = "15",

doi = "10.1016/j.jmaa.2015.02.065",

language = "English",

volume = "427",

pages = "557--580",

journal = "Journal of Mathematical Analysis and Applications",

issn = "0022-247X",

publisher = "Academic Press Inc.",

number = "2",

}

TY - JOUR

T1 - Parabolic BMO estimates for pseudo-differential operators of arbitrary order

AU - Kim, Ildoo

AU - Kim, Kyeong Hun

AU - Lim, Sungbin

PY - 2015/7/15

Y1 - 2015/7/15

N2 - In this article we prove the BMO-L∞ estimate(-δ)γ/2u BMO(Rd+1)≤N∂∂tu-A(t)uL∞(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/2uLp(Rd+1)≤N ut - A(t)u Lp(Rd+1), where p∈(1, ∞) and the constant N is independent of u.

AB - In this article we prove the BMO-L∞ estimate(-δ)γ/2u BMO(Rd+1)≤N∂∂tu-A(t)uL∞(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/2uLp(Rd+1)≤N ut - A(t)u Lp(Rd+1), where p∈(1, ∞) and the constant N is independent of u.

KW - L-estimate

KW - Non-local operator

KW - Parabolic BMO estimate

KW - Pseudo-differential operator

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DO - 10.1016/j.jmaa.2015.02.065

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EP - 580

JO - Journal of Mathematical Analysis and Applications

JF - Journal of Mathematical Analysis and Applications

SN - 0022-247X

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