Parabolic BMO estimates for pseudo-differential operators of arbitrary order

Research output: Contribution to journalArticle

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 languageEnglish
Pages (from-to)557-580
Number of pages24
JournalJournal of Mathematical Analysis and Applications
Volume427
Issue number2
DOIs
Publication statusPublished - 2015 Jul 15

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Pseudodifferential Operators
Arbitrary
Coefficient
Estimate
Class

Keywords

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

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics

Cite this

Parabolic BMO estimates for pseudo-differential operators of arbitrary order. / Kim, Ildoo; Kim, Kyeong Hun; Lim, Sungbin.

In: Journal of Mathematical Analysis and Applications, Vol. 427, No. 2, 15.07.2015, p. 557-580.

Research output: Contribution to journalArticle

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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",
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AU - Kim, Kyeong Hun

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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.

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KW - Pseudo-differential operator

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