An L q(L p)-Theory for Parabolic Pseudo-Differential Equations: Calderón-Zygmund Approach

Ildoo Kim; Sungbin Lim; Kyeong Hun Kim

doi:10.1007/s11118-016-9552-3

An L _q(L _p)-Theory for Parabolic Pseudo-Differential Equations: Calderón-Zygmund Approach

Ildoo Kim, Sungbin Lim, Kyeong Hun Kim

Department of Mathematics

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

7 Citations (Scopus)

Abstract

In this paper we present a Calderón-Zygmund approach for a large class of parabolic equations with pseudo-differential operators A(t) of arbitrary order γ∈ (0 , ∞). It is assumed that (t) is merely measurable with respect to the time variable. The unique solvability of the equation∂u∂t=Au−λu+f,(t,x)∈R^d+1 and the L_q(R,L_p)-estimate ∥ut∥Lq(R,Lp)+∥(−Δ)γ/2u∥Lq(R,Lp)+λ∥u∥Lq(R,Lp)≤N∥f∥Lq(R,Lp)are obtained for any λ > 0 and p, q∈ (1 , ∞).

Original language	English
Pages (from-to)	463-483
Number of pages	21
Journal	Potential Analysis
Volume	45
Issue number	3
DOIs	https://doi.org/10.1007/s11118-016-9552-3
Publication status	Published - 2016 Oct 1

Keywords

Calderón-Zygmund approach
L(L)-estimate
Parabolic Pseudo-differential equations

ASJC Scopus subject areas

Analysis

Access to Document

10.1007/s11118-016-9552-3

Cite this

@article{c31c8dea8876428594d8774ae88c77db,

title = "An L q(L p)-Theory for Parabolic Pseudo-Differential Equations: Calder{\'o}n-Zygmund Approach",

abstract = "In this paper we present a Calder{\'o}n-Zygmund approach for a large class of parabolic equations with pseudo-differential operators A(t) of arbitrary order γ∈ (0 , ∞). It is assumed that (t) is merely measurable with respect to the time variable. The unique solvability of the equation∂u∂t=Au−λu+f,(t,x)∈Rd+1 and the Lq(R,Lp)-estimate ∥ut∥Lq(R,Lp)+∥(−Δ)γ/2u∥Lq(R,Lp)+λ∥u∥Lq(R,Lp)≤N∥f∥Lq(R,Lp)are obtained for any λ > 0 and p, q∈ (1 , ∞).",

keywords = "Calder{\'o}n-Zygmund approach, L(L)-estimate, Parabolic Pseudo-differential equations",

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

note = "Funding Information: This work was supported by Samsung Science and Technology Foundation under Project Number SSTF-BA1401-02 Publisher Copyright: {\textcopyright} 2016, Springer Science+Business Media Dordrecht.",

year = "2016",

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doi = "10.1007/s11118-016-9552-3",

language = "English",

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T1 - An L q(L p)-Theory for Parabolic Pseudo-Differential Equations

T2 - Calderón-Zygmund Approach

AU - Kim, Ildoo

AU - Lim, Sungbin

AU - Kim, Kyeong Hun

N1 - Funding Information: This work was supported by Samsung Science and Technology Foundation under Project Number SSTF-BA1401-02 Publisher Copyright: © 2016, Springer Science+Business Media Dordrecht.

PY - 2016/10/1

Y1 - 2016/10/1

N2 - In this paper we present a Calderón-Zygmund approach for a large class of parabolic equations with pseudo-differential operators A(t) of arbitrary order γ∈ (0 , ∞). It is assumed that (t) is merely measurable with respect to the time variable. The unique solvability of the equation∂u∂t=Au−λu+f,(t,x)∈Rd+1 and the Lq(R,Lp)-estimate ∥ut∥Lq(R,Lp)+∥(−Δ)γ/2u∥Lq(R,Lp)+λ∥u∥Lq(R,Lp)≤N∥f∥Lq(R,Lp)are obtained for any λ > 0 and p, q∈ (1 , ∞).

AB - In this paper we present a Calderón-Zygmund approach for a large class of parabolic equations with pseudo-differential operators A(t) of arbitrary order γ∈ (0 , ∞). It is assumed that (t) is merely measurable with respect to the time variable. The unique solvability of the equation∂u∂t=Au−λu+f,(t,x)∈Rd+1 and the Lq(R,Lp)-estimate ∥ut∥Lq(R,Lp)+∥(−Δ)γ/2u∥Lq(R,Lp)+λ∥u∥Lq(R,Lp)≤N∥f∥Lq(R,Lp)are obtained for any λ > 0 and p, q∈ (1 , ∞).

KW - Calderón-Zygmund approach

KW - L(L)-estimate

KW - Parabolic Pseudo-differential equations

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DO - 10.1007/s11118-016-9552-3

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