TY - JOUR

T1 - ON the Lp-BOUNDEDNESS of the STOCHASTIC SINGULAR INTEGRAL OPERATORS and ITS APPLICATION to Lp-REGULARITY THEORY of STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS

AU - Kim, Ildoo

AU - Kim, Kyeong Hun

N1 - Funding Information:
Received by the editors May 8, 2018, and, in revised form, December 15, 2019. 2010 Mathematics Subject Classification. Primary 60H15, 42B20, 35S10, 35K30, 35B45. Key words and phrases. Stochastic Calderón-Zygmund theorem, stochastic Hörmander condition, stochastic singular integral operator, stochastic partial differential equation, maximal Lp-regularity. Kyeong-Hun Kim is the corresponding author. The first author was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIP) (No. 2017R1C1B1002830). The second author was supported by Samsung Science and Technology Foundation under Project Number SSTF-BA1401-02.
Publisher Copyright:
© 2020 American Mathematical Society. All rights reserved.

PY - 2020/8

Y1 - 2020/8

N2 - In this article we introduce a stochastic counterpart of the Hörmander condition and Calderón-Zygmund theorem. Let Wt be a Wiener process in a probability space Ω and let K(ω, r, t, x, y) be a random kernel which is allowed to be stochastically singular in a domain O ⊂ Rd in the sense that E______t 0_|x-y|<ϵ |K(ω, s, t, y, x)|dydWs_____p =∞ ∀t, p, ϵ > 0, x ∈ O. We prove that the stochastic integral operator of the type Tg(t, x) :=_t 0_O (0.1) K(ω, s, t, y, x)g(s, y)dydWs is bounded on Lp = Lp (Ω × (0,∞);Lp(O)) for all p ∈ [2,∞) if it is bounded on L2 and the following (which we call stochastic Hörmander condition) holds: There exists a quasi-metric ρ on (0,∞)× O and a positive constant C0 such that for X = (t, x), Y = (s, y), Z = (r, z) ∈ (0,∞)×O, sup ω∈Ω,X,Y_∞ 0__ρ(X,Z)≥C0ρ(X,Y ) |K(r, t, z, x)-K(r, s, z, y)| dz_2 dr < ∞. Such a stochastic singular integral naturally appears when one proves the maximal regularity of solutions to stochastic partial differential equations (SPDEs). As applications, we obtain the sharp Lp-regularity result for a wide class of SPDEs, which includes SPDEs with time measurable pseudo-differential operators and SPDEs defined on non-smooth angular domains.

AB - In this article we introduce a stochastic counterpart of the Hörmander condition and Calderón-Zygmund theorem. Let Wt be a Wiener process in a probability space Ω and let K(ω, r, t, x, y) be a random kernel which is allowed to be stochastically singular in a domain O ⊂ Rd in the sense that E______t 0_|x-y|<ϵ |K(ω, s, t, y, x)|dydWs_____p =∞ ∀t, p, ϵ > 0, x ∈ O. We prove that the stochastic integral operator of the type Tg(t, x) :=_t 0_O (0.1) K(ω, s, t, y, x)g(s, y)dydWs is bounded on Lp = Lp (Ω × (0,∞);Lp(O)) for all p ∈ [2,∞) if it is bounded on L2 and the following (which we call stochastic Hörmander condition) holds: There exists a quasi-metric ρ on (0,∞)× O and a positive constant C0 such that for X = (t, x), Y = (s, y), Z = (r, z) ∈ (0,∞)×O, sup ω∈Ω,X,Y_∞ 0__ρ(X,Z)≥C0ρ(X,Y ) |K(r, t, z, x)-K(r, s, z, y)| dz_2 dr < ∞. Such a stochastic singular integral naturally appears when one proves the maximal regularity of solutions to stochastic partial differential equations (SPDEs). As applications, we obtain the sharp Lp-regularity result for a wide class of SPDEs, which includes SPDEs with time measurable pseudo-differential operators and SPDEs defined on non-smooth angular domains.

UR - http://www.scopus.com/inward/record.url?scp=85090533518&partnerID=8YFLogxK

U2 - 10.1090/tran/8089

DO - 10.1090/tran/8089

M3 - Article

AN - SCOPUS:85090533518

SN - 0002-9947

VL - 373

SP - 5653

EP - 5684

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

IS - 8

ER -