A weighted $L_p$-regularity theory for parabolic partial differential equations with time measurable pseudo-differential operators

Abstract

We obtain the existence, uniqueness, and regularity estimates of the following Cauchy problemin (Muckenhoupt) weighted Lp-spaces with time-measurable pseudo-differential operatorsMore precisely, we find sufficient conditions of the symbol ψ(t, ξ) (especially depending on the smoothness of the symbol with respect to ξ) to guarantee that equation (0.1) is well-posed in (Muckenhoupt) weighted Lp-spaces. Here the symbol ψ(t, ξ) is merely measurable with respect to t and the sufficient smoothness of ψ(t, ξ) with respect to ξ is characterized by a property of each weight. In particular, we prove the existence of a positive constant N such that for any solution u to equation (0.1),where p, q ∈ (1, ∞), −d − 1 < α < (d + 1)(p − 1), −1 < α1 < q − 1, −d < α2 < d(p − 1), and γ is the order of the operator ψ(t, i∇).

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“…For a more detail, see [4,Theorem A.1]. Then by the almost orthogonality of ∆ j , the Cauchy-Schwartz inequality, Hölder's inequality, and (A.12), for all…”

“…To estimate D β 0 ξ ψ(t, ξ) m ξ α |ξ| εγ exp ´t s ψ(r, ξ)dr , we borrow the lemma in [4] and introduce it below.…”

“…Hence if y ∈ B 2 −j b (0) and |z| < 2 −j b, then |y − z| ≤ 2 × 2 −j b and f (y − z) = 0. By [4,Lemma 6.6] and Hölder's inequality, ˆRd (∆ j P ε (t, 0, y 0 − z) − ∆ j P ε (t, 0, y 1 − z))f (z)dz…”

“…We refer readers to [19,20,21,22] treating data and solutions in various L p -spaces without weights. For a weighted theory, see recent result of the first and second authors [4]. Moreover, if the order γ of the operator ψ(t, −i∇) is less than or equal to 2, then these operators become generators of stochastic processes so called additive processes which is a time-direction generalization of a Lévy process.…”