Let $A$ be the $L^p$ realization ($1\<p\<\infty$) of a differential operator $P(D_x,D_t)$ on ${ℝ}^n\times {ℝ}^{+}$ with general boundary conditions $B_k(D_x,D_t)u(x,0)=0$ ($1\le k\le m$). Here $P$ is a homogeneous polynomial of order $2m$ in $n+1$ complex variables that satisfies a suitable ellipticity condition, and for $1\le k\le m$$B_k$ is a homogeneous polynomial of order...

Let {Tt}t>0 be the semigroup of linear operators generated by a Schrödinger operator -A = Δ - V, where V is a nonnegative potential that belongs to a certain reverse Hölder class. We define a Hardy space HA1 by means of a maximal function associated with the semigroup {Tt}t>0. Atomic and Riesz transforms characterizations of HA1 are shown.

In this article, we shall extend the formalization of [10] to discuss higher-order partial differentiation of real valued functions. The linearity of this operator is also proved (refer to [10], [12] and [13] for partial differentiation).

In L2(ℝd; ℂn), we consider a wide class of matrix elliptic second order differential operators $𝒜$ε with rapidly oscillating coefficients (depending on x/ε). For a fixed τ > 0 and small ε > 0, we find approximation of the operator exponential exp(− $𝒜$ετ) in the (L2(ℝd; ℂn) → H1(ℝd; ℂn))-operator norm with an error term of order ε. In this approximation, the corrector is taken...