On the $$L^p$$-Theory of Vector-Valued Elliptic Operators

Lp SPECTRAL THEORY OF HIGHER-ORDER ELLIPTIC DIFFERENTIAL OPERATORS

Bilateral composition operators on vector-valued Hardy spaces

Let $T$ be a bounded operator on the Banach space $X$ and $ph$ be an analytic self-map of the unit disk $Bbb{D}$‎. ‎We investigate some operator theoretic properties of‎ ‎bilateral composition operator $C_{ph‎, ‎T}‎: ‎f ri T circ f circ ph$ on the vector-valued Hardy space $H^p(X)$ for $1 leq p leq‎ ‎+infty$.‎ ‎Compactness and weak compactness of $C_{ph‎, ‎T}$ on $H^p(X)$‎ ‎are characterized an...

On the Lp-theory of C0-semigroups associated with second order elliptic operators. II

We study positive C0-semigroups on Lp associated with second order uniformly elliptic divergence type operators with singular lower order terms, subject to a wide class of boundary conditions. We obtain an interval (pmin, pmax) in the Lp-scale where these semigroups can be defined, including the case 2 6∈ (pmin, pmax). We present an example showing that the result is optimal. We also show that ...

On the Lp-theory of C0-semigroups associated with second order elliptic operators. I

We study Lp-theory of second order elliptic divergence type operators with measurable coefficients. To this end, we introduce a new method of constructing positive C0-semigroups on Lp associated with sesquilinear (not necessarily sectorial) forms in L2. A precise condition ensuring that the elliptic operator is associated with a quasi-contractive C0-semigroup on Lp is established.

POINT DERIVATIONS ON BANACH ALGEBRAS OF α-LIPSCHITZ VECTOR-VALUED OPERATORS

The Lipschitz function algebras were first defined in the 1960s by some mathematicians, including Schubert. Initially, the Lipschitz real-value and complex-value functions are defined and quantitative properties of these algebras are investigated. Over time these algebras have been studied and generalized by many mathematicians such as Cao, Zhang, Xu, Weaver, and others. Let  be a non-emp...