The heat equation with rough boundary conditions and holomorphic functional calculus

In this paper we consider the Laplace operator with Dirichlet boundary conditions on a smooth domain. We prove that it has a bounded H^∞-calculus on weighted L^p-spaces for power weights which fall outside the classical class of A_p-weights. Furthermore, we characterize the domain of the operator and derive...

The weyl calculus for group generators satisfying the canonical commutation relations

We generalise the classical Weyl pseudo-differential calculus on R^d to the setting of two d-tuples of operators A = (A₁,..., A_d) and B = (B₁,..., B_d) acting on a Banach space generating bounded C0-groups satisfying the Weyl canonical commutation relations. We show that the resulting Weyl...

The Weyl calculus with respect to the Gaussian measure and restricted Lp-Lq boundedness of the Ornstein-Uhlenbeck semigroup in complex time

In this paper, we introduce a Weyl functional calculus a↦a(Q,P) for the position and momentum operators Q and P associated with the Ornstein-Uhlenbeck operator L=−Δ+x⋅∇, and give a simple criterion for restricted Lp-Lq boundedness of operators in this functional calculus. The analysis of this non-commutative functional calculus is simpler than...

L^p-Analysis of the Hodge–Dirac Operator Associated with Witten Laplacians on Complete Riemannian Manifolds.

We prove R-bisectoriality and boundedness of the (Formula presented.)-functional calculus in (Formula presented.) for all (Formula presented.) for the Hodge–Dirac operator associated with Witten Laplacians on complete Riemannian manifolds with non-negative Bakry–Emery Ricci curvature on k-forms.

Maximal regularity for parabolic equations with measurable dependence on time and applications

The subject of this thesis is the study of maximal L^p-regularity of the Cauchy problem u'(t)+A(t)u(t)=f(t), t∈ (0,T), u(0)=x. <br/>We assume (A(t))_{t∈ (0,T)} to be a family of closed operators on a Banach space X₀, with constant domain D(A(t))=X₁ for every t∈ (0,T). Maximal L^p-regularity means that...

Cylindrical continuous martingales and stochastic integration in infinite dimensions

In this paper we define a new type of quadratic variation for cylindrical continuous local martingales on an infinite dimensional spaces. It is shown that a large class of cylindrical continuous local martingales has such a quadratic variation. For this new class of cylindrical continuous local martingales we develop a stochastic integration...

Maximal Regularity for Non-autonomous Equations with Measurable Dependence on Time

In this paper we study maximal Lp-regularity for evolution equations with time-dependent operators A. We merely assume a measurable dependence on time. In the first part of the paper we present a new sufficient condition for the Lp-boundedness of a class of vector-valued singular integrals which does not rely on Hörmander conditions in the time...

Functional Calculus via Transference, Double Operator Integrals and Applications

This thesis is dedicated to the study of several aspects of the theory of functional calculus. This theory considers the combination of an operator A and a function f(z) of a variable z, resulting in an operator f(A). One then attempts to study properties of the operator f(A) in terms of properties of the operator A and the function f. A...

Maximal ? -regularity

In this paper, we prove maximal regularity estimates in “square function spaces” which are commonly used in harmonic analysis, spectral theory, and stochastic analysis. In particular, they lead to a new class of maximal regularity results for both deterministic and stochastic equations in L p -spaces with 1<p<?. For stochastic equations, the...

Topological Extensions of Holomorphic Functional Calculus for Sectorial and Half-plane Type Operators

The theory of functional calculus deals with the idea of 'inserting operators into functions'. The holomorphic (or Dunford-Riesz) functional calculus is based on a class of functions can be represented by the Cauchy-integral formula. A topological extension means that the primary calculus is extended to a wider class of functions and the...

Maximal Lp-Regularity for Stochastic Evolution Equations