We present a survey of the theory of γ-radonifying operators and its applications to stochastic integration in Banach spaces.

This article combines three components corresponding to real analysis, algebra and operator theory. A simple and universal way of finding a Hermite interpolation (quasi)polynomial and Gel'fond's formula covering both real and complex cases and based on the study of the sufficient conditions for the continuity of divided differences with respect to two types of the simultaneous convergence of nodes is presented. Relying on it, we establish both algebraic and topological Jordan decomposition for algebraic operators and algebraic and topological properties of their calculus, such as a constructive algebraic characterization of the solvability and an explicit estimate of the related a priori constant and a new characterization of bounded algebraic operators in terms of orbits. Further applications include explicit relations between the barycentric or uniform distributions on convex polyhedra and, correspondingly, $B$-splines or Steklov splines and short analytic proofs of some classical results from the polynomial arithmetics, wavelet theory and discrete Fourier transform. We also provide correct deffinitions, typical properties and representations for the holomorphic calculus of the closed operators with mixed spectra including both (double)sectorial and bounded components.

We demonstrate a family of Strichartz estimates for the conformally invariant Klein-Gordon equation on a class of asymptotically de Sitter spaces with $C^2$ metrics by using well-known local Strichartz estimates and a rescaling argument. This class of metrics includes de Sitter space. We also give an application of the estimates to a semilinear Klein-Gordon equation on these spaces.

Suppose that $1 \lt p \geq \infy, (\Omega, \mu)$ is a $\sigma$-finite measure space and $E$ is a closed subspace of a Lebesgue-Bochner space $L^p(\Omega; X)$, consisting of functions on $\Omega$ that take their values in some complex Banach space $X$. Suppose also that $- A$ is injective and generates a grounded holomorphic semigroup ${T_z}$ on $E$. If $0 \lt \alpha \lt 1$ and $f$ belongs to the domain of $A^\alpha$ then the maximal function $\sup_z \|T_zf\|_x$, where the supremum is taken over any given sector contained in the sector of holomorphy, belongs to $L^p$. A similar result holds for generators that are not injective. This extends earlier work of Blower and Doust.

For long range perturbations of the Laplacian in divergence form, we prove low frequency estimates for powers of the resolvent as the spectral parameter goes to zero.

On a fixed smooth compact Reimann surface with boundary $(M_o, g)$, we show that the Cauchy data space (or Dirichlet-to-Neumann map $N$) of the Schrödinger operator $\Delta + V$ with $V \in C^\infty(M_0)$ determines uniquely the potential $V$.

In this note we present a different approach to the $A_\infty$ extrapolation via Carleson measures developed in [HM] and we illustrate the use of this technique by reproving a well known result of [FKP].

We study certain differential operators of the form $AD$ arising from a first-order approach to the Kato square root problem. We show that if such operators are$R$-bisectorial in $L^p$, they remain $R-bisectorial in $L^q$ for all $q$ close to $p$.In combination with our earlier results with Portal, which required such $R$-bisectoriality in different $L^q$ spaces to start with, this shows that the $R$-bisectoriality in just one $L^p$ actually implies bounded $H^\infty$-calculus in $L^q$ for all $q$ close to $p$. We adapt the approach to related second-order results developed by Auscher, Hofmann and Martell, and also employ abstract extrapolation theorems due to Kalton and Mitrea

Let $A_1, A_2$ be bounded linear operators acting on a Banach space $E$. A pair $(\mu_1, \mu_2)$ of continuous probability measures on $[0,1]$ determines a functional calculus $f \rightarrowtail f_{\mu1,|mu2}(A_1, A_2)$ for analytic functions $f$ by weighting all possible orderings of operator products of $A_1$ and $A_2$ via the probability measures $\mu_1$ and $\mu_2$. For example, $f \rightarrowtail f_{\mu,\mu}(A_1, A_2)$ is the Weyl functional calculus with equally weighted operator products. Replacing $\mu_1$ by Lebesque measure $\lambda$ on $[0,t]$ and $\mu_2$ by stochastic integration with respect to a Winer process $W$, we show that there exists a functional calculus $f \rightarrowtail f_{\lamda,w;t}(A + B)$ for bounded holomorphic functions $f$ if $A$ is a densely defined Hilbert space operator with a bounded holomorphic functional calculus and B is small compared to $A$ relative to a square function norm. By this means, the solution of the stochastic evolution equation $dX_t = AX_tdt + BX_tdW_t, X_0 = x$, is represented as $t \rightarrowtail e_{\lambda,W;t}^{A+B}x, t \geq 0$.

We construct holomorphic functional calculi and introduce local quadratic estimates for operators in a reflexive Banach space that are bisectorial except possibly in a neighbourhood of the origin. The main result is an equivalence of local quadratic estimates with bounded holomorphic functional calculi. For operators with spectrum in a neighbourhood of the origin, the results are weaker than those for bisectorial operators. For operators with a spectral gap in a neighbourhood of the origin, the results are stronger. In each case, however, local quadratic estimates are a more appropriate tool than standard quadratic estimates for establishing that our functional calculi are bounded. This shows that in certain applications it suffices to establish local quadratic estimates.

We obtain Strichartz estimates for the linear Schrödinger equation associated with the twisted sub-Laplacian on $\mathbb C^n$.As a consequence, we prove the local wellposedness for semilinear Schrödinger equation with polynomial nonlinearity in certain magnetic field.