Interpolation Spaces

We introduce a notion of generalized Triebel-Lizorkin spaces associated with sectorial operators in Banach function spaces. Our approach is based on holomorphic functional calculus techniques. Using the concept of \(\mathcal {R}_{s}\)-sectorial operators, which in turn is based on the notion of \(\mathcal {R}_{s}\)-bounded sets of operators introduced by Lutz Weis, we obtain a neat theory including equivalence of various norms and a precise description of real and complex interpolation spaces. Another main result of this article is that an \(\mathcal {R}_{s}\)-sectorial operator always has a bounded H∞-functional calculus in its associated generalized Triebel-Lizorkin spaces.

That the classical interpolation theorems of Riesz-Thorin and Marcinkiewicz are of great importance in harmonic analysis has long since been realized. The functional analytic generalization of these interpolation theorems — abstract interpolation space theory — also is now proving to be of importance in harmonic analysis (cf., for instance, [1]). In this paper we reverse the flow in the sense that a circle of ideas from Lp-multiplier theory and tensor products of Banach spaces is used to provide answers to a number of questions that arise naturally in the general theory of abstract interpolation spaces. While some of these questions have been answered before (though not always published), the unified approach used in this paper introduces techniques which may well be of wider utility in interpolation space theory. We shall be interested in both the real interpolation spaces (X o, X 1)θq;J, (X o, X 1)θ, q;k of Peetre and the complex interpolation spaces (X o,X 1)θ of Lions and Calderon. For all unexplained notation and terminology see [2], [5], [6] or [11].

We prove that the fractional BMO space on a one-dimensional manifold is an interpolation space between C and $C^1$. We also prove that $BMO^1$ is an interpolation space between C and $C^2$. The proof is based on some nonclassical interpolation constructio

This paper develops a unified analytical framework for non-separable Banach spaces (NSBS) grounded in the concepts of approximate Schauder bases, weakly compact approximation operators, and local–global interpolation structures. The traditional limitations associated with the absence of separability and sequential compactness are overcome by replacing sequences with directed nets and by localising all analytical arguments within separable hulls. Within this setting, the notions of approximate interpolation couples and approximate real and complex interpolation spaces are introduced and analysed in detail. We establish that boundedness, compactness, duality, and stability properties extend naturally from the classical separable case, providing new generalisations of the Lions–Peetre and Riesz–Thorin theorems. Spectral theory is reformulated for bounded linear operators in NSBS, proving spectral stability under approximate compactness and continuity of spectra under analytic perturbations. Applications to operator theory and mathematical physics are discussed, including spectral–interpolation correspondences, weak compactness criteria, and models for quantum and statistical systems where non-separable structures arise intrinsically. These results yield a coherent generalisation of fundamental principles of functional analysis and open new avenues for research in operator algebras, evolution equations, and non-separable

We prove results on complex interpolation of vector‐valued Sobolev spaces over the half‐line with Dirichlet boundary condition. Motivated by applications in evolution equations, the results are presented for Banach space‐valued Sobolev spaces with a power weight. The proof is based on recent results on pointwise multipliers in Bessel potential spaces, for which we present a new and simpler proof as well. We apply the results to characterize the fractional domain spaces of the first derivative operator on the half line.

We analyze the complexity of the sparse-grid interpolation and sparse-grid quadrature of countably-parametric functions which take values in separable Banach spaces with unconditional bases. Assuming a suitably quantified holomorphic dependence on the parameters, we establish dimension-independent convergence rate bounds for sparse-grid approximation schemes. Analogous results are shown in the case that the parametric families are obtained as approximate solutions of corresponding parametric-holomorphic, nonlinear operator equations as considered in [A. Cohen and A. Chkifa and Ch. Schwab: Breaking the curse of dimensionality in sparse polynomial approximation of parametric PDEs, J. Math. Pures Appl. 103 (2015) 400–428], for example by means of stable, finite-dimensional approximations. We discuss in detail nonlinear Petrov–Galerkin projections. Error and convergence rate bounds for constructive and explicit multilevel, sparse tensor approximation schemes combining sparse-grid interpolation in the parameter space and general, multilevel discretization schemes in the physical domain are proved. The present results unify and generalize earlier works in terms of the admissible multilevel approximations in the physical domain (comprising general stable Petrov–Galerkin and discrete Petrov–Galerkin schemes, collocation and stable domain approximations) and in terms of the admissible operator equations (comprising smooth, nonlinear locally well-posed operator equations). Additionally, a novel computational strategy to localize sequences of nested index sets for the anisotropic Smolyak interpolation in parameter space is developed which realizes best [Formula: see text]-term benchmark convergence rates. We also consider Smolyak-type quadratures in this general setting, for which we establish improved convergence rates based on cancellations in the integrands’ gpc expansions by symmetries of quadratures and the probability measure [J.Z̃ech and Ch.S̃chwab: Convergence rates of high dimensional Smolyak quadrature, Report 2017-27, SAM ETH Zürich (2017)]. Several examples illustrating the abstract theory include domain uncertainty quantification (UQ) for general, linear, second-order, elliptic advection–reaction–diffusion equations on polygonal domains, where optimal convergence rates of FEM are known to require local mesh refinement near corners. Further applications of the presently developed theory comprise evaluations of posterior expectations in Bayesian inverse problems.

The field theory of strings is developed using the functional calculus as a practical technique. A Schr\"odinger-type equation and its eigensolutions are derived in light-cone coordinates. Vertices and amplitudes are represented entirely in a Fock space of second-quantized functionals. Two different open-string interactions are explicitly investigated. The three- and four-string vertex functionals are calculated for arbitrary times and compared with the asymptotic expressions obtained by other authors. We discuss how more exotic string configurations can be incorporated into the theory and emphasize the utility of the functional calculus in describing these topologies.

We study the Stokes operator with Hodge, Navier, and Robin boundary conditions on domains $$\Omega \subseteq \mathbb {R}^d$$ Ω ⊆ R d that are uniformly $$C^{2,1}$$ C 2 , 1 . Starting with the Hodge Laplacian we etablish a bounded Hörmander functional calculus for the Stokes operator with Hodge boundary conditions. This entails a Hörmander functional calculus and boundedness of the $$H^\infty $$ H ∞ -calculus in spaces of soleniodal vector fields for the Stokes operator with Hodge boundary conditions. We then establish boundedness of the $$H^\infty $$ H ∞ -calculus for Stokes operators with Navier type conditions via Robin type perturbations of Hodge boundary conditions. This implies maximal $$L^p$$ L p -regularity for these operators and results on fractional domain spaces. Our results cover certain non-Helmholtz domains.

In this paper, using the recently discovered notion of the S-spectrum, we prove the spectral theorem for a bounded or unbounded normal operator on a Clifford module (i.e., a two-sided Hilbert module over a Clifford algebra based on units that all square to be \(-1\)). Moreover, we establish the existence of a Borel functional calculus for bounded or unbounded normal operators on a Clifford module. Towards this end, we have developed many results on functional analysis, operator theory, integration theory and measure theory in a Clifford setting which may be of an independent interest. Our spectral theory is the natural spectral theory for the Dirac operator on manifolds in the non-self adjoint case. Moreover, our results provide a new notion of spectral theory and a Borel functional calculus for a class of n-tuples of commuting or non-commuting operators on a real or complex Hilbert space. Moreover, for a special class of n-tuples of operators on a Hilbert space our results provide a complementary functional calculus to the functional calculus of J. L. Taylor.

Extensions of an [formula omitted] functional calculus

Sampling and interpolation in Bargmann–Fock spaces of polyanalytic functions

Interpolation in spaces of measurable functions

In this paper, we study the K-envelopes of the real interpolation methods with function space parameters in the sense of Brudnyi and Kruglyak [Y. A. Brudnyi and N. Ja. Kruglyak, Interpolation functors and interpolation spaces (North-Holland, Amsterdam, Netherlands, 1991)]. We estimate the upper bounds of the K-envelopes and the interpolation norms of bounded operators for the KΦ-methods in terms of the fundamental function of the rearrangement invariant space related to the function space parameter Φ. The results concerning the quasi-power parameters and the growth/continuity envelopes in function spaces are obtained.

Multilinear interpolation between adjoint operators

We show how the geometrical properties of uniform convexity and uniformly non‐𝓁 are inherited by real interpolation spaces for infinite families.