Perturbation determinants in Banach spaces — with an application to eigenvalue estimates for perturbed operators

In the first part of this paper we provide a self‐contained introduction to (regularized) perturbation determinants for operators in Banach spaces. In the second part, we use these determinants to derive new bounds on the discrete eigenvalues of compactly perturbed operators, broadly extending some recent results by Demuth et al. In addition, we also establish new bounds on the discrete eigenvalues of generators of C₀‐semigroups.     

Weak type estimates for Calderón‐Zygmund operators on Herz spaces at critical indexes

We define weak Herz spaces (?ⁿ) which are the weak version of the ordinary Herz spaces (?ⁿ). We consider the boundedness of Calderón‐Zygmund operators from to at critical indexes α = ?n/q, n(1? 1/q) and q = 1. We also consider weighted estimates. (© 2003 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Calderón – Zygmund Operators on Weighted Weak Hardy Spaces in Locally Compact Vilenkin Groups

Let G be a bounded locally compact Vilenkin group. We study the atomic decom‐position of weighted weak Hardy space. We also define several Calderón – Zygmund type operators and study their boundedness on, spaces like weighted Hardy spaces, weighted weak Hardy spaces and weighted weak Lebesgue spaces. Sharpness of some of our results is also discussed.     

Parabolic Littlewood–Paley operators

In this paper, weak (1,1) and L^p estimates for the parabolic Littlewood–Paley operators on some homogeneous spaces are established, which are extensions of known results. (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Admissible perturbations of α‐ψ‐pseudocontractive operators: convergence theorems

We prove some convergence theorems for α‐ψ‐pseudocontractive operators in real Hilbert spaces, by using the concept of admissible perturbation. Our results extend and complement some theorems in the existing literature. Copyright © 2016 John Wiley & Sons, Ltd.     

Existence of solutions for some non‐Fredholm integro‐differential equations with the bi‐Laplacian

We prove the existence of solutions for some semilinear elliptic equations in the appropriate H⁴ spaces using the fixed‐point technique where the elliptic equation contains fourth‐order differential operators with and without Fredholm property, generalizing the previous results.     

The calculation of the sum of the spaces of the K‐method of real interpolation

It is shown that the calculation of the sum of the spaces of the K‐Peetre interpolation method can be reduced to the calculation of the sum of the cones of the concave functions included by the parameter in the definition of the spaces of the K‐Peetre interpolation method. As an example, a new extrapolation theorem for operators in Lipschitz spaces is obtained.     

Besov‐Morrey spaces and Triebel‐Lizorkin‐Morrey spaces for nondoubling measures

We define Morrey type Besov‐Triebel spaces with the underlying measure non‐doubling. After defining the function spaces, we investigate boundedness property of some class of the singular integral operators (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the trace problem for Lizorkin–Triebel spaces with mixed norms

The subject is traces of Sobolev spaces with mixed Lebesgue norms on Euclidean space. Specifically, restrictions to the hyperplanes given by x₁ = 0 and x_n = 0 are applied to functions belonging to quasi‐homogeneous, mixed norm Lizorkin–Triebel spaces ; Sobolev spaces are obtained from these as special cases. Spaces admitting traces in the distribution sense are characterised up to the borderline cases; these are also covered in case x₁ = 0. For x₁ the trace spaces are proved to be mixed‐norm Lizorkin–Triebel spaces with a specific sum exponent; for x_n they are similarly defined Besov spaces. The treatment includes continuous right‐inverses and higher order traces. The results rely on a sequence version of Nikol'skij's inequality, Marschall's inequality for pseudodifferential operators (and Fourier multiplier assertions), as well as dyadic ball criteria. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On singular mono‐energetic transport equations in slab geometry

In this paper we establish the well posedness of the Cauchy problem associated to transport equations with singular cross‐sections (i.e. unbounded collisions frequencies and unbounded collision operators) in L₁ spaces for specular reflecting boundary conditions. In addition, we discuss the weak compactness of the second‐order remainder term of the Dyson–Phillips expansion. This allows us to estimate the essential type of the transport semigroup from which the asymptotic behaviour of the solution is derived. The case of singular transport equations with periodic boundary conditions is also discussed. The proofs make use of the Miyadera perturbation theory of positive semigroups on AL‐spaces. Copyright © 2002 John Wiley & Sons, Ltd.     

Paley–Littlewood decomposition for sectorial operators and interpolation spaces

We prove Paley–Littlewood decompositions for the scales of fractional powers of 0‐sectorial operators A on a Banach space which correspond to Triebel–Lizorkin spaces and the scale of Besov spaces if A is the classical Laplace operator on We use the ‐calculus, spectral multiplier theorems and generalized square functions on Banach spaces and apply our results to Laplace‐type operators on manifolds and graphs, Schrödinger operators and Hermite expansion. We also give variants of these results for bisectorial operators and for generators of groups with a bounded ‐calculus on strips.     

Multivalued fractals in <Emphasis Type="Italic">b</Emphasis>-metric spaces

Fractals and multivalued fractals play an important role in biology, quantum mechanics, computer graphics, dynamical systems, astronomy and astrophysics, geophysics, etc. Especially, there are important consequences of the iterated function (or multifunction) systems theory in several topics of applied sciences. It is known that examples of fractals and multivalued fractals are coming from fixed point theory for single-valued and multivalued operators, via the so-called fractal and multi-fractal operators. On the other hand, the most common setting for the study of fractals and multi-fractals is the case of operators on complete or compact metric spaces. The purpose of this paper is to extend the study of fractal operator theory for multivalued operators on complete b-metric spaces.     

Multiple summing operators on<Emphasis Type="Italic">C(K)</Emphasis> spaces

In this paper, we characterize, for 1≤p<∞, the multiple (p, 1)-summing multilinear operators on the product ofC(K) spaces in terms of their representing polymeasures. As consequences, we obtain a new characterization of (p, 1)-summing linear operators onC(K) in terms of their representing measures and a new multilinear characterization ofL _∞ spaces. We also solve a problem stated by M.S. Ramanujan and E. Schock, improve a result of H. P. Rosenthal and S. J. Szarek, and give new results about polymeasures. Both authors were partially supported by DGICYT grant BMF2001-1284.     

Higher‐order Sobolev‐type embeddings on Carnot–Carathéodory spaces

A sufficient condition for higher‐order Sobolev‐type embeddings on bounded domains of Carnot–Carathéodory spaces is established for the class of rearrangement‐invariant function spaces. The condition takes form of a one‐dimensional inequality for suitable integral operators depending on the isoperimetric function relative to the Carnot–Carathéodory structure of the relevant sets. General results are then applied to particular Sobolev spaces built upon Lebesgue, Lorentz and Orlicz spaces on John domains in the Heisenberg group. In the case of the Heisenberg group, the condition is shown to be necessary as well.     

Pseudodifferential operators on mixed‐norm Besov and Triebel–Lizorkin spaces

We prove boundedness of pseudodifferential operators on anisotropic mixed‐norm Besov and Triebel–Lizorkin spaces. Our proof relies only on general maximal function estimates and provides a new perspective even in the case of spaces without mixed norms. Moreover, we cover the case of Fourier multipliers on the above mentioned spaces. As application we establish boundedness of pseudodifferential operators and Fourier multipliers on anisotropic mixed‐norm Sobolev spaces.     

Interpolation of nonlinear operators in weighted <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>-spaces

We show that Peetre’s classical interpolation theorem in weighted L _p -spaces is carried over to some classes of nonlinear operators containing in particular the Lipschitz operators and operators close to them in the properties satisfying less restrictive conditions than Lipschitz in each of the spaces of a Banach pair.     

Signatures for ‐hermitians and ‐unitaries on Krein spaces with Real structures

For J‐hermitian operators on a Krein space satisfying an adequate Fredholm property, a global Krein signature is shown to be a homotopy invariant. It is argued that this global signature is a generalization of the Noether index. When the Krein space has a supplementary Real structure, the sets of J‐hermitian Fredholm operators with Real symmetry can be retracted to certain of the classifying spaces of Atiyah and Singer. Secondary ‐invariants are introduced to label their connected components. Related invariants are also analyzed for J‐unitary operators.     

On the <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript>-boundedness of pseudo-differential operators with non-regular symbols

In this paper, we consider the continuity property of pseudo-differential operators with symbols whose Fourier transforms have compact support. As applications, we obtain the L ^p -boundedness for symbols in Besov spaces and in modulation spaces.     

Trace Formulas for Some Singular Differential Operators and Applications

We characterize the convergence of the series ∑ λ^–1_n, where λⁿ are the non‐zero eigenvalues of some boundary value problems for degenerate second order ordinary differential operators and we prove a formula for the above sum when the coefficient of the zero‐order term vanishes. We study these operators both in weighted Hilbert spaces and in spaces of continuous functions. After investigating the boundary behaviour of the eigenfunctions, we give applications to the regularity of the generated semigroups.