一类半轴上的锥Sobolev空间

Meromorphic Mellin symbols arise in questions of characterizing elliptic regularity and the asymptotics of solutions to differential equations on spaces with conical singularities, and other singularities as well, and the construction of pseudo-differential parametrices to elliptic operators.     

Spectral operators generated by damped hyperbolic equations

We announce a series of results on the spectral analysis for a class of nonselfadjoint opeators, which are the dynamics generators for the systems governed by hyperbolic equations containing dissipative terms. Two such equations are considered: the equation of nonhomogeneous damped string and the 3-dimensional damped wave equation with spacially nonhomogeneous spherically symmetric coefficients. Nonselfadjoint boundary conditions are imposed at the ends of a finite interval or on a sphere centered at the origin respectively. Our main result is the fact the aforementioned operators are spectral in the sense of N. Dunford. The result follows from the fact that the systems of root vectors of the above operators form Riesz bases in the corresponding energy spaces. We also give asymptotics of the spectra and state the Riesz basis property results for the nonselfadjoint operator pencils associated with these operators.     

Interpolation of Bilinear Operators Between Quasi-Banach Spaces

We study interpolation, generated by an abstract method of means, of bilinear operators between quasi-Banach spaces. It is shown that under suitable conditions on the type of these spaces and the boundedness of the classical convolution operator between the corresponding quasi-Banach sequence spaces, bilinear interpolation is possible. Applications to the classical real method spaces, Calderón-Lozanovsky spaces, and Lorentz-Zygmund spaces are presented. The author is supported by the National Science Foundation under grant DMS 0099881. The author is supported by KBN Grant 1 P03A 013 26.     

Boundary value problems for a class of elliptic operator pencils

In this paper operator pencilsA(x, D, ) are studied which act on a manifold with boundary and satisfy the condition of N-ellipticity with parameter, a generalization of the notion of ellipticity with parameter as introduced by Agmon and Agranovich-Vishik. Sobolev spaces corresponding to the Newton polygon are defined and investigated; in particular it is possible to describe their trace spaces. With respect to these spaces, an a priori estimate is proved for the Dirichlet boundary value problem connected with an N-elliptic pencil.Supported in part by the Deutsche Forschungsgemeinschaft and by Russian Foundation of Fundamental Research, Grant 00-01-00387.     

Boundary-contact problems for domains with edge singularities

We study boundary-contact problems for elliptic equations (and systems) with interfaces that have edge singularities. Such problems represent continuous operators between weighted edge spaces and subspaces with asymptotics. Ellipticity is formulated in terms of a principal symbolic hierarchy, containing interior, transmission, and edge symbols. We construct parametrices, show regularity with asymptotics of solutions in weighted edge spaces and illustrate the results by boundary-contact problems for the Laplacian with jumping coefficients.     

Boundary-contact problems for domains with conical singularities

We study boundary-contact problems for elliptic equations (and systems) with interfaces that have conical singularities. Such problems represent continuous operators between weighted Sobolev spaces and subspaces with asymptotics. Ellipticity is formulated in terms of extra transmission conditions along the interfaces with a control of the conormal symbolic structure near conical singularities. We show regularity and asymptotics of solutions in weighted spaces, and we construct parametrices. The result will be illustrated by a number of explicit examples.     

Sobolev type embeddings in the limiting case

We use interpolation methods to prove a new version of the limiting case of the Sobolev embedding theorem, which includes the result of Hansson and Brezis-Wainger for W _n ^k/k as a special case. We deal with generalized Sobolev spaces W _A ^k , where instead of requiring the functions and their derivatives to be in L_n/k, they are required to be in a rearrangement invariant space A which belongs to a certain class of spaces “close” to L_n/k. We also show that the embeddings given by our theorem are optimal, i.e., the target spaces into which the above Sobolev spaces are shown to embed cannot be replaced by smaller rearrangement invariant spaces. This slightly sharpens and generalizes an, earlier optimality result obtained by Hansson with respect to the Riesz potential operator. In memory of Gene Fabes. Acknowledgements and Notes This research was supported by Technion V.P.R. Fund-M. and C. Papo Research Fund.     

Some new classes of Banach-Mackey spaces

Some weakenings of property (K) of Antosik for locally convex spaces are introduced: local property (K) and, for spaces with Schauder-type decompositions, two variants of property (K) defined in terms of block- and tail-sequences. It is shown that if a space enjoys any of these new properties, then it is Banach-Mackey. An application to the barrelledness of the spaces of Pettis integrable functions is given, and examples are provided to distinguish between the various K-type properties.     

Invariance of the cone algebra without asymptotics

Let B be a manifold with conical singularities, and denote by the smooth bounded manifold with cylindrical ends obtained by blowing up near the singularities.B.-W. Schulze has developed a framework for a pseudodifferential calculus on B by defining various classes of distribution spaces and operator algebras, working in fixed coordinates on the manifold . I am showing here that the Mellin Sobolev spaces without asymptotics, the cone algebra without asymptotics, and its ideal of smoothing operators are independent of the choice of coordinates and therefore may be considered intrinsic objects for manifolds with conical singularities.     

Phase space analysis of semilinear parabolic equations

We present a wave packet analysis of a class of possibly degenerate parabolic equations with variable coefficients. As a consequence, we prove local wellposedness of the corresponding Cauchy problem in spaces of low regularity, namely the modulation spaces, assuming a nonlinearity of analytic type. As another application, we deduce that the corresponding phase space flow decreases the global wave front set. We also consider the action on spaces of analytic functions, provided the coefficients are analytic themselves.     

Type I Toeplitz algebras

The class of Toeplitz algebras associated to ordered groups is important in the analysis of Toeplitz operators on the generalised Hardy spaces defined by such groups. The conditions under which these Toeplitz algebras are Type I C^*-algebras are investigated.     

Existence and maximal Lp-regularity of solutions for the porous medium equation on manifolds with conical singularities

We consider the porous medium equation on manifolds with conical singularities and show existence, uniqueness, and maximal L^p-regularity of a short-time solution. In particular, we obtain information on the short time asymptotics of the solution near the conical point. Our method is based on bounded imaginary powers results for cone differential operators on Mellin–Sobolev spaces and R-sectoriality perturbation techniques.     

Limits of higher-order Besov spaces and sharp reiteration theorems

We compute the limits of higher-order Besov norms and derive the sharp constants for certain forms of the Sobolev embedding theorem. Our results extend to higher-order spaces the recent work by Brézis-Bourgain-Mironescu and Maz’ya-Shaposhnikova. The interpolation methods we develop are of interest on their own and could have applications to related inequalities.     

Traces of functions in Hardy and Besov spaces on Lipschitz domains with applications to compensated compactness and the theory of Hardy and Bergman type spaces

In this paper we study conditions guaranteeing that functions defined on a Lipschitz domain Ω have boundary traces in Hardy and Besov spaces on ∂Ω. In turn these results are used to develop a new approach to the theory of compensated compactness and the theory of non-locally convex Hardy and Bergman type spaces.     

Maximum principles for weak solutions of degenerate elliptic equations with a uniformly elliptic direction

For second order linear equations and inequalities which are degenerate elliptic but which possess a uniformly elliptic direction, we formulate and prove weak maximum principles which are compatible with a solvability theory in suitably weighted versions of L²-based Sobolev spaces. The operators are not necessarily in divergence form, have terms of lower order, and have low regularity assumptions on the coefficients. The needed weighted Sobolev spaces are, in general, anisotropic spaces defined by a non-negative continuous matrix weight. As preparation, we prove a Poincaré inequality with respect to such matrix weights and analyze the elementary properties of the weighted spaces. Comparisons to known results and examples of operators which are elliptic away from a hyperplane of arbitrary codimension are given. Finally, in the important special case of operators whose principal part is of Grushin type, we apply these results to obtain some spectral theory results such as the existence of a principal eigenvalue.     

A distance between orbits that controls commutator estimates and invertibility of operators

A distance between orbit spaces generated by a single element is introduced and it is shown that if an operator is invertible in one orbit it is also invertible in nearby orbits, thus proving a version of Shneiberg's theorem for orbital methods. The same machinery is used to extend the celebrated Rochberg-Weiss commutator theorem to the setting of orbital methods. It is shown that these results apply to the real and complex methods of interpolation by proving that these methods can be suitably obtained as orbits of a single element.     

Higher-order Sobolev embeddings and isoperimetric inequalities

Optimal higher-order Sobolev type embeddings are shown to follow via isoperimetric inequalities. This establishes a higher-order analogue of a well-known link between first-order Sobolev embeddings and isoperimetric inequalities. Sobolev type inequalities of any order, involving arbitrary rearrangement-invariant norms, on open sets in Rⁿ${\mathbb{R}}^n$, possibly endowed with a measure density, are reduced to much simpler one-dimensional inequalities for suitable integral operators depending on the isoperimetric function of the relevant sets. As a consequence, the optimal target space in the relevant Sobolev embeddings can be determined both in standard and in non-standard classes of function spaces and underlying measure spaces. In particular, our results are applied to any-order Sobolev embeddings in regular (John) domains of the Euclidean space, in Maz'ya classes of (possibly irregular) Euclidean domains described in terms of their isoperimetric function, and in families of product probability spaces, of which the Gauss space is a classical instance.     

Nonlinear degenerate elliptic equation with Hardy potential and critical parameter

Some embedding inequalities in Hardy–Sobolev spaces with general weight functions were proved, and a positive answer to an open problem raised by Brezis–Vázquez was given. In the weighted Hardy–Sobolev spaces, the existence of nontrivial (many) solutions to the corresponding nonlinear degenerated elliptic equations with Hardy potential and critical parameter under conditions weaker than Ambrosetti–Rabinowitz condition, was obtained.     

Pointwise behaviour of <Emphasis Type="Italic">M</Emphasis><Superscript>1,1</Superscript> Sobolev functions

Our main objective is to study Haj?asz type Sobolev functions with the exponent one on metric measure spaces equipped with a doubling measure. We show that a discrete maximal function is bounded in the Haj?asz space with the exponent one. This implies that every such function has Lebesgue points outside a set of capacity zero. We also show that every Haj?asz function coincides with a Hölder continuous Haj?asz function outside a set of small Hausdorff content. Our proofs are based on Sobolev space estimates for maximal functions.