A priori bounds and a Liouville theorem on a half-space for higher-order elliptic Dirichlet problems

We consider the 2m-th order elliptic boundary value problem Lu = f (x, u) on a bounded smooth domain with Dirichlet boundary conditions on ∂Ω. The operator L is a uniformly elliptic operator of order 2m given by . For the nonlinearity we assume that , where are positive functions and q > 1 if N ≤ 2m, if N > 2m. We prove a priori bounds, i.e, we show that for every solution u, where C > 0 is a constant. The solutions are allowed to be sign-changing. The proof is done by a blow-up argument which relies on the following new Liouville-type theorem on a half-space: if u is a classical, bounded, non-negative solution of ( − Δ) ^m u  =  u ^q in with Dirichlet boundary conditions on and q > 1 if N ≤ 2m, if N > 2m then .      

Distances to spaces of Baire one functions

Given a metric space X and a Banach space (E, ||·||) we use an index of σ-fragmentability for maps to estimate the distance of f to the space B ₁(X, E) of Baire one functions from X into (E, ||·||). When X is Polish we use our estimations for these distances to give a quantitative version of the well known Rosenthal’s result stating that in the pointwise relatively countably compact sets are pointwise relatively compact. We also obtain a quantitative version of a Srivatsa’s result that states that whenever X is metric any weakly continuous function belongs to B ₁(X, E): our result here says that for an arbitrary we have

Inequalities of the Brunn–Minkowski type for Gaussian measures

Let be an integer, let γ be the standard Gaussian measure on , and let . Given this paper gives a necessary and sufficient condition such that the inequality is true for all Borel sets A ₁,...,A _m in of strictly positive γ-measure or all convex Borel sets A ₁,...,A _m in of strictly positive γ-measure, respectively. In particular, the paper exhibits inequalities of the Brunn–Minkowski type for γ which are true for all convex sets but not for all measurable sets.      

Analytic properties in the spectrum of certain Banach algebras

We show a sufficient condition for a domain in to be a H ^∞-domain of holomorphy. Furthermore if a domain has the Gleason property at a point and the projection of the n − 1th order generalized Shilov boundary does not coincide with Ω then is schlicht. We also give two examples of pseudoconvex domains in which the spectrum is non-schlicht and satisfy several other interesting properties.      

Mean-value theorems and extensions of the Elliott–Daboussi theorem on Beurling’s generalized integers (I)

Let be the generalized integers n _j associated with a set of generalized primes p _i in Beurling’s sense. On the basis of the general mean-value theorems, established in our previous work, for multiplicative function f(n _j ) defined on , we prove extensions, in functional form and in mean-value form, of the Elliott–Daboussi theorem to high order mean-values. For the main result, let α,ρ, and τ be positive real constants such that α > 1,ρ≥1 and . Then a multiplicative function f satisfies the following conditions, with some constant , (1) All four series

Weights of twisted exponential sums

Let k be a finite field of characteristic p, l a prime number different from p, a nontrivial additive character, and a character on . Then ψ defines an Artin-Schreier sheaf on the affine line , and χ defines a Kummer sheaf on the n-dimensional torus . Let be a Laurent polynomial. It defines a k-morphism . In this paper, we calculate the weights of under some non-degeneracy conditions on f. Our results can be used to estimate sums of the form

Gaps in the differential forms spectrum on cyclic coverings

We are interested in the spectrum of the Hodge–de Rham operator on a -covering X over a compact manifold M of dimension n + 1. Let Σ be a hypersurface in M which does not disconnect M and such that M − Σ is a fundamental domain of the covering. If the cohomology group H ^n/2(Σ) is trivial, we can construct for each a metric g = g _N on M, such that the Hodge–de Rham operator on the covering (X, g) has at least N gaps in its (essential) spectrum. If , the same statement holds true for the Hodge–de Rham operators on p-forms provided .     

Norm-preserving extensions of functionals and denting points of convex sets

Let W and Z be Banach spaces, and let and be closed subspaces. Let be a subspace of , the Banach space of bounded linear operators from W* to Z**, containing . We describe, for and , all norm-preserving extensions of to the space in terms of convergence of convex combinations. We also characterize denting points of bounded convex subsets of Banach spaces in similar terms. Various applications are presented. Supported by Estonian Science Foundation Grant 5704.     

<Emphasis Type="Italic">L</Emphasis><Superscript>∞</Superscript>-estimates for Sobolev functions in terms of ∇ and Δ

Several L ^∞-estimates are obtained for in terms of and , where are determined by m. If p = 2, then the estimates are given with explicit constants. However, if p ≠ 2, it is difficult to derive explicit constants except in two simple cases. Applicability to PDE’s is illustrated.     

Nearly tight frames and space-frequency analysis on compact manifolds

Let M be a smooth compact oriented Riemannian manifold of dimension n without boundary, and let Δ be the Laplace–Beltrami operator on M. Say , and that f (0)  =  0. For t  >  0, let K _t (x, y) denote the kernel of f (t ² Δ). Suppose f satisfies Daubechies’ criterion, and b  >  0. For each j, write M as a disjoint union of measurable sets E _j,k with diameter at most ba ^j , and measure comparable to if ba ^j is sufficiently small. Take x _j,k ∈ E _j,k . We then show that the functions form a frame for (I  −  P)L ²(M), for b sufficiently small (here P is the projection onto the constant functions). Moreover, we show that the ratio of the frame bounds approaches 1 nearly quadratically as the dilation parameter approaches 1, so that the frame quickly becomes nearly tight (for b sufficiently small). Moreover, based upon how well-localized a function F ∈ (I  −  P)L ² is in space and in frequency, we can describe which terms in the summation are so small that they can be neglected. If n  =  2 and M is the torus or the sphere, and f (s)  =  se ^−s (the “Mexican hat” situation), we obtain two explicit approximate formulas for the φ _j,k , one to be used when t is large, and one to be used when t is small. A. Mayeli was partially supported by the Marie Curie Excellence Team Grant MEXT-CT-2004-013477, Acronym MAMEBIA.     

$$(O( V \oplus F), O( V))$$ is a Gelfand pair for any quadratic space V over a local field F

Let V be a quadratic space with a form q over an arbitrary local field F of characteristic different from 2. Let with the form Q extending q with Q(e) = 1. Consider the standard embedding and the two-sided action of on . In this note we show that any -invariant distribution on is invariant with respect to transposition. This result was earlier proven in a bit different form in van Dijk (Math Z 193:581–593, 1986) for , in Aparicio and van Dijk (Complex generalized Gelfand pairs. Tambov University, 2006) for and in Bosman and van Dijk (Geometriae Dedicata 50:261–282, 1994) for p-adic fields. Here we give a different proof. Using results from Aizenbud et al. (arXiv:0709.1273 (math.RT), submitted), we show that this result on invariant distributions implies that the pair (O(V), O(W)) is a Gelfand pair. In the archimedean setting this means that for any irreducible admissible smooth Fréchet representation (π, E) of we have A stronger result for p-adic fields is obtained in Aizenbud et al. (arXiv:0709.4215 (math.RT), submitted).     

On the Multiplicity of Nodal Solutions to a Singularly Perturbed Neumann Problem

We consider the problem

Positive-definite functions on infinite-dimensional groups

This paper concerns positive-definite functions on infinite-dimensional groups G. Our main results are as follows: first, we claim that if G has a σ-finite measure μ on the Borel field whose right admissible shifts form a dense subgroup G ₀, a unique (up to equivalence) unitary representation (H, T) with a cyclic vector corresponds to through a method similar to that used for the G–N–S construction. Second, we show that the result remains true, even if we go to the inductive limits of such groups, and we derive two kinds of theorems, those taking either G or G ₀ as a central object. Finally, we proceed to an important example of infinite-dimensional groups, the group of diffeomorphisms on smooth manifolds M, and see that the correspondence between positive-definite functions and unitary representations holds for under a fairy mild condition. For a technical reason, we impose condition (c) in Sect. 2 on the measure space throughout this paper. It is also a weak condition, and it is satified, if G is separable, or if μ is Radon. This research was partially supported by a Grant-in-Aid for Scientific Research (No.18540184), Japan Socieity of the Promotion of Science.     

Domestic canonical algebras and simple Lie algebras

For each simply-laced Dynkin graph Δ we realize the simple complex Lie algebra of type Δ as a quotient algebra of the complex degenerate composition Lie algebra of a domestic canonical algebra A of type Δ by some ideal I of that is defined via the Hall algebra of A, and give an explicit form of I. Moreover, we show that each root space of has a basis given by the coset of an indecomposable A-module M with root easily computed by the dimension vector of M. Dedicated to Professor Claus Michael Ringel on the occasion of his 60th birthday.     

Projective curves of degree=codimension+2

In this article we study nondegenerate projective curves of degree d which are not arithmetically Cohen-Macaulay. Note that for a rational normal curve and a point . Our main result is about the relation between the geometric properties of X and the position of P with respect to . We show that the graded Betti numbers of X are uniquely determined by the rank of P with respect to . In particular, X satisfies property N _2,p if and only if . Therefore property N _2,p of X is controlled by and conversely can be read off from the minimal free resolution of X. This result provides a non-linearly normal example for which the converse to Theorem 1.1 in (Eisenbud et al., Compositio Math 141:1460–1478, 2005) holds. Also our result implies that for nondegenerate projective curves of degree d which are not arithmetically Cohen–Macaulay, there are exactly distinct Betti tables.     

Calderón–Zygmund estimates for higher order systems with p(x) growth

For weak solutions of higher order systems of the type , for all , with variable growth exponent p : Ω → (1,∞) we prove that if with , then . We should note that we prove this implication both in the non-degenerate (μ > 0) and in the degenerate case (μ = 0).     

On <Emphasis Type="Italic">Γ</Emphasis>-convergence of quadratic functionals in the plane

In this paper, we prove that if a sequence of homeomorphisms , with bounded planar domains, of Sobolev space has uniformly equibounded distortions in EXP(Ω) and weakly converges to f in then the matrices A(x, f _j ) of the corresponding Laplace-Beltrami operators Γ-converge in the Orlicz–Sobolev space , where Q(t) = t ²log(e + t), to the matrix A(x, f) of the Laplace-Beltrami operator associated to f.      

Singular Integrals and Potential Theory

In this paper I consider a class of non-standard singular integrals motivated by potential theoretic and probabilistic considerations. The probabilistic applications, which are by far the most interesting part of this circle of ideas, are only outlined in Section 1.5: They give the best approximation of the solution of the classical Dirichlet problem in a Lipschitz domain by the corresponding solution by finite differences. The potential theoretic estimate needed for this gives rise to a natural duality between the L _p functions on the boundary ∂Ω and a class of functions A on Ω that was first considered by Dahlberg. The actual duality is given by ∫_Ω S f(x)A(x)dx = (f, A) where S f(x) = ∫_∂Ω |x − y|¹⁻ⁿ f(y)dy is the Newtonian potential. We can identify the upper half Lipschitz space with in the obvious way and express for an appropriate kernel K. It is the boundedness properties of the above (for , ) that is the essential part of this work. This relates with more classical (but still “rough”) singular integrals that have been considered by Christ and Journé. Lecture held in the Seminario Matematico e Fisico on March 14, 2005 Received: April 2007     

Efficient convolution with the Newton potential in <Emphasis Type="Italic">d</Emphasis> dimensions

The paper is concerned with the evaluation of the convolution integral in d dimensions (usually d = 3), when f is given as piecewise polynomial of possibly large degree, i.e., f may be considered as an hp-finite element function. The underlying grid is locally refined using various levels of dyadically organised grids. The result of the convolution is approximated in the same kind of mesh. If f is given in tensor product form, the d-dimensional convolution can be reduced to one-dimensional convolutions. Although the details are given for the kernel the basis techniques can be generalised to homogeneous kernels, e.g., the fundamental solution of the d-dimensional Poisson equation.