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We prove the equivalence of Hardy-and Sobolev-type inequalities, certain uniform bounds on the heat kernel and some spectral regularity properties of the Neumann Laplacian associated with an arbitrary region of finite measure in Euclidean space. We also prove that if one perturbs the boundary of the region within a uniform Hölder category then the eigenvalues of the Neumann Laplacian change by a small and explicitly estimated amount. Relationship between the Sobolev and Hardytype InequalitiesLet Ω ⊂ R N be a region with a finite measure |Ω|, for x ∈ Ω, d(x) be the distance of the point x from the boundary ∂Ω of Ω and, for ε > 0,The Minkowski dimension of ∂Ω relative to Ω (briefly, the Minkowski dimension of ∂Ω) is the following quantity M(∂Ω) = inf {λ > 0 : M λ (∂Ω) < ∞}, where M λ (∂Ω) = lim sup ε→0+ |∂ ε Ω| ε N −λ . Obviously M(∂Ω) ≤ N. However there exist Ω such that M λ (∂Ω) = ∞ for all λ ∈ (0, N), [9]. It can be proved that M(∂Ω) ≥ N − 1, [12]. If Ω satisfies the cone condition, then M(∂Ω) = N − 1, [12]. The proof uses that {(x, φ(x) + C} ∩ V = Ø because otherwise (x, φ(x)) ∈ V. Proof of lemma 19 1. For any subset J ⊂ {1, ..., s} put V J = {x ∈ R N : J(x) = J}, so R N is the disjoint union of allṼ J : R N = JṼ J , and Ω \ T ε (Ω) = J (Ṽ J ∩ (Ω \ T ε (Ω))).

We prove sharp stability results for the dependence of the eigenvalues of second order uniformly elliptic linear operators with homogeneous Dirichlet boundary conditions upon domain perturbation. The main results concern estimates for the variation of the eigenvalues via the Hausdorff distance between the domains or the Lebesgue measure of their symmetric difference. Our analysis includes domains with Lipschitz boundaries as well as domains with boundary degenerations of power type

Abstract. The problem of boundedness of the Hardy-Littewood maximal operator in local and global Morrey-type spaces is reduced to the problem of boundedness of the Hardy operator in weighted L p -spaces on the cone of non-negative non-increasing functions. This allows obtaining sufficient conditions for boundedness for all admissible values of the parameters. Moreover, in case of local Morrey-type spaces, for some values of the parameters, these sufficient conditions are also necessary.For x ∈ R n and r > 0, let B(x, r) denote the open ball centred at x of radius r. Definition 1. Let 0 < p, θ ≤ ∞ and let w be a non-negative measurable function on (0, ∞). We denote by LM pθ,w and GM pθ,w the local and global Morrey-type spaces respectively, defined to be the spaces of all functions f ∈ L loc p (R n ) with finite quasinormsrespectively.Lemma 1. Let 0 < p, θ ≤ ∞ and let w be a non-negative measurable function on (0, ∞).

We prove a stability theorem for the eigenvalues of general non-negative self-adjoint linear operators with compact resolvents and by applying it we prove a sharp stability result for the dependence of the eigenvalues of second order uniformly elliptic linear operators with homogeneous Neumann boundary conditions upon domain perturbation