Rm Sup Research Articles

Examples of incomplete normed barrelled spaces

In this note we investigate spaces of the type \( L_{\varepsilon}^{p}(\mu)=\lbrace f\in L^{p}(\mu);{\rm supp}f\in \varepsilon \rbrace \) where ε is an ideal of “small” measurable sets with certain properties. Typically, these spaces endowed with the p-norm are not complete and thus, classical Banach space theory cannot be used.However, we prove that for good ideals ε the normed space \(L_\varepsilon ^{p}(\mu)\) is ultrabornological and hence barrelled and therefore many theorems of functional analysis like the closed graph theorem or the uniform boundedness principle are indeed applicable.

On Nehari disks and the inner radius

Let D be a simply connected plane domain and B the unit disk. The inner radius of D, \sigma (D) , is defined by \sigma ({\rm D}) = {\rm sup}\left\{a: a \geq 0, \Vert{\rm S}_{f}\Vert_{\rm D} \le a\,{\rm implies}\,f\, {\rm is\,univalent\,in\,D} \right\} . Here Sf is the Schwarzian derivative of f, \rho_{\rm D} the hyperbolic density on D and \Vert{\rm S}_{f}\Vert_{\rm D} = {\rm sup}_{z \in {\rm D}}\mid {\rm S}_{f}(z)\mid \rho_{\rm D}^{-2} (z) . Domains for which the value of \sigma {\rm (D)} is known include disks, angular sectors and regular polygons, as well as certain classes of rectangles and equiangular hexagons. All of the mentioned domains except non-convex angular sectors have an interesting property in common, namely that \sigma {\rm (D)} = 2 - \Vert{\rm S}_h \Vert_{\rm B} , where h maps B conformally onto D. Because of the importance of this property for computing \sigma {\rm(D)} , we say that D is a Nehari disk if \sigma {\rm (D)} = 2 - \Vert{\rm S}_h \Vert_{\rm B} holds.¶This paper is devoted to the problem of characterizing Nehari disks. We give a necessary and sufficient condition for a domain to be a Nehari disk provided it is a regulated domain with convex corners.

All time C∞-regularity of the interface in degenerate diffusion: a geometric approach

We study the connection between the geometry and all time regularity of the interface in degenerated diffusion. Our model considers the porous medium equation ut=Δum, m>1, with initial data u0 nonnegative, integrable, and compactly supported. We show that if the initial pressure f0=u0m− is smooth up to the interface and in addition it is root-concave and also satisfies the nondegeneracy condition |Df0|≠0 at $\partial\overline {\rm supp}$f0, then the pressure fm−1 remains C∞-smooth up to the interface and root-concave, for all time $0 < t < ∞$. In particular, the free boundary is C∞-smooth for all time.

Approximating Planar Rotations

We study the problem of approximating a rotation of the plane, α :R 2 $ \rightarrow $ R 2 , α (x,y)=(x cos θ + y sin θ , y cos θ - x sin θ ), by a bijection β :Z 2 $\rightarrow$ Z 2 . We show by an explicit construction that β may be chosen so that $ {\rm sup}_{z \in Z^2} |\alpha (z)-\beta (z)| \leq ({1}/{\sqrt{2}}) (({1+r})/{\sqrt{1+r^2}}) $ , where r= tan ( θ /2). The scheme is based on those invented and patented by the second author in 1994.

Numerical ranges of elements of involutive Banach algebras and commutativity

Let $ {\cal A} $ be a unital involutive Banach algebra. Bonsall and Duncan defined the numerical range for each element x in $ {\cal A} $ by $ V(x) = \{f(x):f \in {\cal A'}, f(e) = 1 =|| f || \} $ , where e is the unit. We introduce another numerical range by $ W (x) = \{ f(x) : f \in {\cal A'}, f \geq 0, f(e) = 1 \} $ , and we call $ w(x) = {\rm sup} \{|z|: z \in W (x) \} $ the numerical radius of x. We give a few conditions for $ {\cal A} $ to be a C *-algebra, and we see that some mapping theorems for numerical ranges of elements of $ {\cal A} $ hold. We show that if w (x) ≤ 1 implies $ w (\varphi (x) ) \leq 1 $ for every Mobius transform $ \varphi $ of the unit disk, then $ {\cal A} $ is commutative.

Periodic Korteweg de Vries equation with measures as initial data

The main result of the paper is that the periodic KdV equation $y_t + \partial^3_x y + yy_x = 0$ has a unique global solution for initial data y(0) given by a measure $\mu\in M({\Bbb T})$ of sufficiently small norm $\parallel\mu\parallel$ . There are two main ingredients in the proof. The first is the study of the local well-posedness problem in terms of the space-time Fourier-norms as exploited in [Bo] and also subsequent work such as [K-P-V2]. At the end the estimates eventually depend on a uniform estimate in terms of the Fourier coefficients¶¶ $ {{\rm sup}\atop{n\in{\Bbb Z},\,t\in{\Bbb R}}}|\hat{y}(n)(t)| < C .$ .¶¶Such a priori bound (in the space of pseudo-measures) on the solution may be derived from spectral theory and more precisely from the preservation of the periodic spectrum of a potential evolving according to KdV, which is the second ingredient. Thus the result at this stage depends heavily on integrability features of this particular equation. We also sketch an argument establishing almost periodicity properties of these solutions. This work is in spirit closely related to [Bo]. Natural questions suggested by these investigations is an extension of the result (at least for the IVP local in time) to a more general nonintegrable setting as well as to what extent the estimates on Fourier coefficients by spectral invariants and vice versa remains valid in distributional spaces.

Characterizations of character sheaves for complex reductive algebraic groups

Character sheaves are geometric objects associated to a reductive algebraic group. They are closely related to irreducible characters of finite Chevalley groups. The existence of character sheaves was conjectured by G. Lusztig in the book Characters of Reductive Groups over Finite Fields (1984). There he described what one should expect character sheaves to be for complex reductive groups. A definition of character sheaves for an arbitrary characteristic, quite different from this description, was given by him later (1985). It is shown in this thesis, in particular, that the original (1984) Lusztig's suggestion for describing character sheaves is equivalent to the later definition. We use alternative characterizations of character sheaves for a connected complex reductive algebraic group G that are obtained in this thesis and also the one (conjectured by G. Lusztig and G. Laumon) that was obtained in 1988 by V. Ginsburg and independently by I. Mirkovic and K. Vilonen. These characterizations involve microsupport and microlocalization of a sheaf. A microsupport $SS({\cal F})$ of a complex ${\cal F}$ of sheaves on G is a closed conic subset of $T\sp*G = G\times{\bf g};$ it tells us about singularities of ${\cal F}$. A microlocalization $\mu\sb{M}{\cal F}$ of ${\cal F}$ along a submanifold M of G is a complex of sheaves on the conormal bundle $T\sbsp{M}{*}G$ which allows us to analyze ${\cal F}$ on the infinitesimally small neighbourhood of M. Theorem. Let G be a complex connected reductive algebraic group, ${\cal F}$ be a G-equivariant (for conjugation) irreducible perverse sheaf on G. Let ${\cal N}$ be the nilpotent cone of the Lie algebra g of G. S may denote either G or the set of all semisimple elements of G. The following are equivalent: (i) ${\cal F}$ is a character sheaf; (ii) $SS({\cal F})\vert\sb{S}\subset S\times{\cal N};$ (iii) supp $\mu\sb{\cal O}{\cal F}\subset{\cal O}\times{\cal N},$ where ${\cal O}\subset S$ is a conjugacy class; (iv) supp $\mu\sb{x}{\cal F}\subset{\cal N}, x\in S.$ Characterization (ii) when S = G is the one given by Ginsburg, Mirkovic and Vilonen. Characterization (iii) with S being the set of semisimple elements was conjectured by J. Bernstein (for arbitrary characteristic). This kind of characterizations essentially simplifies the development and our understanding of the theory. Currently there is still no such description for positive characteristic. An important intermediate result is the description of the microsupport of a $\doubc$-constructible sheaf ${\cal F}$ on a manifold X in terms of microlocalizations to the points of this manifold: $SS({\cal F}) = \overline{\bigcup\limits\sb{x\in X} {\rm supp} \mu\sb{x}{\cal F}}.$