Symmetrization and norm of the Hardy-Littlewood maximal operator on Lorentz and Marcinkiewicz spaces

We prove that when a function on the real line is symmetricallyrearranged, the distribution function of its uncentered Hardy–Littlewoodmaximal function increases pointwise, while it remains unchangedonly when the function is already symmetric. Equivalently, if is the maximal operator and the symmetrization, then f(x)f(x)for every x, and equality holds for all x if and only if, upto translations, f(x) = f(x) almost everywhere. Using theseresults, we then compute the exact norms of the maximal operatoracting on Lorentz and Marcinkiewicz spaces, and we determineextremal functions that realize these norms.     

Periodic Solutions of Lienard Equations with Asymmetric Nonlinearities at Resonance

The existence of 2-periodic solutions of the second-order differentialequation where a, b satisfy and p(t)=p(t+2),t R, is examined. Assume that limits lim_x±F(x)=F(±)(F(x)=) and lim_x±g(x)=g(±)exist and are finite. It is proved that the equation has atleast one 2-periodic solution provided that the zeros of thefunction ₁ are simple and the zeros of the functions ₁, ₂ aredifferent and the signs of ₂ at the zeros of ₁ in [0,2/n) donot change or change more than two times, where ₁ and ₂ aredefined as follows: Moreover, it is also proved that the given equation has at leastone 2-periodic solution provided that the following conditionshold: with 1 p < q 2.     

Weighted Integral Inequalities for the Hardy Type Operator and the FractionalMaximal Operator

Let w(x), u(x) and (x) be weight functions. In this paper, underappropriate conditions on Young's functions ₁, ₂ we characterizethe inequality for the Hardy-typeoperator T defined in [1] and the inequality for the fractional maximal operator M_{, ;} definedin [8], as well as the corresponding weak-type inequalities.     

Constructible Functions on Artin Stacks

Let be an algebraically closed field, let X be a -variety,and let X() be the set of closed points in X. A constructibleset C in X() is a finite union of subsets Y() for subvarietiesY in X. A constructible function f : X() has f(X()) finiteand f^–1(c) constructible for all c 0. Write CF(X) forthe vector space of such f. Let : X Y and : Y Z be morphismsof -varieties. MacPherson defined a linear pushforward CF(): CF(X) CF(Y) by ‘integration’ with respect tothe topological Euler characteristic. It is functorial, thatis, CF( ) = CF() CF(). This was extended to of characteristiczero by Kennedy. This paper generalizes these results to -schemes and Artin -stackswith affine stabilizer groups. We define the notions of Eulercharacteristic for constructible sets in -schemes and -stacks,and pushforwards and pullbacks of constructible functions, withfunctorial behaviour. Pushforwards and pullbacks commute inCartesian squares. We also define pseudomorphisms, a generalizationof morphisms well suited to constructible functions problems.     

A Priori Estimates and Existence of Positive Solutions for a Quasilinear Elliptic Equation

On the basis of some new Liouville theorems, under suitableconditions, a priori estimates are obtained of positive solutionsof the problem where R^N (N 2) is a bounded smooth domain, p>1 and isa parameter, , q are given constants such that p–1<<p*–1, <q, p*=Np/(N–p) if N > p and p*= when N p, and a(x) is a continuous nonnegative function. Makinguse of the Leray–Schauder degree of a compact mappingand a priori estimates, the paper finds that the problem abovepossesses at least one positive solution. It also discussesthe corresponding perturbed problem, where a(x) is replacedby a(x)+, >0. The results are strikingly different from thoseobtained for the case =p–1.     

Majorations Uniformes de Normes D'Inverses Dans Les Algebres de Beurling

The Beurling algebras l¹(D,)(D=N,Z) that are semi-simple, withcompact Gelfand transform, are considered. The paper gives anecessary and sufficient condition (on ) such that l¹(D,) possessesa uniform quantitative version of Wiener's theorem in the sensethat there exists a function :]0,+[]0,+ such that, for everyinvertible element x in the unit ball of l¹(D,), we have ||x^–1||(r(x^–1)) r(x^–1) is the spectral radiusof x^–1.     

Properties of Removable Singularities for Hardy Spaces of Analytic Functions

Removable singularities for Hardy spaces H^p() = {f Hol(): |f|^p u in for some harmonic u}, 0 < p < are studied. A setE = is a weakly removable singularity for H^p(\E) if H^p(\E) Hol(), and a strongly removable singularity for H^p(\E) if H^p(\E)= H^p(). The two types of singularities coincide for compactE, and weak removability is independent of the domain . The paper looks at differences between weak and strong removability,the domain dependence of strong removability, and when removabilityis preserved under unions. In particular, a domain and a setE that is weakly removable for all H^p, but not strongly removablefor any H^p(\E), 0 < p < , are found. It is easy to show that if E is weakly removable for H^p(\E)and q > p, then E is also weakly removable for H^q(\E). Itis shown that the corresponding implication for strong removabilityholds if and only if q/p is an integer. Finally, the theory of Hardy space capacities is extended, anda comparison is made with the similar situation for weightedBergman spaces.     

Piecewise Absolutely Continuous Cocycles Over Irrational Rotations

For an irrational rotation of the circle group T=R/Z and apiecewise absolutely continuous function f:TR, the unitary operatorVh(x)=e^2if(x)h(x+) on L²(T) is studied. It is shown that iff has a single discontinuity with non-integer jump then V is-weakly mixing for some with 0<||<1. In particular Vhas continuous singular spectrum. The property of -weak mixing(with possible change of the value of , 0<||<1) holdsfor all irrational rotations and, given , is stable under perturbationsof f by functions with sufficiently small O(1/n)-norm. On theother hand, there exists a piecewise linear function f withtwo non-integer jumps such that the spectrum of V is continuoussingular for one value of and Lebesgue for another.     

Coherent Products in a Finite Group along a Linear Ordering

Let C = (C, ) be a linear ordering, E a subset of {(x, y):x< y in C} whose transitive closure is the linear orderingC, and let :E G be a map from E to a finite group G = (G, •).We showed with M. Pouzet that, when C is countable, there isF E whose transitive closure is still C, and such that (p) = (x_o, x₁)•(x₁, x₂)•....•(x_{n– 1}, x_n) G depends only upon the extremities x₀, x_n ofp, where p = (x_o, x₁...,x_n) (with 1 n < ) is a finite sequencefor which (x_i, x_{i + 1}) F for all i < n. Here, we show thatthis property does not hold if C is the real line, but is stilltrue if C does not embed an ₁-dense linear ordering, or evena 2-dense linear ordering when Martin's Axiom holds (it followsin particular that it is independent of ZFC for linear orderingsof size ). On the other hand, we prove that this property isalways valid if E = {(x,y):x < y in C}, regardless of anyother condition on C.     

On Artin's Braid Group And Polyconvexity In The Calculus Of Variations

Let R² be a bounded Lipschitz domain and let be a Carathèodory integrand such that F(x,·) is polyconvex for L²-a.e. x . Moreover assume thatF is bounded from below and satisfies the condition as det for L²-a.e. x . The paper describes the effect of domain topologyon the existence and multiplicity of strong local minimizersof the functional wherethe map u lies in the Sobolev space W_id^1,p (, R²) with p 2and satisfies the pointwise condition u(x) >0 for L²-a.e.x . The question is settled by establishing that F[·]admits a set of strong local minimizers on that can be indexed by the group P_n Zⁿ, the directsum of Artin's pure braid group on n strings and n copies ofthe infinite cyclic group. The dependence on the domain topologyis through the number of holes n in and the different mechanismsthat give rise to such local minimizers are fully exploitedby this particular representation.     

Real Interpolation and Two Variants of Gehring's Lemma

Let be a fixed open cube in Rⁿ. For r[1, ) and [0, ) we define where Q is a cube in Rⁿ (with sides parallel to the coordinateaxes) and _Q stands for the characteristic function of the cubeQ. A well-known result of Gehring [5] states that if (1.1) for some p(1, ) and c(0, ), then there exist q(p, ) and C=C(p,q, n, c)(0, ) such that for all cubes Q, where |Q| denotes the n-dimensional Lebesguemeasure of Q. In particular, a function fL¹() satisfying (1.1)belongs to L^q(). In [9] it was shown that Gehring's result is a particular caseof a more general principle from the real method of interpolation.Roughly speaking, this principle states that if a certain reversedinequality between K-functionals holds at one point of an interpolationscale, then it holds at other nearby points of this scale. Usingan extension of Holmstedt's reiteration formulae of [4] andresults of [8] on weighted inequalities for monotone functions,we prove here two variants of this principle involving extrapolationspaces of an ordered pair of (quasi-) Banach spaces. As an applicationwe prove the following Gehring-type lemmas.     

On the Compactness and Approximation Numbers of Hardy-Type Integral Operators in Lorentz Spaces

We characterize the mapping properties such as the boundedness,compactness and measure of noncompactness for those real weightfunctions , , u0, v0, for which the Hardy-type integral operatorof the form acts from to , when the parameters are restricted to the range 1 < max (r,s) min (p, q) < and the kernel k(x, y) 0 satisfies theOinarov condition (see (2) below). For the case k(x, y) = 1,we obtain lower and upper estimates of the approximation numbers,extending the result of [5].     

On the Total Coloring of Graphs Embeddable in Surfaces

The paper shows that any graph G with the maximum degree (G) 8, which is embeddable in a surface of Euler characteristic() 0, is totally ((G)+2)-colorable. In general, it is shownthat any graph G which is embeddable in a surface and satisfiesthe maximum degree (G) (20/9) (3–())+1 is totally ((G)+2)-colorable.     

An Access Theorem for Continuous Functions

Let f be a continuous function on an open subset of R² suchthat for every x there exists a continuous map : [–1,1] with (0) = x and f increasing on [–1, 1]. Thenfor every there exists a continuous map : [0, 1) suchthat (0) = y, f is increasing on [0; 1), and for every compactsubset K of , max{t : (t) K} < 1. This result gives an answerto a question posed by M. Ortel. Furthermore, an example showsthat this result is not valid in higher dimensions.     

Support Varieties of Non-Restricted Modules over Lie Algebras of Reductive Groups

Let G be a connected semisimple group over an algebraicallyclosed field K of characteristic p>0, and g=Lie (G). Fixa linear function g* and let Z_g() denote the stabilizer of in g. Set N_p(g)={xg|x^[p]=0}. Let C(g) denote the category offinite-dimensional g-modules with p-character . In [7], Friedlanderand Parshall attached to each MOb(C(g)) a Zariski closed, conicalsubset V_g(M)N_p(g) called the support variety of M. Suppose thatG is simply connected and p is not special for G, that is, p2if G has a component of type B_n, C_n or F₄, and p3 if G has acomponent of type G₂. It is proved in this paper that, for anynonzero MOb(C(g)), the support variety V_g(M) is contained inN_p(g)Z_g(). This allows one to simplify the proof of the Kac–Weisfeilerconjecture given in [18].     

On Borel Sets in Function Spaces with the Weak Topology

It is proved that the duality map ,:(, weak)x(()*, weak*)R isnot Borel. More generally, the evaluation e:(C)(K),x KR, e(f,x) = f(x), is not Borel for any function space C(K) on a compactF-space. It is also shown that a non-coincidence of norm-Boreland weak-Borel sets in a function space does not imply thatthe duality map is non-Borel.     

Theta-characteristics on singular curves

On a smooth curve a theta-characteristic is a line bundle L,the square of which is the canonical line bundle . The equivalentcondition om(L, ) L generalizes well to singular curves, asapplications show. More precisely, a theta-characteristic isa torsion-free sheaf of rank 1 with om(, ) . If the curvehas non-ADE singularities, then there are infinitely many theta-characteristics.Therefore, theta-characteristics are distinguished by theirlocal type. The main purpose of this article is to compute thenumber of even and odd theta-characteristics (that is withh⁰(C, ) 0 and h⁰(C, ) 1 modulo 2, respectively) in terms ofthe geometric genus of the curve and certain discrete invariantsof a fixed local type.     

The Isolated Points of G{rho} and the w*-Strongly Exposed Points of P{rho}(G)0

Let G be a separable locally compact group and let be its dualspace with Fell's topology. It is well known that the set P(G)of continuous positive-definite functions on G can be identifiedwith the set of positive linear functionals on the group C^*-algebraC^*(G). We show that if is discrete in , then there exists anonzero positive-definite function associated with such that is a w^*-strongly exposed point of P(G)₀, where P(G)₀={f P(G):f(e)1. Conversely, if some nonzero positive-definite function associatedwith is a w^*-strongly exposed point of P(G)₀, then is isolatedin . Consequently, G is compact if and only if, for every ,there exists a nonzero positive-definite function associatedwith that is a w^*-strongly exposed point of P(G)₀. If, in addition,G is unimodular and , then is isolated in if and only if somenonzero positive-definite function associated with is a w^*-stronglyexposed point of P(G)₀, where is the left regular representationof G and is the reduced dual space of G. We prove that if B(G)has the Radon–Nikodym property, then the set of isolatedpoints of (so square-integrable if G is unimodular) is densein . It is also proved that if G is a separable SIN-group, thenG is amenable if and only if there exists a closed point in. In particular, for a countable discrete non-amenable groupG (for example the free group F₂ on two generators), there isno closed point in its reduced dual space .     

Some Remarks on the Cone of Completely Positive Maps between Von Neumann Algebras

The close relationship between the notions of positive formsand representations for a C*-algebra A is one of the most basicfacts in the subject. In particular the weak containment ofrepresentations is well understood in terms of positive forms:given a representation of A in a Hilbert space H and a positiveform on A, its associated representation is weakly containedin (that is, ker ker ) if and only if belongs to the weak*closure of the cone of all finite sums of coefficients of .Among the results on the subject, let us recall the followingones. Suppose that A is concretely represented in H. Then everypositive form on A is the weak* limit of forms of the typex ^k_{i=1 i}, x_i with the _i in H; moreover if A is a von Neumannsubalgebra of (H) and is normal, there exists a sequence (_i)_{i 1} in H such that (x) = _{i 1 i}, x_i for all x.     

Concavity and B-Concavity of Solutions of Quasilinear Filtration Equations

Spatial concavity properties of non-negative weak solutionsof the filtration equations with absorption u_t = ((u))_xx–(u)in Q = Rx(0, ), '0, 0 are studied. Under certain assumptionson the coefficients , it is proved that concavity of the pressurefunction is a consequence of a ‘weak’ convexityof travelling-wave solutions of the form V(x, t) = (x–t+a).It is established that the global structure of a so-called properset B = {V} of such particular solutions determines a propertyof B-concavity for more general solutions which is preservedin time. For the filtration equation u_t = ((u))_xx a semiconcavityestimate for the pressure, v_xx(t+)^–1'(), due to the B-concavityof the solution to the subset B of the explicit self-similarsolutions (x/t+)) is proved. The analysis is based on the intersection comparison based onthe Sturmian argument of the general solution u(x, t) with subsetsB of particular solutions. Also studied are other aspects ofthe B-concavity/convexity with respect to different subsetsof explicit solutions.