Best Constants for Uncentred Maximal Functions

We precisely evaluate the operator norm of the uncentred Hardy–Littlewoodmaximal function on L^p(R¹). Consequently, we compute the operatornorm of the ‘strong’ maximal function on L^p(Rⁿ),and we observe that the operator norm of the uncentred Hardy–Littlewoodmaximal function over balls on L^p(Rⁿ) grows exponentially asn. 1991 Mathematics Subject Classification 42B25.     

Singular Integrals and Maximal Functions Associated to Surfaces of Revolution

We obtain L^p estimates for singular integrals and maximal functionsassociated to hypersurfaces in Rⁿ⁺¹, n 2, which are obtainedby rotating a curve around one of the coordinate axes.     

A Relation between Hausdorff Dimension and a Condition on Time Sets for the Image by the Brownian Sheet to Possess Interior Points

We prove that if W_{N, d} is a Brownian sheet mapping to R^d and E is a set in (0, )^N of Hausdorff dimensiongreater than , then for almost every rotation about a point x and translation _x such that _x(E) (0, )^N, the set _x(E) is such that almost surely W(E) containsinterior points. The techniques are adapted from Kahane andRosen and generalize to higher dimensional time and range.     

Natural Tiling, Lattice Tiling and Lebesgue Measure of Integral Self-Affine Tiles

In the existing theory of self-affine tiles, one knows thatthe Lebesgue measure of any integral self-affine tile correspondingto a standard digit set must be a positive integer and everyintegral self-affine tile admits some lattice Zⁿ as a translationtiling set of Rⁿ. In this paper, we give algorithms to evaluatethe Lebesgue measure of any such integral self-affine tile Kand to determine all of the lattice tilings of Rⁿ by K. Moreover,we also propose and determine algorithmically another type oftranslation tiling of Rⁿ by K, which we call natural tiling.We also provide an algorithm to decide whether or not Lebesguemeasure of the set K (K+j), j–Zⁿ, is strictly positive.     

On the Topology of Singularities of the Set of Supporting Hyperplanes of a Smooth Submanifold in an Affine Space

Many new universal relations are obtained between the Eulernumbers of manifolds of singular supporting hyperplanes of anarbitrary generic smooth closed k-dimensional submanifold inRⁿ where n 7 or k = 1. These relations are applied to Barner-convexcurves in an odd-dimensional space Rⁿ. A universal (nontrivial)linear relation is established between the numbers of singularsupporting hyperplanes of various types but of the same totalmultiplicity n of tangency with a given generic smooth closedconnected Barner-convex curve in Rⁿ. The coefficients of thisrelation are defined by Catalan numbers.     

Kato Class Potentials for Higher Order Elliptic Operators

Our goal in this paper is to determine conditions on a potentialV which ensure that an operator such as H:=(–)^m+V (1) acting on L²(R^N) defines a semigroup in L^p(R^N) for various valuesof p including p=1. The operator is defined as a quadratic formsum. That is, we put for (all integrals are on R^N and are with respect to Lebesgue measure), and note thatthe closure of the form is non-negative and has domain equalto the Sobolev space W^m,2. We then assume that the potentialhas quadratic form bound less than 1 with respect to Q₀, anddefine This form is closed and is associated with a semibounded self-adjointoperator H in L² (see [17, p. 348; 5, Theorem 4.23]). One canthen ask whether the semigroup e^–Ht defined on L² fort0 is extendable to a strongly continuous one-parameter semigroupon L^p for other values of p, and if so whether one can describethe domain and spectrum of its generator.     

Packing, Tiling, Orthogonality and Completeness

Let R^d be an open set of measure 1. An open set DR^d is calleda ‘tight orthogonal packing region’ for if D–Ddoes not intersect the zeros of the Fourier transform of theindicator function of , and D has measure 1. Suppose that isa discrete subset of R^d. The main contribution of this paperis a new way of proving the following result: D tiles R^d whentranslated at the locations if and only if the set of exponentialsE = {exp 2i, x: } is an orthonormal basis for L²(). (This resulthas been proved by different methods by Lagarias, Reeds andWang [9] and, in the case of being the cube, by Iosevich andPedersen [3]. When is the unit cube in R^d, it is a tight orthogonalpacking region of itself.) In our approach, orthogonality ofE is viewed as a statement about ‘packing’ R^d withtranslates of a certain non-negative function and, additionally,we have completeness of E in L²() if and only if the above-mentionedpacking is in fact a tiling. We then formulate the tiling conditionin Fourier analytic language, and use this to prove our result.2000 Mathematics Subject Classification 52C22, 42B99, 11K70.     

Shapiro's Cyclic Sum

Shapiro's cyclic sum is defined by , If K is the cone in Rⁿ of points withnon-negative coordinates, it is shown that the minimum of Ein K is a fixed point of T², where T is the non-linear operatordefined by (Tx)_i = x_n–i+1/(x_n–i+2 + x_n–i+3)²for i = 1,2,...,n. It is conjectured that Tx = S^kx, where Sis the shift operator in Rⁿ, and a proof is given under someadditional hypotheses. One of the consequences is a simple proofthat at the minimum point, a_i(x) = a_n–i+1–k(x) fori = 1,2,...,n.     

The Nemitskii Operator in Holder Spaces: Some Necessary and Sufficient Conditions

The Nemitskii operator in the Hölder spaces C^0, () with an open bounded subset of Rⁿ, is studied; necessary and sufficientconditions are given for boundedness, uniform continuity, anduniformly continuous differentiability on bounded sets.     

Existence of Solutions To Weak Parabolic Equations For Measures

Let A = (a^ij) be a Borel mapping on [0, 1] x R^d with valuesin the space of non-negative operators on R^d and let b = (bⁱ)be a Borel mapping on [0, 1] x R^d with values in R^d. Let Under broad assumptions on A and b, we construct a family µ= (µ_t)_t [0, 1] of probability measures µ_t on R^dwhich solvesthe Cauchy problem L* µ = 0 with initial conditionµ₀ = , where \nu is a probability measure on R^d, in thefollowing weak sense: and Such an equation is satisfied by transition probabilities ofa diffusion process associated with A and b provided such aprocess exists. However, we do not assume the existence of aprocess and allow quite singular coefficients, in particular,b may be locally unbounded or A may be degenerate. An infinite-dimensionalanalogue is discussed as well. Main methods are L^p-analysiswith respect to suitably chosen measures and reduction to theelliptic case (studied previously) by piecewise constant approximationsin time. 2000 Mathematics Subject Classification 35K10, 35K12,60J35, 60J60, 47D07.     

Essential norms and localization moduli of Sobolev embeddings for general domains

We introduce new measures of non-compactness for the embeddingoperator E_p,q():L_p¹() L_q() and describe their relations withthe essential norm of E_{p, q}(), ‘local’ isoperimetricand isocapacitary constants. An explicit formula for the essentialnorm of E_{p, q}() is obtained for domains with a power cusp onthe boundary and bounded C¹ domains. The Neumann problem fora particular Schrödinger operator is discussed on domainswith a power cusp.     

Interpolation by Polynomials in z and z-1 on an Annulus

A polynomial of degree n in z^–1 and n–1 in z isdefined by an interpolation projection from the space of functionsanalytic in the annulus r|z|R and continuous on its boundary.The points of interpolation are chosen to coincide with then roots of zⁿ=Rⁿeⁱⁿ (0<<2/n) and the n roots of zⁿ=rⁿ.The behaviour of the corresponding Lebesgue function is studied,and an estimate for the operator norm is obtained. The resultsof the present paper give a partial affirmative answer to twoconjectures suggested earlier by Mason on the basis of numericalcomputations.     

The Baire Method for the Prescribed Singular Values Problem

The paper investigates the vectorial Dirichlet problem definedby S_j( u(x))=1,xnO; a.e., j=1,...,n, u(x)=\(x),\,x\in|O. end{cases}Here O is an open bounded subset of Rⁿ with boundary |O, and_j(A) (j=1,...,n) denote the singular values of the gradient u(x). The existence of solutions is established under one ofthe following assumptions: : O – Rⁿ is continuous on Oand locally contractive on O, or : |O – Rⁿ is contractiveon |O. This extends a result due to Dacorogna and Marcellini.The approach is based on the Baire category method developedearlier by the authors.     

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.     

A Remez-Type Theorem for Homogeneous Polynomials

Remez-type inequalities provide upper bounds for the uniformnorms of polynomials p on given compact sets K, provided that|p(x)| 1 for every x K\E, where E is a subset of K of smallmeasure. In this paper we prove sharp Remez-type inequalitiesfor homogeneous polynomials on star-like surfaces in R^d. Inparticular, this covers the case of spherical polynomials (whend = 2 we deduce a result of Erdélyi for univariate trigonometricpolynomials).     

Intersections of Symbolic Powers of Prime Ideals

Let (R,m) be a local ring with prime ideals p and q such that. If R is regular and containsa field, and dim(R/p)+dim(R/q)=dim(R), then it is proved thatp^(m) q⁽ⁿ⁾ m^m+n for all positive integers m and n. This isproved using a generalization of Serre's Intersection Theoremwhich is applied to a hypersurface R/fR. The generalizationgives conditions that guarantee that Serre's bound on the intersectiondimension (R/p)+(R/q)dim(R) holds when R is nonregular.     

K-Theory for Algebras of Operators on Banach Spaces

It is proved that, for each pair (m, n) of non-negative integers,there is a Banach space X for which K₀(B(X))Z^m and K₁(B(X))Zⁿ.The K-groups of all closed ideals of operators contained inthe ideal of strictly singular operators are computed, and someresults about the existence of splittings of certain short exactsequences are derived.     

A Note on Two-Weight Inequalities for Maximal Functions and Singular Integrals

We deal with weighted inequalities of the type [formula] where: T is either the Hardy–Littlewood maximal operator or asingular integral operator; G is any measurable subset of Rⁿ; f is any measurable function, vanishing outside G, such thatTf is well-defined; v and w are weights, that is, nonnegativelocally integrable functions on G; p, q(1, ). 1991 Mathematics Subject Classification 42B20, 42B25.     

On the Boundedness and Compactness of a Class of Integral Operators

Let > 0. The operator of the form is considered, where the real weight function v(x) is locallyintegrable on R⁺ := (0, ). In case v(x) = 1 the operator coincideswith the Riemann–Liouville fractional integral, L^p L^qestimates of which with power weights are well known. This workgives L^p L^qboundedness and compactness criteria for the operatorT in the case 0 < p, q < , p > max(1/, 1).     

Choquet Integrals, Hausdorff Content and the Hardy-Littlewood Maximal Operator

Using the BMO-H¹ duality (among other things), D. R. Adams provedin [1] the strong type inequality whereC is some positive constant independent of f. Here M is theHardy–Littlewood maximal operator in Rⁿ, H is the -dimensionalHausdorff content, and the integrals are taken in the Choquetsense. The Choquet integral of 0 with respect to a set functionC is defined by Precise definitionsof M and H will be given below. For an application of (1) tothe Sobolev space W^{1, 1} (Rⁿ), see [1, p. 114]. The purpose of this note is to provide a self-contained, directproof of a result more general than (1). 1991 Mathematics SubjectClassification 28A12, 28A25, 42B25.