Limiting weak-type behavior for singular integral and maximal operators

The following limit result holds for the weak-type (1,1) constant of dilation-commuting singular integral operator in : for , ,

For the maximal operator , the corresponding result is

     

Limiting weak-type behaviors for Riesz transforms and maximal operators in Bessel setting

We establish the limiting weak type behaviors of Riesz transforms associated to the Bessel operators on IR+. which are closely related to the best constants of the weak type (1,1) estimates for such operators. Meanwhile, the corresponding results for Hardy-Littlewood maximal operator and fractional maximal operator in Bessel setting are also obtained.     

Two-weight weak-type inequalities for potential type operators

For the potential type operator
TФf（x）=∫RnФ（x-y）f（y）dy,
where Ф is a non-negative locally integrable function on R^n and satisfies weak growth condition, a two-weight weak-type （p,q） inequality for TФ is obtained.     

A note on the existence of zeroes of convexly regularized sums of maximal monotone operators

Many algorithms for solving the problem of finding zeroes of a sum of two maximal monotone operators T₁ and T₂, have regularized subproblems of the kind 0T₁(x)+T₂(x)+∂D(x), where D is a convex function. We develop an unified analysis for existence of solutions of these subproblems, through the introduction of the concept of convex regularization, which includes several well-known cases in the literature. Finally, we establish conditions, either on D or on the operators, which assure solvability of the subproblems.     

A note on flux limiting for diffusion discretizations

In this note, a limiting technique is presented to enforcemonotonicity for higher-order spatial diffusion discretizations.The aim is to avoid spurious oscillations and to improve thequalitative behaviour on coarse grids. The technique is relatedto known ones for convection equations, using limiters to boundthe numerical fluxes. Applications arise in pattern formationproblems for reaction–diffusion equations.     

A note on limiting distributions for spacings statistics

Summary Hájek's projection method is used to prove asymptotic normality for a class of spacings statistics.The author thanks Dr. John Petkau for useful discussions.     

A note on maximal estimates for stochastic convolutions

In stochastic partial differential equations it is important to have pathwise regularity properties of stochastic convolutions. In this note we present a new sufficient condition for the pathwise continuity of stochastic convolutions in Banach spaces.     

A note on limiting infisup theorems

We show that the limiting infisup theorem of Blair, Duffin and Jeroslow (1982) is a consequence of the classical bifunctional duality. By doing so we generalize their results and prove another limiting infisup theorem for convex quasi-concave functions.     

A note on restricted weak-type estimates for Bochner-Riesz operators with negative index in

It is shown that the Bochner-Riesz operator on of negative order is of restricted weak type in the critical points and , where , for in the two-dimensional case and , for if .

     

A note on meromorphic operators

Let be a complex Banach space and a bounded linear operator on . is called meromorphic if the spectrum of is a countable set, with the only possible point of accumulation, such that all the nonzero points of are poles of . By means of the analytical core we give a spectral theory of meromorphic operators. Our results are a generalization of some results obtained by Gong and Wang (2003).

     

A note on nice operators

In this note, we characterize nice operators in a class of Banach spaces, which includes spaces and L₁(μ), as those operators that preserve extreme points.     

A note on maximal progression-free sets

Erd?s et al [Greedy algorithm, arithmetic progressions, subset sums and divisibility, Discrete Math. 200 (1999) 119-135.] asked whether there exists a maximal set of positive integers containing no three-term arithmetic progression and such that the difference of its adjacent elements approaches infinity. This note answers the question affirmatively by presenting such a set in which the difference of adjacent elements is strictly increasing. The construction generalizes to arithmetic progressions of any finite length.     

A note on maximal triangle-free graphs

We show that a maximal triangle-free graph on n vertices with minimum degree δ contains an independent set of 3δ ? n vertices which have identical neighborhoods. This yields a simple proof that if the binding number of a graph is at least 3/2 then it has a triangle. This was conjectured originally by Woodall. © 1993 John Wiley & Sons, Inc.     

A limiting weak type estimate for capacitary maximal function

A capacitary analogue of the limiting weak type estimate of P. Janakiraman for the Hardy–Littlewood maximal function of an L¹(Rⁿ)$L^1({\mathbb{R}}^n)$-function (cf.  and ) is discovered.     

Weighted weak-type inequalities for the maximal function of nonnegative integral transforms over approach regions

The relation between approach regions and singularities of nonnegative kernels is studied, where , , , and is a homogeneous space. For , a sufficient condition on approach regions () is given so that the maximal function

is weak-type with respect to a pair of measures and . It is shown that this condition is also necessary for operators of potential type in the sense of Sawyer and Wheedon (Amer. J. Math. 114 (1992), 813-874).

     

A note on the functional calculus for unbounded self-adjoint operators

Two theorems of Riesz and Lorch (1936) are used to pass directly from the functional calculus for bounded symmetric operators to that for unbounded self-adjoint operators, thereby considerably shortening the passage via the spectral resolution for unbounded self-adjoint operators, and making particularly transparent the manner in which properties of the functional calculus for bounded operators are inherited by those which are unbounded.