Singular Integrals and Potential Theory (II)

In this note I give an extension of the T1-Theorem of David and Journé. In this extension the standard estimates on the kernel are replaced by “average standard estimates” but the classical proof is essentially identical. I then use that extension to prove the L^p-boundedness of some natural rough singular integrals. The applications of these singular integrals in Potential Theory has already been described in my previous paper in the area. Received: September 2008     

Spherical Singular Integrals, Monogenic Kernels and Wavelets on the Three-Dimensional Sphere

The construction of wavelets relies on translations and dilations which are perfectly given in . On the sphere translations can be considered as rotations but it is difficult to say what are dilations. For the 2-dimensional sphere there exist two different approaches to obtain wavelets which are worth to be considered. The first concept goes back to W. Freeden and collaborators who define wavelets by means of kernels of spherical singular integrals. The other concept developed by J.P. Antoine and P. Vandergheynst is a purely group theoretical approach and defines dilations as dilations in the tangent plane. Surprisingly both concepts coincides for zonal functions. We will define singular integrals and kernels of singular integrals on the three dimensional sphere which are also approximate identities. In particular the Cauchy kernel in Clifford analysis is a kernel of a singular integral, the singular Cauchy integral, and an approximate identity. Furthermore, we will define wavelets on the 3-dimensional sphere by means of kernels of singular integrals. This paper is dedicated to the memory of our friend and colleague Jarolim Bureš Received: October, 2007. Accepted: February, 2008.     

Quadrature Domains with Infinite Volume

We discuss quadrature domains for subharmonic functions and prove the existence of core quadrature domains for certain positive measures. The core quadrature domains are the smallest quadrature domains as measures and inherit good properties from quadrature domains with finite volume. We next discuss new balayage for the class of harmonic functions integrable in a neighborhood of ∞. We give several estimates of balayage measures. The new balayage is introduced to construct quadrature domains for harmonic functions. Submitted: June 26, 2008. Accepted: July 24, 2008.     

Boundedness of Commutators of Marcinkiewicz Integral with Rough Variable Kernel

In this paper the authors give the boundedness of the commutator of Marcinkiewicz integral with rough variable kernel, which is an extension of the result in [5] . Research supported by NSF of China (Grant: 10571015) and SRFDP of China (Grant: 20050027025).     

Operators with Singular Trace Conditions on a Manifold with Edges

We establish a new calculus of pseudodifferential operators on a manifold with smooth edges and study ellipticity with extra trace and potential conditions (as well as Green operators) at the edge. In contrast to the known scenario with conditions of that kind in integral form we admit in this paper ‘singular’ trace and Green operators. In contrast to standard conditions in the theory of elliptic boundary value problems (like Dirichlet or Neumann conditions) our singular trace conditions, in general, do not act on functions that are smooth up to the boundary, but admit a more general asymptotic structure. Their action is now associated with the Laurent coefficients of the meromorphic Mellin transforms of functions with respect to the half-axis variable, the distance to the edge.     

On the Norms of Singular Integral Operators on Contours with Intersections

The exact bounds are obtained for the norm of the singular integral operator S on the family of rays originating at the same point. These bounds, with the use of the localization technique, are then extended to the essential norm of S on piecewise smooth curves with finitely many points of self intersection. Submitted: April 13, 2007. Accepted: September 19, 2007.     

Multiply Connected Quadrature Domains and the Bergman Kernel Function

This paper derives general analytical formulae for the conformal maps from multiply connected circular preimage domains to multiply connected quadrature domains by considering the Bergman kernel functions of the preimage and target domains. The new formulae are expressed in terms of the Schottky–Klein prime function. They generalize, to the case of arbitrary connectivity, a formula relevant to doubly connected domains derived by Y. Avci in 1977. Submitted: September 17, 2007. Accepted: June 5, 2008.     

Young Integrals and SPDEs

In this note, we study the non-linear evolution problem $dY_t = -A Y_t dt + B(Y_t) dX_t,$ where $XIn this note, we study the non-linear evolution problem where is a -H?lder continuous function of the time parameter, with values in a distribution space, and the generator of an analytical semigroup. Then, we will give some sharp conditions on in order to solve the above equation in a function space, first in the linear case (for any value of in ), and then when satisfies some Lipschitz type conditions (for ). The solution of the evolution problem will be understood in the mild sense, and the integrals involved in that definition will be of Young type.     

Elliptic Problems with Singular Potential and Double-Power Nonlinearity

We prove the existence of positive symmetric solutions to the semilinear elliptic problem

Matrix Functions Consimilar to the Identity and Singular Integral Operators

In this paper we establish the connection between singular integral operators with conjugation and matrix functions consimilar to the identity. We show that any matrix function consimilar to the identity is factorable (in some space L _p ) if and only if it admits a special factorization, that we call antisymmetric, and that this antisymmetric factorization has a direct connection with the factorization of singular integral operators with conjugation. Submitted: April 27, 2007. Accepted: January 23, 2008.     

An Existence Result for a Problem with Critical Growth and Lack of Strict Convexity

We prove the existence of a nontrivial solution for a quasilinear elliptic equation involving a nonlinearity having critical growth and a convex principal part, which is not required to be strictly convex. Supported by MURST, Project “Variational Methods and Nonlinear Differential Equations”.     

Sharp Two Weight Inequalities for Commutators of Riemann-Liouville and Weyl Fractional Integral Operators

Let b be a BMO function, and the commutator of order k for the Weyl fractional integral. In this paper we prove weighted strong type (p, p) inequalities (p > 1) and weighted endpoint estimates (p = 1) for the operator and for the pairs of weights of the type (w, ), where w is any weight and is a suitable one-sided maximal operator. We also prove that, for weights, the operator is controlled in the L ^p (w) norm by a composition of the one-sided fractional maximal operator and the one-sided Hardy-Littlewood maximal operator iterated k times. These results improve those obtained by an immediate application of the corresponding two-sided results and provide a different way to obtain known results about the operators . The same results can be obtained for the commutator of order k for the Riemann-Liouville fractional integral This research has been partially supported by Spanish goverment Grant MTM2005-8350-C03-02. The first author was also supported by CONICET, ANPCyT and CAI+D-UNL. The second author was also supported by Junta de Andalucía Grant FQM 354.     

Central Points of the Complete Quadrangle

Generalizing the classical geometry of the triangle in the Euclidean plane E, we define a central point of an n-gon as a symmetric function E ⁿ → E which commutes with all similarities. We first review various geometrical characterizations of some well-known central points of the quadrangle (n = 4) and show how a look at their mutual positions produces a morphologic classification (cyclic, trapezoidal, orthogonal etc.). From a basis of four central points, full information on the quadrangle can be retrieved. This generalizes a problem first faced by Euler for the triangle. Reconstructing a quadrangle from its central points is a geometric analogue of solving an algebraic equation of degree 4: here the diagonal triangle plays the role of a Lagrange resolvent and the determination of loci for the central points replaces the examination of discriminants for real roots.

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Received: March 2007     

On Some Categories and Functors in the Theory of Quaternionic Bergman Spaces

Basic facts are presented about the theory of quaternionic Bergman spaces with special emphasis on what is happening with them under conformal transformations of the domains. Constructing a series of categories of quaternion-valued functions as well as functors acting between them we show that the arising spaces and operators have conformally covariant or invariant characters in terms of the theory of categories. The second-named and the third-named authors were partially supported by CONACYT projects as well as by IPN in the framework of COFAA and SIP programs.     

From the Mahler Conjecture to Gauss Linking Integrals

We establish a version of the bottleneck conjecture, which in turn implies a partial solution to the Mahler conjecture on the product v(K)  = (Vol K)(Vol K°) of the volume of a symmetric convex body and its polar body K°. The Mahler conjecture asserts that the Mahler volume v(K) is minimized (non-uniquely) when K is an n-cube. The bottleneck conjecture (in its least general form) asserts that the volume of a certain domain is minimized when K is an ellipsoid. It implies the Mahler conjecture up to a factor of (π/4) ⁿ γ _n , where γ _n is a monotonic factor that begins at 4/π and converges to . This strengthens a result of Bourgain and Milman, who showed that there is a constant c such that the Mahler conjecture is true up to a factor of c ⁿ . The proof uses a version of the Gauss linking integral to obtain a constant lower bound on Vol K ^◇, with equality when K is an ellipsoid. It applies to a more general conjecture concerning the join of any two necks of the pseudospheres of an indefinite inner product space. Because the calculations are similar, we will also analyze traditional Gauss linking integrals in the sphere S ^n-1 and in hyperbolic space H ^n-1. Received: December 2006, Accepted: January 2007     

Large Toeplitz Systems with Quasi-polynomial Right-hand Sides

We consider the asymptotics of the solutions of large linear systems with Toeplitz matrices generated by a complex valued symbol which is infinitely differentiable, has no zeros on the unit circle, and whose winding number about the origin is zero. The emphasis is on quasi-polynomials as right-hand sides, in which case we show that the central fragment of the solution is asymptotically also a quasi-polynomial. Moreover, we establish asymptotic formulas that give specific components of the solution independently of the other components. We are greatly indebted to the referees for suggesting substantial simplifications in our original proofs and the constructive advice which helped to improve the exposition.     

Estimates for Littlewood-Paley Functions and Extrapolation

We prove certain L ^p -estimates for Littlewood-Paley functions arising from rough kernels. The estimates are useful for extrapolation to prove L ^p -boundedness of the Littlewood-Paley functions under a sharp kernel condition.      

Hardy-Littlewood system and representations of integers by an invariant polynomial

Let f be an integral homogeneous polynomial of degree d, and let be the level set for each . For a compact subset in ), set

Singular integral operators on Riemann surfaces and elliptic differential systems

The derivatives of the Cauchy kernels on compact Riemann surfaces generate singular integral operators analogous to the Calderón-Zigmund operators with the kernel (t - z)² on the complex plane. These operators play an important role in studying elliptic differential equations, boundary value problems, etc. We consider here the most important case of the multi-valued Cauchy kernel with real normalization of periods. In the opposite plane case, such an operator is not unitary. Nevertheless, its norm in L₂ is equal to one. This result is used to study multi-valued solutions of elliptic differential systems.     

Comparison theorem and uniqueness of positive solutions for sublinear elliptic equations

In semilinear elliptic equations, we prove that the necessary and sufficient condition for the comparison theorem of positive solutions to be valid is that the nonlinear term is sublinear. Our theorem needs neither any regularity of the nonlinear term nor the smoothness of the boundary. Applying this theorem, we prove the uniqueness of positive solutions for the Dirichlet problem. Received: 9 April 2008