Paranormal spaces under

We prove that paranormal spaces of character are -
collectionwise Hausdorff assuming the set-theoretic principle . This gives an affirmative answer to problem 197 in Problems I wish I could solve, by W. S. Watson (Open Problems in Topology (1990), 37-76).

     

The nonarchimedean theta correspondence for

In this paper we consider the theta correspondence between the sets and when is a nonarchimedean local field and . Our main theorem determines all the elements of that occur in the correspondence. The answer involves distinguished representations. As a corollary, we characterize all the elements of that occur in the theta correspondence between and . We also apply our main result to prove a case of a new conjecture of S.S. Kudla concerning the first occurrence of a representation in the theta correspondence.

     

Summability of Fourier orthogonal series for Jacobi weight functions on the simplex in

We study the Fourier expansion of a function in orthogonal polynomial series with respect to the weight functions

on the standard simplex in . It is proved that such an expansion is uniformly summable on the simplex for any continuous function if and only if . Moreover, it is shown that means define a positive linear polynomial identity, and the index is sharp in the sense that means are not positive for .

     

Copies of in topological Riesz spaces

The paper is concerned with order-topological characterizations of topological Riesz spaces, in particular spaces of measurable functions, not containing Riesz isomorphic or linearly homeomorphic copies of or .

     

No submaximal topology on a countable set is -complementary

Two -topologies and given on the same set , are called transversal if their union generates the discrete topology on . The topologies and are -complementary if they are transversal and their intersection is the cofinite topology on . We establish that for any connected Tychonoff topology there exists a connected Tychonoff transversal one. Another result is that no -complementary topology exists for the maximal topology constructed by van Douwen on the rational numbers. This gives a negative answer to Problem 162 from Open Problems in Topology (1990).

     

-compactness and automatic continuity in ultrametric spaces of bounded continuous functions

In this paper (weakly) separating maps between spaces of bounded continuous functions over a nonarchimedean field are studied. It is proven that the behaviour of these maps when is not locally compact is very different from the case of real- or complex-valued functions: in general, for -compact spaces and , the existence of a (weakly) separating additive map implies that and are homeomorphic, whereas when dealing with real-valued functions, this result is in general false, and we can just deduce the existence of a homeomorphism between the Stone-Cech compactifications of and . Finally, we also describe the general form of bijective weakly separating linear maps and deduce some automatic continuity results.

     

An a priori estimate for the Tricomi equation in the upper half space

We establish an a priori estimate for the Tricomi equation. Our main tool is Mihlin's multiplier theorem combined with well-known estimates of the Newtonian potential.

     

Boundedness of the Cesàro operator on mixed norm spaces

In this note, the boundedness of the Cesàro operator on mixed norm space , , is proved.

     

Commuting relations for Hausdorff operators and Hilbert transforms on real Hardy spaces

We consider Hausdorff operators generated by a function ϕ integrable in Lebesgue"s sense on either R or R ², and acting on the real Hardy space H ¹(R), or the product Hardy space H ¹¹(R×R), or one of the hybrid Hardy spaces H ¹⁰(R ²) and H ⁰¹(R ²), respectively. We give a necessary and sufficient condition in terms of ϕ that the Hausdorff operator generated by it commutes with the corresponding Hilbert transform. This revised version was published online in June 2006 with corrections to the Cover Date.     

Characterizations of weakly compact operators on

Let be a locally compact Hausdorff space and let , is continuous and vanishes at infinity} be provided with the supremum norm. Let and be the -rings generated by the compact subsets and by the compact subsets of , respectively. The members of are called -Borel sets of since they are precisely the -bounded Borel sets of . The members of are called the Baire sets of . denotes the dual of . Let be a quasicomplete locally convex Hausdorff space. Suppose is a continuous linear operator. Using the Baire and -Borel characterizations of weakly compact sets in as given in a previous paper of the author's and combining the integration technique of Bartle, Dunford and Schwartz, we obtain 35 characterizations for the operator to be weakly compact, several of which are new. The independent results on the regularity and on the regular Borel extendability of -additive -valued Baire measures are deduced as an immediate consequence of these characterizations. Some other applications are also included.

     

The Toda molecule equation and the -algorithm

One of the well-known convergence acceleration methods, the -algorithm is investigated from the viewpoint of the Toda molecule equation. It is shown that the error caused by the algorithm is evaluated by means of solutions for the equation. The acceleration algorithm based on the discrete Toda molecule equation is also presented.

     

Best bounds for the approximate units for certain ideals of

We compute the best bound for the approximate units of the augmentation ideal of the group algebra of a locally compact amenable group . More generally such a calculation is performed for the kernel of the canonical map from onto , being a closed amenable subgroup of . Analogous results involving certain ideals of the Fourier algebra of an amenable group are also discussed.

     

Annihilating a subspace of with the sign of a continuous function

Let be a -finite, nonatomic, Baire measure space. Let be a finite dimensional subspace of . There is a bounded, continuous function, , defined on , such that

(1) for all , and (2) almost everywhere.

     

On the wave-front set of traces of CR functions on maximally real submanifolds

We prove that, in a locally integrable structure, the wave-front set of the trace of a CR function at a point in a totally real submanifold of maximal dimension is independent of the maximally real submanifold passing through the point .

     

Atomic characterization of the Hardy space of one-dimensional Schrödinger operators with nonnegative potentials

Given a Schrödinger operator on with nonnegative potential , we present an atomic characterization of the associated Hardy space .

     

Endpoint Properties of Localized Riesz Transforms and Fractional Integrals Associated to Schrödinger Operators

Let \({\mathcal L}\equiv-\Delta+V\) be the Schrödinger operator in \({{\mathbb R}^n}\), where V is a nonnegative function satisfying the reverse Hölder inequality. Let ρ be an admissible function modeled on the known auxiliary function determined by V. In this paper, the authors characterize the localized Hardy spaces \(H^1_\rho({{\mathbb R}^n})\) in terms of localized Riesz transforms and establish the boundedness on the BMO-type space \({\mathop\mathrm{BMO_\rho({\mathbb R}^n)}}\) of these operators as well as the boundedness from \({\mathop\mathrm{BMO_\rho({\mathbb R}^n)}}\) to \({\mathop\mathrm{BLO_\rho({\mathbb R}^n)}}\) of their corresponding maximal operators, and as a consequence, the authors obtain the Fefferman–Stein decomposition of \({\mathop\mathrm{BMO_\rho({\mathbb R}^n)}}\) via localized Riesz transforms. When ρ is the known auxiliary function determined by V, \({\mathop\mathrm{BMO_\rho({\mathbb R}^n)}}\) is just the known space \(\mathop\mathrm{BMO}_{\mathcal L}({{\mathbb R}^n})\), and \({\mathop\mathrm{BLO_\rho({\mathbb R}^n)}}\) in this case is correspondingly denoted by \(\mathop\mathrm{BLO}_{\mathcal L}({{\mathbb R}^n})\). As applications, when n?≥?3, the authors further obtain the boundedness on \(\mathop\mathrm{BMO}_{\mathcal L}({{\mathbb R}^n})\) of Riesz transforms \(\nabla{\mathcal L}^{-1/2}\) and their adjoint operators, as well as the boundedness from \(\mathop\mathrm{BMO}_{\mathcal L}({{\mathbb R}^n})\) to \(\mathop\mathrm{BLO}_{\mathcal L}({{\mathbb R}^n})\) of their maximal operators. Also, some endpoint estimates of fractional integrals associated to \({\mathcal L}\) are presented.