Convolution operators induced by approximate identities and pointwise convergence in spaces

Given a sequence of kernels for which the operators converge a.e. in all spaces, , a perturbation method is provided with the property that the modified convolution operators converge pointwise only in selective spaces.

     

A note on Gabor orthonormal bases

The study of Gabor bases of the form for has interested many mathematicians in recent years. Alex Losevich and Steen Pedersen in 1998, Jeffery C. Lagarias, James A. Reeds and Yang Wang in 2000 independently proved that, for any fixed positive integer , is an orthonormal basis for if and only if is a tiling of . Palle E. T. Jorgensen and Steen Pedersen in 1999 gave an explicit characterization of such for , , . Inspired by their work, this paper addresses Gabor orthonormal bases of the form for and some other related problems, where is as above. For a fixed , the generating function of a Gabor orthonormal basis for corresponding to the above is characterized explicitly provided that , which is new even if ; a Shannon type sampling theorem about such is derived when , ; for an arbitrary positive integer , an explicit expression of the with being an orthonormal basis for is obtained under the condition that .

     

Frame wavelets in subspaces of

Let be a real expansive matrix. We characterize the reducing subspaces of for -dilation and the regular translation operators acting on We also characterize the Lebesgue measurable subsets of such that the function defined by inverse Fourier transform of generates through the same -dilation and the regular translation operators a normalized tight frame for a given reducing subspace. We prove that in each reducing subspace, the set of all such functions is nonempty and is also path connected in the regular -norm.

     

Normalizers of the congruence subgroups of the Hecke group II

Let . Let be an ideal of and let be the maximal ideal of such that . Then . In particular, if is square free, then is self-normalized in .

     

Conservativeness of diffusion processes with drift

We show the conservativeness of the Girsanov transformed diffusion process by drift with or 4d/(d+2)$">, or if is of the Hardy class with sufficiently small coefficient of energy . Here 0$"> is the lower bound of the symmetric measurable matrix-valued function appearing in the given Dirichlet form. In particular, our result improves the conservativeness of the transformed process by when .

     

A character of the gradient estimate for diffusion semigroups

Let be the semigroup of the diffusion process generated by on . It is proved that there exists and an -valued function such that holds for all 0$"> and all if and only if satisfies the formula for all

     

Estimates of Gromov's box distance

In 1999, M. Gromov introduced the box distance function on the space of all mm-spaces. In this paper, by using the method of T. H. Colding, we estimate and , where is the -dimensional unit sphere in and is the -dimensional complex projective space equipped with the Fubini-Study metric. In particular, we give the complete answer to an exercise of Gromov's green book. We also estimate from below, where is the special orthogonal group.

     

Necessary conditions for Schatten class localization operators

We study time-frequency localization operators of the form , where is the symbol of the operator and are the analysis and synthesis windows, respectively. It is shown in an earlier paper by the authors that a sufficient condition for , the Schatten class of order , is that belongs to the modulation space and the window functions to the modulation space . Here we prove a partial converse: if for every pair of window functions with a uniform norm estimate, then the corresponding symbol must belong to the modulation space . In this sense, modulation spaces are optimal for the study of localization operators. The main ingredients in our proofs are frame theory and Gabor frames. For and , we recapture earlier results, which were obtained by different methods.

     

Non-zero boundaries of Leibniz half-spaces

It is proved that for any , there exists a norm and two points , in such that the boundary of the Leibniz half-space has non-zero Lebesgue measure. When , it is known that the boundary must have zero Lebesgue measure.

     

On the normal bundle of submanifolds of

Let be a submanifold of dimension of the complex projective space . We prove results of the following type.i) If is irregular and , then the normal bundle is indecomposable. ii) If is irregular, and , then is not the direct sum of two vector bundles of rank . iii) If , and is decomposable, then the natural restriction map is an isomorphism (and, in particular, if is embedded Segre in , then is indecomposable). iv) Let and , and assume that is a direct sum of line bundles; if assume furthermore that is simply connected and is not divisible in . Then is a complete intersection. These results follow from Theorem 2.1 below together with Le Potier's vanishing theorem. The last statement also uses a criterion of Faltings for complete intersection. In the case when this fact was proved by M. Schneider in 1990 in a completely different way.

     

On zeros of Eisenstein series for genus zero Fuchsian groups

Let SL be a genus zero Fuchsian group of the first kind with as a cusp, and let be the holomorphic Eisenstein series of weight on that is nonvanishing at and vanishes at all the other cusps (provided that such an Eisenstein series exists). Under certain assumptions on and on a choice of a fundamental domain , we prove that all but possibly of the nontrivial zeros of lie on a certain subset of . Here is a constant that does not depend on the weight, is the upper half-plane, and is the canonical hauptmodul for

     

-algebras whose torsion part is not cyclic

We consider algebras over a Dedekind domain with the property and generalize Schultz' structure theory of the case to Dedekind domains. We construct examples of mixed -algebras, which are non-split extensions of the submodule of elements infinitely divisible by the relevant prime ideals. This is also new in the case .

     

Coarse embeddings of metric spaces into Banach spaces

There are several characterizations of coarse embeddability of locally finite metric spaces into a Hilbert space. In this note we give such characterizations for general metric spaces. By applying these results to the spaces , we get their coarse embeddability into a Hilbert space for . This together with a theorem by Banach and Mazur yields that coarse embeddability into and into are equivalent when . A theorem by G.Yu and the above allow us to extend to , , the range of spaces, coarse embeddings into which is guaranteed for a finitely generated group to satisfy the Novikov Conjecture.

     

Maximality theorems for Fréchet algebras

The algebra of unbounded holomorphic functions that is contained in the algebra is studied. For in but not in , we show that the algebra generated by and is dense in for all .

     

An elementary proof of the law of quadratic reciprocity over function fields

Let and be relatively prime monic irreducible polynomials in (). In this paper, we give an elementary proof for the following law of quadratic reciprocity in :

where is the Legendre symbol.

     

The Fefferman-Stein type inequality for the Kakeya maximal operator in Wolff's range

Let , , be the Kakeya (Nikodým) maximal operator defined as the supremum of averages over tubes of eccentricity . The (so-called) Fefferman-Stein type inequality:

is shown in the range , where and are some constants depending only on and the dimension and is a weight. The result is a sharp bound up to -factors.

     

Characterization of scaling functions in a multiresolution analysis

We characterize the scaling functions of a multiresolution analysis in a general context, where instead of the dyadic dilation one considers the dilation given by a fixed linear map such that and all (complex) eigenvalues of have absolute value greater than In the general case the conditions depend on the map We identify some maps for which the obtained condition is equivalent to the dyadic case, i.e., when is a diagonal matrix with all numbers in the diagonal equal to There are also easy examples of expanding maps for which the obtained condition is not compatible with the dyadic case. The complete characterization of the maps for which the obtained conditions are equivalent is out of the scope of the present note.

     

Counting cusps of subgroups of

Let be a number field with real places and complex places, and let be the ring of integers of . The quotient has cusps, where is the class number of . We show that under the assumption of the generalized Riemann hypothesis that if is not or an imaginary quadratic field and if , then has infinitely many maximal subgroups with cusps. A key element in the proof is a connection to Artin's Primitive Root Conjecture.

     

On Schwarz type inequalities

We show Schwarz type inequalities and consider their converses. A continuous function is said to be semi-operator monotone on if is operator monotone on . Let be a bounded linear operator on a complex Hilbert space and be the polar decomposition of . Let and for . (1) If a non-zero function is semi-operator monotone on , then for , where . (2) If are semi-operator monotone on , then for . Also, we show converses of these inequalities, which imply that semi-operator monotonicity is necessary.

     

A general functional equation and its stability

Suppose that and are vector spaces over or and are scalar such that whenever We prove that if for and

then each is a ``generalized' polynomial map of ``degree' at most

In case and we show that if some is bounded on a set of positive inner Lebesgue measure, then it is a genuine polynomial function.

Our main aim is to establish the stability of (in the sense of Ulam) in case is a Banach space.

We also solve a distributional analogue of and prove a mean value theorem concerning harmonic functions in two real variables.