Bounded, compact and Schatten class weighted composition operators on the Bergman space are characterized by the use of generalized Berezin transforms. An estimate of the essential norms of weighted composition operators on the Bergman space is given. Most of the results remain true for Hardy spaces and weighted Bergman spaces.

The paper introduces the notion of definable compactness and within the context of o-minimal structures proves several topological properties of definably compact spaces. In particular a definable set in an o-minimal structure is definably compact (with respect to the subspace topology) if and only if it is closed and bounded. Definable compactness is then applied to the study of groups and rings in o-minimal structures. The main result proved is that any infinite definable group in an o-minimal structure that is not definably compact contains a definable torsion-free subgroup of dimension 1. With this theorem, a complete characterization is given of all rings without zero divisors that are definable in o-minimal structures. The paper concludes with several examples illustrating some limitations on extending the theorem.

Riemann–Stieltjes integrals are considered as linear operators on weighted Bloch and Bergman spaces of the open unit ball in several complex variables. For weighted Bloch spaces, boundedness, compactness and weak compactness of Riemann–Stieltjes operators are characterized by means of certain growth properties of holomorphic symbols. For weighted Bergman spaces, some criteria are given for Riemann–Stieltjes operators with holomorphic symbols to be bounded, compact and of Schatten–von Neumann's ideal.

A uniform version of the Schanuel conjecture is discussed that has some model-theoretical motivation. This conjecture is assumed, and it is proved that any ‘non-obviously-contradictory’ system of equations in the form of exponential sums with real exponents has a solution.

Inequalities involving classical operators of harmonic analysis, such as maximal functions, fractional integrals and singular integrals of convolutive type have been extensively investigated in various function spaces. Results on weak and strong type inequalities for operators of this kind in Lebesgue spaces are classical and can be found for example in [4, 20, 24]. Generalizations of these results to Zygmund spaces are presented in [4]. An exhaustive treatment of the problem of boundedness of such operators in Lorentz and Lorentz–Zygmund spaces is given in [3]. See also [8, 9] for further extensions in the framework of generalized Lorentz–Zygmund spaces.

As far as Orlicz spaces are concerned, a characterization of Young functions A having the property that the Hardy–Littlewood maximal operator or the Hilbert and Riesz transforms are of weak or strong type from the Orlicz space LA into itself is known (see for example [13]). In [17, 23] conditions on Young functions A and B are given for the fractional integral operator to be bounded from LA into LB under some restrictions involving the growths and certain monotonicity properties of A and B.

The main purpose of this paper is to find necessary and sufficient conditions on general Young functions A and B ensuring that the above-mentioned operators are of weak or strong type from LA into LB. Our results for (fractional) maximal operators are presented in Section 2, while Section 3 deals with fractional and singular integrals. In particular, we re-cover a result concerning the standard Hardy–Littlewood maximal operator which has recently been proved in [2, 11, 12]. Finally, in Section 4, the resolvent operator of some differential problems is taken into account and a priori bounds for Orlicz norms of solutions to elliptic boundary value problems in terms of Orlicz norms of the data are established. Let us mention that part of the results of the present paper were announced in [6].

Using critical point theory, some sufficient conditions are obtained for the existence of nonconstant $T$-periodic solutions of a class of second order self-adjoint difference equations.

Time-variant filters based on Calderón and Gabor reproducing formulas are important tools in time-frequency analysis. The paper studies the behavior of the eigenvalues of these filters. Optimal two-sided estimates of the number of eigenvalues contained in the interval $(\delta_1,\delta_2)$ , where $0<\delta_1<\delta_2<1$ , are obtained. The estimates cover large classes of localization domains and generating functions.

The paper proves several weighted imbedding theorems for domains with fractal boundaries. The weights considered are distances to the boundary to certain powers, and the domains are so-called s-John domains. The paper also proves, in the general setting, that the existence of an imbedding implies compactness of the imbedding for lower exponents. Moreover, following Maz'ya, the paper reformulates the imbedding theorems in the language of local isoperimetric and capacity estimates.

Let d be a square-free number and let CL(−d) denote the ideal class group of the imaginary quadratic number field ℚ(√−d). Further let h(−d) = #CL(−d) denote the class number. For integers g [ges ] 2, we define [Nscr ]g(X) to be the number of square-free d [ges ] X such that CL(−d) contains an element of order g. Gauss' genus theory demonstrates that if d has at least two odd prime factors (in particular, for almost all d) then CL(−d) contains ℤ2 as a subgroup. Thus N2(X) ∼ 6X/π2. The behaviour of [Nscr ]g(X) is not understood for any other value of g. It is believed that [Nscr ]g(X) ∼ CgX for some positive constant Cg. For odd primes g, H. Cohen and H. Lenstra [3] conjectured that

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N. Ankeny and S. Chowla [1] first showed that [Nscr ]g(X) → ∞ as X → ∞. Although they did not point this out, their method demonstrates that [Nscr ]g(X) [Gt ] X1/2. Recently, M. R. Murty [11] improved this to [Nscr ]g(X) [Gt ] X1/2+1/g. Hitherto this represented the best known lower bounds for [Nscr ]g(X) except in the cases g = 4 and g = 8. In the cases g = 4 or 8, P. Morton [9] used class field theory techniques to show that [Nscr ]g(X) [Gt ] X1−ε. In fact, he demonstrated the elegant result that given any non-negative integers r, s and t, there are ‘many’ d with CL(−d)/CL(−d)8 = ℤr2 × ℤs4 × ℤt8 (see [9] for a precise statement). The complementary question of finding d with p [nmid ] h(−d) has also attracted a lot of attention. H. Davenport and H. Heilbronn [5] proved the striking result that the proportion of d with 3 [nmid ] h(−d) is at least 1/2. For larger primes p, recently W. Kohnen and K. Ono [7] have shown that there are [Gt ] √X/log X square- free integers d [les ] X such that p [nmid ] h(−d).

In this paper, we sharpen Murty's lower bounds on [Nscr ]g(X) for all values of g; see Theorem 1 below. We also offer a simple proof that [Nscr ]4(X) [Gt ] X/√log X; see Proposition 2 below. In §5 we express the hope that these methods may lead to [Nscr ]3(X) [Gt ] X1−ε.

Fuglede's conjecture states that a set $\Omega\subset{\mathbf R}^n$ tiles ${\mathbf R}$ by translations if and only if $L^2(\Omega)$ has an orthogonal basis of exponentials. New partial results are obtained supporting the conjecture in dimension 1.

A large number of papers written over the last ten years have concerned the spectral theory of Laplace–Beltrami operators on complete Riemannian manifolds, and of other self-adjoint second order elliptic operators. Much of the interest has centred on the relationship between various types of Sobolev inequality, parabolic Harnack inequalities and the Liouville property on the one hand, and Gaussian heat kernel bounds on the other. For manifolds of bounded geometry there is an important connection between this problem and a corresponding one for discrete Laplacians on graphs. Standard references are [9, 37] and more recent literature can be traced via [5, 16, 32].

A tangent process of a random process X at a point z is defined to be the limit in distribution of some sequence of scaled enlargements of X about z. The main result of the paper is that a tangent process must be self-similar with stationary increments, at almost all points z where the tangent process is essentially unique. The consequences for tangent processes of certain classes of process are examined, including stable processes and processes with independent increments where unique tangent processes are Lévy processes.

Let [ ] be the open unit disk in the complex plane [Copf ]. The Bergman space L2a([ ]) is the Hilbert space of analytic functions f in [ ] such that

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where dA is the normalized area measure on [ ]. If

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are two functions in L2a([ ]), then the inner product of f and g is given by

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We study multiplication operators on L2a([ ]) induced by analytic functions. Thus for φ ∈ H ∞([ ]), the space of bounded analytic functions in [ ], we define

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by

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It is easy to check that Mφ is a bounded linear operator on L2a([ ]) with

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The notion of stable isomorphism of bundle gerbes is considered. It has the consequence that the stable isomorphism classes of bundle gerbes over a manifold M are in bijective correspondence with H3(M, ℤ). Stable isomorphism sheds light on the local theory of bundle gerbes and enables a classifying theory for bundle gerbes to be developed using results of Gajer on B[Copf ]× bundles.

Guided by the Hopf fibration, a family (indexed by a positive constant $K$ ) of right invariant Riemannian metrics on the Lie group $S^3$ is singled out. Using the Yasuda–Shimada paper as an inspiration, a privileged right invariant Killing field of constant length is determined for each $K > 1$ . Each such Riemannian metric couples with the corresponding Killing field to produce a $y$ -global and explicit Randers metric on $S^3$ . Employing the machinery of spray curvature and Berwald's formula, it is proved directly that the said Randers metric has constant positive flag curvature $K$ , as predicted by Yasuda–Shimada. It is explained why this family of Finslerian space forms is not projectively flat.

The Schwartz space of rapidly decaying test functions is characterized by the decay of the short-time Fourier transform or cross-Wigner distribution. Then a version of Hardy's theorem is proved for the short-time Fourier transform and for the Wigner distribution.

It is proved that a graph K has an embedding as a regular map on some closed surface if and only if its automorphism group contains a subgroup G which acts transitively on the oriented edges of K such that the stabiliser Ge of every edge e is dihedral of order 4 and the stabiliser Gv of each vertex v is a dihedral group the cyclic subgroup of index 2 of which acts regularly on the edges incident with v. Such a regular embedding can be realised on an orientable surface if and only if the group G has a subgroup H of index 2 such that Hv is the cyclic subgroup of index 2 in Gv. An analogous result is proved for orientably-regular embeddings.

A systematic treatment is given of several classes of singular integrals. Their $L^p$ boundedness is proved when their kernels are given by functions $\Omega$ in $L\log L({\bf S}^{n-1})$ .

The natural geometric setting of quadrics commuting with a Hermitian surface of ${\rm PG}(3,q^2)$, q odd, is adopted and a hemisystem on the Hermitian surface ${\cal H}(3,q^2)$ admitting the group $\mathrm{{P}}\Omega^-(4,q)$ is constructed, yielding a partial quadrangle ${\rm PQ}((q-1)/2,q^2,(q-1)^2/2)$ and a strongly regular graph srg$((q^3+1)(q+1)/2,(q^2+1)(q-1)/2,(q-3)/2,(q-1)^2/2)$. For $q>3$, no partial quadrangle or strongly regular graph with these parameters was previously known, whereas when $q=3$, this is the Gewirtz graph. Thas conjectured that there are no hemisystems on ${\cal H}(3,q^2)$ for $q>3$, so these are counterexamples to his conjecture. Furthermore, a hemisystem on ${\cal H}(3,25)$ admitting $3.A_7.2$ is constructed. Finally, special sets (after Shult) and ovoids on ${\cal H}(3,q^2)$ are investigated.

The paper studies the heat kernel of the Schrödinger operator with magnetic fields and of uniformly elliptic operators with non-negative electric potentials in the reverse Hölder class which includes non-negative polynomials as typical examples. The main aim of the paper is to give a pointwise estimate of the heat kernel of the operators above which is affected by magnetic fields and non-negative degenerate electric potentials. A weighted smoothing estimate for the semigroup generated by the operators above is also given.