Integral Kernel Operators on Fock Space – Generalizations and Applications to Quantum Dynamics

A general theory of operators on Boson Fock space is discussed in terms of the white noise distribution theory on Gaussian space (white noise calculus). An integral kernel operator is generalized from two aspects: (i) The use of an operator-valued distribution as an integral kernel leads us to the Fubini type theorem which allows an iterated integration in an integral kernel operator. As an application a white noise approach to quantum stochastic integrals is discussed and a quantum Hitsuda–Skorokhod integral is introduced. (ii) The use of pointwise derivatives of annihilation and creation operators assures the partial integration in an integral kernel operator. In particular, the particle flux density becomes a distribution with values in continuous operators on white noise functions and yields a representation of a Lie algebra of vector fields by means of such operators.     

Hardy type spaces associated with compact unitary groups

We investigate Hilbertian Hardy type spaces of complex analytic functions of infinite many variables, associated with compact unitary groups and the corresponding invariant Haar’s measures. For such analytic functions we establish a Cauchy type integral formula and describe natural domains. Also we show some relations between constructed spaces of analytic functions and the symmetric Fock space.     

Singular integral operators on tent spaces over spaces of homogeneous type: Fefferman–Stein box maximal functions and pointwise Carleson type estimates

In this article we generalize the singular integral operator theory on weighted tent spaces to spaces of homogeneous type. This generalization of operator theory is in the spirit of C. Fefferman and Stein since we use some auxiliary functionals on tent spaces which play roles similar to the Fefferman–Stein sharp and box maximal functions in the Lebesgue space setting. Our contribution in this operator theory is twofold: for singular integral operators (including maximal regularity operators) on tent spaces pointwise Carleson type estimates are proved and this recovers known results; on the underlying space no extra geometrical conditions are needed and this could be useful for future applications to parabolic problems in rough settings.     

The convolution for zero-order Mehler–Fock transform and pseudo-differential operator

The main aim of this paper is to find out the estimates of convolution for zero-order Mehler–Fock transform with various approaches. Pseudo-differential operator in terms of zero-order Mehler–Fock transform is defined and obtained its another integral representation. Further, its estimate in Lebesgue space has been studied. At the end some applications of zero-order Mehler–Fock transform and of convolution are discussed.     

Weighted norm inequalities for integral operators

We consider a large class of positive integral operators acting on functions which are defined on a space of homogeneous type with a group structure. We show that any such operator has a discrete (dyadic) version which is always essentially equivalent in norm to the original operator. As an application, we study conditions of ``testing type,' like those initially introduced by E. Sawyer in relation to the Hardy-Littlewood maximal function, which determine when a positive integral operator satisfies two-weight weak-type or strong-type estimates. We show that in such a space it is possible to characterize these estimates by testing them only over ``cubes'. We also study some pointwise conditions which are sufficient for strong-type estimates and have applications to solvability of certain nonlinear equations.

     

A Convolution Operator Related to the Generalized Mehler–Fock and Kontorovich–Lebedev Transforms

In this paper we study a generalization of an index integral involving the product of modified Bessel functions and associated Legendre functions. It is applied to a convolution construction associated with this integral, which is related to the classical Kontorovich–Lebedev and generalized Mehler–Fock transforms. Mapping properties and norm estimates in weighted L _p -spaces, 1 ≤ p ≤ 2, are investigated. An application to a class of convolution integral equations is considered. Necessary and sufficient conditions are found for the solvability of these equations in L ₂.     

Pointwise Estimates for the Bergman Kernel of the Weighted Fock Space

We prove upper pointwise estimates for the Bergman kernel of the weighted Fock space of entire functions in L ²(e ^−2φ ) where φ is a subharmonic function with Δφ a doubling measure. We derive estimates for the canonical solution operator to the inhomogeneous Cauchy-Riemann equation and we characterize the compactness of this operator in terms of Δφ.     

On the convergence of functions from a Sobolev space satisfying special integral estimates

For sequences of functions from a Sobolev space satisfying special integral estimates, we, in one case, establish a lemma on the choice of pointwise convergent subsequences and, in a different case, prove a theorem on convergence of the corresponding sequences of generalized derivatives in measure. These results are applied to the problem of existence of the entropy solutions of nonlinear equations with degenerate coercivity and L ¹-data. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 2, pp. 168–183, February, 2006.     

Analytic and arithmetic properties of the $$(\Gamma ,\chi )$$ ( Γ , χ ) -automorphic reproducing kernel function and associated Hermite–Gauss series

We consider the reproducing kernel function of the theta Bargmann–Fock Hilbert space associated with given full-rank lattice and pseudo-character, and we deal with some of its analytical and arithmetical properties. Specially, the distribution and the discreteness of its zeros are examined. The analytic sets of zeros of the theta Bargmann–Fock space inside a given fundamental cell is characterized and shown to be finite and of cardinal less or equal to its dimension. Moreover, we obtain some remarkable lattice sums by evaluating the so-called complex Hermite–Gauss coefficients. Some of them generalize some of the arithmetic identities given by Perelomov in the framework of coherent states for the specific case of von Neumann lattice. Such complex Hermite–Gauss coefficients are nontrivial examples of the so-called lattice’s functions according the Serre terminology. The perfect use of the basic properties of the complex Hermite polynomials is crucial in this framework.

     

Some function spaces on spaces of homogeneous type

We introduce Campanato, Morrey, BMO and Sobolev-type spaces for mappings from a space of homogeneous type into a complete metric space which possess properties comparable to their classical analogues. In particular we show integral characterizations, the validity of the John–Nirenberg theorem, Poincarè and Sobolev inequalities, Sobolev's embedding theorem and estimates on the pointwise behavior of Sobolev-type mappings. Received: 4 December 2000 / Revised version: 5 July 2001     

Error estimates for approximate approximations with Gaussian kernels on compact intervals

The aim of this paper is the investigation of the error which results from the method of approximate approximations applied to functions defined on compact intervals, only. This method, which is based on an approximate partition of unity, was introduced by Maz’ya in 1991 and has mainly been used for functions defined on the whole space up to now. For the treatment of differential equations and boundary integral equations, however, an efficient approximation procedure on compact intervals is needed.In the present paper we apply the method of approximate approximations to functions which are defined on compact intervals. In contrast to the whole space case here a truncation error has to be controlled in addition. For the resulting total error pointwise estimates and L₁-estimates are given, where all the constants are determined explicitly.     

Abstract capacitary estimates and the completeness and separability of certain classes of non-locally convex topological vector spaces

We are concerned with establishing completeness and separability criteria for large classes of topological vector spaces which are typically non-locally convex, including Lebesgue-like spaces, Lorentz spaces, Orlicz spaces, mixed-normed spaces, tent spaces, and discrete Triebel–Lizorkin and Besov spaces. For vector spaces of measurable functions we also derive pointwise convergence results. Our approach relies on abstract capacitary estimates and works in certain cases of interest even in the absence of a background measure space and/or of a vector space structure.     

Rate of approximation for functions of bounded variation by integral operators

We estimate pointwise convergence rates of approximation for functions of bounded variation and for functions which are exponentially bounded and locally of bounded variation. The approximation is through the operation of a sequence of integral operators with not necessarily positive kernel functions. The general result is then applied to deduce estimates for particular operators, such as Beta operators, Fourier–Legendre operators, Picard operators, and Gauss–Weierstrass operators.     

Generalized solutions for a class of non-uniformaly elliptic equations in divergence form

We study a general class of quasilinear non-uniformly elliptic pdes in divergence from with linear growth in the gradient. We examine the notions of BV and viscosity solutions and derive for such generalized solutions various a priori pointwise and integral estimates, including a Harnack inequality. In particular we prove that viscosity solutions are unique (on strictly convex domains), are contained in the space BV ^loc and are C ^1,α almost everywhere.     

Dimension-Independent Estimates for Heat Operators and Harmonic Functions

We establish dimension-independent estimates related to heat operators e ^tL on manifolds. We first develop a very general contractivity result for Markov kernels which can be applied to diffusion semigroups. Second, we develop estimates on the norm behavior of harmonic and non-negative subharmonic functions. We apply these results to two examples of interest: when L is the Laplace–Beltrami operator on a Riemannian manifold with Ricci curvature bounded from below, and when L is an invariant subelliptic operator of Hörmander type on a Lie group. In the former example, we also obtain pointwise bounds on harmonic and subharmonic functions, while in the latter example, we obtain pointwise bounds on harmonic functions when a generalized curvature-dimension inequality is satisfied.     

The Slice Hyperholomorphic Bergman Space on $$\varvec{\mathbb {B}_{R}}$$: Integral Representation and Asymptotic Behavior

The aim of the present paper is threefolds. Firstly, we complete the study of the weighted hyperholomorphic Bergman space of the second kind on the ball of radius R centred at the origin. The explicit expression of its Bergman kernel is given and can be written in terms of special hypergeometric functions of two non-commuting (quaternionic) variables. Secondly, we introduce and study some basic properties of an associated integral transform, the quaternionic analogue of the so-called second Bargmann transform for the holomorphic Bergman space. Finally, we establish the asymptotic behavior as R goes to infinity. We show in particular that the reproducing kernel of the weighted slice hyperholomorphic Bergman space gives rise to its analogue for the slice hyperholomorphic Bargamann–Fock space.     

POINTWISE CONVERGENCE FOR EXPANSIONS IN SPHERICAL MONOGENICS

We offer a new approach to deal with the pointwise convergence of FourierLaplace series on the unit sphere of even-dimensional Euclidean spaces. By using spherical monogenics defined through the generalized Cauchy-Riemann operator, we obtain the spherical monogenic expansions of square integrable functions on the unit sphere. Based on the generalization of Fueter＇s theorem inducing monogenic functions from holomorphic functions in the complex plane and the classical Carleson＇s theorem, a pointwise convergence theorem on the new expansion is proved. The result is a generalization of Carleson＇s theorem to the higher dimensional Euclidean spaces. The approach is simpler than those by using special functions, which may have the advantage to induce the singular integral approach for pointwise convergence problems on the spheres.     

On the Fock space of metaanalytic functions

We develop the theory on the Fock space of metaanalytic functions, a generalization of some recent results on the Fock space of polyanalytic functions. We show that the metaanalytic Bargmann transform is a unitary mapping between vector-valued Hilbert spaces and metaanalytic Fock spaces. A reproducing kernel of the metaanalytic Fock space is derived in an explicit form. Furthermore, we establish a complete characterization of all lattice sampling and interpolating sequence for the Fock space of metaanalytic functions.     

Maximal bilinear singular integral operators associated with dilations of planar sets

We obtain square function estimates and bounds for maximal singular integral operators associated with bilinear multipliers given by characteristic functions of dyadic dilations of certain planar sets. As a consequence, we deduce pointwise almost everywhere convergence for lacunary partial sums of bilinear Fourier series with respect to methods of summation determined by the corresponding planar sets.     

Pointwise and integral estimates for the B-Riesz potential in terms of B-maximal and B-fractional maximal functions

We study the maximal and fractional maximal functions and Riesz potentials that are generated by the generalized shift operator associated with the Laplace-Bessel operator. We obtain some pointwise and integral estimates that give a relation between the B-maximal and B-fractional maximal functions and B-Riesz potentials and extend the available results to the objects of a more general nature. Basing on these results, we prove interpolation theorems for the B-fractional maximal functions and B-Riesz potentials.