Radial Differential Operators Research Articles

Embedding and compact embedding of weighted Bergman spaces

We use embeddings of Bergman spaces and fractional radial differential operators to study the Lαp–Lβq mapping properties of Bergman-type projections on the open unit ball of Cn. We recover and extend several existing results in this direction. In particular, this approach allows us to treat the cases when 0<q<1 and α≤−1. We also characterize when the embedding of one weighted Bergman space into another is compact.

Harmonic Besov spaces with small exponents

ABSTRACTWe study harmonic Besov spaces on the unit ball of , where 0<p<1 and . We provide characterizations in terms of partial and radial derivatives and certain radial differential operators that are more compatible with reproducing kernels of harmonic Bergman–Besov spaces. We show that the dual of harmonic Besov space is weighted Bloch space under certain volume integral pairing for 0<p<1 and . Our other results are about growth at the boundary and atomic decomposition.

Weighted Harmonic Bloch Spaces on the Ball

We study the family of weighted harmonic Bloch spaces \(b_\alpha , \alpha \in {\mathbb {R}}\), on the unit ball of \({\mathbb {R}}^n\). We provide characterizations in terms of partial and radial derivatives and certain radial differential operators that are more compatible with reproducing kernels of harmonic Bergman–Besov spaces. We consider a class of integral operators related to harmonic Bergman projection and determine precisely when they are bounded on \(L^\infty _\alpha \). We define projections from \(L^\infty _\alpha \) to \(b_\alpha \) and as a consequence obtain integral representations. We solve the Gleason problem and provide atomic decomposition for all \(b_\alpha , \alpha \in {\mathbb {R}}\). Finally we give an oscillatory characterization of \(b_\alpha \) when \(\alpha >-1\).

Aspects of multivariable operator theory on weighted symmetric Fock spaces

We obtain all Dirichlet spaces ℱq, q ∈ ℝ, of holomorphic functions on the unit ball of ℂNas weighted symmetric Fock spaces over ℂN. We develop the basics of operator theory on these spaces related to shift operators. We do a complete analysis of the effect of q ∈ ℝ in the topics we touch upon. Our approach is concrete and explicit. We use more function theory and reduce many proofs to checking results on diagonal operators on the ℱq. We pick out the analytic Hilbert modules from among the ℱq. We obtain von Neumann inequalities for row contractions on a Hilbert space with respect to each ℱq. We determine the commutants and investigate the almost normality of the shift operators. We prove that the C*-algebras generated by the shift operators on the ℱqfit in exact sequences that are in the same Ext class. We identify the groups K0and K1of the Toeplitz algebras on the ℱqarising in K-theory. Radial differential operators are prominent throughout. Some of our results, especially those pertaining to lower negative values of q, are new even for N = 1. Many of our results are valid in the more general weighted symmetric Fock spaces ℱbthat depend on a weight sequence b.

Holomorphic mean Lipschitz spaces and Hardy–Sobolev spaces on the unit ball

We study two classes of holomorphic functions in the unit ball 𝔹 n of ℂ n : mean Lipschitz spaces and Hardy–Sobolev spaces. Main results include new characterizations in terms of fractional radial differential operators and various comparisons between these two classes.

Calculation of renormalized fermion effective actions in radially symmetric non-Abelian backgrounds

Our recent method to calculate renormalized functional determinants, the partial wave cutoff method, is extended for the evaluation of 4-D fermion one-loop effective action with arbitrary mass in certain types of radially symmetric, non-Abelian, background gauge fields (including instanton-like and instanton-antiinstanton-like configurations). A detailed study on functional determinants for matrix-valued radial differential operators is presented, explicating both our analytic treatment on the high partial wave contribution and the application of the generalized Gel'fand-Yaglom formula to determine the low partial wave contribution. In general, some numerical work is needed for the low partial wave part. In the massless limit, however, the factorizable nature of our partial-wave radial differential operators can be exploited to evaluate semi-analytically even the low partial wave part, and we thus have the full fermion effective action calculated explicitly in a class of non-Abelian background gauge fields. With nonzero mass, we also perform necessary numerical analysis as regards the low partial wave contribution to produce numerically exact results for the massive effective action. Comparing these against the results of the large mass expansion, the validity range of the large mass expansion is addressed. Also studied is the fermion mass dependence of the effective instanton-antiinstanton interaction.

Reproducing Kernels and Radial Differential Operators for Holomorphic and Harmonic Besov Spaces on Unit Balls: a Unified View

We investigate some relations between the reproducing kernels of Hilbert spaces of holomorphic and harmonic functions on the unit balls and the radial differential operators acting on the spaces that allow their characterization via integrals of their derivatives on the balls. We compare and contrast the holomorphic and harmonic cases.

Precise Quark-Mass Dependence of the Instanton Determinant

The fermion determinant in an instanton background for a quark field of arbitrary mass is determined exactly using an efficient numerical method to evaluate the determinant of a partial-wave radial differential operator. The bare sum over partial waves is divergent but can be renormalized in the minimal subtraction scheme using the result of WKB analysis of the large partial-wave contribution. Previously, only a few leading terms in the extreme small and large mass limits were known for the corresponding effective action. Our approach works for any quark-mass and interpolates smoothly between the analytically known small and large mass expansions.

Two-body problem on two-point homogeneous spaces, invariant differential operators and the mass center concept

We consider the two-body problem with central interaction on two-point homogeneous spaces from the point of view of the invariant differential operators theory. The representation of the two-particle Hamiltonian in terms of the radial differential operator and invariant operators on the symmetry group is found. The connection of different mass center definitions for these spaces to the obtained expression for Hamiltonian operator is studied.

Besov spaces and Bergman projections on the ball

A class of radial differential operators are investigated yielding a natural classification of diagonal Besov spaces on the unit ball of C N . Precise conditions are given for the boundedness of Bergman projections from certain L p spaces onto Besov spaces. Right inverses for these projections are also provided. Applications to complex interpolation are presented. To cite this article: H.T. Kaptanoğlu, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 729–732.

Two-body problem on spaces of constant curvature: II. Spectral properties of the Hamiltonian

We consider the problem of two bodies with a central interaction on simply connected constant-curvature spaces of arbitrary dimension. We construct the self-adjoint extension of the quantum Hamiltonian, which was explicitly expressed through the radial differential operator and the generators of the isometry group of a configuration space in Part I of this paper. Exact spectral series are constructed for several potentials in the space $$\mathbb{S}^3 $$ .

Two-body problem on spaces of constant curvature: I. Dependence of the Hamiltonian on the symmetry group and the reduction of the classical system

We consider the problem of two bodies with central interaction that propagate in a simply connected space with a constant curvature and an arbitrary dimension. We obtain the explicit expression for the quantum Hamiltonian via the radial differential operator and generators of the isometry group of a configuration space. We describe the reduced classical mechanical system determined on the homogeneous space of a Lie group in terms of orbits of the coadjoint representation of this group. We describe the reduced classical two-body problem.

Reduced dynamic model of a fixed bed reactor

AbstractThe solution of dynamic distributed parameter models describing complex highly exothermic reactions in fixed bed reactors usually requires excessive computation times which are too demanding for detailed explorations. A lumping technique has been applied to a dynamic distributed parameter model, in which the radial differential operators have been approximated and effectively replaced by simple algebraic expressions by redefining the dependent variable. The resulting reduced model retains the accuracy of description and the necessary structure which is implicit in the full set of equations. This has been achieved by relating a suitably defined set of pseudoparameters to the chemical and physical processes taking place. The direct benefit of this is that the computation time and storage requirements have been reduced to a more realistic level for design purposes.