Graduate Texts in Mathematics
Abstract
* Basic Properties of Harmonic Functions * Bounded Harmonic Functions * Positive Harmonic Functions * The Kelvin Transform * Harmonic Polynomials * Harmonic Hardy Spaces * Harmonic Functions on Half-Spaces * Harmonic Bergman Spaces * The Decomposition Theorem * Annular Regions * The Dirichlet Problem and Boundary Behavior * Volume, Surface Area, and Integration on Spheres * Harmonic Function Theory and Mathematica * References * Symbol Index * Index
Chapters (11)
Harmonic functions, for us, live on open subsets of real Euclidean spaces. Throughout this book, n will denote a fixed positive integer greater than 1 and Ω will denote an open, non-empty subset of R
n
. A twice continuously differentiate, complex-valued function u defined on Ω is harmonic on Ω if $$\Delta u \equiv 0 $$ where Δ = D
12 + ... + D
n2 and D
j2 denotes the second partial derivative with respect to the j
th coordinate variable. The operator Δ is called the Laplacian, and the equation Δu = 0 is called Laplace’s equation. We say that a function u defined on a (not necessarily open) set E ⊂ R
n
is harmonic on E if u can be extended to a function harmonic on an open set containing E.
Liouville’s Theorem in complex analysis states that a bounded holomorphic function on C is constant. A similar result holds for harmonic functions on R
n
. The simple proof given below is taken from Edward Nelson’s paper [13], which is one of the rare mathematics papers not containing a single mathematical symbol.
This chapter focuses on the special properties of positive harmonic functions. We will describe the positive harmonic functions defined on all of R
n
(Liouville’s Theorem), show that positive harmonic functions cannot oscillate wildly (Harnack’s Inequality), and characterize the behavior of positive harmonic functions near isolated singularities (Bôcher’s Theorem).
The Kelvin transform performs a role in harmonic function theory analogous to that played by the transformation f(z) ↦ f(1/z) in holomorphic function theory. For example, it transforms a function harmonic inside the unit sphere into a function harmonic outside the sphere. In this chapter, we introduce the Kelvin transform and use it to solve the Dirichlet problem for the exterior of the unit sphere and to obtain a reflection principle for harmonic functions. Later, we will use the Kelvin transform in the study of isolated singularities of harmonic functions.
Recall the Dirichlet problem for the ball in R
n: given f ε C(S),find such that u is harmonic on B and u|s = f. We know from Chapter 1 that for x ε B. To prove that P[f] is harmonic on B,we computed its Laplacian by differentiating under the integral sign in the equation above and noting that for each fixed ζ ε S, the Poisson kernel is harmonic as a function of x.
In Chapter 1 we defined the Poisson integral of a function f ∈ C(S) to be the function P[f] on B given by
(6.1)
We now extend this definition: for μ a complex Borel measure on S, the Poisson integral of μ, denoted μ[p], is the function on B defined by
(6.2)
Differentiating under the integral sign in 6.2, we see that P [μ] is harmonic on B.
In this chapter we study harmonic functions defined on the upper half-space H. Harmonic function theory on H has a distinctly different flavor from that on B. One advantage of H over B is the dilationinvariance of H. We have already put this to good use in the section Limits Along Rays in Chapter 2. Some disadvantages that we will need to work around: əH is not compact, and Lebesgue measure on əH is not finite.
Throughout this chapter, p denotes a number satisfying 1 ≤ p < ∞. The Bergman space
b
p
(Ω) is the set of harmonic functions u on Ω such that $$||u||{b^p} = {\left({\int_\Omega {|u{|^p}dV} } \right)^{1/p}} <\infty
If K ⊂ Ω is compact and u is harmonic on Ω \ K, then u might be badly behaved near both ∂K and ∂Ω; see, for example, Theorem 11.18. In this chapter we will see that u is the sum of two harmonic functions, one extending harmonically across ∂K, the other extending harmonically across ∂Ω. More precisely, u has a decomposition of the form $$u = \nu + w$$ on Ω \ K, where v is harmonic on Ω and w is harmonic on R
n
\ K. Furthermore, there is a canonical choice for w that makes this decomposition unique.
An annular region is a set of the form {x ∈ R
n
: r
0 < |x| < r
1}; here r
0 ∈ [0, ∞) and r
1 ∈ (0, ∞]. Thus an annular region is the region between two concentric spheres, or is a punctured ball, or is the complement of a closed ball, or is R
n
\ {0}.
In this chapter we construct harmonic functions on Ω that behave in a prescribed manner near
. Here we are interested in general domains
; the techniques we developed for the special domains B and H will not be available. Most of this chapter will concern the Dirichlet problem. In the last section, however, we will study a different kind of boundary behavior problem—the construction of harmonic functions on Ω that cannot be extended harmonically across any part of
.
... The solutions are called harmonic functions and the theory is well elaborated, see e.g. [3,13,14]. The next natural step is to just require the condition ℓ = 0 to be fulfilled, in which case we are in the case presented by Alexander Weinstein, see [21] and also [4,11]. ...
... We denote elements x = (x ′ , x n ) ∈ R n + , where x n > 0. We observe that keeping x n fixed, the operator L admits with respect to the variable x ′ the same invariance properties as the Laplacian in R n−1 , i.e. invariance under the Euclidean rigid motions (cf. [3]). ...
In this article, we study the extended Weinstein equation where u is a sufficiently smooth function defined in with and . We find a detailed construction for a fundamental solution for the operator L. The fundamental solution is represented by the associated Legendre functions .
ERRORS: p. 352, Proposition 4.3 the second \mu should be \nu and in f(\lambda) should be \Gamma(\mu), and all the cases in rest of the paper.
... This method of XY measurement originated in analog oscilloscopes [13]. Classical harmonic analysis is based on harmonic functions [2]. A harmonic function is a twice continuously differentiable function : ...
This paper describes a technique for using time series data to monitor and diagnose faults in production systems. Predictive maintenance is the focus of research in order to assess equipment condition and anticipate potential failures. The method reduces the need for user intervention by combining Automatic Diagnostic Systems (ADS) with continuous monitoring systems to detect equipment failures early on. Three distinct techniques were tested on temperature sensor time series data: Isolation Forest (IF), Local Outlier Factor (LOF), and One-Class Support Vector Machine (OCSVM). Graphs showing how each method distinguished between normal and abnormal sensor behavior were used to evaluate each method’s efficacy in detecting anomalies. According to the results, IF is best used for ongoing monitoring, OCSVM for more thorough anomaly detection, and LOF for distinguished outlier identification. The findings are intended to improve problem detection and maintenance oversight in production systems.
... (We assume in the sequel that n > 2.) Spherical harmonics of degree j, Y j,k , k = 1, ..., N j , j ∈ N 0 , define the space H j (S n−1 ). Its dimension N j satisfies 2 j n−2 /(n − 2)! < N j ≤ nj n−2 , j ≥ 1. See [3] for the comprehensive survey concerning harmonic function theory. ...
In the first part we analyze space and its dual through Laguerre expansions when these spaces correspond to a general sequence , where * is a common notation for the Beurling and Roumieu cases of spaces. In the second part we are solving equation of the form where f belongs to the tensor product of ultradistribution spaces over compact manifolds without boundaries as well as ultradistribution spaces on and ; , and are operators whose eigenfunctions form orthonormal basis of corresponding space. The sequence space representation of solutions enable us to study the solvability and the hypoellipticity in the specified spaces of ultradistributions.
... Among many corollaries of this formula in the study of potential theory, we single out the equation Ω ∆vdA = ∂Ω ∂v ∂n ds and the two versions of mean-value property for median functions given in the classical case for example in Brelot [6] and Axler, Bourdon, Ramey [3]. For a detailed study of discrete potential theory on infinite networks, see Saordi [19]. ...
In the context of random walks whose states are the vertices of an infinite tree, a classification of random walks is given as transient or recurrent. On the infinite homogeneous trees with the assumption that the transition probability between any two neighboring states are the same, a form of the classical Green’s formula is derived. As a consequence, two versions of the mean-value property for median functions are obtained.
... for some constant C, where f is a holomorphic function in |ζ | < ρ −1 and g a holomorphic function in |ζ | > ρ such that g(ζ ) → 0 as |ζ | → ∞ (see, for example, [20]). Because of the third condition in (LC-C-A), we have C = 0. Note that D is symmetric with respect to the imaginary axis. ...
If two conducting or insulating inclusions are closely located, the gradient of the solution may become arbitrarily large as the distance between inclusions tends to zero, resulting in high concentration of stress in between two inclusions. This happens if the bonding of the inclusions and the matrix is perfect, meaning that the potential and flux are continuous across the interface. In this paper, we consider the case when the bonding is imperfect. We consider the case when there are two circular inclusions of the same radii with the imperfect bonding interfaces and prove that the gradient of the solution is bounded regardless of the distance between inclusions if the bonding parameter is finite. This result is of particular importance since the imperfect bonding interface condition is an approximation of the membrane structure of biological inclusions such as biological cells.
... For every m ≥ 0, consider the slice regular power x m : H → H. Then it holds(x m ) s =Z m−1 (x), whereZ k (x) := ⎧ ⎨ ⎩ Z k (x, 1) k + 1 if k ≥ 0 0 if k = −1and Z k (x, 1) is the real valued zonal harmonic of R 4 with pole 1 (see[4, Ch. 5]). ...
In this paper we propose an Almansi-type decomposition for slice regular functions of several quaternionic variables. Our method yields distinct and unique decompositions for any slice function with domain in . Depending on the choice of the decomposition, every component is given explicitly, uniquely determined and exhibits desirable properties, such as harmonicity and circularity in the selected variables. As consequences of these decompositions, we give another proof of Fueter’s Theorem in , establish the biharmonicity of slice regular functions in every variable and derive mean value and Poisson formulas for them.
During the era of NASA’s Apollo missions, Keith S. Runcorn proposed an explanation of discrepancy between the Moon’s negligible global magnetic field and magnetized samples of lunar regolith, based on identical vanishing of external magnetic field of a spherical shell, magnetized by an internal source which is no longer present. We revisit and generalize the Runcorn’s result, showing that it is a consequence of a (weighted) orthogonality of gradients of harmonic functions on a spherical shell in arbitrary number of dimensions. Furthermore, we explore bounds on external magnetic field in the case when the idealized spherical shell is replaced with a more realistic geometric shape and when the thermoremanent magnetization susceptibility deviates from the spherical symmetry. Finally, we analyse a model of thermoremanent magnetization acquired by crustal inward cooling of a spherical astrophysical body and put some general bounds on the associated magnetic field.
In this paper, we study the fully fractional master equation 0.1First we prove a Liouville type theorem for the homogeneous equation 0.2where . When u belongs to the slowly increasing function space and satisfies an additional asymptotic assumption in the case , we prove that all solutions of (0.2) must be constant. This result includes the previous Liouville theorems on harmonic functions [1] and on s-harmonic functions [7] as special cases. Then we establish the equivalence between nonhomogeneous pseudo-differential equations (0.1) and the corresponding integral equations. We believe that these integral equations will become very useful tools in further analysing qualitative properties of solutions, such as regularity, monotonicity, and symmetry. In the process of deriving the Liouville type theorem, through very delicate calculations, we obtain an optimal estimate on the decay rate of . This sharp estimate will become a key ingredient and an important tool in investigating master equations.
Recently an algorithm was given in [Garde & Hyvönen, SIAM J. Math. Anal., 2024] for exact direct reconstruction of any L2 perturbation from linearized data in the two-dimensional linearized Calderón problem. It was a simple forward substitution method based on a two-dimensional Zernike basis. We now consider the three-dimensional linearized Calderón problem in a ball, and use a three-dimensional Zernike basis to obtain a method for exact direct reconstruction of any L3 perturbation from linearized data. The method is likewise a forward substitution, hence making it very efficient to numerically implement. Moreover, the three-dimensional method only makes use of a relatively small subset of boundary measurements for exact reconstruction, compared to a full L2 basis of current densities.
Boundary conforming coordinates are commonly used in plasma physics to describe the geometry of toroidal domains, for example, in three-dimensional magnetohydrodynamic equilibrium solvers. The magnetohydrodynamic equilibrium configuration can be approximated with an inverse map, defining nested surfaces of constant magnetic flux. For equilibrium solvers that solve for this inverse map iteratively, the initial guess for the inverse map must be well-defined and invertible. Even if magnetic islands are to be included in the representation, boundary conforming coordinates can still be useful, for example to parametrize the interface surfaces in multi-region, relaxed magnetohydrodynamics or as a general-purpose, field-agnostic coordinate system in strongly shaped domains. Given a fixed boundary shape, finding a valid boundary conforming mapping can be challenging, especially for the non-convex boundaries from recent developments in stellarator optimization. In this work, we propose a new algorithm to construct such a mapping, by solving two Dirichlet–Laplace problems via a boundary integral method. We can prove that the generated harmonic map is always smooth and has a smooth inverse. Furthermore, we can find a discrete approximation of the mapping that preserves this property.
As a refinement of the global invertibility problem, we address the issue of estimating the cardinality of a prescribed fiber F - 1 ( q ) of a locally invertible map solely in terms of objects that are naturally associated to q itself. The following is a prototypical result. Let F : R n → R n be a local diffeomorphism, n ≥ 3 , and q ∈ F ( R n ) . We show that q is assumed exactly once by F if the pre-image of every 2-plane containing q , when viewed as a geometric surface in Euclidean n -space, is conformally diffeomorphic to R 2 . The proofs of this and other theorems involve geometric constructions, the Poincaré-Hopf theorem, the Bôcher theorem on positive harmonic functions, condensers on Riemann surfaces, and elliptic estimates. We conclude with a section that is devoted to invertibility problems related to various aspects of dynamics, algebraic and differential geometry, real and complex analysis. The paper is written in a semi-expository style, as an invitation to global injectivity.
We characterize the critical points of the double bubble problem in R n \mathbb {R}^n and the triple bubble problem in R 3 \mathbb {R}^3 , in the case the bubbles are convex.
Vibration is an inevitable phenomenon in vehicle dynamics. In this chapter, we review the principles of vibrations, analysis methods, and their applications, along with the frequency and time responses of vibrating systems. Special attention is devoted to frequency response analysis, because most of the optimization methods for vehicle suspensions and vehicle vibrating components are based on frequency responses.
Vehicles are multiple DOF vibrating systems as is shown in Fig. 12.1. Vibration behavior of a vehicle, which is called ride, is highly dependent on the natural frequencies and mode shapes of the vehicle. In this chapter, we review and examine the applied methods of determining the equations of motion, natural frequencies, and mode shapes of different models of vehicles.
A full car vibrating model of a vehicle
We study the low-temperature (2+1) D solid-on-solid model on with zero boundary conditions and nonnegative heights (a floor at height 0 ). Caputo et al. (2016) established that this random surface typically admits either or many nested macroscopic level line loops for an explicit , and its top loop has cube-root fluctuations: For example, if is the vertical displacement of from the bottom boundary point (x,0) , then over . It is believed that rescaling by and by would yield a limit law of a diffusion on . However, no nontrivial lower bound was known on for a fixed (e.g., ), let alone on in , to complement the bound on . Here, we show a lower bound of the predicted order : For every , there exists such that with probability at least . The proof relies on the Ornstein–Zernike machinery due to Campanino–Ioffe–Velenik and a result of Ioffe, Shlosman and Toninelli (2015) that rules out pinning in Ising polymers with modified interactions along the boundary. En route, we refine the latter result into a Brownian excursion limit law, which may be of independent interest. We further show that in a box with boundary conditions (i.e., on the bottom side and elsewhere), the limit of as is a Ferrari–Spohn diffusion.
This paper contains a heuristic presentation of a new method for testing whether the mapping f from a simply connected domain O from compactified n-dimensional Euclidean space to compactified n-dimensional Euclidean space, is constant or when it has a good boundary property. Namely, two theorems have been proved which show that the mapping will be constant or have some good boundary property depending on where the fixed points and points of attraction of the continuous automorphism g of domain O, which generates the cyclic group with respect to which f is normal, are located.
The classical problem of removable singularities is considered for solutions to the stationary Navier–Stokes system in dimension and an old theorem of Shapiro (TAMS 187:335–363, 1974) is recovered and extended to solutions in a half ball vanishing on the flat boundary. Moreover, for n=4 it is proved that there are not distributional solutions, smooth away from the singularity and such that .
For λ ≥ 0 , a function f of z=x+iy z = x + i y defined on a domain of the complex plane C , symmetric about y -axis, is said to be λ -analytic if ( D x + i ∂ y ) f = 0 , where D x is the Dunkl operator on the real line given by D x φ ( x ) = ∂ x φ ( x ) + ( λ / x ) φ ( x ) - φ ( - x ) . In the paper we study the Bergman space B λ p ( C + ) associated with λ -analytic functions on the upper half-plane C + , and also its harmonic analog b λ p ( C + ) . The reproducing kernels of B λ p ( C + ) and b λ p ( C + ) are determined and the reproducing formulas of functions in B λ p ( C + ) and b λ p ( C + ) are proved. The associated Bergman projections are proved to be bounded for 1 < p < ∞ , and the completeness for 1 ≤ p < ∞ and the duality for 1 < p < ∞ of B λ p ( C + ) and b λ p ( C + ) are also considered.
In this paper, we prove that every bounded linear operator on a separable Hilbert space has a non-trivial invariant subspace. This answers the well-known invariant subspace problem.
An example of Cornalba and Shiffman from 1972 disproves in dimension two or higher a classical prediction that the count of zeros of holomorphic self-mappings of the complex linear space should be controlled by the maximum modulus function. We prove that such a bound holds for a modified coarse count inspired by the theory of persistence modules originating in topological data analysis.
We establish resolvent estimates in spaces for the Stokes operator in a bounded domain in . As a corollary, it follows that the Stokes operator generates a bounded analytic semigroup in for any and . The case of an exterior domain is also studied.
Sphere tracing is a fast and high-quality method for visualizing surfaces encoded by signed distance functions (SDFs). We introduce a similar method for a completely different class of surfaces encoded by harmonic functions , opening up rich new possibilities for visual computing. Our starting point is similar in spirit to sphere tracing: using conservative Harnack bounds on the growth of harmonic functions, we develop a Harnack tracing algorithm for visualizing level sets of harmonic functions, including those that are angle-valued and exhibit singularities. The method takes much larger steps than naïve ray marching, avoids numerical issues common to generic root finding methods and, like sphere tracing, needs only perform pointwise evaluation of the function at each step. For many use cases, the method is fast enough to run real time in a shader program. We use it to visualize smooth surfaces directly from point clouds (via Poisson surface reconstruction) or polygon soup (via generalized winding numbers) without linear solves or mesh extraction. We also use it to visualize nonplanar polygons (possibly with holes), surfaces from architectural geometry, mesh "exoskeletons", and key mathematical objects including knots, links, spherical harmonics, and Riemann surfaces. Finally we show that, at least in theory, Harnack tracing provides an alternative mechanism for visualizing arbitrary implicit surfaces.
By the dimension reduction idea, overshoot for random walks, coupling and martingale arguments, we obtain a simpler and easily computable expression for the first-order correction constant between discrete harmonic measures for random walks with rotationally invariant step distribution in Rd(d≥2) and the corresponding continuous counterparts. This confirms and extends a conjecture in Jiang and Kennedy (J Theor Probab 30(4):1424–1444, 2017), and simplifies the related expression of Wang et al. (Bernoulli 25(3):2279–2300, 2019). Furthermore, we propose a universality conjecture on high-order corrections for error estimation between generalized discrete harmonic measures and their continuous counterparts, which generalizes the universality conjecture of the first-order correction in Kennedy (J Stat Phys 164(1):174–189, 2016); and we prove this conjecture heuristically for the rotationally invariant case, and also provide several examples of second-order error corrections to check the conjecture by a numerical simulation argument.
We carry out complete membership to Kaplan classes of functions given by formula { ζ ∈ C : | ζ | < 1 } ∋ z ↦ ∏ k = 1 n ( 1 - z e - i t k ) p k , where n ∈ N , t k ∈ [ 0 ; 2 π ) and p k ∈ R for k ∈ N ∩ [ 1 ; n ] . In this way we extend Sheil-Small’s, Jahangiri’s and our previous results. Moreover, physical and geometric applications of the obtained gap condition are given. The first one is an interpretation in terms of mass and density. The second one is a visualization in terms of angular inequalities between vectors in R 2 .
In this paper we consider steady inviscid three-dimensional stratified water flows of finite depth with a free surface and an interface. The interface plays the role of an internal wave that separates two layers of constant and different density. We study two cases separately: when the free surface and the interface are functions of one variable and when the free surface and the interface are functions of two variables. In both cases, considering effects of surface tension, we prove that the bounded solutions to the three-dimensional equations are essentially two-dimensional. More specifically, assuming that the vorticity vectors in the two layers are constant, non-vanishing and parallel to each other we prove that their third coordinate vanishes in both layers. Also we prove that the free surface, the interface, the pressure and the velocity field present no variations in the direction orthogonal to the direction of motion.
It is a classical theorem that if a function on the unit circle is integrable, then it is the nontangential limit of a holomorphic function on the open disc (subject to a certain growth condition) if and only if its Fourier coefficients for nonnegative integers are zero. In this article we generalize this result to higher complex dimensions by proving that for an integrable function on the unit sphere, it is a “boundary trace” of a holomorphic function on the open unit ball if and only if two particular families of integral equations are satisfied. To do this, we use the theory of Hardy spaces as well as the invariant Poisson and Cauchy integrals.
In this paper, we establish three circles theorem for volume of conformal metrics whose scalar curvatures are integrable in a critical (scaling invariant) norm. As applications, we analyze the asymptotic behavior of such metrics near isolated singularities and use it to show the residual terms of the Chern–Gauss–Bonnet formula are integers. Such strong rigidity implies a vanishing theorem on the integral value of the curvature, with application to the bi-Lipschitz equivalence problem for conformal metrics.
We show that if f is a positive harmonic function on a biregular tree which has maximal growth along an infinite path in the tree, then every harmonic function g on the tree with 0 ≤ g ≤ f is a multiple of f, thus generalizing a result of Cartier about regular trees.
A bstract
A new approach is presented to compute entropy for massless scalar quantum fields. By perturbing a skewed correlation matrix composed of field operator correlation functions, the mutual information is obtained for disjoint spherical regions of size r at separation R , including an expansion to all orders in r / R . This approach also permits a perturbative expansion for the thermal field entropy difference in the small temperature limit ( T ≪ 1/ r ).
We discuss to what extent certain results about totally ramified values of entire and meromorphic functions remain valid if one relaxes the hypothesis that some value is totally ramified by assuming only that all islands over some Jordan domain are multiple. In particular, we prove a result suggested by Bloch which says that an entire function of order less than 1 has a simple island over at least one of two given Jordan domains with disjoint closures.
We study the Bloch and the little Bloch spaces of harmonic functions on the real hyperbolic ball. We show that the Bergman projections from L ∞ ( B ) to B , and from C 0 ( B ) to B 0 are onto. We verify that the dual space of the hyperbolic harmonic Bergman space B α 1 is B and its predual is B 0 . Finally, we obtain atomic decompositions of Bloch functions as series of Bergman reproducing kernels.
In this chapter, we provide the main notation and facts from real and functional analysis. We introduce the spaces of smooth functions, the Sobolev spaces, and the spaces of divergence-free vector functions, which play a fundamental role in studying the Navier–Stokes equations. Additionally, we describe some results from general topology, geometric measure theory, and harmonic analysis that are used in this book. Finally, we formulate some basic properties of solutions to elliptic equations. Our aim is not to present an exhaustive treatment of the auxiliary results. Most of these facts are given without proof, and for more detailed study, we provide references to specialized literature. However, we do present proofs of selected assertions from geometric measure theory and harmonic analysis, which, in our opinion, are not as commonly found in the classical theory of differential equations. We have endeavored to present proofs that are not overly complicated, but the choice of statements to be proved depends on our preferences.
A bosonic Laplacian is a conformally invariant second order differential operator acting on smooth functions defined on domains in Euclidean space and taking values in higher order irreducible representations of the special orthogonal group, in this case, the irreducible representation spaces of homogeneous harmonic polynomials. In this paper, we study boundary value problems involving bosonic Laplacians in the upper-half space and the unit ball. Poisson kernels in the upper-half space and the unit ball are constructed, which give us solutions to the Dirichlet problems with Lp boundary data, 1≤p≤∞. We also prove the uniqueness for solutions to the Dirichlet problems with continuous data for bosonic Laplacians and provide analogs of some properties of harmonic functions for null solutions of bosonic Laplacians, for instance, Cauchy’s estimates, the mean-value property, Liouville’s Theorem, etc.
The main purpose of this paper is to investigate characterizations of composition operators on Bloch and Hardy type spaces. Initially, we use general doubling weights to study the composition operators from harmonic Bloch type spaces on the unit disc D to pluriharmonic Hardy spaces on the Euclidean unit ball Bn. Furthermore, we develop some new methods to study the composition operators from harmonic Bloch type spaces on D to pluriharmonic Bloch type spaces on D. Additionally, some application to new characterizations of the composition operators between pluriharmonic Lipschitz type spaces to be bounded or compact will be presented. The obtained results of this paper provide the improvements and extensions of the corresponding known results.
We extend the recent classification of five-dimensional, supersymmetric asymptotically flat black holes with only a single axial symmetry to black holes with Kaluza–Klein asymptotics. This includes a similar class of solutions for which the supersymmetric Killing field is generically timelike, and the corresponding base (orbit space of the supersymmetric Killing field) is of multi-centred Gibbons–Hawking type. These solutions are determined by four harmonic functions on R 3 with simple poles at the centres corresponding to connected components of the horizon, and fixed points of the axial symmetry. The allowed horizon topologies are S 3 , S 2 × S 1 , and lens space L ( p , 1), and the domain of outer communication may have non-trivial topology with non-contractible 2-cycles. The classification also reveals a novel class of supersymmetric (multi-)black rings for which the supersymmetric Killing field is globally null. These solutions are determined by two harmonic functions on R 3 with simple poles at centres corresponding to horizon components. We determine the subclass of Kaluza–Klein black holes that can be dimensionally reduced to obtain smooth, supersymmetric, four-dimensional multi-black holes. This gives a classification of four-dimensional asymptotically flat supersymmetric multi-black holes first described by Denef et al.
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