Harmonic Function Theory

Axler, Sheldon; Bourdon, Paul S.; Ramey, Wade

doi:10.1007/b97238

Cited by 365 publications

(255 citation statements)

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“…, M d,l , is an orthonormal basis of L 2 (µ). For further properties of harmonic polynomials, see [1,7]. Recall that the Jacobi polynomials are defined as…”

Section: Unit Ball and Its Complementmentioning

confidence: 99%

Fractional Laplace Operator and Meijer G-function

2016

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We significantly expand the number of functions whose image under the fractional Laplace operator can be computed explicitly. In particular, we show that the fractional Laplace operator maps Meijer G-functions of |x| 2 , or generalized hypergeometric functions of −|x| 2 , multiplied by a solid harmonic polynomial, into the same class of functions. As one important application of this result, we produce a complete system of eigenfunctions of the operator (1 − |x| 2 ) α/2 + (−∆) α/2 with the Dirichlet boundary conditions outside of the unit ball. The latter result will be used to estimate the eigenvalues of the fractional Laplace operator in the unit ball in a companion paper [9].

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“…, M d,l , is an orthonormal basis of L 2 (µ). For further properties of harmonic polynomials, see [1,7]. Recall that the Jacobi polynomials are defined as…”

Section: Unit Ball and Its Complementmentioning

confidence: 99%

Fractional Laplace Operator and Meijer G-function

2016

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“…Hence / is finite or countable. Thus by applying Bocher's theorem, the 'minimum' and Harnack's principles (see [1]) we obtain (4A} " (! ' )=o+ 5^i…”

Section: Harmonic Morphismsmentioning

confidence: 99%

“…(i) Riemannian foliations locally generated by Killing fields and 1 The author gratefully acknowledges the support of the O.R.S. Scheme Awards, the School of Mathematics of the University of Leeds, the Tetley and Lupton Scholarships and the Edward Boyle Bursary.…”

Section: Introductionmentioning

confidence: 99%

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Harmonic morphisms with 1-dimensional fibres on 4-dimensional Einstein manifolds

Pantilie¹

2002

Communications in Analysis and Geometry

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“…Recall that (see [2] or Axler et al [10]) given a harmonic function h outside a compact set in R n , there exists a harmonic function H on R n and a constant α such that outside a compact set,…”

Section: H * Functions Defined Near Infinitymentioning

confidence: 99%