Math 2.0

Potential theory Definition

Harmonic majorant

Builds on Open setHarmonic functionSubharmonic functionLiouville's theorem for harmonic functionsHardy space

Definition

Let n≥2n \ge 2, let Ω⊆ℝn\Omega \subseteq \R^n be an open set, and let u ⁣:Ω→[−∞,∞)u \colon \Omega \to [-\infty, \infty) be a function, usually subharmonic. A harmonic majorant of uu is a harmonic function hh on Ω\Omega such that u(x)≤h(x)u(x) \le h(x) for every x∈Ωx \in \Omega.

Remarks

Symmetrically, a harmonic minorant of vv is a harmonic hh with h≤vh \le v, so hh is a harmonic minorant of vv exactly when −h-h is a harmonic majorant of −v-v. For a holomorphic ff on the unit disc and 0<p<∞0 < p < \infty, ff lies in the Hardy space HpH^p if and only if the subharmonic function ∣f∣p\abs{f}^p has a harmonic majorant.

Example: on the unit disc 𝔻⊆ℂ\mathbb{D} \subseteq \C, let u(z)=∣z∣2u(z) = \abs{z}^2. The constant h=1h = 1 is harmonic and ∣z∣2<1\abs{z}^2 < 1 on 𝔻\mathbb{D}, so hh is a harmonic majorant of uu.

Non-example: on all of ℂ\C, u(z)=∣z∣2u(z) = \abs{z}^2 has no harmonic majorant. A harmonic hh with h≥∣z∣2≥0h \ge \abs{z}^2 \ge 0 would be a harmonic function bounded below on ℂ\C, hence constant by Liouville's theorem for harmonic functions. No constant is at least ∣z∣2\abs{z}^2 for all zz.

Reading path

To understand this page from the ground up, read these 77 pages in order. The foundations come first.

  1. Complex numbers object
  2. Euclidean space object
  3. Function concept
  4. Hardy space concept
  5. Infimum concept
  6. Open set concept
  7. Real numbers object
  8. Set bounded above definition
  9. Subharmonic function concept
  10. Supremum concept
  11. Unit disc object
  12. Complex exponential definition
  13. Contour integral definition
  14. Power series in the complex plane definition
  15. Real and imaginary parts definition
  16. Fréchet derivative definition
  17. Total derivative concept
  18. Function of several variables concept
  19. Vector-valued function concept
  20. Jordan measure definition
  21. Supremum definition
  22. Circle integral definition
  23. Disc of convergence definition
  24. Limit of a function of several variables concept
  25. Vector field concept
  26. Differentiable function of several variables concept
  27. Iterated integral definition
  28. Partial derivative concept
  29. Completeness axiom statement
  30. Continuity in R^n definition
  31. Differentiable function of several variables definition
  32. Partial derivative definition
  33. Divergence concept
  34. Gradient concept
  35. Existence of square roots of nonnegative reals statement
  36. Differentiable on an open set definition
  37. Higher partial derivatives definition
  38. Gradient definition
  39. Double integral definition
  40. Jacobian matrix definition
  41. Laplacian concept
  42. Square root definition
  43. Iterated derivative definition
  44. Fubini's theorem statement
  45. Modulus of a complex number definition
  46. Triple integral definition
  47. Harmonic function concept
  48. C^k function characterised by continuous derivatives statement
  49. Complex differentiability definition
  50. Poisson kernel definition
  51. Holomorphic function definition
  52. Cauchy–Riemann equations statement
  53. Parametrised curve definition
  54. Vector field definition
  55. Harnack's inequality statement
  56. Cauchy's integral formula statement
  57. Power series are holomorphic in the disc of convergence statement
  58. Parametrised surface definition
  59. Closed curve definition
  60. Rectifiable curve definition
  61. Divergence definition
  62. Holomorphic functions are analytic statement
  63. Tangent plane and normal vector of a surface definition
  64. Line integral of a vector field definition
  65. Holomorphic functions are infinitely differentiable statement
  66. Green's theorem statement
  67. Surface area definition
  68. The real part of a holomorphic function is harmonic statement
  69. Surface integral definition
  70. Divergence theorem on a rectangle in the plane statement
  71. Mean value property statement
  72. Orientable surface definition
  73. Flux through a surface definition
  74. Divergence theorem statement
  75. Gradient estimates for harmonic functions statement
  76. Liouville's theorem for harmonic functions statement
  77. Harmonic majorant definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Subharmonic_function
About this page
Tag6GWZN8FXC6
URIm2://6GWZN8FXC6
Kinddefinition
WritingW0 stub
MSC31B05, 31A05
DefinesHarmonic majorant
UsesOpen set, Harmonic function, Subharmonic function, Liouville's theorem for harmonic functions, Hardy space

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