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The kernel can be understood as the derivative of the Green's function for the Laplace equation. It is named for Siméon Poisson. Poisson kernels commonly find applications in control theory and two-dimensional problems in electrostatics. In practice, the definition of Poisson kernels are often extended to n-dimensional problems.
The Dirichlet problem for harmonic functions always has a solution, and that solution is unique, ... The integrand is known as the Poisson kernel; ...
Each function above will yield another harmonic function when multiplied by a constant, rotated, and/or has a constant added. The inversion of each function will yield another harmonic function which has singularities which are the images of the original singularities in a spherical "mirror". Also, the sum of any two harmonic functions will ...
The higher Riesz transforms of the Poisson kernel can be computed: = (| | +) / + for k ≥ 1 and the complex conjugate for − k. Indeed, the right hand side is a harmonic function F(x,y,s) of three variable and for such functions [18]
the Poisson kernel is the real part of the integrand above; the real part of a holomorphic function is harmonic and determines the holomorphic function up to addition of a scalar; the above formula defines a holomorphic function, the real part of which is given by the previous theorem
The Martin kernels are positive harmonic functions and every positive harmonic function can be expressed as an integral of functions on the boundary, that is for every positive harmonic function there is a measure , on such that a Poisson-like formula holds:
The function F defined on the unit disk by F(re iθ) = (f ∗ P r)(e iθ) is harmonic, and M f is the radial maximal function of F. When M f belongs to L p (T) and p ≥ 1, the distribution f "is" a function in L p (T), namely the boundary value of F. For p ≥ 1, the real Hardy space H p (T) is a subset of L p (T).
where P is the Poisson kernel. Any function f on the disc determines a function on the group of Möbius transformations of the disc by setting F(g) = f(g(0)). Then the Poisson formula has the form = | | = ^ () where m is the Haar measure on the boundary. This function is then harmonic in the sense that it satisfies the mean-value property with ...