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In mathematics, and specifically inpotential theory, thePoisson kernel is anintegral kernel, used for solving the two-dimensionalLaplace equation, givenDirichlet boundary conditions on theunit disk. The kernel can be understood as thederivative of theGreen's function for the Laplace equation. It is named forSiméon Poisson.

Poisson kernels commonly find applications incontrol theory and two-dimensional problems inelectrostatics.In practice, the definition of Poisson kernels are often extended ton-dimensional problems.

In thecomplex plane, the Poisson kernel for the unit disc^[1] is given by

This can be thought of in two ways: either as a function ofr andθ, or as a family of functions ofθ indexed byr.

If is the openunit disc inC,T is the boundary of the disc, andf a function onT that lies inL¹(T), then the functionu given byisharmonic inD and has a radial limit that agrees withfalmost everywhere on the boundaryT of the disc.

That the boundary value ofu isf can be argued using the fact that asr → 1, the functionsP_r(θ) form anapproximate unit in theconvolution algebraL¹(T). As linear operators, they tend to theDirac delta function pointwise onL^p(T). By themaximum principle,u is the only such harmonic function onD.

Convolutions with this approximate unit gives an example of asummability kernel for theFourier series of a function inL¹(T) (Katznelson 1976). Letf ∈L¹(T) have Fourier series {f_k}. After theFourier transform, convolution withP_r(θ) becomes multiplication by the sequence {r^|k|} ∈ℓ¹(Z).^{[further explanation needed]} Taking the inverse Fourier transform of the resulting product {r^|k|f_k} gives theAbel meansA_rf off:$${\displaystyle A_{r}f(e^{2\pi ix})=\sum _{k\in \mathbb{Z}}{f}_k{r}^{|k|}{e}^{2\pi ikx}.}$$

Rearranging thisabsolutely convergent series shows thatf is the boundary value ofg +h, whereg (resp.h) is aholomorphic (resp.antiholomorphic) function onD.

When one also asks for the harmonic extension to be holomorphic, then the solutions are elements of aHardy space. This is true when the negative Fourier coefficients off all vanish. In particular, the Poisson kernel is commonly used to demonstrate the equivalence of the Hardy spaces on the unit disk, and the unit circle.

The space of functions that are the limits onT of functions inH^p(z) may be calledH^p(T). It is a closed subspace ofL^p(T) (at least forp ≥ 1). SinceL^p(T) is aBanach space (for 1 ≤ p ≤ ∞), so isH^p(T).

Theunit disk may beconformally mapped to theupper half-plane by means of certainMöbius transformations. Since the conformal map of a harmonic function is also harmonic, the Poisson kernel carries over to the upper half-plane. In this case, the Poisson integral equation takes the form$${\displaystyle u(x+iy)={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{P}_y(x-t)f(t)dt,y>0.}$$

The kernel itself is given by$${\displaystyle P_y(x)=\frac{1}{\pi }\frac{y}{x^2+y^2}.}$$

Given a function${\displaystyle f\in L^p(\mathbb{R})}$, theL^p space of integrable functions on the real line,u can be understood as a harmonic extension off into the upper half-plane. In analogy to the situation for the disk, whenu is holomorphic in the upper half-plane, thenu is an element of the Hardy space,${\displaystyle H^p,}$ and in particular,$${\displaystyle \Vert u{\Vert }_{H^p}=\Vert f{\Vert }_{L^p}}$$

Thus, again, the Hardy spaceH^p on the upper half-plane is aBanach space, and, in particular, its restriction to the real axis is a closed subspace of${\displaystyle L^p(\mathbb{R}).}$ The situation is only analogous to the case for the unit disk; theLebesgue measure for the unit circle is finite, whereas that for the real line is not.

For the ball of radius${\displaystyle r,B_r\subset {\mathbb{R}}^n,}$ the Poisson kernel takes the form$${\displaystyle P(x,\zeta )=\frac{r^2-|x{|}^2}{r{\omega }_{n-1}|x-\zeta {|}^n}}$$where${\displaystyle x\in B_r,\zeta \in S}$ (the surface of${\displaystyle B_r}$), and${\displaystyle {\omega }_{n-1}}$ is thesurface area of the unit (n − 1)-sphere.

Then, ifu(x) is a continuous function defined onS, the corresponding Poisson integral is the functionP[u](x) defined by$${\displaystyle P[u](x)={\int }_{S}u(\zeta )P(x,\zeta )d\sigma (\zeta ).}$$

It can be shown thatP[u](x) is harmonic on the ball${\displaystyle B_r}$ and thatP[u](x) extends to a continuous function on the closed ball of radiusr, and the boundary function coincides with the original function u.

An expression for the Poisson kernel of anupper half-space can also be obtained. Denote the standard Cartesian coordinates of${\displaystyle {\mathbb{R}}^{n+1}}$ by$${\displaystyle (t,x)=(t,x_1,\dots ,x_n).}$$ The upper half-space is the set defined by$${\displaystyle H^{n+1}=\left\{(t;x)\in {\mathbb{R}}^{n+1}\mid t>0\right\}.}$$The Poisson kernel forHⁿ⁺¹ is given by$${\displaystyle P(t,x)=c_n\frac{t}{{\left(t^2+{\Vert x\Vert }^2\right)}^{(n+1)/2}}}$$where$${\displaystyle c_n=\frac{\mathrm{\Gamma }[(n+1)/2]}{{\pi }^{(n+1)/2}}.}$$

The Poisson kernel for the upper half-space appears naturally as theFourier transform of theAbel transformin whicht assumes the role of an auxiliary parameter. To wit,$${\displaystyle P(t,x)=\mathcal{F}(K(t,\cdot ))(x)={\int }_{{\mathbb{R}}^n}{e}^{-2\pi t|\xi |}{e}^{-2\pi i\xi \cdot x}d\xi .}$$In particular, it is clear from the properties of the Fourier transform that, at least formally, the convolution$${\displaystyle P[u](t,x)=[P(t,\cdot )\ast u](x)}$$is a solution of Laplace's equation in the upper half-plane. One can also show that ast → 0,P[u](t,x) →u(x) in a suitable sense.

• Schwarz integral formula

• Katznelson, Yitzhak (1976),An introduction to Harmonic Analysis, Dover,ISBN 0-486-63331-4
• Conway, John B. (1978),Functions of One Complex Variable I, Springer-Verlag,ISBN 0-387-90328-3.
• Axler, S.; Bourdon, Paul; Ramey, Wade (1992),Harmonic Function Theory, Springer-Verlag,ISBN 0-387-95218-7.
• King, Frederick W. (2009),Hilbert Transforms Vol. I, Cambridge University Press,ISBN 978-0-521-88762-5.
• Stein, Elias;Weiss, Guido (1971),Introduction to Fourier Analysis on Euclidean Spaces, Princeton University Press,ISBN 0-691-08078-X.
• Weisstein, Eric W."Poisson Kernel".MathWorld.
• Gilbarg, D.;Trudinger, N. (12 January 2001),Elliptic Partial Differential Equations of Second Order, Springer,ISBN 3-540-41160-7.

• Fourier analysis
• Harmonic functions
• Potential theory

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