PoissonSemigroup

The Poisson Semigroup is a family of operators associated with the classical Poisson kernel, widely used in harmonic analysis and partial differential equations. It provides a fundamental tool for understanding harmonic functions and boundary value problems in Euclidean spaces.

Formally, for a function \(f \in L^p(\mathbb{R}^n)\), the Poisson semigroup \(\{P_t\}_{t \geq 0}\) is defined by convolution with the Poisson kernel \(P_t(x)\):

\[

P_t f(x) = (P_t * f)(x) = \int_{\mathbb{R}^n} P_t(x - y)f(y) dy,

\]

where the Poisson kernel is given by

\[

P_t(x) = c_n \frac{t}{(t^2 + |x|^2)^{\frac{n+1}{2}}},

\]

with \(c_n\) being a constant depending on the dimension \(n\).

This family of operators forms a strongly continuous semigroup, satisfying the properties \(P_0 f = f\) and \(P_{s + t} f = P_s (P_t f)\) for all \(s, t \geq 0\). The Poisson semigroup acts as a smoothing operator, extending functions harmonically into the upper half-space and preserving the harmonicity of functions.

In analysis, the Poisson semigroup is instrumental in studying boundary behavior of harmonic and subharmonic functions, as well as in deriving maximal functions and square function estimates. Its connections with harmonic analysis, potential theory, and PDEs make it a central object in mathematical analysis.