Inequalities for the solutions of the heat equation

Abstract

For the solutions u (t, x) of the heat equation

$${\partial _t}u = \Delta u\quad {\text{on}}\quad {\mathbb{R}^ + } \times {\mathbb{R}^n}$$

satisfying the initial condition

$$u\left( {0,x} \right) = F\left( x \right)\quad {\text{on}}\quad {\mathbb{R}^n}$$

for L₂(ℝⁿ, dx) functions F, inequalities for the functions D ^α u (t, x) for any fixed x are derived from both points of view of analyticity in t and integral transforms by the heat kernel.