Inequalities for the solutions of the heat equation

摘要: For the solutions u (t, x) of heat equation $${\partial _t}u = \Delta u\quad {\text{on}}\quad {\mathbb{R}^ + } \times {\mathbb{R}^n}$$ satisfying initial condition $$u\left( {0,x} \right) F\left( x \right)\quad {\mathbb{R}^n}$$ for L2(ℝn, dx) functions F, inequalities for D α any fixed are derived from both points view analyticity in t and integral transforms by kernel.