= Solution
For positive time, convolution with the <heat kernel> gives
$$
w(t,x)=(K_t*g)(x),\qquad K_t(x)=(4\pi t)^{-n/2}e^{-|x|^2/(4t)}.
$$
Differentiation under the integral verifies the <heat equation>, and the <approximate identity> property gives the initial value. Taking absolute values under the integral and using $e^{-|x-y|^2/(4t)}\leq1$ gives
$$
\boxed{|w(t,x)|\leq(4\pi t)^{-n/2}\|g\|_1.}
$$
The printed inequality omits the modulus even though $g$ is complex-valued. A complex number has no compatible order, so the displayed absolute-value estimate is the meaningful correction.
Substitution of $u=e^{it}v$ reduces the forced <partial differential equation> to $(i-\Delta)v=f$. With the <Fourier transform> convention for which $\widehat{\Delta v}=-|\xi|^2\widehat v$, set
$$
\widehat v(\xi)=\frac{\widehat f(\xi)}{i+|\xi|^2}.
$$
The denominator never vanishes for real $\xi$. Every derivative of its reciprocal has at most <polynomial> growth, so multiplying a <Schwartz function> by it gives another <Schwartz function>. The inverse <Fourier transform> therefore supplies the required $v$.
For the prescribed initial value the full solution is $u(t,x)=e^{it}v(x)+K_t*(g-v)(x)$. Applying the <heat kernel> estimate to the <Schwartz function> $g-v$ proves the stronger uniform conclusion
$$
\boxed{\sup_x|u(t,x)-e^{it}v(x)|\leq(4\pi t)^{-n/2}\|g-v\|_1\longrightarrow0.}
$$
Back to article page
