The Duhamel heat potential

Assume Countable Choice. Let $n\ge 1$, $T>0$, $1\le p\le \mathrm{\infty }$, and let $f:[0,T]\to L^p({\mathbb{R}}^n)$ be continuous, where $L^p$ is the quotient space of The space $L^p(\mu )$ as the quotient by null functions. For $0\le t\le T$ the heat potential of $f$ is the Bochner integral $Df(t):={\int }_0^t{H}_{t-s}f(s)ds\in L^p({\mathbb{R}}^n),$ where $H$ is the heat evolution of The heat evolution $H_t$ of initial data and the integrand is the $L^p$-valued function $s\mapsto H_{t-s}f(s)$ interpreted as in Bochner-integrable function and Strongly measurable Banach-valued function.

Well-definedness. The target $L^p$ is Banach by Riesz-Fischer completeness of $L^p$ for $1\le p\le \mathrm{\infty }$ under the declared Countable Choice. For $1\le p<\mathrm{\infty }$ the map $s\mapsto H_{t-s}f(s)$ is norm continuous on $[0,t]$ as a composition of the continuous curve $f$ and the strongly continuous heat flow The heat Cauchy problem for $L^p$ data; its range is therefore separable. For $p=\mathrm{\infty }$ strong continuity at time zero is not available, but the flow is norm continuous on every compact subinterval of $(0,\mathrm{\infty })$ by the $L^1$ continuity of the positive-time kernels (for $a\le \tau \le b$ the explicit kernels converge pointwise with a common integrable Gaussian majorant, so Dominated convergence gives $L^1$ continuity, and Young gives operator-norm continuity), so for fixed $t$ the integrand is norm continuous on every compact subinterval of $[0,t)$ and may be approximated on a countable exhaustion of $[0,t)$ by finite-valued mesh functions, extended by zero on the omitted tail; uniform mesh error at most $1\mathrm{/}k$ on $[0,t-t\mathrm{/}(k+1)]$ gives pointwise convergence at every $s<t$; the single endpoint value at $s=t$ is irrelevant for the integral. In both cases the integrand is strongly measurable, and the contraction estimate of Monotonicity and $L^p$ contractivity of the heat flow gives $\mathrm{\parallel }{H}_{t-s}f(s){\mathrm{\parallel }}_p\le \mathrm{\parallel }f(s){\mathrm{\parallel }}_p$, so ${\int }_0^t\mathrm{\parallel }{H}_{t-s}f(s){\mathrm{\parallel }}_{p}ds\le {\int }_0^t\mathrm{\parallel }f(s){\mathrm{\parallel }}_{p}ds\le T{\sup }_{[0,T]}\mathrm{\parallel }f{\mathrm{\parallel }}_p<\mathrm{\infty }\mathrm{.}$ The Bochner integrability criterion Bochner integrability criterion therefore makes $Df(t)$ a well-defined element of $L^p({\mathbb{R}}^n)$, and the norm inequality for Bochner integrals Bochner integral norm inequality gives the estimate $\mathrm{\parallel }Df(t){\mathrm{\parallel }}_p\le {\int }_0^t\mathrm{\parallel }f(s){\mathrm{\parallel }}_{p}ds\mathrm{.}$

The same definition is used when $f$ is merely strongly measurable with ${\int }_0^T\mathrm{\parallel }f(s){\mathrm{\parallel }}_{p}ds<\mathrm{\infty }$ and $1\le p<\mathrm{\infty }$: approximation by measurable finite-valued simple functions together with the contraction bound gives strong measurability of the integrand: for the $k$-th simple approximation, replace its finitely many positive-time flow curves by finite mesh functions with error at most $1\mathrm{/}k$ on $[0,t-t\mathrm{/}(k+1)]$, and put zero on the omitted tail. At every $s<t$ off the original null set the resulting simple functions converge to $H_{t-s}f(s)$ (Strongly measurable Banach-valued function, Dominated convergence), and the criterion and estimate above apply verbatim.

When $f:{\mathbb{R}}^n\times [0,T]\to \mathbb{R}$ is bounded and jointly continuous, the scalar heat potential is $u(t,x):={\int }_0^t{\int }_{{\mathbb{R}}^n}\mathrm{\Gamma }(x-y,t-s)f(y,s)dyds,$ a scalar potential defined by the convolution Convolution of two functions on ${\mathbb{R}}^n$; the inner integral is absolutely convergent because $\mathrm{\Gamma }(\cdot ,t-s)$ has unit mass Normalisation, parabolic scaling, heat equation and derivative bounds for the heat kernel and $f$ is bounded. Whenever the forcing also defines a Bochner integrable $L^p$-valued map, the scalar potential agrees with the Bochner potential almost everywhere. Indeed, for any bounded measurable test function $\psi$ supported in a bounded set, the pairing $v\mapsto \int \psi v$ is bounded on $L^p$ by Holder's inequality for integrals, including the endpoint cases, and it commutes with the Bochner integral (Bounded linear maps commute with Bochner integration). The iterated scalar integral is absolutely integrable against $\psi$, bounded by $T\mathrm{\parallel }f{\mathrm{\parallel }}_{\mathrm{\infty }}\mathrm{\parallel }\psi {\mathrm{\parallel }}_1$, so Fubini (Tonelli and Fubini for the completed product, with only almost-everywhere section measurability) gives the same pairing for the scalar potential. Equality of all these pairings forces equality almost everywhere: on each bounded box, a positive or negative real or imaginary part of the difference on a measurable set of positive measure would give a nonzero pairing with that set's indicator.

• Why the $p=\mathrm{\infty }$ clause is worded as it is. Positive-time smoothing does not give strong continuity of $H_t$ at $t=0$ in the supremum norm, so the definition claims norm continuity of the integrand only on compact subintervals of $[0,t)$ when $p=\mathrm{\infty }$, and does not need a continuity assertion for $Df$. For $1\le p<\mathrm{\infty }$ the strong continuity of The heat Cauchy problem for $L^p$ data is available and no such caveat is needed.

• Choice accounting. Countable Choice is declared as the ambient hypothesis and enters only through the cited integration theory and heat-flow suppliers; the definition itself makes no selection and no countable exhaustion beyond the explicit intervals used above.