Exercise 1.6.27 (Approximations to the identity)

Define a good kernel to be a measurable function $P : R^{d} \to R^{+}$ which is non-negative, radial (which means that there is a function $\tilde{P} : [0, + \infty) \to R^{+}$ such that $P (x) = \tilde{P} (| x |)$ ), radially non-increasing (so that $\tilde{P}$ is a non-increasing function), and has total mass $\int_{R^{d}} P d m$ equal to $1$ . The functions $P_{t} (x) : = \frac{1}{t^{d}} P (\frac{x}{t})$ for $t > 0$ are then said to be a good family of approximations to the identity.

(i): Show that the heat kernels $P_{t} (x) : = \frac{1}{{(4 π t^{2})}^{d ∕ 2}} e^{- | x |^{2} ∕ 4 t^{2}}$ and Poisson kernels $P_{t} (x) : = c_{d} \frac{t}{{(t^{2} + | x |^{2})}^{(d + 1) ∕ 2}}$ are good families of approximations to the identity, if the constant $c_{d} > 0$ is chosen correctly (in fact one has $c_{d} = Γ ((d + 1) ∕ 2) ∕ π^{(d + 1) ∕ 2}$ , but you are not required to establish this).
(ii): Show that if P is a good kernel, then $c_{d} < \sum_{n = - \infty}^{\infty} 2^{dn} \tilde{P} (2^{n}) \leq C_{d}$
for some constants $0 < c_{d} < C_{d}$ depending only on $d$ .
(iii): Establish the quantitative upper bound $| \int_{R^{d}} f (y) P_{t} (x - y) dy | \leq C_{d}^{'} \sup_{r > 0} \frac{1}{| B (x, r) |} \int_{B (x, r)} | f | d m$
for any absolutely integrable function f and some constant C’_d > 0 depending only on d.
(iv): Show that if $f : R^{d} \to C$ is absolutely integrable and $x$ is a Lebesgue point of $f$ , then the convolution $f * P_{t} (x) : = \int_{R^{d}} f (y) P_{t} (x - y) d m (x)$
converges to $f (x)$ as $t \to 0$ . In particular, $f * P_{t}$ converges pointwise almost everywhere to $f$ .

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Define a good kernel to be a measurable function $P: {\bf R}^d \rightarrow {\bf R}^+$ which is non-negative, radial (which means that there is a function $\tilde P: [0,+\infty) \rightarrow {\bf R}^+$ such that $P(x) = \tilde P(|x|)$), radially non-increasing (so that $\tilde P$ is a non-increasing function), and has total mass $\int_{{\bf R}^d} P\ \mathrm{d}m$ equal to $1$. The functions $P_t(x) := \frac{1}{t^d} P(\frac{x}{t})$ for $t>0$ are then said to be a good family of approximations to the identity.
\begin{enumerate}[label=(\roman*)]
\item Show that the heat kernels $P_t(x) := \frac{1}{(4\pi t^2)^{d/2}} e^{-|x|^2/4t^2}$ and Poisson kernels $P_t(x) := c_d \frac{t}{(t^2+|x|^2)^{(d+1)/2}}$ are good families of approximations to the identity, if the constant $c_d > 0$ is chosen correctly (in fact one has $c_d = \Gamma((d+1)/2)/\pi^{(d+1)/2}$, but you are not required to establish this).

\item Show that if {P} is a good kernel, then
$$\displaystyle c_d < \sum_{n=-\infty}^\infty 2^{dn} \tilde P(2^n) \leq C_d$$
for some constants $0 < c_d < C_d$ depending only on $d$.

\item Establish the quantitative upper bound
$$\displaystyle |\int_{{\bf R}^d} f(y) P_t(x-y)\ dy| \leq C'_d \sup_{r>0} \frac{1}{|B(x,r)|} \int_{B(x,r)} |f|\ \mathrm{d}m$$
for any absolutely integrable function {f} and some constant {C'_d > 0} depending only on {d}.

\item Show that if $f: {\bf R}^d \rightarrow {\bf C}$ is absolutely integrable and $x$ is a Lebesgue point of $f$, then the convolution
$$\displaystyle f*P_t(x) := \int_{{\bf R}^d} f(y) P_t(x-y)\ \mathrm{d}m(x)$$
converges to $f(x)$ as $t \rightarrow 0$. In particular, $f*P_t$ converges pointwise almost everywhere to $f$.
\end{enumerate}

Exercise 1.6.27 (Approximations to the identity)

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