Convolution with the Poisson kernel

Show that if $0 \leq r < 1$ and $t \in \mathbb {R}$, then

Where does the proof of 11.42 fail when $p = \infty$?

For this exercise, for each $r \in [0,1)$ think of $\mathcal{P}_{r}$ as an operator on $L^{2}(\partial \mathbf{D})$.

Suppose $f \in L^{1}(\partial \mathbf{D})$. Prove that $\mathcal{P}_{r}\left(\mathcal{P}_{s} f\right) = \mathcal{P}_{r s} f$ for all $r, s \in [0,1)$.

Suppose $p \in [1, \infty ]$ and $f \in L^{p}(\partial \mathbf{D})$. Prove that if $0 \leq r < s < 1$, then

Suppose $f \in L^{\infty }(\mathbb {R}), x \in \mathbb {R}$, and $f$ is continuous at $x$. Prove that

Suppose $p \in [1, \infty ]$ and $f \in L^{p}(\mathbb {R})$. Prove that $\mathcal{P}_{y}\left(\mathcal{P}_{y^{\prime }} f\right) = \mathcal{P}_{y+y^{\prime }} f$ for all $y, y^{\prime } > 0$.

Suppose $p \in [1, \infty ]$ and $f \in L^{p}(\mathbb {R})$. Prove that if $0 < y < y^{\prime }$, then

For $0 < r < 1$, let $P_r: \partial \mathbb {D} \to [0, \infty )$ be defined by $P_r(z) = \sum_{n=-\infty }^\infty r^{\left|n\right|} z^n$ (the Poisson kernel). Show that $\{ P_r\}$ is an approximate identity for $L^1(\partial \mathbb {D})$ (under convolution).

If $1 \leq p \leq \infty$ and $f \in L^{p}\left(\mathbb {R}^{1}\right)$, prove that $\left(f * h_{\lambda }\right)(x)$ is a harmonic function of $x+i \lambda$ in the upper half plane. ($h_{\lambda }$ is defined in Sec. 9.7 ; it is the Poisson kernel for the half plane).