Is there a known bound on the norm of the standard intertwining operator for the principal series of $G = \operatorname{GL}\_2(\mathbb Q\_p)$?


**Background:**
For a character $\chi = (\chi\_1,\chi\_2)$ of the standard torus $T$ in $G$, extended to a character of the standard Borel subgroup $B = TU$ we can define the induced representation $I(\chi) = \operatorname{Ind}\_B^G \chi$ to be the Banach space of all measurable functions $f: G \rightarrow \mathbb C$ satisfying $f(tng) = \chi(t)\delta\_B^{1/2}(t)f(g)$ and the norm condition
$$\lVert f\rVert^2 = \int\limits\_K \lvert f(k)\rvert^2 dk < \infty$$
where $K = \operatorname{GL}\_2(\mathbb Z\_p)$ (usually, one works with the subspace $I(\chi)^{\infty}$ of locally constant functions in $I(\chi)$). If $w = \begin{pmatrix} & 1 \\ -1 \end{pmatrix}$, the standard intertwining operator $M: I(\chi) \rightarrow I(w(\chi))$ is densely defined on $I(\chi)^{\infty}$ by
$$Mf(g) = \int\limits\_N f(wng)dn.$$
This converges for $\sigma = \operatorname{Re}(\chi\_1\chi\_2^{-1}) > 0$, but can be extended to make sense for most $\chi\_1$, $\chi\_2$ by a process of analytic continuation.


**What I want to do**:


I want to compute, or get an upper bound on, the norm $\lVert M\rVert = \sup\limits\_{\lVert f\rVert=1} \lVert Mf\rVert$. Actually, I'm trying to show that the *global* intertwining operator for $\operatorname{GL}\_2(\mathbb A\_{\mathbb Q})$, densely defined as a product of local intertwining operators, is a bounded linear operator. The norm of the global intertwining operator should be a convergent product of the norms of the local operators.


**Computations for $\lVert{-}\rVert\_1$**:


If we use the $1$-norm instead of the $2$-norm, I can get an estimate for $\lVert M\rVert\_1$. There may be a way to modify these calculations for the $2$-norm, but I haven't been able to do it. For $f$ locally constant,
\begin{gather\*}
\lVert Mf\rVert = \int\limits\_K \lvert Mf(k)\rvert dk \leq \int\limits\_K \int\limits\_N \lvert f(wnk)\rvert dn\,dk= \int\limits\_{N(\mathbb Z\_p)} \int\limits\_K \lvert f(wnk)\rvert dk\,dn + \int\limits\_{n \not\in N(\mathbb Z\_p)} \int\limits\_K \lvert f(wnk)\rvert dk\,dn \\
=\int\limits\_{N(\mathbb Z\_p)} \int\limits\_K \lvert f(k)\rvert dk\,dn + \int\limits\_{x \in \mathbb Q\_p - \mathbb Z\_p} \int\limits\_K \left\lvert f(w \begin{pmatrix} 1 & x \\ & 1 \end{pmatrix}k)\right\rvert dk\,dx = \lVert f\rVert\_1 + \int\limits\_{x \in \mathbb Q\_p - \mathbb Z\_p} \int\limits\_K \left\lvert f(w \begin{pmatrix} 1 & x \\ & 1 \end{pmatrix}k)\right\rvert dk\,dx.
\end{gather\*}
This last integral is
\begin{gather\*}
\int\limits\_{x \in \mathbb Q\_p - \mathbb Z\_p} \int\limits\_K \left\lvert f( \begin{pmatrix} -x^{-1} & 1 \\ & -x\end{pmatrix} \begin{pmatrix} 1 \\ x^{-1} & 1 \end{pmatrix} k)\right\rvert dk\,dx = \int\limits\_{x \in \mathbb Q\_p - \mathbb Z\_p} \int\limits\_K \lvert x\rvert^{-\sigma} \lvert f(k)\rvert dk\,d^{\ast}x = \sum\limits\_{k=1}^{\infty} \lVert f\rVert\_1 \int\limits\_{\lvert x\rvert = p^k} \lvert x\rvert^{-\sigma}d^{\ast}x \\
= \lVert f\rVert\_1 \operatorname{Vol}(\mathbb Z\_p^{\ast}) \frac{p^{-\sigma}}{1- p^{-\sigma}}.
\end{gather\*}


So we see that for $\sigma = \operatorname{Re}(\chi\_1\chi\_2^{-1}) > 0$, the $1$-norm $\lVert M\rVert\_1$ is bounded by $1 + \operatorname{Vol}(\mathbb Z\_p^{\ast}) \frac{p^{-\sigma}}{1- p^{-\sigma}}$.



The norm of the principal series intertwining operator for $\operatorname{GL}_2$
Having a bound on $\|M\|\_1$, you get by duality a bound on $\|M\|\_\infty$
and then you get a bound on $\|M\|\_p$ for every $1\leq p\leq\infty$
by the [Riesz–Thorin theorem](https://en.wikipedia.org/wiki/Riesz%E2%80%93Thorin_theorem).


