$L^p$ boundedness for pseudo-differential operators

Question

Let $\rho, \delta, m$ be real parameters such that $0\le \delta\le \rho\le 1, \delta<1$. The set $S^m_{\rho, \delta}(\mathbb R^{2n})$ is defined as the set of smooth functions $a$ on $\mathbb R^n\times \mathbb R^n$ so that, for all multi-indices $\alpha, \beta\in \mathbb N^n$, $$ \sup_{(x,\xi)\in \mathbb R^{2n}}\bigl\vert(\partial_\xi^\alpha\partial_x^\beta a)(x,\xi)(1+\vert \xi\vert)^{\rho\vert \alpha\vert-\delta\vert \beta\vert-m}\bigr\vert<+\infty. $$ The operator $\text{Op}[a]$ is defined (weakly) by the formula $$ (\text{Op}[a] u)(x)=\int e^{2iπ x\cdot \xi} a(x,\xi) \hat u(\xi) d\xi. $$ It is classical that $\text{Op}[a]$ is bounded on $L^2(\mathbb R^n)$ when $a\in S^0_{\rho, \delta}(\mathbb R^{2n})$ ($\rho, \delta$ satisfy the conditions above) and that for some $a\in S^0_{1, 1}(\mathbb R^{2n})$, $\text{Op}[a]$ fails to be bounded on $L^2(\mathbb R^n)$.

Question 1 . If $\rho, \delta$ satisfy the conditions above, $a\in S^0_{\rho, \delta}(\mathbb R^{2n})$, is it true that $\text{Op}[a]$ is bounded on $L^p(\mathbb R^n)$ for $p\in (1,+\infty)$?

Question 2 . If the answer to Question 1 is positive (which I doubt in fact), what are classical counterexamples of pseudo-differential operators bounded on $L^2$ without being bounded on $L^p$ for $p\not=2$?

For both questions, I would be grateful for references, acknowledging that these matters must be very classical.

Answer 1

I must apologize for answering my own question since I should have checked the classical results in the literature before asking these two questions. However, let me summarize the situation: we always assume that $\rho, \delta$ are real parameters such that $0\le \delta\le \rho\le 1, \delta<1$.

$\bullet$ If $\rho=1>\delta$, $a\in S^0_{\rho, \delta}$, then $\text{Op}[a]$ is bounded on $L^{p}(\mathbb R^{n})$ for all $p\in (1,+\infty)$.

$\bullet$ If $1>\rho\ge \delta\ge 0, p\in (1,+\infty), p\not=2$, there exists $a\in S^0_{\rho, \delta}$, such that $\text{Op}[a]$ fails to be bounded on $L^{p}(\mathbb R^{n})$.

The reader can consult the page 322 of E.M.Stein's book Harmonic Analysis, Princeton Math. Ser., 43, and C.L.Fefferman's article $L^p$ bounds for pseudo-differential operators, Israel J. Math.14(1973), 413–417.