Let's consider the propagator corresponding to the one-dimensional equation $$ u_t=\Lambda^\alpha u_x,\; u(x,0)=f(x) $$ where $$ \widehat{\Lambda^\alpha u}=|\xi|^\alpha\hat{u}(\xi), $$ and $1\leq \alpha\leq 1$.

QUESTION 1) Do you know any good reference in the dispersive estimates for this operator? I mean inequalities of the following form $$ \|u(t)\|_{L^\infty(\mathbb{R}}\leq (1+t)^{-\theta}\|f\|_{X}, $$ for certain $\theta$ and Banach space $X$.

QUESTION 2) Same for the propagator corresponding to $$ u_t=(1-\partial_x^2)^\alpha u_x,\; u(x,0)=f(x) $$ and $1\leq \alpha\leq 1$.

Let's consider the propagator corresponding to the one-dimensional equation $$ u_t=\Lambda^\alpha u_x,\; u(x,0)=f(x) $$ where $$ \widehat{\Lambda^\alpha u}=|\xi|^\alpha\hat{u}(\xi), $$ and $-1< \alpha\leq 1$.

QUESTION 1) Do you know any good reference in the dispersive estimates for this operator? I mean inequalities of the following form $$ \|u(t)\|_{L^\infty(\mathbb{R}}\leq (1+t)^{-\theta}\|f\|_{X}, $$ for certain $\theta$ and Banach space $X$.

QUESTION 2) Same for the propagator corresponding to $$ u_t=(1-\partial_x^2)^\alpha u_x,\; u(x,0)=f(x) $$ and $-1\leq \alpha\leq 1$.