The symbol occurs on Page 22 of Bahouri's book Fourier analysis and nonlinear differential equations.
As defined there,
$$f(D)a:=\mathcal{F}^{-1}\{f\mathcal{F}a\}.$$
The question comes from the definition 1.26 on the same page.
Should I interpret $f(\lambda D)a$ as
$$f(\lambda D)a = f_{\lambda}(D)a=\mathcal{F}^{-1}\{f_{\lambda}\mathcal{F}a\} = \mathcal{F}^{-1}\{f(\lambda\xi)\mathcal{F}a(\xi)\}=\frac{1}{\lambda^d}f^{\lor}(\frac{x}{\lambda})*a(x)?$$
The definition bothers me a bit. I also want to ask if it is a standard way to write it out or just the author's personal preference?
Thanks.
Similar definition for polynomials can be found in the chapters of Fourier transformations from Rudin's Functional analysis.
Yes, that makes sense. The perspective I'm used to is
$$f(x)=\sum_{n=0}^\infty a_nx^n\implies f(D)g(x):=\sum_{n=0}^{\infty}a_n\frac{d^ng}{dx^n}.$$
Since ${\cal F}^{-1}\{x^n{\cal F}a\}$ is equal to the $n$th derivative of $a$ up to rescaling, these two definitions should be expected to agree with each other (modulo any technicalities about where things are defined).
While $f(D)$ is defined in the power series form in contexts I'm familiar with, I haven't been through any specific study of Fourier or functional analysis so I don't know about how standard it is in those two cases or if it's the author's personal preference, if this definition has any advantage of being defined for more functions, etc. Well, clearly it's defined for vector functions of ${\bf R}^d$ instead of just scalar functions so there's a bit more generality right there.