Can I generalize the classical Bernstein inequality?

Question

I have know the proof of the classical Bernstein inequality as follows.
Given some function $ f\in S(\mathbb{R}^d) $, where $ S(\mathbb{R}^d) $ denotes the Schwartz space of functions. Let $ \hat{f} $ denote the Fourier transform of $ f $. Assume that there exists some measurable set $ E $, such that $ supp(\hat{f})\subset E\subset \mathbb{R}^d $. Then for any $ 1\leq p\leq q\leq \infty $, we have the following inequality:$ \left\|f\right\|_{L^q}\leq |E|^{1/p−1/q}\left\|f\right\|_{L^p} $. ($ |E| $ below denotes the Lebesgue measure of $ E $)
I want to ask if we can generalize the Bernstein inequality to the index that $ 0\leq p\leq q\leq 1 $.