Linear bounded operator from $L^p[0,1]$ to itself whose range consists of continuous functions.

Let $$T\colon \mathbb L^p[0,1]\to \mathbb L^p[0,1]$$, $$1, be a linear bounded operator such that $$\operatorname{Im}(T)$$ is contained in the space of continuous functions. It was shown in this thread that $$T$$ is a compact operator.

Some non trivial example of such $$T$$ can be given, for example kernel operators with appropriated conditions on the kernel. The question is whether there exists other example (maybe asking for a characterisation is too ambitious).

• From your comment in the linked post, doesn't it suffice to consider operators from $L_p[0,1]$ to $C[0,1]$? This may help: Operators from $X$ to $C[0,1]$ are characterized by functions from $[0,1]$ to $X^*$ that are continuous when $X^*$ has the weak* topology (c.f. Dunford and Schwartz, Linear Operators, vol 1, Theorem VI.7.1). – David Mitra May 17 at 13:03
• Operator $f\mapsto \int_0^x f(t)dt$ – Boris Bilich May 17 at 13:17
• Another examples: Fractional integrals of the form $(- \partial_{tt}^2 )^{-\beta}$ for any $\beta > 1/p$, by Morrey's Inequality. – Adrián González-Pérez May 18 at 13:34