Norm of a projection in $L_p$

Question

Let $\mu$ be a probability measure on $[0,1]$ and let us define the projection $P$ on $L^p(\mu)$: $$P \; \colon f \mapsto \mathbb{E}(f){\bf 1}$$ (where ${\bf 1}$ is the constant 1 function). What is the norm or a non-trivial estimate of the norm of $I-P$ if $p\neq 2$?

Answer 1

Ok, here we go. Of course, $\|I-P\|_2=1$ because $I-P$ is an orthogonal projection in $L_2[0,1].$ Moreover, a simple calculation shows that $\|I-P\|_1=\|I-P\|_\infty = 2.$ Then a straightforward application of the Riesz-Thorin interpolation theorem gives that $$ \|I-P\|_p \leq 2^{|1-{2\over p}|}.$$

(This reasoning can be found in a paper by S. Rolewicz.) The projection $I-P$ is actually the minimal projection to the hyperlane $X_p = \{ f : \mathbb{E}f = 0\} \subseteq L_p[0,1];$ i.e. it has the minimal norm among the projections with range $X_p.$ Later C. Franchetti showed in his paper 'The norm of the minimal projection onto hyperplanes in $L_p[0,1]$ and the radial constant' that $$ \|I-P\|_p = \max_{0\leq x \leq 1} (x^{p-1} + (1-x)^{p-1})^{1/p}(x^{q-1} + (1-x)^{q-1})^{1/q},$$ where $1/p+1/q =1.$