Carleson-Hunt inequality: changing order of summation

Question

The well-known Carleson-Hunt inequality states that for any $p>1$ and any $f \in L^p(0,2\pi)$, the $L^p$-norm of $sup_n |S_n|$, where $S_n$ is the n-th partial sum of the Fourier series of $f$, is bounded by a constant $c_p$ times the $L^p$-norm of $f$.

Question: if the order of summation is changed (that is, if the frequencies are not summed in the "correct" natural order, but in a permuted order), is a form of the Carleson-Hunt inequality still true or can it fail? Are there any references on this?

(For my purpose, the constant must not be the same for all possible orders of summation, so it's no problem if the constant depends on the permutation.)

Answer 1

No.

It is an old (unpublished) result of Kolmogorov that one can reorder the trigonometric system so that there is an $L^2$ function whose Fourier series diverges on a set of positive measure. This is often refereed to as Kolmogorov's rearrangement theorem. For a proof, see Theorem 2 in Olevskii's book Fourier Series with Respect to General Orthogonal Systems. There he proves the much stronger result:

Given a complete orthonormal systems, there exists an ordering and a continuous function $f$ such that the partial sums of $f$ with respect to the ordering diverge almost everywhere.