Exercise 38, § 1

Pour all $n\in \mathbf{Z}$, we denote by $e_n$ the character $x\mapsto e^{2i\pi nx}$ of $\mathbf{T}$. We denote by P the subspace of $\mathscr{C}(\mathbf{T})$ generated by the $e_n$.

Let $a= (a_h)_{h\in\mathbf{Z}}$ be a family of complex numbers. The conjugate-harmonic family is the family $\widetilde{a}= (-i$ s($h$)$a_h)_{h\in\mathbf{Z}}$, where s($h$) denotes the sign function.

For $f\in P$, we denote by $\widetilde{f}\in P$ the function on $\mathbf{T}$ such that $\mathscr{F}(\widetilde{f}) =\widetilde{\mathscr{F}}(f)$ (conjugate-harmonic function of $f$).

a) Let $f\in P$ be such that $\mathscr{F}(f)(0) = 0$ and such that $f$ is real-valued. For every integer $k\geqslant 1$, there exists a constant $C_k$, independent of $f$, such that

(Observe that the Fourier coefficients of $g=f+i\widetilde{f}$ are supported on $\mathbf{N}^*$, write

and expand using the binomial formula.)

b) Let $1< p <+\infty$. The mapping $f\mapsto \widetilde{f}$ from P into P admits a unique continuous linear extension from $L^p(\mathbf{T})$ into $L^p(\mathbf{T})$ such that $\mathscr{F}(\widetilde{f}) =\widetilde{\mathscr{F}}(f)$ for every $f\in L^p(\mathbf{T})$ (cf. exercise 8). (For $p\geqslant 2$, use the preceding question and the inequality of M. Riesz (INT, IV, §6, exerc. 18); for $1< p <2$, use duality.)

c) The result of the preceding question does not extend to the case $p= 1$. (Use question b) of exercise 30.)