Hausdorff–Young for periodic Fourier coefficients

Let $C$ assign to the almost-everywhere class of a complex finite simple function $s$ on $\mathbb{T}$ with finite-measure nonzero set its coefficient sequence $Cs=(\widehat{s}(k){)}_{k\in \mathbb{Z}}$. This is well defined on classes and complex-linear by [F1]; its target is a space of measurable classes on $\mathbb{Z}$ with counting measure, which is sigma-finite, and its source space is the probability space $\mathbb{T}$.

For such a class $s$ and every $k$, $\mathrm{\mid }\widehat{s}(k)\mathrm{\mid }\le {\int }_{\mathbb{T}}\mathrm{\mid }s\mathrm{\mid }\mathrm{\mid }{e}_{-k}\mathrm{\mid }d{m}_{\mathbb{T}}=\mathrm{\parallel }s{\mathrm{\parallel }}_1$ by [F1] and [F5], so $\mathrm{\parallel }Cs{\mathrm{\parallel }}_{\mathrm{\infty }}\le \mathrm{\parallel }s{\mathrm{\parallel }}_1$: the L^1-to-L-infinity endpoint bound holds with $A=1$.

A finite simple function on a probability space is bounded, hence lies in $L^2(\mathbb{T};\mathbb{C})$, and [F2] gives $\mathrm{\parallel }Cs{\mathrm{\parallel }}_2=\mathrm{\parallel }s{\mathrm{\parallel }}_2$; the L^2-to-L^2 endpoint bound holds with $B=1$.

Applying [F3] to the core operator $C$ of step 1.1 with endpoints $(p_0,q_0)=(1,\mathrm{\infty })$, $A=1$ and $(p_1,q_1)=(2,2)$, $B=1$ yields, for every $1<p<2$, a unique compatible bounded extension $T_p:L^p(\mathbb{T})\to {\mathrm{\ell }}^{p^{\mathrm{\prime }}}(\mathbb{Z})$ with $\mathrm{\parallel }{T}_{p}f{\mathrm{\parallel }}_{p^{\mathrm{\prime }}}\le \mathrm{\parallel }f{\mathrm{\parallel }}_p$, while at $p=1$ and $p=2$ the endpoint estimates of steps 2.1 and 2.2 hold; both measure spaces are sigma-finite.

The $L^1$-extension of $C$ is the coefficient map: the coefficient map is a bounded linear map $L^1(\mathbb{T})\to {\mathrm{\ell }}^{\mathrm{\infty }}(\mathbb{Z})$ agreeing with $C$ on the finite simple classes, which are dense in $L^1(\mathbb{T})$ by [F7], and extensions from a dense core into a Banach space are unique; the same argument identifies the $L^2$-extension with the coefficient map on $L^2(\mathbb{T})$.

Fix $1<p<2$. Since $m_{\mathbb{T}}(\mathbb{T})=1$, [F6] gives $L^p(\mathbb{T})\subseteq L^1(\mathbb{T})$ with $\mathrm{\parallel }f{\mathrm{\parallel }}_1\le \mathrm{\parallel }f{\mathrm{\parallel }}_p$, so the domain intersection of the $L^p$-extension $T_p$ and the $L^1$-extension contains all of $L^p(\mathbb{T})$; by [F4] the two extensions agree as measurable classes there.

By steps 4.1 and 3.2, for $1<p<2$ the sequence $T_{p}f$ is the coefficient sequence $(\widehat{f}(k))$, and since ${\mathrm{\ell }}^{p^{\mathrm{\prime }}}(\mathbb{Z})\subseteq {\mathrm{\ell }}^{\mathrm{\infty }}(\mathbb{Z})$ with $\mathrm{\mid }\widehat{f}(k)\mathrm{\mid }\le \mathrm{\parallel }{T}_{p}f{\mathrm{\parallel }}_{p^{\mathrm{\prime }}}$, step 3.1 gives $\mathrm{\parallel }\widehat{f}{\mathrm{\parallel }}_{p^{\mathrm{\prime }}}\le \mathrm{\parallel }f{\mathrm{\parallel }}_p$; the case $p=2$ is the Parseval equality of [F2].

The case $p=1$ is the endpoint estimate of step 2.1 applied to $L^1$ classes, and Countable Choice is used only through the cited Parseval and interpolation interfaces [A1]; the finite-measure convention $m_{\mathbb{T}}(\mathbb{T})=1$ enters only through [F6] and the probability-space identification of the core.