Norm convergence of partial sum operator on $L^p$, $1 < p < \infty$.

Question

I am working through Mark Pinsky's "Introduction to Fourier Analysis and Wavelets" textbook and have been working through the following problems. Note that here $S_R$ is the partial sum operator defined by $$ (\widehat{S_R f}) = \chi_{B_R} \widehat{f}. $$

1. For any $f \in \mathcal{S}(\mathbb{R})$, the Schwartz space, we have the bound $ \|S_Rf\|_p \leq C_p \|f\|_p $, where $C_p$ is independent of $R$.
2. If $2 < p < \infty$ and $f \in \mathcal{S}(\mathbb{R})$, then $\|S_R f - f\|_p \rightarrow 0$ as $R \rightarrow \infty$.
3. If $f \in L^p(\mathbb{R}), 2 < p < \infty$, then $\|S_R f - f\|_p \rightarrow 0$ as $R \rightarrow \infty$.
4. Use the duality of $L^p$ and $L^{p^\prime}$ to prove that for any $f \in L^p(\mathbb{R}), 1 < p < 2$, then $\|S_R f - f\|_p \rightarrow 0$ as $R \rightarrow \infty$.

As you can see these problems build upon one another. I have figured out the first 3 problems, but find myself struggling to solve 4. using duality. I'm not exactly sure what the author means by duality, but to me I take it that for $f \in L^p, 1 < p < 2$, then $\widehat{f} \in L^{p^\prime}$, where $p^\prime$ is the conjugate exponent of $p$. This aided me in solving 2. using the Hausdorff-Young inequality, however the inequality doesn't seem to help me when considering $f \in L^p$ for $1 < p < 2$. Duality also brings up the following definition: $$ \|S_R f\|_p = \sup \left\{ \left| \int S_R f \cdot g \right|\, : \, \|g\|_{p^\prime} \leq 1 \right\}, $$ however I seem unable to "swap" the $S_R$ operator between $f$ and $g$ the way you can with the Hilbert transform. Thank you for the guidance!

Answer 1

By the definition you noted, it's enough to show that $|\langle S_Rf-f,g\rangle|\rightarrow 0$ for all $g\in L^{p'}(\mathbb{R})$ where $\langle S_Rf-f,g\rangle$ is the pairing $\int_\mathbb{R} (S_Rf(x)-f(x))g(x)\,dx$. We know that $||S_Rg -g||_{L^{p'}(\mathbb{R})}\rightarrow 0$ so the idea is to try to rewrite the above pairing in terms of $S_Rg$. Thus, you should try to rewrite $\int_\mathbb{R}S_Rf(x)g(x)\,dx$ as $\int_\mathbb{R}f(x)T_Rg(x)\,dx$ where $T_R$ is some operator very closely related to $S_R$. In fact, $T_R$ is what is called the adjoint of $S_R$ meaning that for all $f\in L^{p}(\mathbb{R}),g\in L^{p'}(\mathbb{R})$, $\langle S_Rf,g\rangle=\langle f,T_Rg\rangle$. You should see that $\langle f,T_Rg\rangle\rightarrow \langle f,g\rangle $ because of $T_R$'s relationship to $S_R$ and that fact that $S_Rg\rightarrow g$ in $L^{p'}(\mathbb{R})$.

More abstractly, what is meant by "using duality" is proving something about an operator or sequence of operators (usually that it is bounded or that the sequence converges to some limit) by knowing something about the adjoint(s). In the setting of $L^p$ spaces, if $T^*$ is similar to $T$, or even the same, and we know something about $T^*$ on $L^p$ for $2<p<\infty$, we can learn something about $T$ for $1<p'<2$, $1/p'+1/p=1$ since these are the dual spaces.