The Lp range: interpolation below two and adjoint duality above two

Let $1<q<2$ and $f\in L^q\cap L^2$. Then $f\in L^1+L^2$ and $T$ is defined at $f$; sublinearity is automatic for the linear $T$, the weak $(1,1)$ and strong $(2,2)$ hypotheses are those assumed, so [F2] applies with $(X,\mu )=({\mathbb{R}}^n,\lambda )$ and gives $\mathrm{\parallel }Tf{\mathrm{\parallel }}_q\le C_q\mathrm{\parallel }f{\mathrm{\parallel }}_q$ with $C_q=[q(2A\mathrm{/}(q-1)+4B^2\mathrm{/}(2-q)){]}^{1\mathrm{/}q}$.

Consequently, for every $1<q<2$ the operator $T$ has a unique bounded extension to all of $L^q({\mathbb{R}}^n;\mathbb{C})$ with norm at most $C_q$: since $C_c^{\mathrm{\infty }}({\mathbb{R}}^n;\mathbb{C})\subseteq L^q\cap L^2$ is dense in $L^q$ by [F3], step 1.1 applied to differences of test functions shows that $g\mapsto Tg$ is uniformly continuous on this dense subspace, so it extends uniquely to the closure with the same bound.

The adjoint $T^{\ast }$ is a bounded linear operator on $L^2$ with norm $B$ by [F1]; it satisfies the weak $(1,1)$ bound with constant $A$ by hypothesis, and it is linear, hence sublinear. Therefore step 2.1 applies to $T^{\ast }$ at the exponent $p^{\mathrm{\prime }}\in (1,2)$ whenever $2<p<\mathrm{\infty }$: for every $h\in L^{p^{\mathrm{\prime }}}({\mathbb{R}}^n;\mathbb{C})$, $\mathrm{\parallel }{T}^{\ast }h{\mathrm{\parallel }}_{p^{\mathrm{\prime }}}\le C_{p^{\mathrm{\prime }}}\mathrm{\parallel }h{\mathrm{\parallel }}_{p^{\mathrm{\prime }}}$.

Let $2<p<\mathrm{\infty }$, $f\in L^p\cap L^2$ and $g\in L^{p^{\mathrm{\prime }}}\cap L^2$ with $\mathrm{\parallel }g{\mathrm{\parallel }}_{p^{\mathrm{\prime }}}\le 1$. Using $\bar{g}\in L^{p^{\mathrm{\prime }}}\cap L^2$ and the adjoint identity of [F1] applied to the pair $(Tf,\bar{g})$, $\mid {\int }_{{\mathbb{R}}^n}(Tf)gd\lambda \mid =\mid ⟨Tf,\bar{g}⟩\mid =\mid ⟨f,T^{\ast }\bar{g}⟩\mid \le \mathrm{\parallel }f{\mathrm{\parallel }}_p\mathrm{\parallel }{T}^{\ast }\bar{g}{\mathrm{\parallel }}_{p^{\mathrm{\prime }}}\le C_{p^{\mathrm{\prime }}}\mathrm{\parallel }f{\mathrm{\parallel }}_p,$ where the first inequality is Hölder's inequality [F4] and the second is step 3.1 applied to $h=\bar{g}$, which has $\mathrm{\parallel }\bar{g}{\mathrm{\parallel }}_{p^{\mathrm{\prime }}}=\mathrm{\parallel }g{\mathrm{\parallel }}_{p^{\mathrm{\prime }}}\le 1$. To establish $Tf\in L^p$ before using norm recovery, put $u=Tf\in L^2$, $E_N=B(0,N)\cap \{\mathrm{\mid }u\mathrm{\mid }\le N\}$ and $a_N=({\int }_{E_N}\mathrm{\mid }u{\mathrm{\mid }}^p{)}^{1\mathrm{/}p}$. If $a_N>0$, take $g_N={\mathbf{1}}_{E_N}\mathrm{\mid }u{\mathrm{\mid }}^{p-1}{\theta }_u\mathrm{/}{a}_N^{p-1}$, with ${\theta }_u=\bar{u}\mathrm{/}\mathrm{\mid }u\mathrm{\mid }$ on $u\mathrm{\ne }0$ and zero otherwise. This test is bounded on a finite-measure set, hence lies in $L^{p^{\mathrm{\prime }}}\cap L^2$, and satisfies $\mathrm{\parallel }{g}_N{\mathrm{\parallel }}_{p^{\mathrm{\prime }}}=1$, $\int u{g}_N=a_N$. The preceding pairing bound gives $a_N\le C_{p^{\mathrm{\prime }}}\mathrm{\parallel }f{\mathrm{\parallel }}_p$; if $a_N=0$ the same inequality is immediate. Since $E_N$ increases to a full-measure set, monotone convergence gives $\mathrm{\parallel }Tf{\mathrm{\parallel }}_p\le C_{p^{\mathrm{\prime }}}\mathrm{\parallel }f{\mathrm{\parallel }}_p$.

If $2<p<\mathrm{\infty }$, step 4.1 and the density [F3] of $C_c^{\mathrm{\infty }}\subseteq L^p\cap L^2$ in $L^p$ extend the bound to all $f\in L^p$ with the same constant $C_{p^{\mathrm{\prime }}}$, exactly as in step 2.1. Together with step 2.1 for $1<p<2$, this proves the asserted bounds for every $1<p<\mathrm{\infty }$, $p\mathrm{\ne }2$.