two jointly acceptable exponent triplets in the first column.

$$\begin{aligned}
    \rp q + \rp \infty = n\left(1-\rp r-\rp {\tilde r}\right),& &    \rp q + \frac n r = \frac n b,\qquad\\
    \rp q < n\left( \rp p - \rp r \right),\qquad&                        &\rp b < \rp p,\qquad\quad\\
    q=\infty, \quad 1\leq p=r\leq\infty,&                          &a=b=c=p=r,\quad\\
    \operatorname{HM}(p,r)=\operatorname{HM}(\tilde p', \tilde r')&                              &\operatorname{HM}(p,r)=\operatorname{HM}(b,c),\\
    \tilde q=\infty,\quad \trip \text{ is KT-admissible}&               &a=b=c,\qquad\\
    a < r < r^*(a)\qquad&    & \begin{cases}  \quad a < r < r^*(a),&\\
                                        \quad p^*(a) < p < a,&\\
                                        \quad p < b<a<c<r.&
                         \end{cases}\end{aligned}$$

Estimate follows from by duality, see [@O Duality lemma] for proof. Note also that the tricky cases which cannot be treated by duality, e.g. the dual estimate of , where $\tilde p'=\infty$, $\tilde r' < \infty$, are precluded by our definition of joint acceptability in the case when $\tilde q=\infty$.

Estimates for acceptable exponents
==================================

Local Inhomogeneous Estimates
-----------------------------

Following Foschi [@F], we want to find the range of local estimates for $W(t)$ that are invariant to the scaling $$\begin{aligned}
 \label{eq: local scal}
 \normLL{W(t)[\chi_{\lambda J}F]} \lesssim  \lambda^{\bbeta}\normRL{F}, \quad \forall \lambda >0,
 $$ where $I$ and $J$ are two unit intervals separated by a unit distance and $\chi_{\lambda J}$ is the characteristic of the 