natural question is immediate: Are $S_p$, and respectively $T_p$, equivalent for all $1\leq p<\infty.$ Or even in a weaker sense: if both of the conditions $S_p,T_p$ hold simultaneously, could we deduce $S_q$ or $T_q$ for some $q>p?$ By Theorem \[nip1\] the answer is no.

The next proposition will clarify the situation and point out how the counterexample in Theorem \[nip1\] can be constructed.

Recall, that a function $\omega:{\mathbb{R}}^d\to(0,\infty)$ is said to be in the class of $A_p$ weights, $1<p<\infty$, if $$[w]_{A_p} = \sup_{Q} {\langle w\rangle}_Q{\langle w^{-\frac{p'}{p}}\rangle}_Q^{\frac{p}{p'}}<\infty,\quad p' = \frac{p}{p-1},$$ where the supremum is taken over all cubes.

\[prop1\] Given $1<p<q<\infty,$ there exists functions $\phi,\psi\in L^p_{loc}$ satisfying the conditions $S_p,T_p$ and failing the condition $S_q.$

Let $$\psi(x) = x^{-\frac{2}{p+q}}1_{(0,1)},\qquad \phi = \psi^{-1}1_{(0,1)}.$$ We check that the conditions $S_p,T_p$ hold. Let $[a,b)$ be an arbitrary intervall such that $[a,b)\cap[0,1)=[c,d)\not=\emptyset$ (if the intersection is empty, then the claim is trivial). First, $$\frac{1}{b-a}\int_a^b{|\psi-\langle\psi\rangle_{[a,b)}|}^p\frac{1}{b-a}\int_a^b{|\phi-\langle\phi\rangle_{[a,b)}|}^p \leq   \frac{1}{d-c}2^{p+1}\int_c^d\psi^p\frac{1}{d-c}2^{p+1}\int_c^d\phi^p.$$ Then, by the fact (see Grafakos [@CFA]) that $|x|^{-\frac{2p}{p+q}}\in A_2$, we have $$\frac{1}{d-c} \int_c^d\psi^p\frac{1}{d-c} \int_c^d\phi^p\le [|x|^{-\frac{2p}{p+q}}]_{A_2}.$$ It follows that $S_p(\psi,\phi) \lesssim 1.$

By the above estimates and $\phi\psi \leq1,$ it follows for an arbitrary interval $I$ that $$\frac{1}{{|I|}}\int_I{|\psi-\langle\psi\rangle_I|}^p{|\phi-\langle\phi\rangle_I|}^p \leq 4^p  \frac{1}{{|I|}}\left(\int_I\psi^p\phi^p + \int_I\psi^p\langle\phi\rangle_I^p+ \int_I\phi^p\langle\psi\rangle_I^p + \langle\psi\rangle_I^p \langle\phi\rangle_I^p      \right) \lesssim 