we need. It is obvious that $\psi,\phi\in L_{loc}^\infty$. It remains to verify that $(\psi,\phi)$ fails $S_{2+\varepsilon}$ for any $\varepsilon>0$. By Hölder’s inequality we can assume $0<\varepsilon<1.$ Find $\ell\in {\mathbb{N}}$ such that with $k:= 4l+2$ it holds that $(2+\varepsilon)\eta_k >1+(2k)^{-1}.$ Hence, with $I=(k,k+1)$ we get $$\begin{aligned}
        \int_{I}{\Big|\psi-\langle\psi\rangle_{I}\Big|}^{2+\varepsilon} = \int_{0}^1{\Big|\psi_0^k-\langle\psi_0^k\rangle_{(0,1)}\Big|}^{2+\varepsilon} &\ge  c_k^{2+\varepsilon} \int_{c_k^{6k} e^{-100k^2}}^{2c_k^{6k} e^{-100k^2}}{\Big| x^{-\frac{1}{2+k^{-1}}} - \int_{c_k^{6k}e^{-100k^2}}^1 x^{-\frac{1}{2+k^{-1}}} \Big|}^{2+\varepsilon}{\,\mathrm{d}}x \\
&\gtrsim c_k^{2+\varepsilon} \int_{c_k^{6k} e^{-100k^2}}^{2c_k^{6k} e^{-100k^2}}{\Big| x^{-\frac{1}{2+k^{-1}}}   \Big|}^{2+\varepsilon}{\,\mathrm{d}}x\\
&\gtrsim c_k^{\varepsilon-1} e^{50k}.
        \end{aligned}$$ On the other hand, $$\int_{I}{\Big|\phi-\langle\phi\rangle_{I}\Big|}^{2+\varepsilon} = \int_{0}^1{\Big|\phi_0-\langle\phi_0\rangle_{(0,1)}\Big|}^{2+\varepsilon}\sim 1.$$ We conclude the proof by letting $\ell \rightarrow \infty$.

Theorem \[nip1\] leads us to consider weaker joint conditions involving Young functions that can be made stricly weaker than $S_{2+\varepsilon},T_{2+\varepsilon}$ for all $\varepsilon > 0.$

Young functions, their basic properties and the conditions $S_{A,B},T_C$
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We may also define joint conditions involving Young functions. A function $A: [0, \infty)\rightarrow [0, \infty)$ is called a Young function if it is continuous, convex, strictly increasing and satisfies $$A(0)=0,\qquad\lim_{t\rightarrow \infty} A(t)/t =\infty.$$ Given a Young function $A$, the complementary Young function $\bar A$ is defined by $$\bar A(t)= \sup_{s>0} \{st-A(s)\}, \quad t>0.$$ We also have the maximal function associated with a Young function $A:$ $$M_Af(x)= \sup_{Q\ni x}\langle |f| \rangle_{A, Q},$$ where the Luxemburg norm is defined 