I'm stuck on the following exercise 17.9 from Leoni's text A first course in Sobolev Spaces (second edition). This is from the chapter on Besov spaces, but this is really just a integral inequality which should only need basic analytic techniques.

For $1 \leq p < \infty,$ $s \in (0,N/p)$ and $a>0$ and define $$ u(x) = \chi(x) |x|^{s-N/p} \left|\log |x|\right|^a, $$ where $\chi \in C^{\infty}_c(\Bbb R^N)$ is a cutoff supported in $B(0,1)$ such that $\chi \equiv 1$ on $B(0,1/2).$ For which values of $1 \leq q < \infty$ does $u$ lie in the Besov space $B_q^{s,p}(\Bbb R^N)?$

For simplicity I will assume $N/p \leq 1,$ so $s<1.$ In this case the Besov spaces $B_q^{s,p}(\Bbb R^N)$ are defined in the text as the set of functions $u \in L^p(\Bbb R^N)$ such that $$ \lvert u \rvert_{B^{s,p}_q(\Bbb R^N)} = \left( \int_{\Bbb R^N} \lVert \Delta_hu\rVert_{L^p(\Bbb R^N)}^q \frac{\mathrm{d}h}{h^{N+sq}}\right)^{1/q} < \infty.$$ Here $\Delta_hu(x) = u(x+h)-u(x)$ is the first order forward difference quotient.

My thoughts: Due to the cutoff term we have $u$ is smooth away from zero, so it suffices to focus on what happens at the origin, and as is usual for Besov spaces we only need to worry about small values of $h.$ The main difficultly appears to come with how to estimate the following quantity $$ \lVert \Delta_hu\rVert_{L^p(\Bbb R^N)}$$ in terms of $h.$

The naïve approach I tried was to take a first order approximation $$ \Delta_hu(x) \sim \nabla u(x).h, $$ and take the $L^p$ norm on both sides. However this evidently does not hold uniformly in $x$ due to the singular behaviour at $x,$ and worse it doesn't make much use of the logarithmic growth term; in particular such an estimate will bound $\Delta_hu$ by constants independent of $a,$ which is the scale I think we want to capture here. This suggests that a Taylor expansion approximation wouldn't be suitable.

However I'm struggling to see how to estimate the quantity $$ |x+h|^{s-N/p}|\log|x+h||^a - |x|^{s-N/p}|\log|x||^a$$ in terms of something I can evaluate the $L^p$ norm of, and whose bound would depend on $h$ and $a.$ Any suggestions would be appreciated.


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