The paper establishes integral representation formulas in arbitrarily wide Banach spaces $b_{\omega }^p\left({ℝ}^n\right)$ of functions harmonic in the whole ${ℝ}^n$.

Let $M_{\beta }$ be the fractional maximal function. The commutator generated by $M_{\beta }$ and a suitable function $b$ is defined by $[M_{\beta },b]f=M_{\beta }\left(bf\right)-b{M}_{\beta }\left(f\right)$. Denote by $𝒫\left({ℝ}^n\right)$ the set of all measurable functions $p(·):{ℝ}^n\to [1,\infty )$ such that $$1<p_{-}:=\underset{x\in {ℝ}^n}{\mathrm{ess}\inf }p\left(x\right)\text{and}{p}_{+}:=\underset{x\in {ℝ}^n}{\mathrm{ess}\sup }p\left(x\right)<\infty ,$$ and by $ℬ\left({ℝ}^n\right)$ the set of all $p(·)\in 𝒫\left({ℝ}^n\right)$ such that the Hardy-Littlewood maximal function $M$ is bounded on $L^{p(·)}\left({ℝ}^n\right)$. In this paper, the authors give some characterizations of $b$ for which $[M_{\beta },b]$ is bounded from $L^{p(·)}\left({ℝ}^n\right)$ into $L^{q(·)}\left({ℝ}^n\right)$, when $p(·)\in 𝒫\left({ℝ}^n\right)$, $0<\beta <n/p_{+}$ and $1/q(·)=1/p(·)-\beta /n$ with $q(·)(n-\beta )/n\in ℬ\left({ℝ}^n\right)$.

We prove that, if μ>0, then there exists a linear manifold M of harmonic functions in ${ℝ}^N$ which is dense in the space of all harmonic functions in ${ℝ}^N$ and lim‖x‖→∞ x ∈ S ‖x‖μDαv(x) = 0 for every v ∈ M and multi-index α, where S denotes any hyperplane strip. Moreover, every nonnull function in M is universal. In particular, if μ ≥ N+1, then every function v ∈ M satisfies ∫H vdλ =0 for every (N-1)-dimensional hyperplane H, where λ denotes the (N-1)-dimensional Lebesgue measure. On the other...