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Weighted norm inequalities for fractional operators. (English) Zbl 1158.42010

The very interesting paper under review deals with weighted norm inequalities for fractional powers of elliptic operators and their commutators with BMO functions, encompassing what is known for the classical Riesz potentials and elliptic operators with Gaussian domination by the classical heat operator. The authors’ method relies mainly upon good-$\lambda$ technique which do not uses any size or smoothness estimates for the kernels.

The model example considered is the fractional power of an elliptic operator $L$ over ${ℝ}^{n},$ given formally by

${L}^{-\alpha /2}=\frac{1}{{\Gamma }\left(\alpha /2\right)}{\int }_{0}^{\infty }{t}^{\alpha /2}{e}^{-tL}\frac{dt}{t}$

where $\alpha >0,$ $Lf=-\text{div}\phantom{\rule{0.166667em}{0ex}}\left(A\nabla f\right)$ with an elliptic, $n×n$ matrix $A$ of complex and ${L}^{\infty }$-value coefficients. The authors obtain sufficient conditions for the weighted norm boundedness of ${L}^{-\alpha /2}·$ Namely, if ${p}_{-} and $\alpha /n=1/p-1/q,$ then ${L}^{-\alpha /2}$ turns out to be bounded from ${L}^{p}\left({w}^{p}\right)$ into ${L}^{q}\left({w}^{q}\right)$ for $w\in {A}_{1+1/{p}_{-}-1/p}\cap R{H}_{q{\left({p}_{+}/q\right)}^{\text{'}}},$ where ${A}_{p}$ and $R{H}_{q}$ are the standard Muckenhoupt and reverse Hölder classes, respectively. Moreover, estimates for the $k$-th order commutators

${\left({L}^{-\alpha /2}\right)}_{b}^{k}f\left(x\right)={L}^{-\alpha /2}\left({\left(b\left(x\right)-b\right)}^{k}f\right)\left(x\right)$

with BMO functions are obtained. Precisely, if ${p}_{-} and $\alpha /n=1/p-1/q,$ then given $k\in ℕ,$ $b\in \text{BMO}$ and $w\in {A}_{1+1/{p}_{-}-1/p}\cap R{H}_{q{\left({p}_{+}/q\right)}^{\text{'}}},$ one has

$\parallel {\left({L}^{-\alpha /2}\right)}_{b}^{k}{f\parallel }_{{L}^{q}\left({w}^{q}\right)}\le {C\parallel b\parallel }_{\text{BMO}}^{k}{\parallel f\parallel }_{{L}^{p}\left({w}^{p}\right)}·$

##### MSC:
 42B25 Maximal functions, Littlewood-Paley theory 35J15 Second order elliptic equations, general