Change of variables for \(A_{\infty}\) weights by means of quasiconformal mappings: sharp results (Q2860898)

A measurable function \(w:\;\mathbb{R}^n\to\mathbb{R}^n\) is called a weight if it is positive almost everywhere and locally integrable in \(\mathbb{R}^n\). The Muckenhoupt class \(A_\infty\) consists of all weight functions \(w\) satisfying NEWLINE\[NEWLINEA_\infty(w):=\sup_{B}\left(\frac{1}{|B|}\int_B w(x)\,dx\right) \left(\exp\frac{1}{|B|}\int_B\log\frac{1}{w(x)}\,dx\right)<\infty,NEWLINE\]NEWLINE where the supremum is taken over all balls \(B\) of \(\mathbb{R}^n\). \(w\in A_\infty\) is equivalent to the condition that, for every ball \(B\subset\mathbb{R}^n\) and every measurable set \(E\subset B\), NEWLINE\[NEWLINE\frac{|E|}{|B|}\leq M\left[\frac{\int_E w(x)\,dx}{\int_B w(x)\,dx}\right]^\alpha,\eqno{(\ast)}NEWLINE\]NEWLINE where the constants \(0<\alpha\leq 1\leq M\) are independent of \(E\) and \(B\). For \(w\in A_\infty\), the \(\widetilde{A}_\infty\)-constant of \(w\) is defined by NEWLINE\[NEWLINE\widetilde{A}_\infty(w):=\inf\left\{\frac{M}{\alpha}:\;0<\alpha\leq 1\leq M\;\mathrm{and}\;(\ast)\;\mathrm{holds}\right\}.NEWLINE\]NEWLINENEWLINENEWLINEThe Gehring class \(G_1\) consists of all weights \(v\) satisfying NEWLINE\[NEWLINEG_1(v):=\sup_{B}\left(\exp\frac{1}{|B|}\int_B \frac{v(x)}{v_B}\log\frac{v(x)}{v_B}\,dx\right)<\infty,NEWLINE\]NEWLINE where \(v_B:=\frac{1}{|B|}\int_B v(x)\,dx\). \(v\in G_1\) is equivalent to the condition that, for every ball \(B\) of \(\mathbb{R}^n\) and every measurable set \(F\subset B\), NEWLINE\[NEWLINE\frac{\int_F v(x)\,dx}{\int_B v(x)\,dx}\leq L\left(\frac{|F|}{|B|}\right)^\beta,\eqno(\ast\ast)NEWLINE\]NEWLINE where the constants \(0<\beta\leq1 \leq L\) are independent of \(F\) and \(B\). For \(v\in G_1\), an auxiliary constant \(\widetilde{G}_1(v)\) of \(v\) is defined by NEWLINE\[NEWLINE\widetilde{G}_1(v):=\inf\left\{\frac{L}{\beta}:\;0<\beta\leq1 \leq L\;\mathrm{and}\;(\ast\ast)\;\mathrm{holds}\right\}.NEWLINE\]NEWLINENEWLINENEWLINENEWLINENEWLINELet \(\Omega\) be an open subset of \(\mathbb{R}^n\) with \(n\geq 2\). A homeomorphism \(f:\;\Omega\to \mathbb{R}^n\) is called a \(K\)-quasiconformal mapping for a constant \(K\geq 1\) if \(f\in W^{1,n}_{\mathrm{loc}}(\Omega,\mathbb{R}^n)\) and NEWLINE\[NEWLINE|Df(x)|^n\leq K J_f(x)\;\;\;\mathrm{for a.\,e.}\;x\in\Omega,NEWLINE\]NEWLINE where \(Df(x)\) stands for the differential matrix of \(f\), \(J_f(x)\) denotes the Jacobin determinant of \(f\) and \(|Df(x)|:=\sup\{|Df(x)\xi|:\;\xi\in\mathbb{R}^n,\;|\xi|=1\}\). The weakly quasisymmetric constant \(H_f\) of a quasiconformal mapping \(f\) is defined by NEWLINE\[NEWLINEH_f:=\sup\left\{\frac{|f(x)-f(y)|}{|f(x)-f(z)|}:\;x,\,y,\,z\in\Omega,\, x\neq z,\,\frac{|x-y|}{|x-z|}\leq 1\right\}.NEWLINE\]NEWLINENEWLINENEWLINEIn this paper, the authors prove that, if \(f:\;\mathbb{R}^n\to\mathbb{R}^n\) is a quasiconformal mapping with \(n\geq 2\) and \(w\in A_\infty\), then the following estimates hold NEWLINE\[NEWLINE\frac{1}{H^n_{f^{-1}}\widetilde{A}_\infty(J_{f^{-1}})}\widetilde{A}_\infty(w)\leq \widetilde{A}_\infty[(w\circ f)J_f]\leq H^n_f\widetilde{A}_\infty(J_f)\widetilde{A}_\infty(w)NEWLINE\]NEWLINE and NEWLINE\[NEWLINE\frac{1}{H^n_{f^{-1}}}\widetilde{A}_\infty(J_{f^{-1}})\leq\widetilde{G}_1(J_f) \leq H^n_f\widetilde{A}_\infty(J_{f^{-1}}),NEWLINE\]NEWLINE where \(f^{-1}\) denotes the inverse mapping of \(f\). The authors also point out that these estimates are sharp.