Strong \(A_{\infty}\)-weights and scaling invariant Besov capacities. (English) Zbl 1149.46028

This article studies strong \(A_\infty\)-weights and Besov capacities as well as their relationship to Hausdorff measures. A weight \(\omega\) is said to be an \(A_\infty\)-weight if there exist constants \(C\geq1\) and \(q>1\) such that \(\left(\frac{1}{| B| }\int_B \omega(x)^q\,dx\right)^{1/q}\leq\frac{1}{| B| }\int_B \omega(x)\,dx\) for all balls \(B\subset{\mathbb R}^n\). An \(A_\infty\)-weight \(\omega\) is called a strong \(A_\infty\)-weight if there exists a distance function \(\delta^1_\mu\) on \({\mathbb R}^n\) and a positive constant \(C\) such that \(C^{-1}\delta_\mu(x,y)\leq\delta_\mu^1(x,y)\leq C\delta_\mu(x,y)\), where \(\mu\) is the measure on \({\mathbb R}^n\) with density \(\omega\) and \(\delta_\mu(x,y)=\mu(B_{x,y})^{1/n}\), where \(B_{x,y}\) denotes the smallest closed ball which contains the points \(x\) and \(y\).
The author proves that in the Euclidean space \({\mathbb R}^n\) with \(n\geq2\), whenever \(n-1<s\leq n\), a function \(u\) yields a strong \(A_\infty\)-weight of the form \(\omega=e^{nu}\) if the distributional gradient \(\nabla u\) has sufficiently small \(\| .\| _{\mathcal{L}^{n,n-s}({\mathbb R}^n;\, {\mathbb R}^n)}\)-norm, where \(\mathcal{L}^{n,n-s}({\mathbb R}^n)\) denotes the Morrey space of vector-valued measurable functions and
\[ \| \nabla u\| _{\mathcal{L}^{n,n-s}({\mathbb R}^n;\,{\mathbb R}^n)}=\sup_{x\in{\mathbb R}^n}\sup_{r>0}\left(r^{-(n-s)}\int_{B(x,r)}| \nabla u(y)| ^n\, dy\right)^{1/n}. \]
As a corollary of this result, the author obtains strong \(A_\infty\)-weights of the form \(\omega=e^{nu}\), where \(u\) is a distributional solution of \(-\text{div}(| \nabla u| ^{n-2}\nabla u)=\mu\) whenever \(\mu\) is a signed Radon measure with small total variation. Similarly, the author also proves that if \(2\leq n<p<\infty\), then \(\omega=e^{nu}\) is a strong \(A_\infty\)-weight whenever the Besov \(B_p\)-seminorm \([u]_{B_p({\mathbb R}^n)}\) of \(u\) is sufficiently small, where \([u]_{B_p({\mathbb R}^n)}=(\int_{{\mathbb R}^n}\int_{{\mathbb R}^n}\frac{| u(x)-u(y)| ^p}{| x-y| ^{2n}}\,dx\,dy)^{1/p}\).
The author also develops a theory of the Besov \(B_p\)-capacity on \({\mathbb R}^n\) and proves that this capacity is a Choquet set function. Moreover, the author obtains lower estimates of the Besov \(B_p\)-capacities in terms of the Hausdorff content associated with gauge functions \(h\) satisfying the condition \(\int^1_0h(t)^{p'-1}\frac{dt}{t}<\infty\), where \(1/p+1/p'=1\).


46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
31C99 Generalizations of potential theory
30C99 Geometric function theory
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