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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 $$1/p+1/p'=1$$.

##### MSC:
 46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems 31C99 Generalizations of potential theory 30C99 Geometric function theory
##### Keywords:
strong $$A_\infty$$-weight; Besov space; capacity
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