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Эта публикация цитируется в 8 научных статьях (всего в 8 статьях)
Статьи
The fractional Riesz transform and an exponential potential
B. Jayea, F. Nazarova, A. Volbergb a Kent State University, Department of Mathematics, Kent, OH
b Michigan State University, Department of Mathematics, East Lansing, MI
Аннотация:
In this paper we study the $s$-dimensional Riesz transform of a finite measure $\mu$ in $\mathbf R^d$, with $s\in(d-1,d)$. We show that the boundedness of the Riesz transform of $\mu$ yields a weak type estimate for the Wolff potential $\mathcal W_{\Phi,s}(\mu)(x)=\int_0^\infty\Phi\bigl(\frac{\mu(B(x,r))}{r^s}\bigl)\frac{dr}r$, where $\Phi(t)=e^{-1/t^\beta}$ with $\beta>0$ depending on $s$ and $d$. In particular, this weak type estimate implies that $\mathcal W_{\Phi,s}(\mu)$ is finite $\mu$-almost everywhere. As an application, we obtain an upper bound for the Calderón–Zygmund capacity $\gamma_s$ in terms of the non-linear capacity associated to the gauge $\Phi$. It appears to be the first result of this type for $s>1$.
Ключевые слова:
Riesz transform, Calderón–Zygmund capacity, nonlinear capacity, Wolff potential, totally lower irregular measure.
Поступила в редакцию: 11.07.2012
Образец цитирования:
B. Jaye, F. Nazarov, A. Volberg, “The fractional Riesz transform and an exponential potential”, Алгебра и анализ, 24:6 (2012), 77–123; St. Petersburg Math. J., 24:6 (2013), 903–938
Образцы ссылок на эту страницу:
https://www.mathnet.ru/rus/aa1310 https://www.mathnet.ru/rus/aa/v24/i6/p77
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