arXiv subjects

Suile Liu

Publications and source records attributed to Suile Liu.

2 recordsLinked to original sources

arXiv2013

Boundedness of Maximal Calderón-Zygmund Operators on Non-homogeneous Metric Measure Spaces

Let $(\cx,\,d,\,μ)$ be a metric measure space and satisfy the so-called upper doubling condition and the geometrically doubling condition. In this paper, the authors show that for the maximal Calderón-Zygmund operator associated with a singular integral whose kernel satisfies the standard size condition and the Hörmander condition, its $L^p(μ)$ boundedness with $p\in(1,\infty)$ is equivalent to its boundedness from $L^1(μ)$ into $L^{1,\infty}(μ)$. Moreover, applying this, together with a new Cotlar type inequality, the authors show that if the Calderón-Zygmund operator $T$ is bounded on $L^2(μ)$, then the corresponding maximal Calderón-Zygmund is bounded on $L^p(μ)$ for all $p\in(1,\infty)$, and bounded from $L^1(μ)$ into $L^{1,\infty}(μ)$. These results essentially improve the existing results.

math.CA↗

arXiv2010

Boundedness of Calderón-Zygmund Operators on Non-homogeneous Metric Measure Spaces

Let $({\mathcal X}, d, μ)$ be a separable metric measure space satisfying the known upper doubling condition, the geometrical doubling condition and the non-atomic condition that $μ(\{x\})=0$ for all $x\in{\mathcal X}$. In this paper, we show that the boundedness of a Calderón-Zygmund operator $T$ on $L^2(μ)$ is equivalent to that of $T$ on $L^p(μ)$ for some $p\in (1, \infty)$, and that of $T$ from $L^1(μ)$ to $L^{1,\,\infty}(μ).$ As an application, we prove that if $T$ is a Calderón-Zygmund operator bounded on $L^2(μ)$, then its maximal operator is bounded on $L^p(μ)$ for all $p\in (1, \infty)$ and from the space of all complex-valued Borel measures on ${\mathcal X}$ to $L^{1,\,\infty}(μ)$. All these results generalize the corresponding results of Nazarov et al. on metric spaces with measures satisfying the so-called polynomial growth condition.

math.CA↗