arXiv 1602.008202016-02-02math.FA

带核Hardy型算子的有界性：$0<q<1\le p<\infty$情形的积分加权条件

Boundedness of Hardy-type operators with a kernel: integral weighted conditions for the case $0<q<1\le p<\infty$

Martin Křepela

摘要：

设$U:[0,\infty)^2 \to [0,\infty)$是一个~可测核，满足：（i）$U(x,y)$在$x$中不增，在$y$中不减;（ii）存在一个~常数$θ>0$，使得对所有$0\le x<y<z<\infty$都有$U(x,z) \le θ\left( U(x,y)+U(y,z) \right)$;（iii）对所有$y>0$都有$U(0,y)>0$.令$0<q<1< p <\infty$。证明了加权不等式PDMATHBMW对PDMATHBMY上的所有非负可测函数PDMATHBMX成立当且仅当PDMATHBMZ和\[ \left( \int_0^\infty \left( \int_t^\infty w(x) U^q(t,x) dx \right)^\frac{r}{p} w(t) \sup_{z\in(0,t)} U^q(z,t)\left( \int_0^z v^{1-p'}(s) ds \right)^\frac{r}{p'} dt \right)^\frac 1r <\infty, \]成立，其中$p':=\frac{p}{p-1}$和$r:=\frac{pq}{p-q}$.类似的情况下$p=1$和对偶版本的不等式的条件。

英文摘要：

Let $U:[0,\infty)^2 \to [0,\infty)$ be a~measurable kernel satisfying: (i) $U(x,y)$ is nonincreasing in $x$ and nondecreasing in $y$; (ii) there exists a~constant $θ>0$ such that $U(x,z) \le θ\left( U(x,y)+U(y,z) \right)$ for all $0\le x<y<z<\infty$; (iii) $U(0,y)>0$ for all $y>0$. Let $0<q<1< p <\infty$. We prove that the weighted inequality \[ \left( \int_0^\infty \left( \int_0^t f(x)U(x,t) dx \right)^q w(t) dt \right)^\frac 1q \le C \left( \int_0^\infty f^p(t)v(t)dt \right)^\frac 1p \] holds for all nonnegative measurable functions $f$ on $(0,\infty)$ if and only if \[ \left( \int_0^\infty \left( \int_t^\infty w(x)dx \right)^\frac{r}{p} w(t) \left( \int_0^t U^{p'}(z,t)v^{1-p'}(z) dy \right)^\frac{r}{p'} dt \right)^\frac 1r <\infty \] and \[ \left( \int_0^\infty \left( \int_t^\infty w(x) U^q(t,x) dx \right)^\frac{r}{p} w(t) \sup_{z\in(0,t)} U^q(z,t)\left( \int_0^z v^{1-p'}(s) ds \right)^\frac{r}{p'} dt \right)^\frac 1r <\infty, \] where $p':=\frac{p}{p-1}$ and $r:=\frac{pq}{p-q}$. Analogous conditions for the case $p=1$ and for the dual version of the inequality are also presented.

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