Boundedness of Lusin-area and $g_λ^\ast$ Functions on Localized BMO Spaces over Doubling Metric Measure Spaces

Details Boundedness of Lusin-area and $g_λ^\ast$ Functions on Localized BMO Spaces over Doubling Metric Measure Spaces Boundedness of Lusin-area and $g_λ^\ast$ Functions on Localized BMO Spaces over Doubling Metric Measure Spaces

2009-03-26

arxiv.org/abs/0903.4587v2

Mathematics

Classical Analysis and ODEs

30 pages; Bull. Sci. Math. (to appear).

Scientific paper

Let ${\mathcal X}$ be a doubling metric measure space. If ${\mathcal X}$ has the $\delta$-annular decay property for some $\delta\in (0,\,1]$, the authors then establish the boundedness of the Lusin-area function, which is defined via kernels modeled on the semigroup generated by the Schr\"odinger operator, from localized spaces ${\rm BMO}_\rho({\mathcal X})$ to ${\rm BLO}_\rho({\mathcal X})$ without invoking any regularity of considered kernels. The same is true for the $g^\ast_\labda$ function and unlike the Lusin-area function, in this case, ${\mathcal X}$ is not necessary to have the $\delta$-annular decay property. Moreover, for any metric space, the authors introduce the weak geodesic property and the monotone geodesic property, which are proved to be respectively equivalent to the chain ball property of Buckley. Recall that Buckley proved that any length space has the chain ball property and, for any metric space equipped with a doubling measure, the chain ball property implies the $\delta$-annular decay property for some $\delta\in (0,1]$. Moreover, using some results on pointwise multipliers of ${\rm bmo}({\mathbb R})$, the authors construct a counterexample to show that there exists a nonnegative function which is in ${\rm bmo}({\mathbb R})$, but not in ${\rm blo}({\mathbb R})$; this further indicates that the above boundedness of the Lusin-area and $g^\ast_\lambda$ functions even in ${\mathbb R}^d$ with the Lebesgue measure or the Heisenberg group also improves the existing results.

Affiliated with

Also associated with

No associations

Rating

Boundedness of Lusin-area and $g_λ^\ast$ Functions on Localized BMO Spaces over Doubling Metric Measure Spaces does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Boundedness of Lusin-area and $g_λ^\ast$ Functions on Localized BMO Spaces over Doubling Metric Measure Spaces, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Boundedness of Lusin-area and $g_λ^\ast$ Functions on Localized BMO Spaces over Doubling Metric Measure Spaces will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-64828