• East Asian mathematical journal
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• v.23 no.1
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• pp.37-43
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• 2007

We study a maximal operator defined on spaces of homogeneous type, and we prove that this operator is of weak type (1,1). As a consequence we show that the maximal operator belongs to the Muckenhoupt's class $A_1$.

• Bulletin of the Korean Mathematical Society
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• v.36 no.1
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• pp.25-31
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• 1999

Let $\nu$ be a positive Borel measure on a space of homogeneous type (X, d, $\mu$), satisfying the doubling property. A condition on a weight $\omega$ for whixh a maximal operator $M\nu f$(x) defined by M$mu$f(x)=supr>0{{{{ { 1} over {ν(B(x,r)) } INT _{ B(x,r)} │f(y)│d mu (y)}}}}, is of weak type (p,p) with respect to (ν, $omega$), is that there exists a constant C such that C $omega$(y) for a.e. y$\in$B(x, r) if p=1, and {{{{( { 1} over { upsilon (B(x,r) } INT _{ B(x,r)}omega(y) ^ (-1/p-1) d mu (y))^(p-1)}}}} C, if 1$infty$.

• Journal of the Korean Mathematical Society
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• v.55 no.3
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• pp.671-694
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• 2018

In this paper, we investigate the weighted vector-valued bounds for a class of multilinear singular integral operators, and its commutators, from $L^{p_1}(l^{q_1};\;{\mathbb{R}}^n,\;w_1){\times}{\cdots}{\times}L^{p_m}(l^{q_m};\;{\mathbb{R}}^n,\;w_m)$ to $L^p(l^q;\;{\mathbb{R}}^n,\;{\nu}_{\vec{w}})$, with $p_1,{\cdots},p_m$, $q_1,{\cdots},q_m{\in}(1,\;{\infty})$, $1/p=1/p_1+{\cdots}+1/p_m$, $1/q=1/q_1+{\cdots}+1/q_m$ and ${\vec{w}}=(w_1,{\cdots},w_m)$ a multiple $A_{\vec{P}}$ weights. Our argument also leads to the weighted weak type endpoint estimates for the commutators. As applications, we obtain some new weighted estimates for the $Calder{\acute{o}}n$ commutator.

• Communications of the Korean Mathematical Society
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• v.10 no.4
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• pp.881-887
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• 1995

Let f be a measurable function on the unit ball B in $C^n$, then we define a maximal function $M_p(f), 1 \leq p < \infty$, by $$ M_p(f)(\zeta ) = \sup_{\delta > 0}(\frac{1}{\sigma(\beta(\zeta, \delta))} \int_{T(\beta(\zeta, \delta))} $\mid$f(z)$\mid$^p \frac{d\nu(z)}{(1-$\mid$z$\mid$^n})^{1/p} $$ where $\sigma$ denotes the surface area measure on S, the boundary of B, and $T(\beta(\zeta, \delta))$ denotes the tent over the ball $\beta(\zeta, \delta)$. We prove that the maximal operator $M_p$ belongs to the Muckenhoupt class $A_1$.

• Applied Biological Chemistry
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• v.18 no.4
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• pp.194-202
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• 1975

This study was conducted to study the properties of the water-soluble natural chelating agents from garbage compost and activated sewage sludge responsible for Fe chelation, which is closely associated with the effectiveness in correcting iron chlorosis in plant. The water-soluble fraction of these materials was fractionated by menas of Sephadex gel filtration and the fractions of Fe chehates were traced by radioactive $^{59}Fe$. The fractions were examined by ultraviolet and infrared. spectroscopy and stability constants for Fe. The water-soluble fraction from garbage compost was separated by Sephadex G-25 into approximately four fractions. Most of the added $^{59}Fe$ was associated with fraction I, which appeared at the void volume. Further fractionation by Sephadex G-50 indicated that the molecular weight of water-soluble chelating agents is in the approximate range of 5000 to 10,000. The water-soluble fraction from activated sewage sludge gave six fractions by Sephadex G-25. Most of the added $^{59}Fe$ was found in the fraction I,II, and III, The molecular weights of most chelating agents associated with $^{59}Fe$ appeared to be less than 5,000 and those of fraction I that appeared at the void volume was in the range of 5,000 to 1,000. Discrepancy between radio activity count and UV absorption indicated the heterogeneity of the fractions obtained by Sephadex gel filtration. Ultraviolet absorption spectra of all fractions separated by Sephadex G-25 and containing chelating agents showed no differences. Fraction IV and V of sewage extract showed absorption maxima and shifting similar to nucleic acid components suggesting the presence of decomposition products of nucleic acid. Similarity fraction VI contained phenolic type amino acid groups. Fraction I of compost extract contained most of the added $^{59}Fe$ and showed weak but extra definite absorption in the 1230, and $1270cm^{-1}$ region, suggesting that extra oxygen groups in polyphenolic structure were probably involved in Fe chelation. In sewage extract, fraction I,II, and III in which most of the $^{59}Fe$ was found, showed strong definite polypeptide absorption in the region of $1540cm^{-1}$ due to NH deformation and C-N stretching of amide groups in the peptidebond. These extra functional groups in fraction I, II, and III appeared to be associated with Fe chelation. The other fractions, not associated with $^{59}Fe$, still have carboxyl and hydroxyl groups, suggesting that these functional groups in these water extracts may not independently form the Fe chelates. Precipitation of ferric hydroxide precluded measuring the stability constants for Fe-chelates. However, the formation constants for Zn chelates as log K values for compost extract and sewage extract at pH 4.0 from which the strength of chelation with Fe could be presumed, were 8.23, and 9.75, respectively, indicating strong complexation with metals. The chelating capacity of compost extract containing 6.5 g organic matter per liter was 0.82 mM, and that of sewage extract containing 5.3 g per liter was 0. 64 mM.