The hardy-littlewood maximal function of a sobolev function

We prove that the Hardy-Littlewood maximal operator is bounded in the Sobolev spaceW ^1,p (R ⁿ ) for 1<p≤∞. As an application we study a weak type inequality for the Sobolev capacity. We also prove that the Hardy-Littlewood maximal function of a Sobolev function is quasi-continuous.     

Observation of topical catalysis by sphingomyelinase coupled to microspheres

Sphingomyelinase, SMase (EC 3.1.4.12), was coupled onto amino-derivatized acrylate microspheres and was shown to retain its catalytic activity. The immobilized enzyme allows one to carry out topical enzymatic reaction in a controlled manner. Accordingly, these spheres were held with a micropipet and using micromanipulator brought into contact with a giant liposome membrane composed of phosphatidylcholine and sphingomyelin (SOPC/C16:0-SM, 0.75:0.25, molar ratio), representing the substrate for the immobilized enzyme. The macroscopic consequences of the enzyme reaction were visualized using fluorescence microscopy as well as differential interference contrast microscopy. The surface contact of the giant vesicle and immobilized enzyme causes membrane microdomain formation and domain clustering (capping) in the membrane and subsequent shedding of small vesicles from the membrane into the interior of the giant liposome. The method described represents a novel approach to study enzymatic reactions and allows manipulating giant vesicles as well as cultured cells in a spatially controlled manner.