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

Let (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 ∈ 𝒳. In this paper, we show that the boundedness of a Calderón–Zygmund operator T on L2(μ) is equivalent to that of T on Lp(μ) for some p ∈ (1,∞), and that of T from L1(μ) to L1,∞(μ). As an application, we prove that if T is a Calderón–Zygmund operator bounded on L2(μ), then its maximal operator is bounded on Lp(μ) for all p ∈ (1,∞) and from the space of all complex-valued Borel measures on 𝒳 to L1,∞(μ). All these results generalize the corresponding results of Nazarov et al. on metric spaces with measures satisfying the so-called polynomial growth condition.

A study of catalytic actions at solid surfaces. Part VII.— The influence of pressure on the rate of hydrogenation of liquids in presence of Nickel

In Parts I, II and V of this series we have indicated the reasons which lead us to believe that during the catalytic hydrogenation of liquids the function of the metallic catalyst is to combine with both agents—the unsaturated organic compound and hydrogen—and produce an unstable intermediate complex. The experimental evidence has furthermore impressed upon us the conviction that the determining factor in hydrogenation is the degree of affinity displayed between nickel and the unsaturated compound; whilst we have had ample opportunity to observe the selective nature of the process as exemplified by the widely-varying rates of absorption of hydrogen characteristic of various definite types of organic compounds, we have not been fortunate enough to obtain experimental proof of the mode in which nickel effects the actual introduction of hydrogen to an organic molecule. We have, therefore, directed our attention to the influence of the concentration of hydrogen on the rate of hydrogenation of liquids, and have examined a wide range of unsaturated organic compounds with respect to the rates at which they absorb hydrogen under varying pressures. Existing data on this subject are scanty, but indicate, in the case of the hydrogenation of fatty oils, that the rate of action is simply proportional to the pressure. On the other hand, it has been suggested that the action of nickel is to adsorb the hydrogen (at the same time dissociating it into the atomic condition) and that it is the atomic hydrogen which interacts with the ethylenic linkage; if this were the case the acceleration induced by increasing the gas-pressure should be proportional to the square root of the latter.