Weighted Inequalities for Hardy–Steklov Operators

2007 ◽  
Vol 59 (2) ◽  
pp. 276-295 ◽  
Author(s):  
A. L. Bernardis ◽  
F. J. Martín-Reyes ◽  
P. Ortega Salvador

AbstractWe characterize the pairs of weights (v, w) for which the operator with s and h increasing and continuous functions is of strong type (p, q) or weak type (p, q) with respect to the pair (v, w) in the case 0 < q < p and 1 < p < ∞. The result for the weak type is new while the characterizations for the strong type improve the ones given by H. P. Heinig and G. Sinnamon. In particular, we do not assume differentiability properties on s and h and we obtain that the strong type inequality (p, q), q < p, is characterized by the fact that the functionbelongs to Lr(gqw), where 1/r = 1/q – 1/p and the supremum is taken over all c and d such that c ≤ x ≤ d and s(d) ≤ h(c).

Author(s):  
A. L. Bernardis ◽  
F. J. Martín-Reyes

We consider the maximal operator defined on the real line by which is related to the Cesàro convergence of the singular integrals. We characterize the weights w for which Mα is of weak type, strong type and restricted weak type (p, p) with respect to the measure w(x) dx.


Author(s):  
Lina Karlova ◽  
◽  
Olena Lesnovska ◽  
Roman Mylostyvyi ◽  
◽  
...  

The aim of the research was to evaluate the influence of the type of the nervous system on the productive and reproductive qualities of animals. The type of nervous system (strong, weak) of cows of the Ukrainian red dairy breed was determined by the index of the nervous system, which is based on the variability of the fat content in the morning milk yield after changing the conditions of cows keeping. Under the new conditions of keeping cows with a strong type of nervous activity, the coefficients of variability of the morning milk yield and the fat content in milk practically remained at the same level (their slight increase was noted at the level of 4.5 and 3.1%). In cows of a weak type under similar conditions of keeping, a significant increase in the variability of milk yield (by 53.6%) and fat content in milk (by 266.7%) was found. In animals of both types, in the first days of summer-camp keeping, in comparison with winter-stall keeping, there was an increase in milk yield and a decrease in fat content in milk. In cows of the strong type, the fat content decreased by 0.08% and of the weak type by 0.12%. The average index of the nervous system type in animals of the strong type is 1.18, and in cows of the weak type 2.71 (P> 0.999). Before and during milking, the pulse rate in the cows of weak type was higher by 8.0 and 10.8 beats / min. (P> 0.999), compared with the strong type. The animals of strong type of the nervous system, in comparison with the weak type, had a smaller thickness of the concha and skin, a higher pulse and respiratory rate, and a lower rectal body temperature. In cows of a strong type of the nervous system in postnatal ontogenesis, the formation of productive traits took place more rapidly and they better adapted to the conditions of the external environment. This is evidenced by their high coefficient of relative decline in the growth at the age of one (85.0) than in animals with a weak type of nervous system (78.5). The advantage of cows with a strong type of nervous activity over a weak one in milk yield was 348 kg (P>0.95) and in milk fat 8.0 kg. The service period in animals with a strong type of nervous activity was 6.3 days shorter. The animals with a strong type of nervous activity during the first lactation had a higher adaptation index compared with animals of a weak type. The difference for this indicator is 0.49 (or 3.7%). They also had a higher productivity index for the first (3.1%) and second (4.0%) lactation. Thus, the typological features of the nervous activity of cows affect the formation of the most important functions of the body, which can be used in breeding work to improve the herd.


2021 ◽  
Vol 58 (2) ◽  
pp. 216-229
Author(s):  
Yanbo Ren ◽  
Congbian Ma

Let ɣ and Φ1 be nondecreasing and nonnegative functions defined on [0, ∞), and Φ2 is an N -function, u, v and w are weights. A unified version of weighted weak type inequality of the formfor martingale maximal operators f ∗ is considered, some necessary and su@cient conditions for it to hold are shown. In addition, we give a complete characterization of three-weight weak type maximal inequality of martingales. Our results generalize some known results on weighted theory of martingale maximal operators.


1995 ◽  
Vol 47 (2) ◽  
pp. 225-245
Author(s):  
Nakhlé Asmar ◽  
Earl Berkson ◽  
T. A. Gillespie

AbstractIn the context of a locally compact abelian group, we establish maximal theorem counterparts for weak type (1,1) multipliers of the classical de Leeuw theorems for individual strong multipliers. Special methods are developed to handle the weak type (1,1) estimates involved since standard linearization methods such as Lorentz space duality do not apply to this case. In particular, our central result is a maximal theorem for convolutions with weak type (1,1) multipliers which opens avenues of approximation. These results complete a recent series of papers by the authors which extend the de Leeuw theorems to a full range of strong type and weak type maximal multiplier estimates in the abstract setting.


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