Sentence examples for estimates of weak from inspiring English sources

An extension of the Marcinkiewicz interpolation theorem is proved, yielding a necessary and sufficient condition for every quasilinear operator, satisfying given endpoint estimates of weak type, to be bounded from an Orlicz space into another.

In this section, we give the decay estimates of weak solutions obtained by Corollary 4.1.

Very recently, Xiang [14] established the asymptotic estimates of weak solutions for p-Laplacian equation with Hardy term and critical Sobolev exponent.

Therefore, this estimate also bounds the minimal solution, so that the outcome of the two methods is the same, as far as existence and estimates of weak solutions is concerned.

Therefore, our aim of this paper is to attain the (L^{p} -theory for quasiL^{p} -theoryiptic equations, whose conclusions and approach are diforrent from the Hölder estimates of weak solutions derived in [2, 38, 39].

The crucial point of the proof of our result is to obtain an L∞-estimate of weak solutions and to verify a certain "Pohozaev-type inequality" for weak solutions belonging to P.

For the second result, we have the estimate of weak solutions in L∞tLpv, p>1, which extends part of results by Fournier and Guerin [10] and Alexandre, Liao and Lin [1].

This paper deals with some global in time a priori estimates of the spatially homogeneous Landau equation for soft potentials γ∈[−2,0). For the first result, we obtain the estimate of weak solutions in LαtL3−εv for α="2(3−ε)3(2−ε) and 0<ε<1, which is an improvement over estimates by Fournier and Guerin [10].

The Commerce Department reported that the nation's total economic output decreased at a 1% annual rate from January through March, down significantly from the government's first estimate of weak, but positive, 0.1% growth for the period.

Apple Computer said that its fourth-quarter earnings would fall far below estimates because of weak September sales.

The DuPont Company, posting a second-quarter loss, said yesterday that it would eliminate more jobs and warned that its third-quarter earnings would fall short of estimates because of weak demand for chemicals and plastics.