First and second order Opial inequalities
Studia Mathematica, Tome 126 (1997) no. 1, pp. 27-50
Voir la notice de l'article provenant de la source Institute of Mathematics Polish Academy of Sciences
Let $T_γ f(x) = ʃ_0^x k(x,y)^γ f(y)dy$, where k is a nonnegative kernel increasing in x, decreasing in y, and satisfying a triangle inequality. An nth-order Opial inequality has the form $ʃ_0^∞ (∏_{i=1}^n |T_{γ_i}f(x)|^{q_i}|) |f(x)|^{q_0} w(x)dx ≤ C(ʃ_0^∞ |f(x)|^p v(x)dx)^{(q_0+…+q_n)/p}$. Such inequalities can always be simplified to nth-order reduced inequalities, where the exponent $q_0 = 0$. When n = 1, the reduced inequality is a standard weighted norm inequality, and characterizing the weights is easy. We also find necessary and sufficient conditions on the weights for second-order reduced Opial inequalities to hold.
@article{10_4064_sm_126_1_27_50,
author = {Steven Bloom},
title = {First and second order {Opial} inequalities},
journal = {Studia Mathematica},
pages = {27--50},
publisher = {mathdoc},
volume = {126},
number = {1},
year = {1997},
doi = {10.4064/sm-126-1-27-50},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.4064/sm-126-1-27-50/}
}
Steven Bloom. First and second order Opial inequalities. Studia Mathematica, Tome 126 (1997) no. 1, pp. 27-50. doi: 10.4064/sm-126-1-27-50
Cité par Sources :