First and second order Opial inequalities

Steven Bloom

Studia Mathematica (1997)

  • Volume: 126, Issue: 1, page 27-50
  • ISSN: 0039-3223

Abstract

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Let T γ f ( x ) = ʃ 0 x k ( x , y ) γ f ( y ) d y , 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 ) d x ≤ C ( ʃ 0 ∞ | f ( x ) | p v ( x ) d x ) ( 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.

How to cite

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Bloom, Steven. "First and second order Opial inequalities." Studia Mathematica 126.1 (1997): 27-50. <http://eudml.org/doc/216442>.

@article{Bloom1997,
abstract = {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.},
author = {Bloom, Steven},
journal = {Studia Mathematica},
keywords = {weighted norm inequality; second-order reduced Opial inequalities},
language = {eng},
number = {1},
pages = {27-50},
title = {First and second order Opial inequalities},
url = {http://eudml.org/doc/216442},
volume = {126},
year = {1997},
}

TY - JOUR
AU - Bloom, Steven
TI - First and second order Opial inequalities
JO - Studia Mathematica
PY - 1997
VL - 126
IS - 1
SP - 27
EP - 50
AB - 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.
LA - eng
KW - weighted norm inequality; second-order reduced Opial inequalities
UR - http://eudml.org/doc/216442
ER -

References

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