# First and second order Opial inequalities

Studia Mathematica (1997)

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

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topBloom, 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 -

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