On bilinear Hardy inequality and corresponding geometric mean inequality
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On bilinear Hardy inequality and corresponding geometric mean inequality Amiran Gogatishvili1
· Pankaj Jain2 · Saikat Kanjilal2
Received: 15 July 2020 / Accepted: 11 September 2020 © Università degli Studi di Napoli "Federico II" 2020
Abstract The main aim of this paper to provide several scales of equivalent conditions for the bilinear Hardy inequalities in the case 1 < q, p1 , p2 < ∞ with q ≥ max( p1 , p2 ). Keywords Hardy inequality · Bilinear Hardy inequality · Integral conditions · Equivalent conditions · Geometric mean inequality Mathematics Subject Classification 26D10 · 46E35
1 Introduction Let M denote the set of all Lebesgue measurable functions on (a, b), −∞ ≤ a < b ≤ ∞, M+ ⊂ M is the subset of all non-negative functions. p and write Let u, v, ∈ M+ , 0 < p, q ≤ ∞, p ≥ 1. Denote p := p−1
b
U (x) =
u(t) dt, V (x) =
x
x
v 1− p (t) dt,
(1)
a
and assume that U (x) < ∞, V (x) < ∞ for almost everywhere (a.e.) x ∈ (a, b).
B
Amiran Gogatishvili [email protected] Pankaj Jain [email protected]; [email protected] Saikat Kanjilal [email protected]; [email protected]
1
Institute of Mathematics of the Czech Academy of Sciences, Žitná 25, 115 67 Prague 1, Czech Republic
2
Department of Mathematics, South Asian University, Chanakya Puri, New Delhi 110021, India
123
A. Gogatishvili et al.
Consider the one dimensional Hardy inequality
b
a
x
q f (t) dt
q1 u(x) d x
b
≤C
a
f (x)v(x) d x p
1p
,
f ∈ M+ .
(2)
a
It is known that the inequality (2) is characterized by the Muckenhoupt condition [9] in the case 1 < p ≤ q < ∞ which is given by 1
A M := sup A M (x) = sup U q (x)V a
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