Mixed Norm Inequalities for Lebesgue Spaces
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RESEARCH ARTICLE
Mixed Norm Inequalities for Lebesgue Spaces Pankaj Jain1 • Santosh Kumari2 • Monika Singh3
Received: 8 April 2016 / Revised: 8 April 2016 / Accepted: 24 May 2019 Ó The National Academy of Sciences, India 2019
Abstract We give necessary and sufficient condition for the two-dimensional mixed norm Hardy inequality to hold with functions that are non-increasing in each variable. The corresponding inequalities involving adjoint operator and the mixed Hardy operators are also discussed. Keywords Weighted Lebesgue space Mixed norm Hardy operator Adjoint Hardy operator
in one dimension and Z xZ y f ðs; tÞdtds ðH2 f Þðx; yÞ :¼ 0
in two dimensions. For weights w1 , w2 , v1 and v2 consider the two-dimensional mixed norm inequality Z q1 !q1 Z 1
1
w1 ðxÞ
0
C
Z
1
v1 ðxÞ 0
By a weight function, we shall mean a function which is measurable, positive, integrable and finite almost everywhere (a.e.) on ð0; 1Þ. Consider the Hardy operator Z x ðH1 f ÞðxÞ :¼ f ðtÞdt 0
Santosh Kumari [email protected] Monika Singh [email protected] 1
Department of Mathematics, South Asian University, Akbar Bhawan, Chanakya Puri, New Delhi 110021, India
2
Department of Mathematics, Bhagini Nivedita College (University of Delhi), Kair, Delhi 110043, India
3
Department of Mathematics, Lady Shri Ram College For Women (University of Delhi), Lajpat Nagar, New Delhi 110 024, India
1
dx
Z
1
f p2 ðx; yÞv2 ðyÞdy
!p1
pp1 2
ð1:1Þ
1
dx
0
and the following two one-dimensional inequalities: Z 1 q1 Z 1 p1 1 1 q1 p1 ðH1 gÞ ðxÞw1 ðxÞdx C g ðxÞv1 ðxÞdx 0
0
ð1:2Þ and Z 0
& Pankaj Jain [email protected]; [email protected]
q2
ðH2 f Þq2 ðx; yÞw2 ðyÞdy
0
Mathematics Subject Classification 26D10
1 Introduction
0
1
q2
ðH1 hÞ ðyÞw2 ðyÞdy
q1
2
C
Z
p1
1 p2
h ðyÞv2 ðyÞdy
2
:
0
ð1:3Þ For nonnegative functions, Appell and Kufner [1] studied (1.1) in terms of (1.2) and (1.3). In fact, they proved that for the inequality (1.1) to hold, it is necessary that at least one of (1.2) and (1.3) holds while for sufficiency of (1.1), both (1.2) and (1.3) must hold. In [2], Fiorenza, Gupta and Jain remarked that interesting inequalities are those in which the RHS is finite. With this in mind, they proved that both (1.2) and (1.3) are necessary and sufficient for (1.1) to hold. For more mixed norm inequalities involving different operators, one may refer to [3–5].
123
P. Jain et al.
In this paper, we re-investigate (1.1) for those functions f which are non-increasing in each variable. We prove that such inequality holds if and only if both (1.2) and (1.3) hold for all non-increasing functions g and h. We point out that for necessity, we use modified arguments as used in [2]. However, sufficiency does not follow from those arguments. We make use of level functions to obtain sufficiency. We also obtain characterization when the operators H1 and H2 are replaced by their adjoint i.e., Z 1 ðH1 f ÞðxÞ :¼ f ðtÞdt x
and ðH2 f Þðx; yÞ
Z
1
Z
:¼ x
1
f ðs; tÞdtds:
y
In the ineq
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