J. of Inequal. & Appl., 1997, Vol. 1, pp. 183-197
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Weighted Modular Inequalities
for Monotone Functions
P. DR/i, BEKa, H.P. HEINIG ’,* and A. KUFNERc
Department of Mathematics, University of West Bohemia,
P. O. Box 314, 306 14 Pilsen, Czech Republic
b Department of Mathematics and
Statistics, McMaster University,
Hamilton, Ontario, L8S 4K1, Canada
Mathematical Institute, Czech Academy of Sciences,
Zitna 25, 11567 Prague, Czech Republic
(Received 4 June 1996)
Weight characterizations of weighted modular inequalities for operators on the cone of monotone
functions are given in terms of composition operators on arbitrary non-negative functions
with changes in weights. The results extend to modular inequalities, those corresponding to
weighted Lebesgue spaces given by E.T. Sawyer [15]. Application to Hardy and fractional
integral operators on monotone functions are given.
Keywords: Weighted modular inequalities; Orlicz spaces; norms; duality; Hardy type
inequalities.
1991 Mathematics Subject Classification: Primary 26D 15, 42B22; Secondary 46E30.
1 INTRODUCTION
The object of this note is to characterize weight functions
v, for which the modular inequality
Q[Wl (X) Tf (x)lwo(x) dx < p-1
Wi,
0, 1 and
P[C f (x)]v(x) dx
(1.1)
*Author for correspondence.
183
E DR/BEK et al.
184
holds, where T is a linear operator defined on non-negative monotone
functions. P and Q are N-functions, where no order relations such as
Q o P- convex- are assumed, and C > 0 is a constant. The characterizations,
via duality arguments are given in terms of modular inequalities of the form
(1.1) where T is replaced by the composition operators T o I and T o I*,
defined on arbitrary non-negative functions and with changes of weights.
Here I and I* are defined
(Ih)(x)
fo
x
h,
and
(I*h)(x)
h
x > O.
Our results generalize to modulars the duality principle for weighted
L P-spaces and operators defined on non-negative non-increasing functions
Ixlq/q,
given by E.T. Sawyer [15]. His results follow on taking Q(x)
P(x)
Ixl p/p, 1 < p, q < cx, while the case p q and T the Hardy
operator yields a result of M. Arifio and B. Muckenhoupt [2].
Characterizations of weights for which (1.1) holds and T is a Volterra
convolution type operator defined on monotone functions was given recently
by J.Q. Sun [19] [20] in the case when Q e-1 is convex. Our methods
are different from those and except for some special cases, we do not obtain
these results for the general Volterra convolution operator. Conversely our
results (Proposition 2.2) for the Riemann-Liouville operator on increasing
functions and the Weyl fractional integral operator defined on decreasing
functions does not follow from his.
Our main result (Theorem 2.3) depends strongly on a duality principle
given in [7] and on an extension of a result by C. Herz [8] and S. Bloom and
R. Kerman [3], establishing an equivalence of a weighted modular inequality
and certain weighted Orlicz-Luxemburg norm inequality (Proposition 2.1).
This result may be of independent interest.
We conclude this section by giving some definitions and notation required
in the sequel.
DEFINITION 1.1 (a) A convex function P IR+ -+ IR+ is called a Young
function if P(0) 0 and limx P(x) o.
(b) A continuous Young function P is called an N-function if it has the
form
Ixl
P(x)
p(t) dt
WEIGHTED MODULAR INEQUALITIES
185
where p is non-decreasing, right continuous, p(0+) 0 and p(o)
and p(t) > 0 if > 0. If p-1 denotes the right continuous inverse of p, then
the complementary function P of P is defined by
(x)
p-l(t)dt
If P is an N-function, so is/5.
(c) If v is a non-negative measurable (weight) function defined on a
measure space X and P is an N-function, then the Orlicz space L P(v)
consists of those measurable f on X for which the (Luxemburg) norm
IlfllP(v)
inf
{Z fxP [ If(x)l]
.
>O
v(x) dx < 1
(1.2)
is finite. If v(x) dx is replaced by a positive Borel measure/z we write
also P(dlx), and if v 1 simply P, in (1.2).
For further properties of N-functions and Orlicz spaces we refer to
[9, 10, 141.
DEFINITION 1.2 (a) An N-function P satisfies the A2 condition (P e A2),
if there is a constant D > 0, such that P(2x) < D P(x), for all x > 0.
(b) The order relation -< is defined by P -< Q, (P, Q, N-functions) if
there is a constant C > 0, such that
-QP-l(ai)<-CQe-l(i
for every non-negative sequence {ai }.
Note that if Q o P- is convex, then P -< Q. For other properties see [5,
Lemma 1.1 ].
Finally we denote by )e the characteristic function of the set E, and define
i0 by io(x) x. Constants are denoted by A, B, C, D and inequalities,
such as (1.1) are interpreted to mean that if the right side is finite, so is
the left side and the inequality holds. Weight functions are non-negative
measurable functions on a measure space X (usually IR+) and are denoted
by w0, w, w, u, v. Non-negative non-increasing _= decreasing, respectively,
non-decreasing increasing functions f are denoted by 0 _< f$, respectively
0 < f’.
E DRA,BEK et al.
186
2 RESULTS AND APPLICATIONS
We require the following result whose special case P Q and u w 1
was proved by C. Herz in the preprint [8] (see also [3, Proposition 2.5]).
PROPOSITION 2.1 Suppose (X, dtx), (Y, dr) are or-finite measure spaces and
T is a linear operator mapping measurable functions on X to measurable
functions on Y. If P and Q are N-functions, then the modular inequality
Q-l
{ fy
Q[W(y) l(Tf)(Y)l]dv(Y)
is satisfied,
} <- P-l { fx
if and only if, for every e
Ilw Tflla(eodv)
holds, where ea
1/Q(1/e), ep
> O,
<
inf
--inf
(2.2)
CllufllP(,d)
1/P(1/e).
-
Proof Suppose (2.2) holds. Define e by P ()
IlufllP(d
}
P[Cu(x) lf (x)l]dtz(x)
(2.1)
fx
then
{Z fxP [Ufz ]ePd <- }
> O"
P
>0"
I
dl <
1
P[uf]dl
and therefore w rfll (d) < C. From the homogeneity of the norm and
the linearity of T it follows that
IlwT(f/C)llQQd=inf{L>o" fyQ[WT(f/C)leQdu<
l
_<1.
Hence
Q p-1
a[w T(f/C)]dv <
which implies (2.1) after replacing f/C by f.
Conversely, if (2.1) is satisfied, fix f and let ot
homogeneity of the norms shows that
l
llu f/llP(epdlz)
inf
P[u f]dtx
Ilu flleu), then the
{) fxP [U-]epdlz }
> O"
< l
WEIGHTED MODULAR INEQUALITIES
187
and therefore,
Now (2.1) and the linearity of T imply
=8QQ
()
1.
Hence
< otC
CllufllP(eedx)
and the result follows.
The (weighted) duality principle in Orlicz spaces may be written in the
following form:
o<_f
IlfllP(o)
v
where v is a weight function and/5 is the complementary function of the
N-function P.
We also require the following known duality principle for monotone
functions:
TIEOREM 2.1 ([7, Theorems 2.2, 2.3]) Suppose P and [’ are N-functions
satisfying the A2 condition.
v, with (Iv)(oo) ; then
(i) If (Ig)(x) f)x g, g > O; (Iv)(x) =
f
o_<f,
llfllpo)
(2.4)
:()
E DRABEK et al.
188
(ii)
If (I*g)(x)
fx
g, g > O; (I*v)(x)
fx
v, with (I*v)(O)
then
sup
fx f (x)g(x) dx
Here the symbol is defined to mean that the quotient of the left and right
side of (2.4) and (2.5) is bounded above and below by positive constants.
Remark 2.1 It follows from (2.3) and (2.4) that if T is a linear operator
defined on decreasing functions on IR+ then the inequality IITfll a(w) <_
c f P (v) (0 <_ f,) is equivalent to
w O(w)
g>0
where T* is the adjoint of T. Similarly, the above norm inequality for
increasing functions is by (2.3) and (2.5) equivalent to
I*(T*g)
I*u
w
f’(v)
g
(2(w)’
>- O.
Our main result is now the following:
are N-functions, and T is a positive
THEOREM 2.2 Suppose P, P
A2
linear operator.
(i) IfO < f $ and (Iv)(o)
cx then the modular inequalities
Q-1
Q[wl (x)(Tf)(x)lwo(x)dx < p-1
P[C f (x)lv(x)dx
(2.6)
and
Q-1
Ifo
Q[Wl (x)T(I*h)(x)lwo(x)dx
< p-1
h > O, are equivalent.
{ fo [
]
p C(Iv)(x)h(x) v(x)dx
v(x)
}
(2.7)
WEIGHTED MODULAR INEQUALITIES
(ii)
If 0 < f’ and (l’v)(0)
Q-1
{fo
189
cx, then (2.6) and
Q[Wl (x)T(Ih)(x)]wo(x)dx
< p-1
]
{ fo [
p C(I*v)(x)h(x) v(x)dx
v(x)
(2.8)
h > O, are equivalent.
Proof (i) By Proposition 2.1 with wo(x)dx dr(x), v(x)dx dl(X),
(2.6) is equivalent to IIw Tflla(eQwo) < CIIfllP) where e > 0. Writing
T1 f W Tf, this estimate has the form
0 < f,
But then by (2.3) and (2.4) (cf. Remark 2.1, with w
(Iv)(x) becomes et,(Iv)(x)) (2.9) is equivalent to
(2.9)
EQtoO,
V
,pV SO
<C
EQ toO
-
is the adjoint of T1. Now, for any linear operator T2 and weight
where
functions or,/3, to and v
T*
IlaZ2gllp(0)
Cll/3gll0(o
==
<C
Otto
Q
P(w)
where T2* is the adjoint of T2.
Define T2 by T2g
I Tg, then T2*
TII* and hence by (2.11) with
and
ot
1/(spIv), w sev,
1/(eawo) v =sawo, (2.10) is equivalent
to
(Iv)h
P(eev)
But by Proposition (2.1) this is equivalent to (2.7) since Th TI (l’h)
wiT(I’h).
(ii) The proof for 0 < f’ is quite similar. Again we obtain (2.9) from (2.6)
for 0 < f’. But then by (2.3) and (2.5) this is equivalent to
P. DP-BEK et al.
190
l*(Tg)
<C
ep(Iv)
g >_ O.
eQWO
(2.12)
Now, applying (2.11) with T2g
1/(ep(I*v)), w
I*T{g,
1/(eQWO) and v eQWO, (2.12) is equivalent to
IIZhllaowo)
epv,
C
By Proposition 2.1 this is equivalent to (2.8) since Th
11)1 T(Ih).
Remark 2.2 As noted earlier, there are no order relations assumed on the
N-functions P and Q in Theorem 2.2 and if for example Q -< P, then in
general weight characterizations for which the modular inequality (2.7) and
(2.8) hold seem not to be known. However, if P(x) xP/p, Q(x) xq/q,
1 < p, q < x then Theorem 2.2 shows that IITfllq,w < CIIfllp,v holds for
h(Iv)
0 _< f$ ifand only if IIT(i*h)llq,w < CII--;IIp,o holds for0 _< h. But this
I (T...*..g)
< C IIq’, where p’ and q’
norm inequality is equivalent to
are the conjugate indices of p and q. However, this is exactly the formulation
obtained by E. Sawyer 15]. In the case 0 < f ’, one obtained in a similar
way, using now Theorem 2.2(ii), the corresponding result of V.D. Stepanov
[81.
If P -< Q, then for suitable operators T I* and T I weight characterizations
for which (2.7) and (2.8) holds are known. We begin with our applications
of Theorem 2.2 when T is the identity operator.
,
COROLLARY 2.1 Suppose P and Q are N-functions, such that P -< Q. Let
p,/5 E A2 and (Iv)(x) o. If 0 <_ f then thefollowing are equivalent:
Q-1
Q[wl (x) f (x)]wo(x)dx < p-1
P[Cf (x)]v(x)dx
(2.13)
There exists B > O, such that for all r > O, e > 0
Q-1
holds.
{for [
Q
(X)g oo)(ii))
Wl
X(r,
]
()
(2.14)
WEIGHTED MODULAR INEQUALITIES
191
There exists a B > O, such that, for all r > 0, e > 0,
Q-1
Ifo
r
Q[ewl (x)]wo(x)dx <
P-l[p(Be)(Iv)(r)]
(2.15)
holds.
Proof Let T be the identity operator, then T(I*h)(x)
(I*h)(x)
Hence by Theorem 2.2(i), (2.13) is equivalent to
Q-l
{ foC [wl fx h]
<p-a
{ fo [
Q
(X)
fxc h.
}
] }
wo(x)dx
p C(Iv)(x)h(x) v(x)dx
But by [5, Corollary 2.2] or [15, Proposition 3] this modular inequality is
equivalent to (2.14).
That (2.13) and (2.15) are equivalent follows from a result of J.Q. Sun
([20, Theorem 3.4]).
"
If 0 < f a result similar to Corollary 2.1 holds. In fact if P and Q are
as in Corollary 2.1 and (I’v)(0) x then (2.13) with 0 < f’ is equivalent
to:
There is a B > 0, such that for all r > 0, e > 0,
(X)
Q Wl
B
Q-1
’(ev)] w(x)dx}
<P-l(1/e)
(2.16)
and there exists a B > 0, such that for all r > 0, e > 0
Q-1
{fr
}
Q[ewl(x)lwo(x)dx <
P-l{p(Be)(I*v)Ir)}.
(2.17)
The proof of this follows (again) from Theorem 2.2(ii) and [5, Theorem 2.1
or [20, Theorem 3.4].
If P(x)
1,
xP/p, Q(x)
xq/q, 1 < p _< q < cxz, l/)l(X)
Corollary 2.1 reduces to a result of E. Sawyer 15], while the case for 0 < f’
may be found in [6].
P. DR/BEK et al.
192
Our next application considers the Hardy operator defined on decreasing
functions.
COROLLARY 2.2 Suppose P and Q are N-functions, P, P
If ( I v ( cx o and 0 <_ f $ then
A2 and P -< Q.
(2.18)
is satisfied, if and only if there are constants Bo, B1 > O, such that for all
r>O,e>O
Q-1
Q
[
Wl (x)
B0
X (0, r) io
e(Iv)
B1
el(v)
and
wo(x) dx < P
(Recall that io (x)
x.)
Proof If Tf(x)
f f, then T(I*h)(x)
f
is
Hence by Theorem 2.2(i), (2.18)
Q-1
< p-1
But since P
+ x fxo h(t)dr.
equivalent to
h(t) dt + x
Q Wl (x)
h(t)dt
h(t) dt
wo(x) dx
[fo [C(Iv)(x)h(X)]v(x)dx}
P
A2, the convexity of Q and P show that this is equivalent to
Q-1
Q
< p-1
Wl (X)
{ foC [
h(t) dt wo(x) dx
]
p Cl (Iv)(x)h(x) v(x) dx
v(x)
}
WEIGHTED MODULAR INEQUALITIES
and
Q-l
{ fo [.Wl fx dt] }
{ fo [C2(Iv)(x)h(X)] v(x)dx}
Q
< p-1
(X)X
193
wo(x) dx
h(t)
p
v(x)
(2.19)
where C1, C2 > 0 and h > O. Since P -< Q the weight characterizations for
which (2.19) hold are shown in [5] or [19] to be equivalent to the conditions
of the corollary.
Note that the proof of Corollary 2.1 shows that the second modular of
(2.19) is also equivalent to
Q-1
r
{fo
}
Q[exwl (x)]wo(x)dx < p-l {p(Be)(iv)(r)}
>O,e>O.
-
Remark 2.3 (i) The equivalence of (2.18) and (2.19) does not require
the order condition P -< Q. Hence if P (x)
x q/q,
xP/p, Q(x)
1 < p,q < xin(2.19) withwl(x)=
weight characterizations are
known (cf. [12]) and one obtained the result of E. Sawyer [15] and for p q
that of M. Arifio and B. Muckenhoupt [2].
(ii) Corollary 2.2 was proved by J.Q. Sun 19] by different methods.
(iii) For 0 < f ’, a result corresponding to Corollary 2.2 may also be
given. Only now one applies Theorem 2.2(ii) and uses the modular estimates
for
(x
r(Ih)(x)
s)h(s) ds
given in [2, 13, 16, 17]. The reduction to the weighted Lebesgue case yield
the results of 1, 6, 11].
Our final example involves the Riemann-Liouville fractional integral
operator of order or, 0 < c < c defined by
(Iaf)(X)
l
r(ot)
foX
(x
t) a-1 f(t) dt
(t
X) a-1 f(t) dt.
and the Weyl fractional integral
(I*af)(x)
1
P. D1LBEK et al.
194
For the case ot > 1 weight characterizations for weighted modular inequalities
or for weighted LP-norm inequalities are known. For the case 0 < ct < 1 a
weighted LP-norm characterization was given recently in [4] while weighted
modular inequalities seem not to be available in the literature. It may therefore
be surprising that in the monotone case more can be said.
PROPOSITION 2.2 (i) Suppose P, Q are N-functions, P, P
P < Q. If(Iv)(o) o and 0 <_ f, thenforO < ot < cx
Q-1
}
Q[wl (x)(I f)(x)]wo(x) dx < P-
if and only iffor all e > O, r
{fo
A2 and
}
P[C f(x)lv(x)dx
(2.20)
> O,
and
(r_x)a]wo(x)dx} ()
< p-1
are satisfied.
(ii) If (I’v)(0)
Q-1
o and 0 <
f
,
then for 0 <
Q[wl (x)(Ia f)(x)lwo(x) dx < p-1
ot
< cx
P[C f (x)lv(x) dx
(2.21)
holds,
Q-1
if and only if, for all r
Q
Wl.(X)
c
> O, e > O,
")X(O,r)
(r
e(l*v)
and
X(O,r)
e(I*v)
(x_r)a]wo(x)dx} <p-l()
WEIGHTED MODULAR INEQUALITIES
195
Proof (i) By Theorem 2.2(i), (2.20) is equivalent to
Q-1
If0
Q[wl (x)i.a (i.h)(x)]wo(x) dx
< p-1
P
C(Iv)(x)h(x)
v(x)dx
v(x)
* h) (x) and since ct + 1 > 1, Theorem 2.2 of
h > 0. But I(I*h)(x) (Ia+
also
[20] (see
[3, 13]) shows that this is equivalent to the conditions of the
proposition.
(ii) By Theorem 2.2(ii), (2.21) is equivalent to
Q-1
Q[Wl (x)la(Ih)(x)lwo(x) dx
<p-l_
{L [
P
C(’*v)(x)h(x)
]
v(x) dx
where h > 0. But Ialh
Ia+lh, and since a -4- 1 > 1 the result follows
from [19, Theorem 2.1]. (See also [3, 13]).
For a > 1 this result was also proved by J.Q. Sun [201.
Note that if P(x) xP/p, Q(x) xq/q, 1 < p < q < x, wi(x) 1,
then Proposition 2.2(i) shows that for 0 < a <
wo(x)
dx)
1/q
<C
lip
(fo
f(x)Pv(x) dx
holds for all 0 < f$, if and only if, for every r > 0
tLrll)o(x)dx)liq(ir (x--r)aP’(Lx )
ex
13
-p’
t(x)
dx)
1/p’
<C
and
(Lr(r
l/q
x)aq too(x)
dx) (Lr dx)
-1/p
v(x)
A similar remark holds for the Riemann-Liouville operator defined on
increasing functions.
196
E DRA,BEK et al.
Acknowledgements
The second author wishes to express his appreciation for the hospitality of
the University of West Bohemia in Pilsen, Czech Republic, and specifically
to Professor P. Drfibek and A. Kufner during his stay April-May 1995, when
this research began.
The second author’s research was supported in part by NSERC grant
A-4837 and by a grant from the University of West Bohemia, Pilsen, while the
research of the first and third authors was supported by the Grant Agency of
the Czech Republic, Grants No. 201/94/1008 and 201/94/1066, respectively.
References
[1] K. Andersen, Weighted generalized Hardy inequalities for non-increasing functions,
Canad. J. Math., 43 1991), 1121-1122.
[2] M. Arifio and B. Muckenhoupt, Maximal functions on classical Lorentz spaces and
Hardy’s inequality with weights for non-increasing functions, Trans. Amen Math. Soc.,
32(2) (1990), 727-735.
[3] S. Bloom and R. Kerman, Weighted L integral inequalities for operators of Hardy type,
Studia Math., 110(1) (1994), 35.
[4] I. Genebashvili, A. Gogatishvili and V. Kokilashvili, Solution of two-weight problems
for integral transforms with positive kernels, Georgian Math. J., 3 (1996), 319-342.
[5] Hans. P. Heinig and Lech Maligranda, Interpolation with weights in Orlicz spaces,
Bolletino U.M.I., $B(7) (1994) 37-55.
[6] H.P. Heinig and V.D. Stepanov, Weighted Hardy inequalities for increasing functions,
Canad. J. Math., 45(1) (1993) 104-116.
[7] H.P. Heinig and A. Kufner, Hardy operators of monotone functions and sequences in
Orlicz spaces, J. Lond. Math. Soc., 53(2) (1996), 256-270.
[8] C. Herz, A best possible Marcinkiewicz interpolation theorem with applications to
martingale inequalities, McGill University, February 1974, 81 pp.
[9] V. Kokilashvili and M. Krbec, Weighted inequalities in Lorentz and Orlicz spaces, World
Scientific, Singapore, 1991.
10] M.A. Krasnosel’skii and Ya. B. Rutickii, Convex functions and Orlicz spaces, Noordhoff,
G6ningen, 1961.
[11] P.J. Martin-Reyes and E. Sawyer, Weighted inequalities for the Riemann-Liouville
fractional integral operator of order one and greater, Proc. Amen Math. Soc., 106 (1989),
727-733.
[12] B. Opic and A. Kufner, Hardy type inequalities, Pitman Research Notes in Math. Ser.
219; Longman Scientific and Technical; Harlow 1990.
13] Quinsheng Lai, Weighted integral inequalities for the Hardy type operator and fractional
maximal operator, J. Lond. Math. Soc., 49(2) (1994) 224-266.
14] M.M. Rao and Z.D. Ren, Theory of Orlicz spaces, Marcel Dekker, 1991.
15] E. Sawyer, Boundedness of classical operators on classical Lorentz spaces, Studia Math.,
96 (1990), 145-158.
16] V.D. Stepanov, Two weighted estimates for the Riemann-Liouville integrals, Math. USSR
Izvestiya, 36 (1991), 669-681.
17] V.D. Stepanov, Weighted inequalities for a class of Volterra convolution operators, J. Lond.
Math. Soc., 45(2) (1992), 232-241.
WEIGHTED MODULAR INEQUALITIES
197
18] V.D. Stepanov, Integral operators on the cone of monotone functions and embeddings of
Lorentz spaces, Soviet Math. Dokl., 43(2) (1991), 620-623.
19] Jim Qile Sun, The modular inequalities for a class of convolution operators on monotone
functions (to appear, Proc. AMS).
[20] Jim Qile Sun, Hardy inequalities on weighted Orlicz spaces, Ph.D. thesis, The University
of Western Ontario, London, Ontario, Canada, March 1995.