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Hardy's inequality on hardy spaces.

1. Introduction. The main theme of this paper is the Hardy inequalities on rearrangement-invariant Hardy spaces including the classical Hardy spaces, the Hardy-Lorentz spaces and the Hardy-Orlicz spaces.

The Hardy inequality is one of the important inequalities in analysis. It is a crucial tool in real interpolation theory [2] and its high dimension generalization provides inspiration on the Hardy inequality for Sobolev functions.

It is impossible to give a detailed review on Hardy's inequality in this short paper, the reader is referred to [4,19,24] for a detailed reference for Hardy's inequality and its applications on analysis.

One of the extensions on the Hardy inequality is the validity of the Hardy inequalities on some non-Lebesgue spaces. For instance, we have the Hardy inequalities on rearrangement-invariant Banach function spaces in [20].

The Hardy inequalities on the Morrey spaces built on rearrangement-invariant Banach function spaces are obtained [13]. In addition, the Hardy inequalities on block spaces are given in [14].

We have the Hardy inequalities on Lebesgue spaces with variable exponents in [3, 8, 9, 21, 25, 26].

The Hardy inequalities on the Hardy-Morrey spaces, Hardy-Morrey spaces with variable exponents and weak Hardy-Morrey spaces are presented in [16 18], respectively.

In this paper, we extend the Hardy inequalities to the classical Hardy spaces and the rearrangement-invariant Hardy spaces in the form given in [3] and [19, p. 6] which are generalizations of the Hardy inequalities in [13, 17, 18].

We use the atomic decompositions of Hardy spaces to obtain the Hardy inequalities on the classical Hardy spaces. With these Hardy inequalities, the Hardy inequalities on rearrangement-invariant Hardy space are established by using the interpolation functor introduced in [15].

2. Hardy's inequality. We establish the Hardy inequalities on the classical Hardy spaces in this section. We begin with the Hardy operator used in this paper.

Let [Z.sub.-] denote the set of non-positive integers. For any [mu] [member of] R and [alpha] [member of] [Z.sub.-], write

[T.sub.[alpha],[mu]f] (x) = [x.sup.[alpha]+[mu]-1] [[integral].sup.x.sub.0] f(y)/[y.sup.[alpha]] dy.

We present the main result of this paper in the following theorem.

Theorem 2.1. Let 0 < p [less than or equal to] 1 and 0 [less than or equal to] [mu] < 1 and [alpha] [member of] [Z.sub.-]. If

1/p = 1/r + [psi],

then there exists a constant C > 0 such that for any f [member of] [H.sup.p] (R) with suppf [subset or equal to] [0,1),

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

As we prove the above theorem by using the atomic decompositions of Hardy spaces, we recall the atomic decompositions in the followings.

Let B(z, r) = {x [member of] R : [absolute value of (x - z)] < r} denote the open ball with center z [member of] R and radius r > 0. Let B = {B(z, r) : z [member of] R, r> 0} and [B.sub.+] = {B [member of] B : B [subset or equal to] (0,1)}.

Definition 2.1. Let 1 < q [less than or equal to] [infinity] and N [member of] N.

A Lebesgue measurable function A is a (q, N)-atom for [H.sup.p] (R) if there exists a B 2 B such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Theorem 2.2. Let 0 < p [less than or equal to] 1 < q [less than or equal to] [infinity]. For any N [member of] N with N [greater than or equal to] [1/p - 1] and f [member of] [H.sup.p] (R), we have a family of (q, N)-atoms {[a.sub.j]} and scalars {[[lambda].sub.j]} such that f = [summation] [[lambda].sub.j][[alpha].sub.j] in [H.sup.p] (R) and

(2.1) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for some C > 0. Furthermore,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The reader is referred to [5, Theorem 7.4] for the proof of the above result.

We now study the action of [T.sub.[alpha],[mu]] on the (q, N)-atom.

Lemma 2.1. Let 0 < r < [infinity], 1 < q [less than or equal to] [infinity], [mu] [member of] R and [alpha] [member of] [Z.sub.-]. If 1/q - 1/r < [mu] [less than or equal to] 1/q, then for any Lebesgue measurable function a satisfying

(2.2) supp a [subset or equal to] [bar.B], B [member of] [B.sub.+],

(2.3) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(2.4) [integral] [x.sup.-[alpha]] a(x) dx = 0,

we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for some C > 0.

Proof. Let supp a = [c, d] = [bar.B]. In view of the support condition (2.2) and the vanishing moment condition (2.4) satisfied by a, we find that

[[integral].sup.x.sub.0] [y.sup.-[alpha]] a(y)dy = 0, x < c and [[integral].sup.x.sub.0] [y.sup.-[alpha]] a(y)dy = 0, x> d.

Therefore, supp([T.sub.[alpha],[mu]] a) [subset or equal to] [c, d].

By the Holder inequality, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for some C > 0.

Consequently,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

As supp([T.sub.[alpha],[mu]] a) [subset or equal to] [c, d], we find that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

As 1/q - 1/r < [mu] [less than or equal to] 1/q, we have 0 < r[mu] - r/q + 1 [less than or equal to] 1.

Hence,

[d.sup.r[mu]-r/q+1] - [c.sup.r[mu]-r/q+1] [less than or equal to] [(d - c).sup.r[mu]-r/q+1].

The size condition (2.3) assures that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

In Theorem 2.1, we consider f [member of] [H.sup.p] (R) with supp f [subset or equal to] [0, [infinity]). Notice that the atomic decomposition given in Theorem 2.2 does not guarantee that the supports of the atoms for the atomic decomposition of f are subsets of [0, [infinity]). In order to tackle this problem, we consider the even part and the odd part of tempered distributions and modify the atomic decomposition obtained in Theorem 2.2.

For any f [member of] S' (R), define f(-*) as <f, [phi]> = <f (-*), [phi](-*)>, [phi] [member of] S(R). For any f [member of] [H.sup.p] (R), the even part and the odd part of f is defined as [f.sub.e] (x) = f(x) + f(-x)/2 and [f.sub.o] (x) = f(x) - 2f(-x)/2, respectively.

Proposition 2.1. Let 0 < p [less than or equal to] 1 < q [less than or equal to] [infinity] and [alpha] [member of] [Z.sub.-]. For any f 2 Hp (R) with suppf [subset or equal to] [0, [infinity]), we have a family of Lebesgue measurable functions {[a.sub.j]} satisfying (2.2) (2.4) and scalars {[[lambda].sub.j]} such that f = [summation] [[lambda].sub.j][a.sub.j] and

(2.5) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for some C > 0.

Proof. We first consider the case when [absolute value of ([alpha])] is even.

According to Theorem 2.2, we have f = [[summation].sub.j[member of]z] [[lambda].sub.j] [a.sub.j] where [{[a.sub.j]}.sub.j[member of]z] are (p, N) atoms with N > [absolute value of ([alpha])].

We consider the even part of f and find that

[f.sub.e] (x) = [summation over (j [member of] Z)] [[lambda].sub.j] [a.sub.j] (x) + [a.sub.j] (-x)/2.

As [a.sub.j] satisfies the vanishing moment condition up to order N and N > [absolute value of ([alpha])], we find that

1/2 [[integral].sub.R] [x.sup.-[alpha]] [a.sub.j] (x) dx = 1/2 [[integral].sub.R] [x.sup.-[alpha]] [a.sub.j] (-x) dx = 0.

If supp [a.sub.j] [subset] [0, [infinity]), [a.sub.j](-x) [equivalent to] 0 on (0,[infinity]). If supp [a.sub.j] [subset or equal to] (-1, 0], [a.sub.j](x) [equivalent to] 0 on (0, [infinity]) and [a.sub.j](-x) is a (p, N) atom. Therefore, they satisfy (2.2) (2.4).

If 0 is an interior point of supp [a.sub.j], we get

[[integral].sub.R] = [x.sup.-[alpha]] [[chi].sub.[0,[infinity])] (x) [a.sub.j] (x) + [[chi].sub.[0,[infinity])] (x) [a.sub.j] (-x)/2 dx = [[integral].sub.R] [x.sup.-[alpha]] [a.sub.j] (x) dx = 0.

Therefore,

[[chi].sub.(Q,[infinity])] (x)[a.sub.j] (x) + [[chi].sub.(0,1)] (x)[a.sub.j](- x)/2

satisfies (2.2)-(2.4).

As supp f [subset or equal to] [0,1), we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Finally, (2.5) is inherited from (2.1). Therefore, we obtain our desired decomposition for f.

For the case when [absolute value of ([alpha])] is odd, we consider the odd part of f. The rest of the proof for this case is almost identical to the proof of the case when [absolute value of ([alpha])] is even. The only modification is that for the odd part

[f.sub.0] (x) = [summation over (j[member of]Z)] [[lambda].sub.j] [a.sub.j] (x) - [a.sub.j] (-x)/2,

when 0 is the interior point of supp [a.sub.j], we have

[[integral].sub.R] = [x.sup.-[alpha]] [[chi].sub.[0,[infinity])] (x)[a.sub.j] (x) - [[chi].sub.[0,[infinity])] [a.sub.j] (-x)/2 dx = [[integral].sub.R] [x.sup.-[alpha]] [a.sub.j] (x) dx = 0.

This is similar to the case when [absolute value of ([alpha])] is even, we find that

f (x) = 2[[chi].sub.[0,[infinity])] (x) [f.sub.0] (x) = 2 [summation over (j [member of] Z)] [[lambda].sub.j] [[chi].sub.[0,[infinity])] (x) [a.sub.j] (x) - [[chi].sub.[0,[infinity])] (x) [a.sub.j] (-x)/2

which is our desired decomposition.

We are now ready to present the proof of Theorem 2.1.

Proof of Theorem 2.1. In view of Proposition 2.1, we have a family of Lebesgue measurable functions {[a.sub.j]} and scalars {[[lambda].sub.j]} satisfying (2.2) (2.5) such that f = [summation] [[lambda].sub.j][a.sub.j].

We consider F = [summation] [[lambda].sub.j] [T.sub.[alpha],[mu]] [a.sub.j]. As 0 < p [less than or equal to] 1 and 0 [less than or equal to] [mu] < 1, there exists a q > 1 such that

1/q - 1/r < 1/p - 1/r = [mu] < 1/q.

When [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] satisfies the triangle inequality. According to Lemma 2.1, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for some C > 0 because p [less than or equal to] r.

When [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is a norm, we find that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Therefore, (2.5) yields that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

In the proof of Theorem 2.1, we find that we need to use the atomic decompositions of Hardy spaces with (q, N)-atoms satisfying 1 < q < 1/[mu]. Notice that a substantial amount of applications of the atomic decomposition can be achieved by considering ([infinity], N)-atoms.

The above result shows that the atomic decompositions of Hardy spaces by (q, N)-atoms with q close to 1 also yield some valuable application which cannot be obtained by ([infinity], N)-atoms.

For the atomic decompositions with atoms defined by non-Lebesgue spaces, the reader is referred to [10,12].

3. Rearrangement-invariant Hardy spaces. In this section, we extend the Hardy inequalities to rearrangement-invariant Hardy spaces. We first recall the definition of rearrangement-invariant quasi-Banach function space (r.i.q.B.f.s.) from [11, Definition 4.1].

Let M(R) be the set of Lebesgue measurable functions on R.

Definition 3.1. A quasi-Banach space X [subset or equal to] M(R) is called a rearrangement-invariant quasi-Banach function space if there exists a quasi-norm [[rho].sub.X] : M(0, [infinity]) [right arrow] [0, [infinity]] satisfying

(a) [[rho].sub.X] (f) = 0 [??] f = 0 a.e.,

(b) [absolute value of (g)] [less than or equal to] [absolute value of (f)] a.e. [??] [[rho].sub.X] (g) [less than or equal to] [[rho].sub.X] (f),

(c) 0 [less than or equal to] [f.sub.n] [up arrow] f a.e. [??] [[rho].sub.X] ([f.sub.n]) [up arrow] [[rho].sub.X] (f) and

(d) [[chi].sub.E] [member of] M(0, [infinity]) and [absolute value of (E)] < [infinity] [??] [[rho].sub.X] ([[chi].sub.E]) < [infinity], so that

(3.1) [[parallel]f[parallel].sub.X] = [[rho].sub.X] ([f.sup.*]), [for all]f [member of] x.

For any s [greater than or equal to] 0 and f [member of] M(0, [infinity]), define ([D.sub.s] f)(t) = f(st), t [member of] (0, [infinity]). Let [[parallel][D.sub.s][parallel].sub.[bar.X][right arrow][bar.X]] be the operator norm of [D.sub.s] on [bar.X]. We recall the definition of Boyd's indices for r.i.q.B.f.s. from [22].

Definition 3.2. Let X be a r.i.q.B.f.s. on R. Define the lower Boyd index of x, [p.sub.X] and the upper Boyd index of X, [q.sub.X] as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

respectively.

As the definition of interpolation functor involves the notion of category and compatible couples, for simplicity, we refer the reader to [27, Section 1.2] for details of category and compatible couples.

We recall the definition of the K-functional from [2, Section 3.1] and [27, Section 1.3.1].

Definition 3.3. Let ([x.sub.0], [x.sub.1]) be a compatible couple of quasi-normed spaces. For any f 2 [X.sub.0] + [X.sub.1], the K-functional is defined as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where the infimum is taking over all f = [f.sub.0] + [f.sub.1] for which [f.sub.i] [member of] [X.sub.i], i = 0,1.

The following interpolation functor is introduced in [15, Definition 4.2].

Definition 3.4. Let 0 < [theta], r < [infinity] and X be a r.i.q.B.f.s. Let ([X.sub.0], [X.sub.1]) be a compatible couple of quasi-normed spaces. The space [([X.sub.0], [X.sub.1]).sub.[theta],r,X] consists of all f in [X.sub.0] + [X.sub.1] such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [[rho].sub.X] is the quasi-norm given in (3.1).

The above interpolation functor is an extension of the interpolation functor given in Marcinkiewicz real interpolation functor and the interpolation functors in [6,7] for the studies of Lorentz-Karamata spaces and Orlicz spaces, respectively.

We recall a function space associated with the above interpolation from [15, Section 3.1].

Definition 3.5. Let [alpha] [greater than or equal to] 0. For any r.i.q.B.f.s. X, [X.sub.[alpha]] consists of those f [member of] M(R) such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Obviously, from (3.1), we have [x.sub.0] = x. In [15], we find that [X.sub.[alpha]] is related to the mapping properties of the fractional integral operators, the convolution operators and the Fourier integral operators in r.i.q.B.f.s.

We find that whenever x is a r.i.q.B.f.s., [X.sub.[alpha]] is also a r.i.q.B.f.s.

Proposition 3.1. Let [alpha] > 0 and X be a r.i.q.B.f.s. If 0 < [p.sub.X] [less than or equal to] [q.sub.X] < 1/[alpha], then [X.sub.[alpha]] is a r.i.q.B.f.s.

For the proof of the above proposition, the reader is referred to [15, Proposition 3.1].

We have the following theorem from [15, Theorem 4.2] which assures that [X.sub.[alpha]] is an interpolation space from Lebesgue spaces by using the functor [(*, *).sub.[theta],r,X].

Theorem 3.1. Let 0 [less than or equal to] [alpha] < [infinity], 0 < [p.sub.0] < [p.sub.1] < [infinity] and X be a r.i.q.B.f.s. with 0 < [p.sub.X] [less than or equal to] [q.sub.X] < 1/[alpha]. Let r, [theta] satisfy

(3.2) 1/[theta] = 1/[p.sub.0] - 1/[p.sub.1] and 1/r = 1/[p.sub.0] + [alpha].

Suppose that [p.sub.1] > [q.sub.X], [p.sub.0] < [p.sub.X] and

(3.3) 1/[p.sub.1] + [alpha]/n < 1/qX [less than or equal to] 1/pX < 1/[p.sub.0] + [alpha].

Then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The reader may consult [15, Theorem 4.2] for the proof of the preceding theorem.

We now turn to the definition of rearrangement-invariant Hardy spaces. Let P denote the class of polynomials on R.

Definition 3.6. Let x be a r.i.q.B.f.s with 0 < [p.sub.X] [less than or equal to] [q.sub.X] < [infinity]. The rearrangement- invariant Hardy space associated with x, [H.sub.x], consists of those f [member of] S'(R)/P such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [[phi].sub.j] (x) = [2.sup.jn] [phi] ([2.sup.j] x), j [member of] Z and [phi] [member of] S(R) satisfy

supp [??] [subset or equal to] {[xi] [member of] [R.sup.n] : 1/2 [less than or equal to] [absolute value of ([xi])] [less than or equal to] 2} and [absolute value of ([??]([xi]))] [greater than or equal to] C, 3/5 [less than or equal to] [absolute value of ([xi])] [less than or equal to] 5/3

for some C > 0.

Notice that [H.sub.X] is not rearrangement-invariant in terms of the condition given in [2, Chapter 2, Definition 1.2]. For simplicity, we use the absurd terminology "rearrangement-invariant" to name [H.sub.X].

If X = [L.sup.p] with 0 < p [less than or equal to] 1, we write [H.sub.X] by [H.sub.p].

When X = [L.sub.p,q] where [L.sub.p,q] is a Lorentz space, then [H.sub.X] becomes the Hardy-Lorentz spaces [H.sub.p,q] studied in [1].

If X is generated by a growth function of lower type [PHI] (see [28, p. 403]), then [H.sub.X] is the Hardy type Orlicz spaces [H.sub.[PHI]] considered in [23,28].

Theorem 3.2. Let X be a r.i.q.B.f.s. with 0 < [p.sub.X] [less than or equal to] [q.sub.X] [less than or equal to] 1/[alpha]. Suppose that 0 < [p.sub.0] < [p.sub.X] [less than or equal to] [q.sub.X] < [p.sub.1] < [infinity] and r,[theta] satisfy (3.2) and (3.3). Then,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

For the proof of Theorem 3.2, the reader is referred to [15, Corollary 8.5].

We are now ready to extend the Hardy inequalities to rearrangement-invariant Hardy spaces.

Theorem 3.3. Let 0 [less than or equal to] [mu] < 1, [alpha] [member of] [Z.sub.-] and X be a r.i.q.B.f.s. with 0 < [p.sub.X] [less than or equal to] [q.sub.X] < 1. Then there exists a constant C > 0 such that for any f [member of] [H.sub.X] with suppf [subset or equal to] [0,1),

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Proof. In view of Theorem 2.1, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

whenever

1/p = 1/r + [mu].

As 0 < [p.sub.X] [less than or equal to] [q.sub.X] < 1, there exist [s.sub.1], [s.sub.0] such that [q.sub.X] < [s.sub.1] < 1 < 1/[mu] and 0 < [S.sub.0] < [p.sub.X].

The mappings [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] with

1/[s.sub.i] = 1/[q.sub.i] + [mu], i = 0,1

are bounded.

Let 1/[theta] = 1/[s.sub.0] - 1/[s.sub.1] = 1/[q.sub.0] - 1/[q.sub.1]. In addition, as

1/[q.sub.1] + [mu] = 1/[s.sub.1] < 1/[q.sub.X] [less than or equal to] 1/[p.sub.X] < 1/[s.sub.0] = 1/[q.sub.0] + [mu],

(3.2) and (3.3) are fulfilled for the interpolations [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Theorems 3.1 and 3.2 yield

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

As some special cases of the above theorem, we have Hardy inequalities on the Hardy-Lorentz spaces [1] and the Hardy-Orlicz spaces [23,28].

2010 Mathematics Subject Classification. Primary 26D10, 42B30, 46E30, 47B38.

doi: 10.3792/pjaa.92.125

References

[1] W. Abu-Shammala and A. Torchinsky, The Hardy-Lorentz spaces [H.sup.p,q]([R.sup.n]), Studia Math. 182 (2007), no. 3, 283 294.

[2] C. Bennett and R. Sharpley, Interpolation of operators, Pure and Applied Mathematics, 129, Academic Press, Boston, MA, 1988.

[3] L. Diening and S. Samko, Hardy inequality in variable exponent Lebesgue spaces, Fract. Calc. Appl. Anal. 10 (2007), no. 1, 1 18.

[4] D. E. Edmunds and W. D. Evans, Hardy operators, function spaces and embeddings, Springer Monographs in Mathematics, Springer, Berlin, 2004.

[5] M. Frazier and B. Jawerth, A discrete transform and decompositions of distribution spaces, J. Funct. Anal. 93 (1990), no. 1, 34 170.

[6] A. Gogatishvili, B. Opic and W. Trebels, Limiting reiteration for real interpolation with slowly varying functions, Math. Nachr. 278 (2005), no. 1 2, 86 107.

[7] J. Gustavsson and J. Peetre, Interpolation of Orlicz spaces, Studia Math. 60 (1977), no. 1, 33 59.

[8] A. Harman, On necessary condition for the variable exponent Hardy inequality, J. Funct. Spaces Appl. 2012, Art. ID 385925.

[9] A. Harman and F. I. Mamedov, On boundedness of weighted Hardy operator in [L.sup.p(*)] and regularity condition, J. Inequal. Appl. 2010, Art. ID 837951.

[10] K.-P. Ho, Characterization of BMO in terms of rearrangement-invariant Banach function spaces, Expo. Math. 27 (2009), no. 4, 363 372.

[11] K.-P. Ho, Littlewood-Paley spaces, Math. Scand. 108 (2011), no. 1, 77 102.

[12] K.-P. Ho, Atomic decomposition of Hardy spaces and characterization of BMO via Banach function spaces, Anal. Math. 38 (2012), no. 3, 173 185.

[13] K.-P. Ho, Hardy's inequality and Hausdorff operators on rearrangement-invariant Morrey spaces, Publ. Math. Debrecen 88 (2016), no. 1 2, 201 215.

[14] K.-P. Ho, Hardy-Littlewood-Polya inequalities and Hausdorff operators on block spaces, Math. Inequal. Appl. 19 (2016), no. 2, 697 707.

[15] K.-P. Ho, Fourier integrals and Sobolev embedding on rearrangement invariant quasi-Banach function spaces, Ann. Acad. Sci. Fenn. Math. 41 (2016), no. 2, 897 922.

[16] K.-P. Ho, Hardy's inequality on Hardy-Morrey spaces. (to appear in Georgian Math. J.).

[17] K.-P. Ho, Hardy's inequality on Hardy-Morrey spaces with variable exponents. (to appear in Mediterr. J. Math.).

[18] K.-P. Ho, Atomic decompositions and Hardy's inequality on weak Hardy-Morrey spaces. (to appear in Sci. China Math.).

[19] A. Kufner and L.-E. Persson, Weighted inequalities of Hardy type, World Sci. Publishing, River Edge, NJ, 2003.

[20] L. Maligranda, Generalized Hardy inequalities in rearrangement invariant spaces, J. Math. Pures Appl. (9) 59 (1980), no. 4, 405 415.

[21] R. A. Mashiyev, B. Cekic, F. I. Mamedov and S. Ogras, Hardy's inequality in power-type weighted [L.sup.p(*)] (0, [infinity]) spaces, J. Math. Anal. Appl. 334 (2007), no. 1, 289 298.

[22] S. J. Montgomery-Smith, The Hardy operator and Boyd indices, in Interaction between functional analysis, harmonic analysis, and probability (Columbia, MO, 1994), 359 364, Lecture Notes in Pure and Appl. Math., 175, Dekker, New York, 1996.

[23] E. Nakai and Y. Sawano, Orlicz-Hardy spaces and their duals, Sci. China Math. 57 (2014), no. 5, 903 962.

[24] B. Opic and A. Kufner, Hardy-type inequalities, Pitman Research Notes in Mathematics Series, 219, Longman Sci. Tech., Harlow, 1990.

[25] H. Rafeiro and S. Samko, Hardy type inequality in variable Lebesgue spaces, Ann. Acad. Sci. Fenn. Math. 34 (2009), no. 1, 279 289.

[26] S. Samko, Hardy inequality in the generalized Lebesgue spaces, Fract. Calc. Appl. Anal. 6 (2003), no. 4, 355 362.

[27] H. Triebel, Interpolation theory, function spaces, differential operators, North-Holland Mathematical Library, 18, North-Holland, Amsterdam, 1978.

[28] B. E. Viviani, An atomic decomposition of the predual of BMO([rho]), Rev. Mat. Iberoamericana 3 (1987), no. 3 4, 401 425.

Kwok-Pun HO

Department of Mathematics and Information Technology, The Education University of Hong Kong, 10 Lo Ping Road, Tai Po, Hong Kong, China

(Communicated by Masaki KASHIWARA, M.J.A., Nov. 14, 2016)
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