Forever Young - DiVA portal706959/FULLTEXT01.pdf · Forever Young Convolution Inequalities in Weighted Lorentz-type Spaces Martin Křepela LICENTIATE THESIS | Karlstad University

Forever YoungConvolution Inequalities in Weighted Lorentz-type Spaces

Martin Křepela

LICENTIATE THESIS | Karlstad University Studies | 2014:21

Mathematics

Faculty of Health, Science and Technology

Martin K

řepela | Forever Young | 2014:21

Forever Young

Convolution plays an important role in both pure mathematics and practical applications. Many important operators in mathematical analysis have a form of a convolution with a fixed kernel, which gives numerous applications to the general theoretical results about convolution. One of the most important theorems involving convolution is the Young inequality, estimating the norm of the convolution in the Lebesgue spaces.

In this licentiate thesis, convolution is investigated in the framework of weighted Lorentz-type spaces. Boundedness of a general convolution operator in these function spaces is characterized in terms of the kernel. Consequently, new Young-type inequalities for the weighted Lorentz-type spaces are proved and their optimality is studied.


ISSN 1403-8099

ISBN 978-91-7063-552-6



Martin Křepela

urn:nbn:se:kau:diva-31754

Distribution:Karlstad University Faculty of Health, Science and TechnologyDepartment of Mathematics and Computer ScienceSE-651 88 Karlstad, Sweden+46 54 700 10 00

© The author

ISBN 978-91-7063-552-6

Print: Universitetstryckeriet, Karlstad 2014

ISSN 1403-8099

Karlstad University Studies | 2014:21

LICENTIATE THESIS

Martin Křepela

Forever Young - Convolution Inequalities in Weighted Lorentz-type Spaces

WWW.KAU.SE

Abstract

This thesis is devoted to an investigation of boundedness of a generalconvolution operator between certain weighted Lorentz-type spaces with theaim of proving analogues of the Young convolution inequality for these spaces.

Necessary and sufficient conditions on the kernel function are given, forwhich the convolution operator with the fixed kernel is bounded betweena certain domain space and the weighted Lorentz space of type Gamma.The considered domain spaces are the weighted Lorentz-type spaces definedin terms of the nondecreasing rearrangement of a function, the maximalfunction or the difference of these two quantities.

In each case of the domain space, the corresponding Young-type con-volution inequality is proved and the optimality of involved rearrangement-invariant spaces is shown.

Furthermore, covering of the previously existing results is also discussedand some properties of the new rearrangement-invariant function spaces ob-tained during the process are studied.

i

List of papers

The main body of the thesis consists of the following papers:

[I] M.Krepela, Convolution inequalities in weighted Lorentz spaces, toappear in Math. Inequal. Appl.

[II] M.Krepela, Convolution in rearrangement-invariant spaces definedin terms of oscillation and the maximal function, to appear in Z. Anal.Anwend.

[III] M.Krepela, Convolution in weighted Lorentz spaces of type Γ, sub-mitted.

ii

Acknowledgements

Above all, I would like to thank my supervisor Sorina Barza for the greatamount of help, support and advice. I also thank Lubos Pick for adviceand numerous discussions about mathematics in my own language, as wellas Lars-Erik Persson for his support.

Moreover, I thank Amiran Gogatishvili who suggested the research ideaof this thesis.

I am especially grateful to Ilie Barza and Martin Lind for their extensivesupport and help on various occasions, as well as the thorough introductioninto the Swedish (and Romanian) culture. I also thank my other colleaguesfrom the Department of Mathematics and Computer Science in Karlstad andthe Department of Mathematical Analysis at MFF UK in Prague.

iii

Contents

1 Introduction 1

2 Lorentz-type spaces 1

3 Convolution inequalities 8

4 Summary of the main papers’ contents 114.1 Paper [I] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124.2 Paper [II] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144.3 Paper [III] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

5 Remarks and further comments 15

References 17

iv

1 Introduction

The aim of the following introductory summary is to offer a basic insight intothe topic of this thesis and place its results in a broader context.

At first, the general framework of function spaces is introduced, sinceall the research carried out here involves various types of function spacesand the definitions and auxiliary results from the function spaces theory areextensively used. The function spaces involved in the thesis are presented,(e.g. the Lebesgue, Lorentz and weighted Lorentz-type spaces) together withtheir basic properties and related observations.

When this framework is prepared, in the next step we introduce the con-volution operator and related results and applications. The focus is laid ondescribing the development of Young-type convolution inequalities, summa-rizing and commenting the existing results and their relations.

Further on, the research questions and results of the three papers makingthe main part of this thesis are briefly presented. Relation between thecontents of these papers and the existing results is also discussed, togetherwith possibilities of future development and applications to related problems.

2 Lorentz-type spaces

The spaces of interest in this thesis are almost exclusively those based oncertain “integral properties” of functions. A typical and in many senses“model” example of such spaces is the Lebesgue Lp space. Let us recall itsdefinition:

Let (Ω,M, µ) be a measure space and let p ∈ (0,∞]. For any real-valuedfunction f measurable on (Ω,M, µ) define

∥f∥p ∶=⎛⎝

∞

∫0

∣f(x)∣pdµ(x)⎞⎠

1p

, if 0 0; ∣f ∣ ≤ c µ-a.e. on Ω , if p =∞.

Then the set Lp is defined by

Lp(Ω,M, µ) ∶= f measurable on (Ω,M, µ); ∥f∥p <∞ .Usually we will consider the underlying measure space to be Rd with thed-dimensional Lebesgue measure. We write only Lp when there is no risk ofconfusion about the underlying measure space.

1

From now on we will also use the symbol M (Ω) to denote the cone ofLebesgue-measurable functions on Ω, where the set Ω will be always specified,usually as Rd or an interval on the real axis.

A special case of the Lp-spaces which is useful for our purposes is theweighted Lebesgue space Lp(v) over (0,∞). It consists of all such f ∈M (0,∞)that

∥f∥Λp(v) ∶=⎛⎝

∞

∫0

∣f(x)∣pv(x)dx⎞⎠

1p

<∞, if 0 0

∣f(x)∣v(x) <∞, if p =∞,

where v is a given nonnegative measurable function on (0,∞).Note that while Lp(v) is always a linear set, the functional ∥ ⋅ ∥Λp(v) is

a norm if and only if p ≥ 1 and v is positive a.e. If 0 < p < 1 and v is positivea.e., then ∥ ⋅ ∥Λp(v) is a quasi-norm but not a norm, since the Minkowski in-equality fails for p < 1.

The Lp spaces possess many “good” properties. This motivated the def-inition of the Banach function spaces1 introduced by W.A.J. Luxemburg in[62]. Roughly speaking, these spaces are a general class of function spaceswhich possesses a set of properties, inspired by the properties of Lp spaces.We shall now present the proper definition here. We are still considering Rd

to be the domain of the functions in the resulting space (for other variantssee [10] or [I]).

Let ϱ ∶M (Rd)→ [0,∞] be a mapping. We call ϱ a Banach function normif for all f, g, fn ∈M (Rd), (n ∈ N), for all constants a ≥ 0 and all measurablesets E ⊂ Rd, the following properties hold:

(P1) ϱ(f + g) ≤ ϱ(f) + ϱ(g),

(P2) ϱ(af) = aϱ(f),

(P3) ϱ(f) = 0 ⇔ f = 0 a.e.,

(P4) 0 ≤ g ≤ f a.e. ⇒ ϱ(g) ≤ ϱ(f),

(P5) 0 ≤ fn ↑ f a.e. ⇒ ϱ(fn) ↑ ϱ(f),1The name Banach function space may be slightly misleading. It does not mean that

the object is just an arbitrary Banach space consisting of functions.

2

(P6) ∣E∣ <∞ ⇒ ϱ(χE) <∞,

(P7) ∣E∣ < ∞ ⇒ ∫E f ≤ CEϱ(f) for some constant CE ∈ (0,∞) dependingon E and ϱ but independent of f ,

If ϱ is a Banach function norm, the collection X = X(ϱ) of all functionsf ∈M (Rd) such that ϱ(∣f ∣) <∞ is called a Banach function space.

For our purposes, the focus is laid on a particular type of Banach functionspaces, namely the rearrangement-invariant ones. In order to describe them,we need the following definition.

Let f ∈M (Rd). The nonincreasing rearrangement of f , denoted by f∗,is defined by

f∗(t) ∶= inf s ≥ 0; ∣x ∈ Rd, ∣f(x)∣ > s∣ ≤ t , t ∈ (0,∞).

The notation ∣x ∈ Rd, ∣f(x)∣ > s∣ used here stands for the (d-dimensional)Lebesgue measure of the set x ∈ Rd, ∣f(x)∣ > s.

Now, a Banach function norm ϱ is called a rearrangement-invariant (r.i.)norm if, for all f, g ∈M (Rd), it satisfies

(P8) f∗ = g∗ on (0,∞) ⇒ ϱ(f) = ϱ(g).

Since being a norm is a rather strong property of a functional, let usintroduce some additional terms for the functionals and structures based onweaker conditions.

We say that the mapping ϱ ∶ M (Rd) → [0,∞] is an r.i. quasi-norm iffor all f, g, fn ∈ M (Rd), (n ∈ N), all a ≥ 0 and all measurable E ⊂ Rd, theconditions (P2)-(P7) and

(P1*) ϱ(f +g) ≤ B(ϱ(f)+ϱ(g)) for a constant B ∈ (1,∞) independent of f, g,

are satisfied. In this case, X(ϱ) is called a quasi-Banach r.i. space.Next, we call the collection of functions X(ϱ) an r.i. lattice if for all

f, g ∈ Rd, all a ≥ 0 and all measurable E ⊂ Rd, the conditions (P2), (P4), (P6)and (P8) are satisfied. If X(ϱ) is an r.i. lattice, for every f ∈ Rd we denote∥f∥X ∶= ϱ(∣f ∣).

3

A simple and well-known example of an r.i. space is the Lorentz spaceLp,q generated by the following functional:

∥f∥p,q ∶=⎛⎝

∞

∫0

(f∗(t))qtqp−1 dt⎞⎠

1q

, 0 < p, q <∞,

∥f∥p,∞ ∶= supt>0

f∗(t)t1p , 0 < p < q =∞.

The functional ∥⋅∥p,q is not necessarily a norm, but if p ∈ (1,∞) and q ∈ (1,∞],then Lp,q is normable. To show this, one introduces the following functional:

∥f∥(p,q) ∶=⎛⎝

∞

∫0

(f∗∗(t))qtqp−1 dt⎞⎠

1q

, 0 < p, q <∞,

∥f∥(p,∞) ∶= supt>0

f∗∗(t)t1p , 0 0.

It can be proved (see e.g. [10]) that ∥ ⋅ ∥p,q is equivalent to ∥ ⋅ ∥(p,q) whenp ∈ (1,∞) and q ∈ (1,∞].

The Lp,q and L(p,q) spaces play a significant role in interpolation theory[10, 11]; the L(p,q) spaces are real interpolation spaces between L1 and L∞.These spaces also often appear in various refinings of classical inequalities(see e.g. the generalizations of the Young inequality in the next section).

Defining and studying new types of function spaces is often motivated bythe fact that these are interpolation spaces between other previously knownones. Among other reasons for a new function space’s emergence is, forinstance, the fact that such a space is the dual or associated space to a knownone, it is the optimal domain/range space for an operator etc.

The question of normability, quasi-normability and even linearity emergesfrequently when a new function “space” is defined. As seen, this issue appearsalready when we consider the Lp and Lp,q “spaces.” However, as a conven-tion, we will call various structures “spaces”, as we did in the case Lp, Lp,q,even though these are not necessarily spaces in the exact sense of the word.

4

In [60, 61], G.G.Lorentz defined a more general class of function spaces–the Λ spaces. Before we present their definition, note that from now on,a weight will mean a function v ∈M+(0,∞) such that

0 <t

∫0

v(s)ds <∞ for all t > 0.

Now let p ∈ (0,∞] and let v be a weight. The weighted Lorentz spaceΛp(v) is the set f ∈M (Rd); ∥f∥Λp(v) <∞, where

∥f∥Λp(v) ∶=⎛⎝

∞

∫0

(f∗(t))pv(t)dt⎞⎠

1p

, if 0 0

f∗(t)v(t), if p =∞.

The spaces Λp(v) with p <∞, usually called classical weighted Lorentz spaces,appeared first in [60, 61]. The weak-type weighted Lorentz spaces, i.e., thosewith p =∞, were introduced in [24] and further treated e.g. in [25, 20, 26, 23].

The class of Λp(v)-spaces encompasses a variety of function spaces, whichare obtained as special cases of Λp(v) by a particular choice of the weight v.These include the aforementioned Lp,q-spaces, Lorentz-Zygmund spaces [9]and their generalizations [70], Lorentz-Karamata spaces [65] and others.

A Λ space is not necessarily a linear set. The main cause of this problemis the fact that the “star-operator” (i.e., the rearrangement) is not sublin-ear, i.e., the inequality (f + g)∗(t) ≤ Cf∗(t) + g∗(t) does not hold for anyC ∈ (0,∞). The questions of linearity and (quasi-)normability of Λp(v) werestudied e.g. in [29].

E.Sawyer [81] first described the associate space to Λp(v). This type ofa function space is called the Γ space and is defined as follows. If p ∈ (0,∞]and v is a weight, the weighted Lorentz space Γp(v) is the set of all f ∈M (Rd)such that ∥f∥Γp(v) <∞, where

∥f∥Γp(v) ∶=⎛⎝

∞

∫0

(f∗∗(t))pv(t)dt⎞⎠

1p

, if 0 0

f∗∗(t)v(t), if p =∞.

5

The classical Γp(v) space with p <∞ is the one introduced in [81], althougha space with a norm involving f∗∗ was explicitly presented already in [18].The weak-type spaces Γ∞(v) appear in [24] and are, as well as their weak-Λcounterparts, further studied e.g. in [25, 20, 26, 23, 39].

There is a strong relation between the Λ and Γ spaces. Besides theassociatedness property we may e.g. mention the following relation (see [81,5]):

If p ∈ (1,∞), then the functional ∥ ⋅ ∥Λp(v) is equivalent to a norm if andonly if v ∈ Bp, where

Bp ∶=⎧⎪⎪⎨⎪⎪⎩v ∈M+(0,∞) ∶ ∃C ∈ (0,∞)∀t > 0 ∶ tp

∞

∫t

v(s)sp

ds ≤ Ct

∫0

v(s)ds⎫⎪⎪⎬⎪⎪⎭.

Note that if v ∈ Bp, then Λp(v) = Γp(v) with equivalent norms. (For moredetails see e.g. [25, 26, 84, 27, 37].)

The question of linearity and normability of Γp(v) is considerably simplerthan in the case of Λp(v). The reason is that the Hardy-Littlewood maximalfunction does satisfy

(f + g)∗∗(t) ≤ f∗∗(t) + g∗∗(t) (1)

for all t > 0 and all locally integrable functions f, g (see e.g. [10]). Thanksto (1), the functional ∥ ⋅ ∥Γp(v) is always at least a quasi-norm, for p ≥ 1 itis a norm by Minkowski inequality. Functional properties of Γ were furtherstudied for example in [49].

There have been further generalizations of Γ-spaces [38, 33, 34, 41], basedon generalized versions of the Hardy-Littlewood maximal operator. In thesecases, normability and other functional properties of such spaces have notyet been studied satisfactorily2.

The last Lorentz-type “space” considered in here is the class S, introducedin [21]. If p ∈ (0,∞] and v is a weight, then the class Sp(v) consists of allf ∈ Rd such that lims→∞ f∗(s) = 0 and

∥f∥Sp(v) ∶=⎛⎝

∞

∫0

(f∗∗(t) − f∗(t))pv(t)dt⎞⎠

1p

, if 0 0(f∗∗(t) − f∗(t))v(t), if p =∞.

2There is, however, an ongoing research by F.Soudsky into this topic.

6

Unlike the Λ and Γ spaces, the class Sp(v) is not even an r.i. lattice (fora detailed study of this and related issues see [21]). The functional f∗∗ − f∗is important in various parts of analysis [6, 7, 8, 10, 15, 21, 40, 51, 53, 54,55, 57, 63, 64, 77] and represents a natural tool to measure oscillation of f[8, 10].

One may compare the class Sp(v) to the L∞,q spaces consisting of func-tions f ∈M (Rd) such that

∥f∥L∞,q ∶= ∥f∥1 +⎛⎝

∞

∫0

(f∗∗(t) − f∗(t))q dtt

⎞⎠

1q

<∞.

(Here we consider q < ∞.) The Sp(v) spaces in a sense generalize the L∞,q

spaces. For details and applications of the L∞,q spaces see e.g. [8, 10].

An extensively studied feature of function spaces is the question of theirinclusion relations or embeddings. Recall that an r.i. lattice X is embeddedinto an r.i. lattice Y , denoted X Y , if

∥f∥Y ≤ C∥f∥X , f ∈X.

Note that this notation means that the inequality is satisfied for all f ∈X withC being a constant independent of f . Such notation will be used in the samesense throughout the text. Notice also that the definition of an embeddingmay be obviously extended to cover Sp(v) as well (see [22] for some details),although in general, Sp(v) is not an r.i. lattice.

As for the weighted Lorentz-type spaces Λ, Γ and S, the existence ofan embedding between either two types of them is, in fact, always equivalentto satisfaction of a certain Hardy-type inequality. The name Hardy inequalityin general refers to the inequality

XXXXXXXXXXXXt↦ 1

t

t

∫0

f

XXXXXXXXXXXXY≤ C∥f∥X , f ∈X,

where X,Y are some given spaces of functions on (0,∞). The original Hardyinequality involved the spaces X = Y = Lp(0,∞), p > 1. By the term Hardy-type inequality one refers to any analogous inequality involving the Hardyoperator 1

t ∫t

0 f or similar integral operators. For a detailed treatment of

7

the history, various forms and applications of Hardy-type inequalities, see[58, 59, 69].

The Hardy-type inequalities related to embeddings of weighted Lorentz-type spaces were investigated e.g. in [5, 81, 85, 24, 25, 26, 20, 23, 38, 83,21, 39, 22, 35, 42, 36]. Worth noticing is the article of A.Gogatishvili andV.D.Stepanov [42], in which a universal reduction method is presented, ap-plicable to characterizing boundedness of quasi-linear operators on the conesof monotone functions, i.e., including the Hardy-type operators. Moreover,in [36] linear positive operators on the cone of quasi-concave functions arestudied, providing results useful to describe embeddings of spaces with normsinvolving the Hardy-Littlewood maximal operator.

3 Convolution inequalities

The central point of this thesis is the notion of convolution. Given twofunctions f, g ∈ L1

loc(Rd), their convolution f ∗ g is defined by

(f ∗ g)(x) ∶= ∫Rd

f(y)g(x − y)dy, x ∈ Rd. (2)

We will usually consider this setting with Rd as the integration domain andfunctions defined on Rd, though definitions with different underlying measurespaces are of course possible.

Convolution has a very broad use both in the theory and practical appli-cations. On the theoretical level, it is present above all in Fourier analysisand the approximation theory (see e.g. [30, 88]). Results from these fieldsof mathematics have direct practical applications, e.g. in signal and imageprocessing, electrical engineering, probability and statistics, etc.

The convolution may be naturally viewed as a bilinear operator. Anotherapproach is to fix the function g ∈ L1

loc(Rd) and consider the linear operator

Tgf(x) ∶= (f ∗ g)(x), f ∈ L1loc(Rd), x ∈ Rd. (3)

The function g in this case is called a kernel. A variety of important oper-ators has exactly this form, when each one of them corresponds to Tg witha particularly chosen kernel g. Examples of such operators include e.g. theRiesz potential or Riemann-Liouville integral [44, 10], Bessel potential [44],Newton potential [31], Hilbert and Riesz transforms [43, 10], Stieltjes trans-form [86], mollifiers [31, 1], etc. In Fourier analysis, convolution with e.g. the

8

Dirichlet, Fejer and Jackson kernels is frequently used in substantial parts ofthe theory (see [88, 30]).

A fundamental question in a study of any operator is whether it is boundedon its domain. In our particular case of the operator Tg, the problem may bestated as follows. Given the domain function space X and the range spaceZ, find conditions on the kernel g, under which it holds

∥f ∗ g∥Z ≤ C(g)∥f∥X , f ∈X, (4)

i.e., under which Tg is bounded between X and Z. These conditions shouldpreferably characterize the boundedness, i.e., be both necessary and suffi-cient. Inequalities of the form of (4) may be in general called convolutioninequalities. Our main focus is laid here on a specific type of them, namelythose in which the term C(g) is equal or equivalent to a norm of the kernelg in another function space Y . More precisely, the convolution inequalitiesof our interest are those of the form

∥f ∗ g∥Z ≤ C∥f∥X∥g∥Y , f ∈X, g ∈ Y, (5)

where the constant C is independent of f and g.The most well-known result of this type is the classical Young inequality.

It reads as follows. If 1 ≤ p, q, r ≤∞ and 1p +

1r = 1 +

1q , then

∥f ∗ g∥q ≤ ∥f∥p∥g∥r, f ∈ Lp, g ∈ Lr.

Note the assumption p ≤ q which may not be dropped (cf. [47] for the un-boundedness of Tg between Lp and Lq when q < p).

The Young inequality is essential whenever convolution is used in connec-tion with function spaces. Besides its classical applications in pure analysisand theory of partial differential equations (see e.g. [43, 1, 88, 10, 11]), wemay, for example, mention its use within the kinetic theory of gases (see e.g.[2, 3, 45] and the references therein). It might be considered a model resultfor many further developments. Not surprisingly, convolution inequalitieshaving the form (5) are often called Young-type (convolution) inequalities.

There has been an extensive research into more Young-type inequalities(5) with spaces X,Y,Z different from Lp. In [67], R.O’Neil proved a theorem,which is sometimes called the Young-O’Neil inequality. It states that, if

9

1 < p, q, r < ∞ and 1 ≤ a, b, c ≤ ∞ are such that 1p +

1r = 1 +

1q and 1

a =1b +

1c ,

then∥f ∗ g∥q,a ≤ C∥f∥p,b∥g∥r,c, f ∈ Lp,b, g ∈ Lr,c. (6)

The article [67] includes a proof of a powerful pointwise inequality statingthat, for any f, g ∈ L1

loc and any t > 0, it holds

(f ∗ g)∗∗(t) ≤ tf∗∗(t)g∗∗(t) +∞

∫t

f∗(s)g∗(s)ds. (7)

We call this result the O’Neil inequality. It is in fact a cornerstone of the prooftechnique used in the articles constituting this thesis. (More information isgiven in the description of the content of the main articles later on.) Theproof of the O’Neil inequality works correctly for the ordinary convolution(as given by (2)) but contains some flaws if used with O’Neil’s more generaldefinition of a convolution operator. This was observed and corrected byL.Y.H.Yap in [87].

In [48, 87, 13], the inequality (6) was shown to hold even for an extendedrange of parameters 0 < a, b, c ≤∞ (while the other conditions on a, b, c, p, q, rremain the same as above). Furthermore, a limiting case of (6) with 1 < p <∞, 1 ≤ b, c ≤ ∞, 1 = 1

p +1r and 1

a =1b +

1c < 1 was studied by H.Brezis and

S.Wainger in [16], leading to the following result:

∥f ∗ g∥BWa ≤ C∥f∥p,b (∥g∥r,c + ∥g∥1) , f ∈ Lp,b, g ∈ Lr,c ∩L1,

where

∥f∥BWa ∶=⎛⎝

1

∫0

( f∗(t)1 + ∣ log t∣

)adt

t

⎞⎠

1a

.

A.P.Blozinski [14] considered another limiting case of the parameters,namely such that p = q > 1, r = 1 and 0 < a, b ≤ ∞. He showed that, withthese parameters, if g ≥ 0 and Tg ∶ Lp,b → Lp,a, then necessarily g = 0 a.e.It is important to recall here that Lp,a = Lp,a(Rd) and Lp,b = Lp,b(Rd). Ifwe consider an underlying measure space with a finite measure, then theprevious result does not have to be true, as we will see shortly.

E.Nursultanov and S.Tikhonov [66] investigated boundedness of convo-lution of 1-periodic functions in Lp,q-spaces. (The spaces are then of coursedefined over the interval [0,1] instead of Rd.) In this case, it was shown that,with the setting 1 ≤ p <∞, 1 ≤ a, b, c ≤∞ and 1

a =1b +

1c , it holds

∥f ∗ g∥L(p,a)[0,1] ≤ C∥f∥L(p,b)[0,1]∥g∥L(1,c)[0,1], f ∈ L(p,b)[0,1], g ∈ L(1,c)[0,1].

10

This contrasts with the aforementioned negative result of Blozinski (recallthat Lp,q = L(p,q) if 1 < p < ∞ and 1 ≤ q ≤ ∞). Among other results of[66] is the Young-O’Neil inequality for spaces L∞,p[0,1], which reads that if0 < 1

a =1b +

1c , then

∥f ∗ g∥L∞,a[0,1] ≤ 4∥f∥L∞,b[0,1]∥g∥L(1,c)[0,1], f ∈ L∞,b[0,1], g ∈ L(1,c)[0,1].

Boundedness of convolution and Young-type inequalities were furtherstudied in the weighted Lp-spaces with power weights [50, 17], Lp-spaces withgeneral Borel measures [4] and Wiener amalgam spaces [48]. In [12, 32, 52]it was investigated, under which conditions the Lp(w) space is a convolutionalgebra, i.e. when the inequality

∥f ∗ g∥Lp(w) ≤ ∥f∥Lp(w)∥g∥Lp(w), f, g ∈ Lp(w),

is satisfied. The convolution algebra property of r.i. spaces and various gen-eral properties of the convolution operator acting on r.i. spaces were alsostudied by E.A.Pavlov in [71]-[76].

Analogues of the Young inequality in the Lebesgue spaces with variableexponent Lp(x) were obtained by S.Samko in [78, 79] (see also [80, 19] andthe references given therein).

Moreover, in [68] R.O’Neil examined the convolution operator in Or-licz spaces, providing a corresponding Young-type convolution inequality forthese spaces.

4 Summary of the main papers’ contents

The three papers which build the main body of this thesis contribute to theresearch into the topic described in the previous section. Their aim is to studyboundedness of the convolution operator and related Young-type convolutioninequalities in the weighted Lorentz-type spaces Γ,Λ and S. These problemswere not investigated before, except for some special cases of weights, e.g.those establishing the Lp,q spaces (see the survey of the existing results in theprevious section). Here we work with considerably more generality, makinglittle or no assumptions on the weights, and implementing a method whichis different from those used in the existing works.

11

4.1 Paper [I]

The research questions of paper [I] read as follows: Let two weighted Lorentzspaces Λp(v) and Γq(w) be given.

(i) Characterize the conditions on the kernel g, the weights and exponents,under which the convolution operator Tg (see (3)) is bounded betweenΛp(v) and Γq(w).

(ii) Find the optimal r.i. (quasi-)space Y such that the Young-type inequal-ity

∥f ∗ g∥Γq(w) ≤ C∥f∥Λp(v)∥g∥Y , f ∈ Λp(v), g ∈ Y, (8)

holds with C independent of f, g.

By optimality of Y it is meant that if we replace the space Y by another r.i.space Y , then necessarily Y = Y with equivalent norms. Or, in other words,Y is the largest r.i. (quasi-)space for which (8) holds.

In order to solve the problem, the following method is implemented: Atfirst, O’Neil’s inequality (7) is used, which gives

∥f ∗ g∥Γq(w) ≤XXXXXXXXXXXXt↦ tf∗∗(t)g∗∗(t) +

∞

∫t

f∗g∗XXXXXXXXXXXXLq(w)

. (9)

Now, if the right-hand side can be estimated by the term ∥f∥Λp(v), thenTg ∶ Λp(v)→ Γq(w). Such estimates correspond with the inequalities

∥t↦ tf∗∗(t)g∗∗(t)∥Lq(w) ≤ A1∥f∥Λp(v), f ∈ Λp(v), (10)

and XXXXXXXXXXXXt↦

∞

∫t


≤ A2∥f∥Λp(v), f ∈ Λp(v). (11)

Both (10) and (11) are Hardy-type inequalities for nonincreasing functions.These inequalities have been systematically studied and the least possibleconstants Ai = Ai(q, v,w, p, q), i = 1,2, such that these inequalities hold, werefound [23, 21, 42]. Using these results, one obtains that a sufficient conditionfor boundedness Tg ∶ Λp(v)→ Γq(w) is

Ai = Ai(q, v,w, p, q) <∞, i = 1,2. (12)

12

To deal with the necessity question, the following inverse to the O’Neilinequality is used: If both f and g are nonnegative and radially decreasing,then

tf∗∗(t)g∗∗(t) +∞

∫t

f∗g∗ ≤ Cd(f ∗ g)∗∗(t), t > 0,

with Cd depending only on the dimension d.3 Thanks to this inequality andthe fact that the condition (12) characterizes the validity of (10) and (11),we obtain this result: If g is nonnegative and radially decreasing, then (12)is a necessary condition for boundedness Tg ∶ Λp(v)→ Γq(w).

In the next step it is observed that the sum of the obtained conditions A1

and A2 is equivalent to an r.i. norm (or quasi-norm) of g, denoted by ∥g∥Y . Itgives directly the Young-type inequality (8). The optimality of Y is grantedthanks to the necessity part of the previous result (for nonnegative radiallydecreasing g’s) and thanks to Y being r.i.

The range of exponents covered in [I] is 0 < p ≤ q ≤∞, 1 ≤ q < p <∞ and0 < q < p =∞. The article is written in such way that the results cover boththe convolution on Rd and on a compact interval for periodic functions. Itis shown that the “classical” results [67, 48, 87, 66] follow as special casesof the presented theorems. In particular, it is pointed out that both theresult of [14], stating that Tg with a nonnegative nontrivial g is not boundedbetween Lp,b(Rd) and Lp,a(Rd), p > 1, and the result of [66], stating that thesame boundedness is possible in case of 1-periodic functions on [0,1], areconsequences of a single theorem of [I]. The proved optimality of the domainspace Y is a key point for the possibility to draw such conclusions.

Furthermore, the last part of [I] deals with the r.i. spaces which appearas the optimal domain space Y . Some basic functional properties of thesespaces are shown there.

The next two papers [II] and [III] further exploit the technique introducedin [I]. The description of their content can therefore be made slightly shorter.

3This reverse inequality is usually labeled as “well-known”. O’Neil [67] mentions itsone-dimensional version without proof. In paper [I], an elementary one-dimensional proofis shown (cf. also [82]). The proof for a general dimension may be found e.g. in [56].

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4.2 Paper [II]

The following question is asked: If the spaces Sp(v) and Γq(w) are given,when is Tg bounded between Sp(v)(R) ∩ L1(R) and Γq(w)(R)? Again, themain goal is to give the result the shape of a Young-type inequality, in thiscase having the following form:

∥f ∗ g∥Γq(w) ≤ C∥f∥Sp(v)∥g∥Y , f ∈ Sp(v), g ∈ Y ∩L1.

The method from [I] is used, supplemented with an additional observation.Namely, the right-hand side of O’Neil’s inequality (7) is equal to

lims→∞

sf∗∗(s)g∗∗(s) +∞

∫t

(f∗∗ − f∗)(g∗∗ − g∗)

for any t > 0. Since f ∈ Sp(v) and g ∈ L1, the first term disappears andthe whole problem is transformed into the problem of characterization of theinequality

XXXXXXXXXXXXt↦

∞

∫t

(f∗∗ − f∗)(g∗∗ − g∗)XXXXXXXXXXXXLq(w)

≤ C∥f∥Sp(v), f ∈ Sp(v).

This is just another example of a Hardy-type inequality, here on the cone ofnondecreasing functions, since the function

t↦ t(f∗∗(t) − f∗(t))

is nondecreasing on (0,∞). Using the known characterizations of that Hardy-type inequality, the initial question of the paper is answered and the desiredYoung-type inequalities are given. The range of parameters covered in [II] is0 < p, q ≤∞.

4.3 Paper [III]

This paper focuses on the problem of boundedness of Tg between spaces Γp(v)and Γq(w). The final result is the Young-type inequality

∥f ∗ g∥Γq(w) ≤ C∥f∥Γp(v)∥g∥Y , f ∈ Γq(w), g ∈ Y,

14

with the r.i. space Y being optimal for the given pair Γp(v) and Γq(w).The method applied in [I] and [II] proves useful in this case as well. The

problem reduces to characterizing the validity of the inequality

XXXXXXXXXXXXt↦ tf∗∗(t)g∗∗(t) +

∞

∫t


≤ C∥f∥Γq(w), f ∈ Γq(w). (13)

Since this inequality involves the Hardy operator on the right-hand side andthe adjoint-Hardy-type operator ∫

∞t f∗g∗ on the left-hand one, a character-

ization of (13) may seem to be a more difficult task (cf. [22]). However,similarly as in [II], it is observed that the right-hand side of (7) may berewritten by different terms. Namely, it is equivalent to the term S(f∗∗)(t)with S being a certain positive linear operator (see [III, Lemma 3.1]). Thenthe inequality (13) takes the form

∥S(f∗∗)∥Lp(w) ≤ C∥f∥Γp(v), f ∈ Γp(v).

Such inequality is then handled using the results of [36] regarding the bound-edness of positive linear operator on the cone of quasi-concave functions. Thisapproach is adopted in the case 1 < p, q <∞. In the other cases presented inthe paper, there are different methods used, involving other known resultsabout Hardy-type inequalities.

The range of exponents covered by [III] is 1 ≤ p, q ≤ ∞, 0 < p < 1 &q ∈ 1,∞ and p =∞ & 0 < q < 1.

5 Remarks and further comments

As observed before, one of the main advantages of the technique applied inthe main papers is the generality of the produced results. If any assumptionson the weights are made, they are only those assuring a certain kind of non-triviality of the corresponding weighted space. Thus, such conditions areusually natural and non-restrictive. There are many technical features in theproofs but the used method is rather straightforward.

An important feature of the presented results is their optimality. Inthe previous research articles about Young-type inequalities, there has oftenbeen no attention paid to the optimality issues. Although the results veryoften are optimal, this fact is not explicitly proved in there. In contrastwith that, the optimality of the domain space Y is proved in this thesis, as

15

connected to the fact that g ∈ Y is a both necessary and sufficient conditionfor boundedness of Tg between the given spaces, provided that the kernel g isradially decreasing and nonnegative. This fact allows, for example, applyingthese general theorems to various known operators of convolution type, andgetting both the “positive” results (i.e., the operator is bounded betweenthe spaces) and the “negative” ones (i.e., it is not bounded). If we knewonly that the related Young-type inequality held or, equivalently, that g ∈ Ywas a sufficient condition for boundedness, only the “positive” part could beobtained.

There are some cases of the exponents p and q which are not covered inthe thesis’ main papers. Often it is the case 0 < q < p < 1, which usually ex-hibits the fiercest resistance to established proof techniques. In the “missingcases”, the needed Hardy-type inequalities are either not yet characterized ortheir characterizations involve e.g. discrete conditions which are not directlyapplicable. In these cases, more research will be required into these topicsthemselves before the secondary questions can be answered.

In all the papers [I]-[III], the spaces appearing as the optimal space Yhave the same form (although they involve different weights). The type ofr.i. spaces having this form are denoted as “K” in [I]. They are generated bythe functional

∥f∥Kp,q(u,v) ∶=⎛⎜⎝

∞

∫0

⎛⎝

∞

∫x

(f∗∗(t))pu(t)dt⎞⎠

qp

v(x)dx⎞⎟⎠

1q

,

with standard modifications in the “weak” variants. A space of this typewith a special choice of weights appeared e.g. in [28]. As said, the “K”spaces play a significant role in the study of convolution operators in [I]-[III],but recently [41] the same type of r.i. spaces appeared in the role of theassociate space to a generalized Γ space. Yet, aside from the short survey oftheir functional properties in [I], these spaces have not been studied yet (upto the author’s knowledge). In particular, the embeddings of “K” spaces andthe other weighted Lorentz-type spaces have not been characterized yet. Theinequalities corresponding to such embeddings seem to have quite promisingapplications in various problems within the function spaces theory. Studyingof such problems is therefore one of the considered directions of researchduring the author’s continued Ph.D. studies.

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Martin Křepela


Mathematics

Faculty of Health, Science and Technology

Martin K

řepela | Forever Young | 2014:21

Forever Young

Convolution plays an important role in both pure mathematics and practical applications. Many important operators in mathematical analysis have a form of a convolution with a fixed kernel, which gives numerous applications to the general theoretical results about convolution. One of the most important theorems involving convolution is the Young inequality, estimating the norm of the convolution in the Lebesgue spaces.

In this licentiate thesis, convolution is investigated in the framework of weighted Lorentz-type spaces. Boundedness of a general convolution operator in these function spaces is characterized in terms of the kernel. Consequently, new Young-type inequalities for the weighted Lorentz-type spaces are proved and their optimality is studied.


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