THÈSE. Présentée pour l obtention du diplôme de Doctorat troisième cycle THÈME

Transcription

1 RÉPUBLIQUE ALGÉRIENNE DÉMOCRATIQUE ET POPULAIRE MINISTÈRE DE L ENEEIGNEMENT SUPÉRIEURE ET DE LA RECHERCHE SCIENTIFIQUE UNIVERSITÉ MOHAMED BOUDIAF- M'SILA FACULTÉ DE MATHÉMATIQUES ET DE L'INFORMATIQUE DÉPARTEMENT DE MATHÉMATIQUES THÈSE Présentée pour l obtention du diplôme de Doctorat troisième cycle Spécialité : Mathématiques Option : Géométrie linéaire et non linéaire des espaces de Banach Présentée par WAFA HEBBACHE THÈME Continuité de certains opérateurs pseudo-différentiels sur les espaces de Besov et Triebel-Lizorkin avec des exposants variables Directeur de thése : Douadi DRIHEM Soutenue le 24/06/2018, devant le jury : d'order: Lahcène MEZRAG Prof Université Mohamed Boudiaf - M'sila Président Douadi DRIHEM Prof Université Mohamed Boudiaf - M'sila Rapporteur Madani MOUSSAI Prof Université Mohamed Boudiaf - M'sila Examinateur Dahmane ACHOUR Prof Université Mohamed Boudiaf - M'sila Examinateur Fares MOKHTARI MCA Université Benyoucef Benkhadda - Alger Examinateur Promotion Promotion: : 2017/ /2018

2 People s Democratic Republic of Algeria Ministry of Higher Education and Scientific Research Mohamed BOUDIAF University M sila Faculty of Mathematics and Informatics Department of Mathematics N 0 d'order: THESIS Submitted for the degree of doctorate LMD Specialty: MATHEMATICS Option: Linear and non-linear geometry of Banach spaces Presented by: WAFA HEBBACHE Entitled Boundedness of some pseudo-differential operators on Besov and Triebel-Lizorkin spaces of variable exponents Thesis director: Douadi DRIHEM Publicly presented on : 24/06/2018, in front of the jury: Lahcène MEZRAG Prof University Mohamed Boudiaf - M'sila President Douadi DRIHEM Prof University Mohamed Boudiaf - M'sila Supervisor Madani MOUSSAI Prof University Mohamed Boudiaf - M'sila Examinator Dahmane ACHOUR Prof University Mohamed Boudiaf - M'sila Examinator Fares MOKHTARI MCA University of Benyoucef Benkhedda - Algiers Examinator Promotion: 2017/2018 Promotion : 2017/2018

3 Acknowledgments All praise to Allah, today we fold the day's tiredness and the errandsumming up between the cover of this humble work My thanks first of all intended for my director thesis the professor Douadi DRIHEM for his patience, motivation,enthusiasm, constant support, guidance and encouragement My sincere thanks to the president of the jury professor Lahcène MEZRAG to accept this task and to give interest to my work Also, my thanks to the professors Madani MOUSSAI, Dahmane ACHOUR to accept being the examiners of my thesis My sincere thanks to the Dr Fares MOKHTARI from Benyoucef Benkhedda university Algiers-to accept being the examiner of my thesis To the people who paved our way of science and knowledge

4 DEDICATE The simplest words are the strongest, I send all my love to my father and my mother thank you for making me what I am today To whose love flows in my veins, and my heart always remembers them, to my brothers and sistersto the most precious to my heart Mehdi,Djana,Mohamed To the spirit of my sister and grandmother ZOUINA AND KALTOM Finally, to the best person in my life To the person who taught me the meaning of patience and respect you're just yours all my life هب وفاء اش

5 Contents Notation 2 Introduction 5 1 Variable Besov spaces 7 11 Variable Lebesgue spaces Modular space 7 12 Lebesgue spaces L p ) 9 13 Besov spaces B s ) 11 2 Boundedness of pseudo-differential operators in the L p ) spaces Definitions and basic properties Fourier multipliers L p ) boundedness 19 3 Boundedness of pseudo-differential operators on variable Besov spaces Auxiliary results Main results 35 4 Boundedness of pseudo-differential operators on variable Triebel-Lizorkin spaces Basic tools Main results 50

6 Notation R n is the n-dimensional real Euclidean space N 0 is the collection of all natural numbers Z is the collection of all integer numbers If E R n is a measurable set, then E stands for the Lebesgue) measure of E For x R n and r > 0 we denote by Bx, r) the open ball in R n with center x and radius r Q will denote an cube in the space R n with sides parallel to the coordinate axes and lq) will denote the side length of the cube Q For all cubes rq and r > 0, rq is the cube concentric with Q having the side length rlq) α = α 1,, α n ) N n 0 stands for some multi-index, whose length is denoted by α = α α n and α = α α 1 x1 x αn n The Euclidean scalar product of x = x 1,, x n ) and y = y 1,, y n ) is given by x y = x 1 y x n y n The expression f g means that f < cg for some independent constant c and non-negative functions f and g) f g means f g f The notation X Y stands for continuous embeddings from X to Y, where X and Y are quasi-normed spaces f g x) = f x y) g y) dy is the product of the convolution of functions f and g R n L 1 loc Rn ) is the collection of all locally integrable functions on R n 2

7 L p R n ) for 0 < p stands for the Lebesgue spaces on R n for which ) 1/p f L p R n ) = f p = f x) p dx <, 0 < p < R n and f L R n ) = f = ess-sup x R n f x) < If 1 p, then p is the conjugate exponent of p given by = 1 p p c 1, c 2, c 3, positive constants, their values may depend on certain parameters and some auxiliary functions, and change from one line to another DR n ) is the space of functions with continuous derivatives of all orders and compact support SR n ) is the Schwartz space of all complex-valued, infinitely differentiable and rapidly decreasing functions on R n By supp f we denote the support of the function f The dual S R n ) is the space of temperate distributions We define the Fourier transform of a function f SR n ) by Ff ξ) = ˆf ξ) = e ixξ f x) dx, ξ R n R n and its inverse Fourier transform by: F 1 f x) = ˇf x) = 2π) n R n e ixξ f ξ) dξ, x R n f its spectrum in B0, r) and r > 0; supp ˆf B0, r) By l q, 0 < q, we denote the space of all complex) sequences {a k } k Z equipped with the quasi-norm {ak } k Z l q = with the usual modification if q = ) k= a k q ) 1 q 3

8 If a : R n R n C is a function, we denote by ax, D) the pseudo-differential operator PDO) of symbol a is noted a x, D) and defined on the class S R n ) by: a x, D) f x) = 2π) n R n e ixξ a x, D) ˆf ξ) dξ, f S R n ) "ie" stands simply for "in other words" "ae" stands simply for "almost everywhere" χ E is the characteristic function of E R n η v,m x) = 2 nv v x ) m, for any x R n, v N 0 and m > 0 4

9 Introduction In recent years there has been a growing interesting in generalizing classical spaces such as Lebesgue and Sobolev spaces to cases with either variable integrability or variable smoothness see [15] The motivation to study such function spaces also comes from applications to other fields of applied mathematics, but also from applications to image processing and PDE with nonstandard growth conditions Some example of these spaces can be mentioned such as: variable Lebesgue space, variable Besov and Tiebel-Lizorkin spaces Pseudo-differential operators play an import role in Harmonic analysis and in nonlinear partial differential The boundedness these operators has been extensively addressed in several works In Lebesgue spaces with symbols in the Hörmander classes can be found in [5-8, 11-12, 21, 23, 37-38] and references therein In another function spaces, such that Besov spaces, Triebel-Lizorkin spaces, BM O spaces and Hardy spaces, see [26, 32-34, 39, 46-48] In [34] J Marschall introduced the class SB m δ r, µ, v; N, λ), which is defined by means of vector-valued Besov spaces, and proved the boundedness of the corresponding pseudodifferential operators on Besov spaces and Triebel-Lizorkin spaces Boundedness of pseudodifferential operators, with symbols in the Hörmander classes, on weighted variable exponent Lebesgue and Bessel potential spaces was studied by VS Rabinovich and S Samko [37-38] and by A Yu Karlovich and I M Spitkovsky in [28] in variable Lebesgue space) Since Besov spaces can be written as a real) interpolation space between appropriate Bessel potential spaces, Almeida and Hästö [2] extend the results of VS Rabinovich and S Samko to Besov spaces with variable integrability B s p ),q The variable Besov spaces B s ), initially appeared in the paper of A Almeida and P Hästo [2], serveral bassic properties where established, such as the Fourier analytical characterization when p, q, s are constants they coincide with the usual function spaces B s p,q studied in detail by H Triebel in [43-45] Diening, Hästö and Roudenko, [14] introduced and investigated Triebel-Lizorkin spaces with variable smoothness and integrability F s ) spaces behaved nicely with respect to the trace operator with s ) 0, and showed that these The aim of the thesis is to study the boundedness of the pseudodifferential operators on variable Besov spaces Triebel-Lizorkin spaces 5

10 Our thesis consists four chapter, in the first chapter, we give some basic properties of variable Lebesgue spaces and mixed variable Lebesgue s sequence space, after this we define variable Besov spaces where all the three parameters are variable and we recall some their basic properties In the second chapter, we investigate the L p ) - boundedness of certain class of pseudodifferential operators with non regular symbols We employ regularisation methods In Chapter 3, we present the boundedness properties of the pseudo-differential operators on Besov spaces with variable smoothness and integrability with symbols in SB m δ where we use the decomposition of Littelwood-Paley r, µ, v; N, λ), In Chapter 4, we present the boundedness properties of the pseudo-differential operators on Triebel-Lizorkin spaces with variable smoothness and integrability with symbols in SB m δ r, µ, v; N, λ) Also we give some technical lemmas needed in the proof of the main statement The Chapter three is a paper published in Journal of Pseudo-Differential operators and applications "Boundedness of non regular pseudodifferential operators on variable Besov spaces, 8 2), ", while the Chapter four is papers in preparation with the advisor In the future we will try to apply these results to study some problems in harmonic analysis and partial differential equations 6

11 Chapter 1 Variable Besov spaces In this chapter we present some results, which remain fixed throughout this theses We recall some conventions and results on modular and variable Lebesgue spaces, we then define the Besov space B s ) with fixed exponents and give some their basic properties in analogy to the Besov spaces 11 Variable Lebesgue spaces The object of this section is to recall some fundamental properties of variable Lebesgue spaces, see [10, Chapters 1 2] and [15, Chapters 1 3] for details and further properties 111 Modular space We will use the modular to define the variable Lebesgue spaces Definition 11 Let X be a k-vector space A function ϱ : X [0, ] is called a semimodular on X if the following properties hold a) ϱ 0) = 0 b) ϱ λx) = ϱ x) for all x X, λ k with λ = 1 c) ϱ is quasi convex d) ϱ is left-continuous e) ϱ λx) = 0 for all λ > 0 implies x = 0 7

12 A semimodular is called a modular if f) ϱ x) = 0 implies x = 0 A semimodular is called a continuous if g) The mapping λ ϱ λx) is continuous on [0, ) for every x X The following examples are from [15] Example 12 a) If 1 p <, then ϱ p f) = f x) p dx R n defines a continuous modular on L 0 R n ) b) Let w L 1 loc Rn ) with w > 0 almost everywhere and 1 p < Then ϱ f) = f x) p w x) dx R n defines a continuous modular on L 0 R n ) c) If 1 p <, then defines a continuous modular on R N ϱ p x j )) = x j p j=0 Remark 13 Let ϱ be a semimodular on X Then by quasi convexity, non-negative of ϱ and ϱ 0) = 0 it follows that, ϱ λx) = ϱ λ x) C λ ϱ x) for all λ 1, 11) ϱ λx) = ϱ λ x) 1 C λ ϱ x) for all λ 1, where C 1 is independent of λ and x Definition 14 If ϱ be a semimodular or modular on X, then X ϱ = { x X : lim ϱ λx) = 0 } λ 0 is called a semimodular space or modular space, respectively, where the limit λ 0 takes place in k 8

13 Since ϱ λx) = ϱ λ x) it is enough to require lim λ 0 ϱ λx) with λ 0, ) From 11) we can alternatively define X ϱ by X ϱ = {x X : ϱ λx) < for some λ > 0}, since for λ < λ we have by 11) that λ ) ϱ λx) = ϱ λ λx C λ λ ϱ λx) 0 as λ 0 and C 1 We equipped this space with the following quasi-norm, for the proof see [15, Theorem 217] Theorem 15 Let ϱ be a semimodular on X Then X ϱ is a quasi-normed k-vector space The quasi-norm, called the Luxemburg quasi-norm, is defined by { ) 1 } x ϱ = inf λ > 0 : ϱ λ x 1 Example 16 Let 1 p < and ϱ p be as in Example 12 Then X ϱp = L p R n ) For an exposition of these concepts we refer to the monographs [15] and [36] 12 Lebesgue spaces L p ) In this subsection we recall the definition and basic properties of the variable Lebesgue spaces They differ from classical L p spaces where the exponent p is not constant but a function from R n to [c, ), c > 0 The variable exponents that we consider are always measurable functions p on R n with range in [c, [ for some c > 0 We denote the set of such functions by P 0 R n ) The subset of variable exponents with range [1, [ is denoted by P R n ) We use the standard notation p = ess-inf px), x R p+ := ess-sup px) n x R n The variable exponent Lebesgue space L p ) is the class of all measurable functions f on R n such that the modular ϱ p ) f) = fx) px) dx R n 9

14 is finite This is a quasi-banach function space equipped with the quasi-norm 1 ) f p ) = inf {µ > 0 : ϱ p ) µ f } 1 If px) = p is constant, then L p ) = L p is the classical Lebesgue space The following classes for the exponents are necessary for us We say that g : R n R is locally log-hölder continuous, abbreviated g C log loc, if there exists c logg) > 0 such that gx) gy) c log g) loge + 1/ x y ) 12) for all x, y R n We say that g satisfies the log-hölder decay condition, if there exists g R and a constant c log > 0 such that for all x R n gx) g c log loge + x ) The constants c log g) and c log are called the locally log-hölder constant and the log-hölder decay constant, respectively We note that all functions g C log loc always belong to L We say that g is globally-log-hölder continuous, abbreviated g C log, if it is locally log- Hölder continuous and satisfies the log-hölder decay condition We define the following class of variable exponents P log R n ) = {p P R n ) : 1 p Clog }, were introduced in [16, Section 2] Note that 1 p p itself can be unbounded although is bounded, the variable exponent Example 17 We set px) = max1 e 3 x, min6/5, max1/23/2 x 2 ))), x R then p C log, see [37] A basic tool to study the variable Besov spaces is the following Lemma 18 If p P log R n ), then the convolution with a radially decreasing L 1 -function is bounded on L p ) : ϕ f p ) c ϕ 1 f p ) 10

15 Proof For the proof, see [15] Lemma 19 If p P R n ), then f p ) 1 and ϱ L p ) f) 1 are equivalent For f L p ) R n ) we have 1 If f p ) 1, then ϱ L p ) f) f p ) 2 If 1 < f p ), then f p ) ϱ L p ) f) Lemma 110 Hölder s inequality) Let p, q, s P R n ) such that 1 s y) = 1 p y) + 1 q y) for every y R n Then fg s ) 2 f p ) g q ), for all f L p ) R n ) and g L q ) R n ) For the proof, see [15] 13 Besov spaces B s ) We recall in this subsection the definition and some properties of mixed Lebesgue sequence and Besov spaces with variable smoothness and integrability Definition 111 Let p, q P 0 R n ) The mixed Lebesgue-sequence space l q ) L p ) ) is defined on sequences of L p ) -functions by the modular ϱ l q ) L p ) ){f v } v ) = v inf The quasi)-norm is defined from this as usual: { λ v > 0 : ϱ p ) {f v } v l q ) L p ) ) = inf {µ > 0 : ϱ l q ) L p ) ) If q + <, then we can replace 13) by the simpler expression fv λ 1/q ) v ) 1 µ {f v} v ) } 1 } 1 13) ϱ l q ) L p ) ){f v }) = v fv q ) p ) q ) 11

16 Furthermore, if p and q are constants, then l q ) L p ) ) = l q L p ) The case p := can be included by replacing the last modular by ϱ l q ) L ){f v } v ) = v fv q ) We define 1 p = lim x 1 px) and we use the convention 1 = 0 These function spaces introduced recently in [1] Now we recall the definition of the spaces B s ), as given in [1] We first need the concept of a smooth dyadic resolution of unity Let Ψ be a function in SR n ) satisfying Ψx) = 1 for x 1 and Ψx) = 0 for x 2 We put Fϕ 0 x) = Ψx) Fϕ 1 x) = Ψx) Ψ2x) and Then {Fϕ v } v N0 Fϕ v x) = Fϕ 1 2 v x) for v = 2, 3, is a smooth dyadic resolution of unity, Fϕ v x) = 1 for all x R n Thus we obtain the Littlewood-Paley decomposition v=0 f = ϕ v f v=0 of all f S R n ) convergence in S R n )) Now, we define the Besov spaces B s ) Definition 112 Let {Fϕ v } v N0 be as resolution of unity For s : R n R and p, q P 0 R n ), the Besov space B s ) consists of all distributions f S R n ) such that f s ) B = { 2 vs ) ϕ v f } v l q ) L p ) ) To the Besov space we can also associate the following modular: < 14) { ϱ ϕ f) = ϱ B s ) l q ) L p ) ) 2 vs ) ϕ v f } v) 12

17 We directly obtain the following simplification in the case when q is constant, see [1] If q is a constant, then f ϕ = { 2 vs ) ϕ B s ) v f } v For any p, q P log 0 R n ) and s C log loc, the space Bs ) smooth dyadic resolution of unity {Fϕ v } v N0 SR n ) B s ) S R n ) p ) l q If p, q P log 0 R n ) and s C log loc, then SRn ) is dense in B s ), does not depend on the chosen in the sense of equivalent quasi-norms) and and they are quasi-banach spaces Moreover, if p, q, α are constants, we re-obtain the usual Besov spaces B s p,q, studied in detail by H Triebel in [44], [45] and [46] See also [20] and [41] for further properties The following theorem gives basic embedding generalize the constant exponent versions Theorem 113 Let α, α 0, α 1 L R n ) and p, q 0, q 1 P 0 R n ) a) If q 0 q 1, then b) If α 0 α 1 ) > 0, then B α ) p ),q 0 ) Rn ) B α ) p ),q 1 ) Rn ) B α 0 ) Rn ) B α 1 ) Rn ) that We next consider embeddings of Sobolev-type For constant exponents it is well-known B α 0 p 0,q B α 1 p 1,q if α 0 n p 0 = α 1 n p 1, where 0 < p 0 p 1, 0 < q, < α 1 α 0 < see eg [44], Theorem 271) For variable case we have the following results, see [1] Theorem 114 Sobolev inequality) Let p 0, p 1, q P 0 R n ) and α 0, α 1 L R n ) with α 0 α 1 If 1 and q α 0 x) n p 0 x) = α 1 x) n p 1 x) are locally log-hölder continuous, then B α 0 ) p 0 ),q ) Rn ) B α 1 ) p 1 ),q ) Rn ) 13

18 Corollary 115 Let p 0, p 1, q 0, q 1 P 0 R n ) and α 0, α 1 L R n ) with α 0 α 1 If α 0 x) is locally log-hölder continuous and ε > 0, then n p 0 x) = α 1 x) n p 1 x) + ε x) B α 0 ) p 0 ),q 0 ) Bα 1 ) p 1 ),q 1 ) The full treatment of the spaces B s ) can be found in [1], [17], [29],[30], [31] and [50] We refer to the papers [50-51], for further results on the variable Besov spaces B s ) p ),q only the case of constant q was considered), see also [3], [4]) 14

19 Chapter 2 Boundedness of pseudo-differential operators in the L p ) spaces In this chapter we present the boundedness of a certain class of pseudodifferential operators in variable Lebesgue spaces 21 Definitions and basic properties We now introduce the basic pseudodifferential symbol class S m There are many other more general or modified symbol classes, which are used in the literature and research for different purposes But the following symbol class is the most simple and most common It is a natural symbol class that contains the symbols of differential operators with smooth coeffi cients Definition 21 Let m, + ) The symbol S m is used in place of the set of all functions a x, ξ) in C R n R n ) such that for any two multi-indices α and β, there is positive constant C α,β, depending on α and β only, such that D α x D β ξ a x, ξ) Cα,β 1 + ξ ) m β, x, ξ R n 21) We call any function a in m R S m a symbol Definition 22 Let a be a symbol The pseudo-differential operator T a associated to a is defined by T a ϕ) x) = 2π) n 2 e ixξ a x, ξ) ˆϕ ξ) dξ, ϕ SR n ) R n 15

20 We give some examples Example 23 Let P x, D) = α m If all the coeffi cients a α x) are C polynomial a α x) D α be a linear partial differential operator on R n and have bounded derivatives of all orders, then the P x, D) = α m a α x) ξ α is in S m and hence P x, D) is a pseudo-differential operator, see [55] Example 24 Let a ξ) = 1 + ξ 2) m 2, < m < Then a S m and hence T a is a pseudo-differential operator, see [55] The proof of the following results can be found in [55, Chapters, 4 and 9] Proposition 25 Let a be a symbol Then T a maps the Schwartz space S into itself Proposition 26 T a is a linear mapping from S R n ) into S R n ) Let us recall the begin definition of the Hardy-Littlewood maximal function, which plays a very important role in harmonic analysis Definition 27 Suppose that f is a locally integrable on R n, ie f L 1 loc Rn ) The Hardy- Littlewood maximal operator M is defined on L 1 loc by 1 Mfx) = sup r>0 Bx, r) Bx,r) fy) dy, where, the supremum is taken over balls B in R n which contain the point x The Hardy-Littelwood maximal operator M, in general, is not a bounded from L 1 R n ) to itself Take fx) = χ [0,1] x), then for any x 1, we have Hence Mfx) 1 2x R n Mfx)dx 1 2x 0 f y) dy = 1 2x Mfx)dx 1 dx = 2x Although M is not a bounded operator on L 1 R n ), however, as its a replacement result we shall see that M is a bounded operator from L 1 R n ) to L 1, R n ), ie the weak L 1 R n ) space 16

21 Remark 28 The maximum function M is bounded on L R n ), Mf f Theorem 29 Let f be a measurable function on R n a) If f L p R n ) for 1 p, then Mf x) < ae x R n b) There exists a constant C = C n, p) > 0 such that for any f L p R n ), Mf p C f p, 1 < p The proof can be found in [23] For the variable case we have, see [13], [15], [16] Theorem 210 Let p P log R n ) with p > 1 Then there exists K > 0 only depending on the dimension n and c log p) such that Mf p ) K p ) f p ) for all f L p ) R n ) 22 Fourier multipliers The purpose of this subsection is to review some known properties of Fourier multipliers Definition 211 A p weights 1 p < ) Let ω > 0 and ω L 1 loc Rn ) We say that ω A p for 1 < p < if there is a constant C > 0 such that ) p sup ω x) dx ω x) dx) 1 p C, 22) Q Q Q where and below, 1 /p + 1 /p that Q Q = 1 We say that ω A 1 if there is a constant C > 0 such Mω x) Cω x) for almost all x R n 23) The smallest constant C for which 22) or 23) hold, denoted by A p As an example, we can take γx) = x α, α R Then γ A p, 1 < p <, if and only if n < α < np 1) 17

22 Clearly, ω A 1 if and only if there is a constant C > 0 such that for any cube Q 1 ω x) dx C inf Q ω x) x Q Some properties of A p weights can be found in [23] Let m be a bounded function on R n and consider the multiplier operator T f defined initially for functions f in the Schwartz space SR n ) by FT f) x) = m x) Ff x) Denote by s a real number greater than or equal to 1 and l a positive integer m M s, l) if ) 1 sup R s α n D α m x) s s dx < for all α l R>0 R< α <2R We say Definition 212 Let 1 < p < Given a measurable set E and a weight γ, we denote the space of all functions f : R n C with finite quasi-norm fw f L p R n,ω) = 1 p L p R n ), by L p R n, ω) We need a weighted version of Hörmender s multiplier theorem, see [32] Theorem 213 Let k > [ ] n 2 and m C k R n {0}) Suppose that m M 2, k), ) α 2 sup r 2 α n m x) x dx B, for α k r x 2r When k < n and n < p <, m is a bounded multiplier from k Lp R n, ω) to L p R n, ω) if ω A pn Finally, if k > n, m is a bounded multiplier from Lp R n, ω) to L p R n, ω), k 1 < p <, the norm of the operator depends only on B, p, w, k and n Let B R n ) be the set of p ) P R n ) such that M is bounded on L p ) R n ) Here F will denote a family of ordered pairs of non-negative, measurable functions f, g) Theorem 214 Given a family F Suppose that for some p 0, 0 < p 0 <, and for every weight w A 1 f x) p 0 w x) dx C 0 R n g x) p 0 w x) dx, R n f, g) F 24) 18

23 where C 0 depends only on p 0 and the A 1 constant of w Let p ) P 0 R n ) be such that p 0 < p, and p ) /p 0 ) B R n ), that is the Hardy-Littlewood maximal operator is bounded on L p )/p 0) Then for all f, g) F such that f L p ) R n ), f p ) C g p ), 25) where the constant C is independent of the pair f, g) Proof See [9] Observe that if P log R n ) with p > 1, then p ) /p 0 ) B R n ) for any p 0 < p 23 L p ) boundedness Let us present some results which are useful for us The following lemmas are from [55] and [22] Set and a m) α,β = sup x,ξ ξ β m Dx α D β ξ a x, ξ) ξ = 1 + ξ, ξ R n Lemma 215 Let Q 0 be the cube with center at the origin and edges of length 1 parallel to the coordinates axes in R n Let η D R n ) be identically 1 on Q 0 Let a S 0, a m x, ξ) = η x m) a x, ξ) for m Z n and â m λ, ξ) = e iλε a m x, ξ)dx R n Then, for all α N n and all N N there exists C > 0 depending only on n, η and N such that for all λ, ξ) R 2n D α ξ â m λ, ξ) C sup a 0) α,β ξ α λ N β N Lemma 216 Let a S 0 and K a be the distribution F 1 ξ z a x, ξ)) in S R2n ) Then 1 for each x R n, K a x, ) is a function defined on R n \{0}, 19

24 2 for each N suffi ciently large there exists a constant c, depending only on N and n such that for all z 0, K a x, z) c sup a 0) α,0 z N α N 3 for each x R n and ϕ S R n ), vanishing in a neighborhood of x ax, D)ϕ x) = K a x, x z) ϕ z) dz R n To prove the main result of this chapter we need the weight version of [55, Theorem 97] and [22] Theorem 217 Let a S 0 and p 0 P log Then, there exists N N and a constant C depending only on n, N and p such that ax, D)ϕ L p 0 R n,ω) C for all ϕ S R n ) and any ω A 1 sup a 0) α,β ϕ L p 0 R n,ω) α+β N Proof We write R n as a union of cube Q m, where Q m is the cube with center m Z m and edge of length 1 Let Q m = 3 2 Q m and Q m = 2Q m It follows that Q m Q m Q m and that for some δ > 0 one has x z δ, for all x Q m and z R n \Q m Let now ψ D R n ) be such that 0 ψ 1, suppψ Q 0 and ψ x) = 1 on a neighborhood of Q 0 It follows that ψ m x) = ψ x m) has support contained in Q m and ψ m x) = 1 on a neighborhood of Q m For each ϕ S R n ) we can write ϕ = ϕ 1,m + ϕ 2,m, where ϕ 1,m = ψ m ϕ and ϕ 2,m = 1 ψ m ) ϕ Therefore, It is clear that ax, D)ϕ = ax, D)ϕ 1,m + ax, D)ϕ 2,m ax, D)ϕ p 0 L p 0 w) = ax, D)ϕ x) p 0 w x) dx 26) m Z m Q m ax, D)ϕ 1,m x) p0 w x) dx m Z m Q m + ax, D)ϕ2,m x) p0 w x) dx Q m m Z m 20

25 We will present the proof on three steps: Step 1 Let us estimate Q m ax, D)ϕ 1,m x) p0 w x) dx Let η D R n ) be identically 1 on Q 0 and a m x, ξ) = η x m) a x, ξ) Hence, ax, D)ϕ1,m x) p0 w x) dx ax, D)ϕ1,m x) p0 w x) dx 27) Q m R n Since a m is compactly supported in x we get ax, D)ϕ 1,m x) = e ixλ e ixξ â m λ, D)ˆϕ 1,m ξ) dξdλ = e ixλ â m λ, D) ) ϕ 1,m x) dλ R n R n R n From Lemma 215 we have that for all N N D α ξ â m λ, ξ) C sup a 0) α,β ξ α λ N, β N where C depends only on n, η and N We apply Theorem 213 to f ξ) = â m λ, ξ) with B = C sup a 0) α,β λ N β N, α [n/2]+1 and obtain that there exists a constant Ć, depending on N, n, η and p such that âm λ, D)ϕ L 1,m p 0 w) Ć sup a m 0) α,β λ N ϕ1,m L β N, α [n/2]+1 p 0 w) 28) for all λ R n for all m Z n and ϕ 1,m S R n ) An application of the Minkowski s inequality in integral from leads from 28) to = Thus, choosing N = n + 1 we get am λ, D)ϕ L 1,m p 0 w) { e ixλ â m λ, D) ) ϕ 1,m x) dλ R n R { n p0 w x) dx } 1 p 0 dλ â m λ, D) ) ϕ 1,m x) p0 w x) dx R n R n â m λ, D)ϕ L 1,m p 0 w) dλ R n Ć sup a m 0) α,β λ N dλ ϕ1,m β N, α [n/2]+1 R n L p 0 w) am λ, D)ϕ L 1,m p 0 w) Ć sup a m 0) α,β ϕ1,ml p 0 w), 29) β n+1, α [n/2]+1 } 1 p 0 21

26 valid for all m Z m and ϕ S R n ) Going back to the estimate 29) combined with 27) yields: Q m ax, D)ϕ1,m x) p0 dx, ax, D)ϕ1,m x) p0 w x) dx am λ, D)ϕ 1,m p0 L p 0 w) Q m C p sup where C p does not depend on m Step 2 We now estimate β n+1, α [n/2]+1 Q m ax, D)ϕ2,m x) p0 w x) dx ) 1 p 0 a m 0) α,β Since ϕ 2,m is identically 0 on Q m Q m from Lemma 216 we have R n \Q m ) 1 p 0 ) p0 ϕ1,m p 0 L p 0 w), ax, D)ϕ2,m x) p0 w x) dx Q m ) 1 p 0 = K a x, x z)ϕ 2,m z) dz w x) dx p0 Q m R n Ka x, x z)ϕ 2,m z) ) 1 p0 p 0 w x) dx dz R n Q m C λ,n sup a 0) α,0 x z 2Np 0 ϕ2,m z) ) 1 p0 p 0 w x) dx dz, α 2N Q m valid for 2N > n Let us fix λ n+1 Since x z δ for all x Q m and all z R n \Q m, there exists a constant C λ,n such that and λ+ x z λ+ m z x m x z 2N λ + x z ) 2N C λ,n 210) λ n 2 ) + m z for all x Q m and all z R n \Q m By 210) and 211) we get Q m ax, D)ϕ 2,m x) p0 w x) dx n +1+ m z = µ+ m z 2 211) ) 1 p 0 22

27 can be estimated by c sup a 0) α,0 α 2N R n \Q m c sup a 0) α,0 α 2N c sup a 0) α,0 α 2N R n \Q m R n \Q m Q m λ + x z ) 2Np 0 ϕ2,m z) p0 w x) dx Np 0 µ + x z ) Q m µ + m z ) Np 0 ϕ2,m z) { µ + m z ) N ϕ 2,m z) p 0 w x) dx ) 1 p 0 dz ) 1 p 0 dz Q m µ + x z ) Np 0 w x) dx } 1 p 0 dz We prove that We have µ + x z ) Np 0 w x) dx < c w z), z R n \Q m Q m µ + x z ) Np 0 w x) dx Q m { x z >δ} µ + x z ) Np 0 w x) dx = x z >δ j=0 2 j δ< x z 2 j+1 δ j=0 2 j δ< x z 2 j+1 δ µ + x z ) Np 0 w x) dx x z Np 0 w x) dx Observe that x z Np 0 2 j δ) Np 0, then the last term is bounded by 2 j δ ) Np 0 j=0 x z 2 j+1 δ w x) dx, = c 2 j δ ) n Np 0 1 B z, 2 j+1 δ) j=0 w z) 2 j δ ) n Np 0, z R n \Q m, j=0 since w A 1 Assume that N > n p 0, then valid for all z R n \Q m Hence Q m µ + x z ) Np 0 w x) dx Q m ax, D)ϕ 2,m x) p0 w x) dx C λ,n,p0 sup α 2N Bz,2 j+1 δ) w x) dx ) 1 p 0 < cw z) 1 p 0 a 0) ) p0 α,0 R n \Q m ϕ 2,m z) p 0 w z) dz µ + m z ) Np 0/2 212) 23

28 for all l Z n Step3 A combination of 26) with 29) and 212) yields ax, D)ϕ L p 0 w) C sup a m 0) α,β m Z n β n+1 α [n/2]+1 +C sup a 0) α,0 α 2N m Z n l m ϕ1,m L p 0 w) Q l ϕ2,m z) p 0 w z) dz, µ + m z ) Np 0/2 with λ n + 1 and Np 0 > 2n p 0 1) From the definition of a m and ϕ 1,m get ax, D)ϕ L p 0 w) C sup a m 0) α,β ϕ L 1,m p 0 w) m Z n β n+1 α [n/2]+1 +C sup a 0) α,0 α 2N m Z n l m Q l ϕ2,m z) p 0 and ϕ 2,m we µ + m z ) Np 0/2 w z) dz ) 1 Arguing as in 211) we have that µ + m z 1 + m l when with z Q l with l m Hence ϕ2,m z) p 0 w z) dz Q l µ + m z ) Np 0/2 1 + m l ) Np 0/2 ϕ2,m z) p0 w z) dz m Z n l m Q l 1 + m l ) Np0/2 ϕ2,m z) p0 w z) dz m Z n l Z n Q l m Z n l m m Z n 1 + m ) Np 0/2 l Z n Q l ϕ2,m z) p 0 w z) dz Choosing Np 0 max 2n p 0 1), 2n) and we obtain ax, D)ϕ L p 0 w) C sup β n+1 α [n/2]+1 valid for all ϕ S R n ) This completes the proof From this theorem and Theorem 214 we obtain a m 0) α,β + sup a 0) α,0 ) ϕ L p 0 w) α 2N Theorem 218 Let a S 0 and p P log Then, there exists N N and a constant C such that for all ϕ S R n ) ax, D)ϕ p ) C sup a 0) α,β ϕ p ) α+β N We would like to mention that this theorem was proved in [38] but here we present a another method p 0 24

29 Chapter 3 Boundedness of pseudo-differential operators on variable Besov spaces In this chapter concerns the boundedness properties of the pseudodifferential operators on Besov spaces with variable smoothness and integrability with symbols in SB m δ r, µ, v; N, λ) The chapter is arranged as follows In section 1 we give some key technical lemmas needed in the proofs of the main statements, where we recall the definition of the Besov spaces with variable smoothness and integrability For making the presentation clearer, we give the proof of the main result of this section With the help of the results of Section 1, we prove the boundedness of non-regular pseudodifferential operators in the space B s ) Let { Fϕ j }j be a resolution of unity For a function a : Rn R n C, we write Let 0 < µ, 1 λ, r n µ distributions a S R n R n ) such that a j x, ξ) = F 1 y x Fϕj y) Fa y, ξ) ) a B r µ,v B N λ, ) = { 2 jr a j x, ) B N λ, } and N > n The space λ Br µ,vbλ, N ) consists of all < l v L µ ) Notice that these spaces are just the spaces SB r p, q with r = N, r), p = λ, µ) and q =, v), see [42] for further properties of these function spaces Let m, r, N R, 0 δ 1, 0 < µ, r > n µ and N > n λ We say that a symbol a belongs to SBm δ sup 2 km a x, 2k ) 2k ) Fϕ k L B N < k λ, dx) sup 2 km+δr) a x, 2k ) 2k ) Fϕ k B r <, k µ,v Bλ, N ) j r, µ, v; N, λ) if 25

30 which, introduced by J Marschall [34] and [35] Choosing µ = v = N = λ =, these symbols include the classical Hörmander classes S m 1,δ Moreover the class SBm 0 r, µ, v;, 1) equal the class S B µ,v 1,,1),r ) m of M Yamazaki [53] Notice that SB m δ r, µ, v; N, λ) SB m δ 1 r 1, µ 1, v; N, λ), 31) if 0 < µ < µ 1, 0 < v, r n µ = r 1 n µ 1 and δr = δ 1 r 1, see [35, Lemma 10] A pseudo-differential operator with symbol a SBδ m r, µ; N, λ) is defined by a x, D) fx) = 1 2π) n e ixξ a x, ξ) Ff ξ) dξ, where f SR n ) Besov estimates, with fixed exponents, for such operators were given by J Marschall [35] 31 Auxiliary results In this section we present some results which are useful for us The following lemma is from [31, Lemma 19], see also [14, Lemma 61] Lemma 31 Let α C log loc α Then and let R c logα), where c log α) is the constant from 12) for 2 vαx) η v,m+r x y) c 2 vαy) η v,m x y) with c > 0 independent of x, y R n and v, m N 0 The next lemma often allows us to deal with exponents which are smaller than 1 Lemma 32 Let r > 0, v N 0 and m > n Then there exists c = cr, m, n) > 0 such that for all g S R n ) with suppfg {ξ R n : ξ 2 v+1 }, we have gx) cη v,m g r x)) 1/r, x R n The following lemma is from A Almeida and P Hästö [1, Lemma 47] we use it, since the maximal operator is in general not bounded on l q ) L p ) ), see [1, Example 41]) Lemma 33 Let p P log, q P log 0 with 0 < q q + < and p > 1 For m > n + c log 1/q), there exists c > 0 such that { η v,m f v } v l c {f q ) L p ) ) v} v l q ) L p ) ) 26

31 We will also make use of the following statement were proved by Franke [21, Theorem 241] in the case of constant p, see [18] for variable case Lemma 34 Let p P log 0, l, k N 0 with k l and ϕ SR n ) Then for all {f l } l N0 S R n ) L p ) with suppff l {ξ R n : ξ 2 l }, we have ϕ k f l p ) c 2 k l)n1 1/ min1,p )) fl p ), where ϕ k = 2 kn ϕ2 k ) and c > 0 is independent of k and l The next lemma is a Hardy-type inequality which is easy to prove Lemma 35 Let 0 < a < 1 and 0 < q Let {ε k } k N0 numbers, such that ε k ) k l q = I < The sequence be a sequence of positive real {δ k : δ k = a k j ε j } k N0 j=0 is in l q with {δ k } k l q c I c depends only on a and q The following proposition plays a fundamental role in this section Proposition 36 Let s C log loc, p 1, p 2, q P log 0, 0 < µ and 1 λ with 1 p 1 ) = 1 p 2 ) + 1 µ Let a : Rn R n C be a bounded and measurable symbol such that suppa x, ) {ξ R n : ξ c2 k } If p 1 1 or if 0 < p 1 < 1 and suppff {ξ R n : ξ c2 k }, and if N > n max { 1, 1, } 1 2 λ p + clog s) + c log 1 ), then q 2 2 ks ) δ 1 q ) a x, D) f p1 ) a, 2k ) B N λ, µ 2 ks ) δ 1 q ) f p2 ) for any k N 0 and any δ [ 2 k, k], with the implicit constant not depending on k 27

32 Proof The proof follows the ideas in [35, Proposition 4], see also [33, Lemma 3] We will do the proof in two steps Step 1 Let us begin by the case p 1 1 Let K x, x y) = 1 2π) n e ix y)ξ a x, ξ) dξ = F 1 ξ a x, ) x y), be the kernel of a x, D) Set 1 = max { 1, 1, } 1 2 λ p It follows from the Hölder inequality 2 2 ksx) δ 1 qx) a x, D) f x) = 2 ksx) δ 1 qx) K x, x y) f y) dy 2 ksx) δ 1 qx) K x, x y) Fϕv y x) f y) dy v= v= Observe that K x, x y) Fϕ v y x) dy) 1 x y 2 v+1 2 ksx) δ 1 K x, x y) Fϕ v y x) = F 1 ξ F ξ K x, ) Fϕ v ))) x y) = F 1 ξ F 1 ξ Fϕ v F ξ a x, )) ) x y) qx) f y) dy ) 1 Let us recall the Hausdorff-Young inequality If f L p R n ) with 1 p 2 then Hence by this inequality, since 1 2, 2 ksx) δ 1 qx) a x, D) f x) c F 1 ξ Fϕ v F ξ a x, )) v= Ff p c f p x y 2 v+1 2 ksx) δ 1 qx) ) 1 f y) dy Since s and 1 q are log-hölder continuous and δ [ 2 k, k] we get 2 ksx) 2 kn η k,clog s)x y)2 ksy) k+v+1) c log s) 2 ksy) and δ 1 qx) 2 kn 1 η k,clog 1 )x ) y)δ qy) k+v+1 c log 1 q ) δ 1 qy) q for any x, y R n such that x y 2 v+1 Therefore, 2 ksx) δ 1 qx) a x, D) f x) 28

33 is bounded by c 2 v+k) n +c logs)+c log 1 q )) H v,k x) + c v= k k 1 v= 2 v+k) n Hv,k x), where H v,k x) = F 1 ξ Fϕv+k F ξ a x, 2 k )) M 2 ks ) δ ) 1 q ) f x) The first sum clearly is bounded by c a x, 2k ) B N, M 2 ks ) δ 1 q ) f ) x) The second sum is bounded by c sup i 0 F 1 ξ where we have used the fact that F 1 ξ ϕi F ξ a x, 2 k )) M 2 ks ) δ ) 1 q ) f x) c a x, 2 k ) M 2 ks ) δ 1 q ) f ) x), ϕi F ξ a x, 2 k )) F 1 ϕ i 1 a x, 2k ) a x, 2k ) Therefore, 2 ks ) δ 1 q ) a x, D) f p1 ) a, 2k ) + a, 2k ) B N, ) 2 ks ) δ 1 q ) f u p2 ), since the the maximal function is bounded in L p 2 )/ Our estimate follows by the fact that a x, 2k ) + a x, 2k ) B N, a x, 2k ) B N λ, Step 2 Let now 0 < p 1 < 1 Here we have in addition that suppff {ξ R n : ξ c2 k } Let us recall the Plancherel-Polya-Nikol skij inequality in a form stated in [44, Sect 132] Let 0 < p q, R > 0 and f L p R n ) S R n ) with suppff {ξ R n : ξ R} Then f q can be estimated by c R n1/p 1/q) f p 29

34 Applying this inequality, we get 2 ksx) δ 1 qx) a x, D) f x) 2 kn 1 1) This expression with power is bounded by c2 kn1 ) v= 2 ksx) δ qx) K x, x y) f y) dy) 1/ sup K x, x y) Fϕ v x y) 2 ksx) δ 1 qx) f y) dy y x y <2 v+1 2 kn1 ) 2 c logs)+c log 1 )) q max0,k+v)+vn sup K x, x y) Fϕ v x y) y v= M 2 ks ) δ ) ) 1 q ) f x) As before, K x, x ) Fϕ v x ) F 1 ξ Fϕ v F ξ a x, )) 1 = 2 kn F 1 ξ Fϕv+k F ξ a x, 2 k )) 1 Hence 2 ksx) δ 1 qx) a x, D) f x) a x, 2k ) n B +c log s)+c log 1 q ) 1, M 2 ks ) δ 1 q ) f ) x) Now our estimate follows by Hölder s inequality and the boundedness of the maximal function in L p 2 )/ The next three lemmas are used in the proof of our result, see [35] for the constant exponents Lemma 37 Let A, B > 0, p, q P log 0 and s C log loc such that q+ < Let {f k } k N0 be a sequence of functions such that suppff 0 {ξ R n : ξ A} and Then it holds that: suppff k { ξ R n : B2 k+1 ξ A2 k+1} k=0 f k B s ) { 2 ks ) f k }k l q ) L p ) ) 30

35 Proof Let { Fϕ j }j N 0 be as resolution of unity Using the support properties of Ff k and Fϕ j, the sum ϕ j k=0 f k become κ 2 l= κ 1 ϕ j f j+l for some natural numbers κ 1, κ 2 N 0 Observe that ϕj f j+l ηj,m f j+l t) 1/t for any m > n + c log s) + c log 1/q) and any t > 0 Therefore, with 0 < t < p, k=0 f k B s ) by Lemmas 31 and 33 κ 2 l= κ 1 κ 2 l= κ 1 { η 2 js )t f j,m clogα) j+l t) 1/t } { 2 js ) f j+l }j l q ) L p ) ), j l q ) L p ) ) Lemma 38 Let A > 0, p, q P log 0 and s C log loc such that 0 < q+ < Let s > nmax{1, 1/p } 1) Let {f k } k N0 be a sequence of functions such that suppff k { ξ R n : ξ A2 k+1} Then it holds that k=0 B f k s ) { 2 ks ) f k }k l q ) L p ) ) Proof By the scaling argument, we see that it suffi ces to consider the case { 2 ks ) f k } k l = 1 q ) L p ) ) and show that the modular of the function on the left-hand side is bounded In particular, we will show that 2 js ) ϕ j j=0 k=0 f k q ) p ) q ) Let 0 < r < min 1 q +, p q + ) Using the support properties of Ff k and Fϕ j, the sum ϕ j k=0 f k become k=j κ ϕ j f k for some natural number κ N 0 The left-hand side of the above expression can be rewritten us c j=0 k=j Using the fact that ϕ SR n ) we obtain 2 js ) ϕ j f k rq ) 1 r 1 p ) rq ) 2 js ) ϕj f k 2 js ) η j,m f k ), m > n 32) 31

36 Since s is log-hölder continuous, we move 2 js ) inside the convolution by Lemma 31, 2 js ) η j,m f k ) η j,m1 2 js ) f k, where m 1 = m c log s) Therefore, 32) is bounded by c η j,m1 2 js ) f k j=0 j=0 k=j k=j 2 j k)nd+s )q r Let us prove that 2 nk j)d η j,m1 2 ks ) f k q ) p ) q ) ) rq ) 1 r p ) rq ) 2 nk j)d η j,m1 2 ks ) f k q ) r p ) q ) 2 ks ) f k q ) p ) + 2 j q ) = δ, k j, where d = 1 max{1, 1/p } This is equivalent to δ 1 q ) 2 nk j)d η j,m1 2 ks ) f k 1 p ) Since 1 q is log-hölder continuous and δ [2 j, j ], we can move δ 1 q ) inside the convolution by Lemma 31, c 2 nk j)d δ 1 q ) ηj,m1 2 ks ) f k 2 nk j)d ηj,h δ 1 q ) 2 ks ) f k, p ) with h = m c log s) c log 1/q) We claim that this expression is bounded by c 2 ks ) δ 1 q ) fk p ) The last quasi-norm is less than or equal to one if and only if: 2 ks ) δ 1 q ) q ) fk p ) 1, q ) p ) ) 1 r 33) which follows immediately from the definition of δ Therefore, the right-hand side of 33) is bounded by c j=0 k=j 2 nj k)nd+s )q r 2 ks ) f k q ) p ) + c q ) k=0 1, 2 ks ) f k q ) r p ) q ) ) 1 r + c 32

37 where we have used Lemma 35 since s > nd) Now we prove our claim Here we use the same arguments of [18] If p 1, then convolution with a radially decreasing L 1 -function is bounded on L p ) : η j,h δ 1 q ) 2 ks ) f k η 1 δ 1 q ) j,h 2 ks ) f k p ) p ) δ 1 q ) 2 ks ) f k, p ) by taking m > n + c log s) + c log 1/q) Let 0 < p < 1 By Lemma 32, we have Since 1 q Clog loc, then f k y) ) 1/p η k,l f k p y), L > n, y R n δ 1 qy) c δ 1 qx) j y x ) c log1/q) c δ 1 qx) k y x ) c log1/q) for any j k and any x, y R n Hence, with L 1 = L c log α) c log 1/q), η j,h δ 1 q ) 2 ks ) f k x) η j,h η k,l1 δ p q ) 2 ks )p f k p ) 1/p x) ) = c η j,h x y) η k,l1 δ p 1/p q ) 2 ks )p f k p y) dy, R n where c > 0 is independent of j and k By the inequalities j x y ) h j x z ) h j y z ) h j x z ) h k y z ) h, x, y, z R n, j k, the last expression can be estimated by c 2 jn n/p ) ) p 1/p η j,hp x z) η k,l1 hp y z) δ qz) 2 ksz)p f k z) p dz dy R n R n Therefore, Minkowiski s inequality gives η j,h δ 1 q ) 2 ks ) f k x) 2 j k)n n/p ) ηk,l1 /p h 1 2 j k)n n/p ) η j,hp δ p q ) 2 ks )p f k p x) ) η j,hp δ p 1/p q ) 2 ks )p f k p x) ) 1/p 33

38 for any L > n + h)p and any x R n Since p ) p decreasing L 1 -function is bounded on L p ) p : η j,h δ 1 q ) 2 js ) f k p ) 2 j k)n n/p ) η j,hp p δ q ) 2 ks )p 1/p f k p The proof is completed 2 j k)n n/p ) η j,hp 1/p 1 P log, then convolution with a radially p δ = c2 j k)n n/p ) δ 1 q ) 2 ks ) f k p ) p ) p q ) 2 ks )p f k p 1/p Lemma 39 Let A > 0, p, q P log 0 such that 0 < q + < Let {f k } k N0 be a sequence of functions such that suppff k { ξ R n : ξ A2 k+1} Let α = n max {1, 1/p } 1) Then it holds that, for some constant c > 0, B f k { 2 kα f α k }k l 34) min1,p ) L p ) ) p ), k=0 Moreover if the right-hand side inequality in 34) is finite, then S R n ) to a distribution k=0 f k satisfying this inequality { N p ) p k=0 f k } N converges in Proof The proof follows the ideas in [35, Lemma 3] First we assume the convergence and putting β = min1, p ) Using the support properties of Ff k and Fϕ j, the sum ϕ j k=0 f k become k=j κ ϕ j f k for some natural number κ N 0 Using Lemma 34, we get B f k = sup 2 αj ϕ α j f k p ), j 0 p ) k=0 sup j 0 k=j κ k=j κ 2 ) αβj ϕj f k β 1/β p ) sup 2 αj 2 k j)αβ f k β p ) j 0 k=j κ { 2 kα f k }k l β L p ) ) { N } Let us now prove the convergence of in S R n ) It follows from the result we just proved above, M k= N f k B α p ), k=0 f k N { 2 kα f k } k=m k= N l β L p ) ) ) 1/β 34

39 { N } which tend to zero if N, M and then k=0 f k N hence a convergent sequence in S R n ) is a Cauchy sequence in B α p ), and We remark that Lemmas 37 and 38 are true in B s ) p ), spaces with the same assumptions on p and s 32 Main results The following theorem concerning the continuity of pseudo-differential operator on variable Besov spaces Theorem 310 Let s C log loc, p, q Plog 0 with 0 < q + < Let a SBδ m r, µ, v; N, λ) be such that 0 < µ, v, r > 0, 1 δ) r n µ and 1 λ Let N > n max { 1 2, 1 λ, 1 p } + c log s) + c log 1 q ) i) If { n max 1, 1 µ + 1 } { 1 n 1 δ) r < s s + < r n max p µ 1 } p, 0, + then a x, D) is a continuous linear mapping from B s )+m ii) If 1 δ) r > n µ, v q < and { 1 s := r n max µ 1 } p, 0, + to Bs ) then a x, D) is a continuous linear mapping from B s+m to Bs iii) We suppose that 1 µ + 1 p 1 or 0 < p + 1 and 1 µ + 1 p > 1 If 1 δ) r > n µ, 0 < q + min {1, p } and { s := n max 1, 1 µ + 1 } n 1 δ) r, p then a x, D) is a continuous linear mapping from B s+m to Bs Proof Let {Fϕ k } k be a resolution of unity We set We decompose the symbol into three parts a j,k x, ξ) = F 1 η x Fϕj η) F x η a, ξ) ) Fϕ k ξ) a x, ξ) = a 1) x, ξ) + a 2) x, ξ) + a 3) x, ξ), 35

40 where a 1) x, ξ) = a 2) x, ξ) = a 3) x, ξ) = k 4 a j,k x, ξ) k=4 j=0 k+3 k=0 j=k 3 k=0 j=k+4 a j,k x, ξ) a j,k x, ξ) As in Marschall [35], we need only to estimate a i), i = 1, 2, 3 in B s ) spaces Step 1 Proof of i) Substep 11 We will prove in this step that there is a constant c > 0 such that for every f B s )+m a 1) x, D)f s ) B c f s )+m B 35) Observe that k 4 j=0 a j,kx, D)f k has its spectrum in { ξ R n : c 1 2 k ξ c 2 2 k} Then we can apply Lemma 37 to obtain a 1) x, D)f B s ) c {2 ks ) k 4 } a j,k x, D)f k j=0 l q ) L ), p ) where f k := ϕ k f Let us show that the last quasi-norm is bounded by k c f s )+m B By the scaling argument, we see that it suffi ces to consider the case 2 ks )+m) f k q ) p ) = 1 q ) k=0 and show that the modular of a constant times the function on the left-hand side is bounded In particular, we will show that k=4 2 ks ) k 4 j=0 This clearly follows from the inequality a j,k x, D) f k q ) p ) q ) C 2 ks ) k 4 j=0 a j,k x, D) f k q ) p ) q ) = δ, 2 ks )+m) f k q ) p ) + 2 k q ) 36

41 which we proceed to prove The claim can be reformulated as showing that which is equivalent to δ 1 2 ks ) δ 1 k 4 j=0 k 4 q) 2 ks ) a j,k x, D) f k q ) p ) q ) 1, a j,k x, D) f k 1 p ) j=0 By Proposition 36, the left-hand side is bounded by k 4 c a j,k, 2k ) 2 ks ) B δ 1 q ) fk N j=0 λ, p ) 2 ks )+m) δ 1 q ) fk, p ) where the implicit constants not depending on k and δ Now the right-hand side is less than or equal to one if and only if 2 ks )+m) δ 1 q ) q ) fk p ) 1, q ) which follows immediately from the definition of δ This finish the proof of 35) Substep 12 We will prove in this step that there is a constant c > 0 such that for every f B s )+m a 2) x, D)f s ) B c f s )+m B 36) We have k+3 j=k 3 a j,kx, D)f k has its spectrum in { ξ R n : ξ c 2 2 k} Let 1 p 1 ) = 1 µ + 1 p ) Since 1 δ) r n, we have the Sobolev embedding µ By Lemma 38, a 2) x, D)f s ) B B s )+1 δ)r p 1 ),q ) B s )+1 δ)r n µ {2 ks )+1 δ)r) k+3 Let us show that the last quasi-norm is bounded by j=k 3 c f s )+m B B s ) a j,k x, D)f k } Again by the scaling argument, we see that it suffi ces to consider the case 2 ks )+m) f k q ) p ) = 1 q ) k=0 l q ) L p 1 ) ) k 37

42 and show that the modular of a constant times the function on the left-hand side is bounded In particular, we will show that k=0 2 ks )+1 δ)r) k+3 j=k 3 a j,k x, D)f k q ) p1 ) q ) C This clearly follows from the inequality: 2 ks )+1 δ)r) k+3 j=k 3 a j,k x, D)f k q ) p1 ) q ) 2 ks )+m) f k q ) p ) + 2 k = δ q ) The claim can be reformulated as showing that By Proposition 36, H = δ 1 q ) 2 ks )+1 δ)r) k+3 H 2 k1 δ)r 1 j=k 3 δ 1 q) 2 ks )+m) f k p ) k+3 j=k 3 a j,k x, D)f k p1 ) 1 a j,k, 2k ) B N λ, µ δ 1 q) 2 ks ) f k p ) Substep 13 We will prove in this step that there is a constant c > 0 such that for every f B s )+m We can apply Lemma 37 to obtain a 3) x, D)f B s ) a 3) x, D)f s ) B c f s )+m B 37) c {2 js ) j 4 } a j,k x, D)f k l q ) L ) p ) k=0 Let us show that the last quasi-norm is bounded by j c f s )+m B By the scaling argument, it suffi ces to suppose that f s )+m B 1 and we will show that j=4 j 4 2 js ) k=0 a j,k x, D) f k q ) p ) q ) 1 38) 38

43 We set µ 1 = maxµ, p + ) Let r 1 > 0 and δ 1 > 0 be such that r n µ = r 1 n µ 1 and δr = δ 1 r 1 Let 0 < σ < min 1 q +, p q + ) The quasi-norm on the right-hand side with power σ is bounded by j 4 2 js ) q ) σ a j,k x, D) f k k=0 Let us prove that 2 ks )+j k)r 1 q ) a j,k x, D) f k p ) q ) p ) q ) 2 ks )+m+1 δ 1)r 1 ) f k q ) p2 + 2 k ) q ) = δ, where 1 = p ) µ 1 p 2 The claim can be reformulated as showing that ) δ 1 2 ks )+j k)r 1 q ) a j,k x, D) f k p ) 1, q ) which is equivalent to δ 1 q) 2 ks )+j k)r 1 a j,k x, D) f k p ) 1 Observe that f k and a j,k x, D) have its spectrum in { ξ R n : ξ c 2 2 k}, by Proposition 36 and the Sobolev embedding 31), the left-hand side is bounded by c2 jr 1 a j,k, 2k ) δ B N λ, µ1 1 q ) 2 ks ) kr 1 f k p2 ) δ 1 q ) 2 ks )+m 1 δ 1 )r 1 ) f k p2 ), where the implicit constants not depending on k and δ Now the right-hand side is less than or equal to one if and only if 2 ks )+m 1 δ 1)r 1 ) δ 1 q ) q ) fk p2 1, ) q ) which follows immediately from the definition of δ Our estimate 38) follows by Lemma 35 and the embeddings B s )+m Bs )+m 1 δ)r+ n µ B s )+m 1 δ 1)r 1 p 2 ),q ) Step 2 Proof of ii) We need only to estimate a 3) x, D) Let µ 1, r 1 and δ 1 be as in Substep 13 Let ϱ = min1, p, q ) If v q <, then a 3) x, D)f B B s = a j,k x, D) f k p ),q s p ),q k=0 j=k+4 k=0 j=k+4 a j,k x, D) f k ϱ B s p ),q ) 1 ϱ 39

44 Using Lemma 37 and Proposition 36 we get q a j,k x, D) f k j=k+4 B s p ),q j=k+4 f k q 2 js a j,k x, D) f k q p 2 ) j=k+4 Since a SB m δ r 1, µ 1, v; N, λ), v q < and s = r 1, Therefore, since 1 δ) r > n µ j=k+4 p ) 2 jsq aj,k, 2k ) B N λ, q 2 jsq aj,k, 2k ) q B 2 km+δ 1r 1 )q N λ, µ 1 a 3) x, D)f B s f m+δ B 1 r 1 p ),q p 2 ),ϱ = f s+m+ n µ 1 δ)r µ n B 1 p 2 ),ϱ f s+m+ n B µ 1 δ)r p ),ϱ f B s+m Step 3 Proof of iii) We need only to estimate a 2) x, D) Let us consider two cases 1 Case Let = µ p p 1 Then we have ) µ p ), µ 1 B 0 p 1 ), B s+1 δ)r n µ p ), B s Applying Lemma 39, we obtain a 2) x, D)f B s a 2) x, D)f B 0 p1 ), By Proposition 36 we get k+3 j=k 3 a j,k x, D) f k p1 ) k+3 k=0 j=k 3 k+3 j=k 3 a j,k x, D) f k p1 ) a j,k, 2k ) u B f N k p ) λ, 2 km 1 δ)r) f k p ) Therefore, a 2) x, D)f B s f B m 1 δ)r p ),1 f B s+m, 40

45 since s = 1 δ)r Case 2 1 µ + 1 p > 1 and 0 < p + 1 Let µ if p + µ µ 1 = max1, µ) if p + > µ Obviously B n p n p ), = n Bs+1 δ)r µ p ), B s Let r 1 > 0 and δ 1 > 0 be such that r n µ = r 1 n µ 1 and δr = δ 1 r 1 Let 1 p ) = 1 µ p 2 ) Then, a SB m δ r 1, µ 1, v; N, λ), p 2 ) p ) and s Applying again Lemma 39, we obtain n p ) = n p n 1 δ 1)r 1 n p 2 ) a 2) x, D)f B s a 2) x, D)f 2 kn1 p ) n p B n p ), k+3 k=0 j=k 3 m+ n p B n 1 δ 1 )r 1, p 2 ),p f by Proposition 36 Our estimate follows by the Sobolev embedding B s+m n p Bm+ n 1 δ 1)r 1 p 2 ),p ) 1/p a j,k x, D) f k p p ) This finish the proof 41

46 Chapter 4 Boundedness of pseudo-differential operators on variable Triebel-Lizorkin spaces This chapter concerns the boundedness properties of the pseudodifferential operators on Triebel-Lizorkin spaces with variable smoothness and integrability with symbols in the class SB m δ r, µ, v; N, λ) Let { Fϕ j }j N 0 be as resolution of unity For a function a : R n R n C, we write a j x, ξ) = F 1 y x Fϕj y) Fa y, ξ) ) Let 0 < µ <, 0 < v, 1 λ, 0 δ 1, 1 δ)r n and N > n The space µ λ F r µ,vb N λ, ) consists of all distributions a S R n R n ) such that a F r µ,v B N λ, ) = { 2 jr a j x, ) B N λ, } < L µ l v ) Let m, r, N R, 0 δ 1, 0 < µ, v, 1 λ, 1 δ)r n and N > n We say µ λ that a symbol a belongs to SF m δ r, µ, v; N, λ) if sup 2 km a x, 2k ) 2k ) Fϕ k L B N < k λ, dx) sup 2 km+δr) a x, 2k ) 2k ) Fϕ k F r <, k µ,v Bλ, N ) which, introduced by J Marschall [34] Notice that j SB m δ r, µ, p; N, λ) SF m δ 1 r 1, p, q; N, λ), 41) 42