Integral Operators of Harmonic Analysis in Local Morrey-Lorentz Spaces

Transcription

1 Integral Operators of Harmonic Analysis in Local Morrey-Lorentz Spaces Ayhan SERBETCI Ankara University, Department of Mathematics, Ankara, Turkey Dedicated to 60th birthday of professor Vagif S. GULİYEV July 10-13, KIRSEHIR/TURKEY

2 Introduction The theory of boundedness of classical operators of harmonic analysis from one weighted Lebesgue space to another one is by now well studied. These results have good applications in the theory of partial differential euations. However, it should be noted that in the theory of partial differential euations, along with weighted Lebesgue spaces, general Morrey-type spaces also play an important role. In a series of papers by Guliyev, Aykol, Kucukaslan and Serbetci (2013, 2016, 2016 and 2017) the local Morrey-Lorentz spaces M loc p,;λ (Rn ) have been introduced and the basic properties of these spaces have been given, and the boundedness of the Hilbert transform H, the Hardy-Littlewood maximal operator M and the Calderón-Zygmund operators T, and Riesz potential I α on the local Morrey-Lorentz spaces M loc p,;λ (Rn ) has been extensively studied, respectively.

3 Introduction This talk is dedicated to the results obtained by the author jointly with his co-authors V.S. Guliyev, C. Aykol, A. Kucukaslan. We will give the basic properties of the local Morrey-Lorentz spaces M loc p,;λ (Rn ) and the boundedness of classical operators of harmonic analysis, such as the Hardy-Littlewood maximal operator M, Hilbert transform H, the Calderón-Zygmund operators T, and Riesz potential I α will be given on the local Morrey-Lorentz spaces Mp,;λ loc (Rn ). Local Morrey-Lorentz spaces M loc p,;λ (Rn ) are generalizations of Lorentz spaces L p, (R n ) such that for the case λ = 0 we get the space L p, (R n ).

4 Introduction Some type of Morrey-Lorentz spaces were studied by some authors (see Mingione 2010, Ho 2014 and Ragusa 2012). Mingione [18] defined the Lorentz-Morrey spaces L p,;λ (R n ) by following: Definition (Mingione 2010) The Lorentz-Morrey spaces L p,;λ (R n ) is the set of all measurable functions f on R n : for 1 p <, 0 < <, and 0 λ n, f L p,;λ (R n ) if and only if f Lp,;λ = sup r λ p χ B(x,r) f Lp, <. x R n, r>0

5 Introduction Mingione studied the boundedness of the restricted fractional maximal operator M β,b : M β,b f (x) = sup B(x, r) β n 1 f (y) dy, x R n B(x,r) B B(x,r) in the restricted Lorentz-Morrey spaces L p,;λ (B), where B is any ball. Mingione derives a general non-linear version, extending a priori estimates and regularity results for possibly degenerate non-linear elliptic problems to the various spaces of Lorentz and Lorentz-Morrey type considered in (Adams, 1981, 1982 and 1996).

6 Introduction Ragusa [21] defined the Lorentz-Morrey spaces L p,;λ (R n ) and studied some embeddings between these spaces. Definition (Ragusa, 2012) The Lorentz-Morrey spaces L p,;λ (R n ) is the set of all measurable functions f on R n : for 1 p <, 0 < <, and 0 λ n, if and only if f Lp,;λ = sup r λ χ B(x,r) f Lp, <. x R n, r>0 Note that the spaces L p,;λ (R n ) and L p,;λ p (Rn ) defined by Mingione and Ragusa respectively, coincide, thus L p,;λ (R n ) = L p,;λ p (Rn ).

7 Preliminaries We shall use the following notation. For a Lebesgue measurable set E R n and 0 < p, L p (E) is the standard Lebesgue space of all functions f Lebesgue measurable on E for which if 0 < p < and ( f Lp(E) := E ) 1 f (y) p p dy <, f L (E) := sup{α : {y E : f (y) α} > 0} <, if p =. Also, for an open set E R n, L loc p (E) is the set of all functions f such that f L p (K) for any compact K E. If E = R n, then, for brevity, we write L p for L p (R n ) and L loc p L loc p (R n ). for

8 Preliminaries The same convention refers to the case of weak Lebesgue spaces WL p (E), the space of all functions f Lebesgue measurable on E for which f WLp(E) := sup t 1/p f (t), 0<t E 1 p < and f WL f L, p =. Here E is the Lebesgue measure of E, and f denotes the right continuous non-increasing rearrangement of f : where f (t) := inf {α > 0 : µ f (α) t}, t (0, ), is the distribution function of f. µ f (α) := {y R : f (y) > α}

9 Preliminaries In the following we give the local Morrey spaces LM p,λ (0, ) (see, Alvarez, Lakey, Guzman-Partida 2000, N. Samko ). Definition Let 1 p < and 0 λ 1. We denote by LM p,λ LM p,λ (0, ) the local Morrey space, the space of all functions ϕ L loc p (0, ) with finite uasinorm ϕ LMp,λ = sup r λ p ϕ Lp(0,r). r>0 Also by WLM p,λ WLM p,λ (0, ) we denote the weak local Morrey space of all functions ϕ WL loc p (0, ) for which ϕ WLMp,λ = sup r λ p ϕ WLp(0,r) <. r>0

10 Preliminaries Lorentz spaces are introduced by Lorentz in These spaces are uasi-banach spaces and generalizations of the more familiar L p spaces, also they are appear to be useful in the general interpolation theory. Definition The Lorentz space L p, L p, (R n ), 0 < p,, is defined as the set of all measurable functions f on R n with finite uasi-norm f Lp, := τ 1 p 1 f (τ) L(0, ). The functional Lp, is a norm if and only if either 1 p or p = =. If p = =, then the space L, (R n ) is denoted by L (R n ). Clearly L p,p L p and L p, WL p.

11 Preliminaries We denote by M(R n ) be the set of all extended real valued measurable functions on R n and M + (0, ) the set of all non-negative measurable functions on (0, ). Definition Let 0 < p and ψ M + (0, ). We denote by Λ p,ψ (R n ) the classical Lorentz spaces, the spaces of all measurable functions with finite uasinorm Λ p,ψ (R n ) := {f M(R n ) : f Λp,ψ := ψf Lp(0, )}. Therefore, for ψ(t) = t 1 p 1, 0 < p, we get the spaces Λ p,t 1 p 1 (Rn ) are eual with norm coincide to the Lorentz spaces L p, (R n ).

12 Definition Let f L loc 1 (Rn ). The space of functions with bounded mean oscillation, BMO BMO(R n ), consists of those functions f for which 1 f BMO := sup f f B dx B R n B B is finite, where the supremum is taken over open balls B R n and f B = 1 fdx. B B

13 Local Morrey-Lorentz type spaces Definition (Aykol, Guliyev, Serbetci, 2013) Let 0 < p, and 0 λ 1. We denote by Mp,;λ loc Mloc p,;λ (Rn ) the local Morrey-Lorentz spaces, the spaces of all measurable functions with finite uasinorm f M loc p,;λ := sup r λ t 1 p 1 f (t) L(0,r). r>0 If λ < 0 or λ > 1, then M loc p,;λ (Rn ) = Θ, where Θ is the set of all functions euivalent to 0 on R n.

14 Basic properties of local Morrey-Lorentz type spaces Lemma 1) In the limiting case λ = 0 the space M loc p,;0 (Rn ) is the Lorentz space L p, (R n ). That is M loc p,;0 (Rn ) = L p, (R n ), 0 < p, and f M loc = f p,;0 Lp, τ 1 p 1 f (τ) L(0, ). 2) In the limiting case λ = 1 the space M loc p,;1 (Rn ) is the classical Lorentz space Λ,t 1 p 1 (Rn ). That is M loc p,;1(r n ) = Λ,t 1 p 1 (Rn ), 0 < p, and f M loc p,;1 = f Λ 1,t p 1 τ 1 p 1 f (τ) L (0, ).

15 Basic properties of local Morrey-Lorentz type spaces Lemma If f be a measurable function, then f M loc is a uasi-norm on p,;λ Mp,;λ loc (Rn ). If 1 p or p = =, then the functional f M loc is a norm. p,;λ

16 Basic properties of local Morrey-Lorentz type spaces Lemma If p =, then we have M,;λ loc = Θ for any 0 < <. Proof. Assume that M,;λ loc Θ. Then there exists a nonzero function f M,;λ loc which means that there exists c > 0 and a positive measurable set A such that f (x) c for all x A. Then f M loc,;λ = sup r λ t 1 f (t) L(0,r) r>0 sup r λ t 1 (f χ A ) (t) L(0,r) r>0 c sup r λ t 1 L(0,min{ A,r}) =. r>0

17 Basic properties of local Morrey-Lorentz type spaces Lemma Let 0 < p <, 1 r = 1 p λ and 0 < λ p. Then and ( p ) 1 f WLr M loc p,;λ(r n ) WL r (R n ) f M loc λ 1 f WLr, f WL r (R n ). p,;λ Corollary Let 0 < p <. Then M loc p,; p (R n ) L (R n ).

18 Basic properties of local Morrey-Lorentz type spaces Lemma Let 0 < p, < and 0 λ 1. Then and for f M loc p,;λ (Rn ) M loc p,;λ(r n ) L p,;nλ (R n ) f Lp,;nλ f M loc. p,;λ

19 Hardy-Littlewood Maximal Operator For x R n and r > 0, we denote by B(x, r) the open ball centered at x of radius r, and by B(x, r) c denote its complement. Let B(x, r) be the Lebesgue measure of the ball B(x, r). Therefore B(x, r) = ω n r n, ω n denotes the volume of unit sphere S n 1 in R n. For f L loc 1 (Rn ), the Hardy-Littlewood maximal function Mf of f is defined by 1 Mf (x) = sup f (y) dy, x R n. r>0 B(x, r) B(x,r)

20 Hardy-Littlewood Maximal Operator It is well known that for the classical Hardy-Littlewood maximal operator the rearrangement ineuality cf (t) (Mf ) (t) Cf (t), t (0, ) (1) holds (Bennett and Sharpley, 1988), where the positive constants c, C are independent of f and t, and f (t) = 1 t t 0 f (s)ds.

21 We will use the following Hardy operator to obtain the boundedness of the maximal operator M, Hilbert operator H, Calderon-Zygmund operator T and maximal Calderon-Zygmund operator T in the local Morrey-Lorentz spaces M loc p,;λ. Definition (N. Samko, 2012) Let f be a measurable function on (0, ) and β be a real number. The weighted Hardy operators A β and A β with power weights acting on f are defined by A β f (t) = t β 1 t 0 f (s) s β ds, A βf (t) = t β f (s) ds. t sβ+1

22 The following theorem was proved by N. Samko, Theorem Theorem A. Let β R, 0 λ < 1 and 1 <. If β < λ + 1 and β > λ 1, then the operators A β and A β are bounded on the local Morrey space LM,λ (0, ), respectively. The following theorem was proved by Aykol, Guliyev, Kucukaslan, Serbetci, Theorem Theorem B. Let β R, 0 λ < 1 and 1 < <. If β = λ + 1 and β = λ 1, then the operators A β and A β are bounded from the lokal Morrey space LM,λ (0, ) to the weak local Morrey space WLM,λ (0, ), respectively.

23 Hardy-Littlewood Maximal Operator In the following theorem we prove the boundedness of maximal operator M on the local Morrey-Lorentz spaces M loc p,;λ (Rn ). Theorem (Guliyev, Aykol, Kucukaslan, Serbetci, 2016) Let 1, 0 λ < 1 and +λ p. (i) If +λ < p <, then the maximal operator M is bounded on the local Morrey-Lorentz space Mp,;λ loc. (ii) If p = +λ, then the operator M is bounded from Mloc p,;λ to the weak local Morrey- Lorentz space WMp,;λ loc. (iii) If p = =, then the operator M is bounded on L (R n ).

24 Proof. (i) Suppose +λ < p < and f Mloc p,;λ. From the definition in local Morrey-Lorentz spaces and ineuality (1) we get Mf M loc p,;λ = sup r λ t 1 p 1 f (t) L(0,r) r>0 C sup r λ t 1 p 1 t 1 r>0 = C A ( 1 p 1 )g LM,λ, (0, ) 0 f (s)ds L(0,r) where g(t) = t 1 p 1 f (t). Since 1 p λ < 1, for β = 1 p 1 the ineuality β < λ + 1 holds. From the boundedness of Hardy operator we get A ( 1 p 1 )g LM,λ (0, ) C g LM,λ (0, ) = C f M loc. p,;λ Therefore we obtain the boundedness of M in M loc p,;λ.

25 Proof. (ii) For the limiting case p = Mf WM loc C sup +λ,;λ r>0 +λ suppose f Mloc p,;λ. We have r λ t λ 1 t 0 = C A β h WLM,λ (0, ), f (s)ds where β = 1 + λ 1 λ 1 1+ and h(t) = t f (t). From the boundedness of Hardy operator we get WL(0,t) A β h WLM,λ (0, ) C h LM,λ (0, ) = C f M loc +λ,;λ. Then we obtain the boundedness of the operator M from the space M loc +λ,;λ to the weak space WMloc,;λ. +λ

26 Proof. (iii) In the limiting case p =, suppose f M,;λ loc. Since the space M,;λ loc is trivial for any 0 < <, we must consider the case =. Since the operator M is bounded on L we get the statement.

27 Hilbert Transform Let f be a locally integrable function on R. The Hilbert transform Hf of f is defined by the principal-value integral Hf (x) = p.v. 1 π f (y) x y dy provided it exists almost everywhere. The modified Hilbert transform H is defined as [ 1 Hf (x) := lim ε 0 + x y + χ(y) ] f (y)dy, y x y >ε where χ(y) is the characteristic function of y > 1. The boundedness of the modified Hilbert transform H from the space L to the BMO space was proved by B. Muckenhoupt and R.L. Wheeden in 1976.

28 Hilbert transform For each measurable function f on (0, ) and each t > 0, the following operator (Sf )(t) = = 1 t min ( 1, s 0 t t 0 f (s)ds + ) ds f (s) s t f (s) ds s was defined by A.P. Calderon (Calderon 1966). It is clear that S is linear. For the aim, its importance based on the fact that it dominates the maximal Hilbert transform.

29 Theorem Theorem C. (Bennett, Sharpley, 1988) Let f L loc 1 (R) and suppose Then ( Sf ) (1) = 1 0 f (s)ds + f (s) ds 1 s <. (2) ( Hf ) (t) CS ( f ) (t), 0 < t <, (3) where C is a constant independent of f and t.

30 Hilbert transform Theorem Theorem D. Let f L loc 1 (R) and f satisfies (2). Then the Hilbert transform Hf (x), x R exists almost everywhere. Furthermore, ( Hf ) (t) CS ( f ) (t), 0 < t <, (4) where C is a constant independent of f and t.

31 Hilbert transform The following is a well known theorem about Hilbert transform. Theorem Theorem E. Let 0 < and 1 p. (i) If 1 < p < 1, then the operator H is bounded in the Lorentz space L p,. (ii) If p = 1 and 0 < 1, then the operator H is bounded from L 1, to the space WL 1. (iii) If p =, then the modified Hilbert operator H is bounded from L to BMO.

32 Hilbert transform The following theorem is the main result of our presentation, in which we get the analogue of Theorem E for the boundedness of the Hilbert transform in the local Morrey-Lorentz spaces M loc p,;λ. Theorem (Aykol, Guliyev, Kucukaslan, Serbetci, 2016) Suppose that f Mp,;λ loc (R) satisfies the euation (2), 0 <, 0 λ < 1 and +λ p λ. Then the Hilbert transform Hf exists almost everywhere. Furthermore, (i) If +λ < p < λ, 1 <, then the operator H is bounded in the local Morrey-Lorentz space Mp,;λ loc. (ii) If p =, 1 < <, then the operator H is bounded from M loc p,;λ +λ to the weak space WMloc p,;λ. (iii) If p = λ, 0 <, then the modified Hilbert operator H is bounded from Mp,;λ loc to BMO.

33 Proof. Since f satisfies (2), from Theorem D the Hilbert transform Hf (x), x R exists almost everywhere. (i) Suppose that 1 <, 0 λ < 1, +λ < p < λ and f Mp,;λ loc. We have Hf M loc p,,λ C sup r λ t 1 p 1 C r>0 ( sup r>0 = I 1 + I 2. = sup r λ t 1 p 1 (Hf ) (t) L(0,r) r>0 ( 1 t f (s)ds + t 0 t r λ t 1 p 1 1 t 0 f ) (s) ds s L(0,r) f (s)ds L(0,r) + sup r λ t 1 p 1 r>0 t f (s) ds s L

34 Proof. Let us estimate I 1 : I 1 = C sup r λ t 1 p 1 t 1 f (s)ds L(0,r) r>0 0 = C A ( 1 p 1 )g LM,λ (0, ), where g(t) = t 1 p 1 f (t). Since 1 p λ < 1, for β = 1 p 1 the ineuality β < λ + 1 holds. From Theorem A the operator A β is bounded in the local Morrey spaces LM,λ (0, ). Then, I 1 A ( 1 p 1 )g L,λ (0, ) g L,λ (0, ) = sup r λ t 1 p 1 f (t) L(0,r) = f M loc. r>0 p,;λ (5)

35 Proof. Now we consider I 2 : I 2 = C sup r λ t 1 p 1 r>0 t = C A ( 1 p 1 )g LM,λ (0, ), f (s) ds s L(0,r) where, again, g(t) = t 1 p 1 f (t). Since 1 p λ > 0, for β = 1 p 1 the ineuality β > λ 1 holds. From Theorem B the operator A β is bounded in the local Morrey spaces LM,λ (0, ). Then, I 2 A ( 1 p 1 )g LM,λ (0, ) g LM,λ (0, ) = sup r λ t 1 p 1 f (t) L(0,r) = f M loc. r>0 p,;λ (6) From the ineualities (5) and (6) we obtain the boundedness of the operator H in M loc p,;λ.

36 Proof. (ii) For the limiting case p = +λ, 1 < <, suppose f Mp,;λ loc. From the definition of norm in weak local Morrey-Lorentz spaces and by using the ineuality (4) and Minkowski s ineuality we get Hf WM loc = sup r λ t +λ 1 (Hf ) (t) WL(0,r) +λ,;λ r>0 = sup r λ t r>0 C sup r λ λ 1 1+ t r>0 C sup r λ t λ 1 r>0 1+ λ 1 (Hf ) (t) WL(0,r) t + C sup r λ λ 1 1+ t r>0 ( 1 t 0 t 0 f (s)ds + t f (s)ds WL(0,r) t f ) (s) ds s WL(0,r) f (s) ds s WL(0,r) = N 1 + N 2.

37 Proof. Let us estimate N 1 : N 1 = C sup r λ t λ 1 t f (s)ds WL(0,r) r>0 0 = C A β h WLM,λ (0, ), where β = 1 + λ 1 from Theorem B and h(t) = t 1+ λ 1 f (t). Therefore we get N 1 A β h WLM,λ (0, ) h LM,λ (0, ) = sup r λ λ 1 1+ t f (t) L(0,r) r>0 = f M loc. (7) +λ,;λ

38 Proof. Now we consider N 2 : N 2 = C sup r λ λ 1 1+ t r>0 t = C A β h WL,λ (0, ), f (s) ds s WL(0,r) where, again, β = 1 + λ 1 λ 1 1+ and h(t) = t f (t). From Lemma 2 (ii) the operator A β is bounded from the Morrey spaces L,λ (0, ) to WL,λ (0, ). Then, N 2 A β h WL,λ (0, ) h L,λ (0, ) = sup r λ λ 1 1+ t f (t) L(0,r) = f M loc. r>0 +λ,;λ (8) From the ineualities (7) and (8) we obtain the boundedness of the operator H from M loc +λ,;λ to WMloc,;λ. +λ

39 Proof. (iii) For the limiting case p = λ, 0 <, it will be convenient to use the modified Hilbert transform H instead of H. The reason for using Hf is that it may exist while Hf may not exist (Dynkin, 1991). Since H is bounded from L to BMO, the ineuality Hf BMO C f L f WL holds (Muckenhoupt, Wheeden, 1976; Dynkin, 1991). Then we get Hf BMO C f M loc λ,;λ, which proves that the modified Hilbert transform H is bounded from M loc to BMO.,;λ λ

40 Calderon-Zygmund operator Suppose that K L loc 1 (Rn \ {0}) and satisfies the following conditions: (i) K(x) C x n, x Rn \ {0}, (ii) K(x)dx = 0, 0 < r 1 < r 2, r 1 < x <r 2 (iii) K(x y) K(x) C y x n+1 for 2 y x. Then K is called the Calderon-Zygmund kernel, where C is a constant independent of x and y. Set T ε f (x) = K(x y)f (y)dy. B(x,ε) c

41 We define the Calderon- Zygmund singular integral associated to K as Tf (x) = (K f )(x) = lim ε 0 T ε f (x). and the maximal singular integral by T f (x) = sup T ε f (x). ɛ>0

42 Calderon-Zygmund operator The Calderon-Zygmund operator T extends to the whole space L p, 1 p <, by continuity. In the case p = we need a renormalization of T (see [Dyn kin]). For this reason let us choose a point x 0 R n and let f L. Set T 0 f (x) = T (f χ 2B )(x) T (f χ 2B )(x 0 )+ [K(x y) K(x 0 y)]f (y)d B(x,r) where x 0 B(x, r). If f L p (R n ), p <, then obviously T 0 f (x) = T (f )(x) T (f )(x 0 ).

43 Calderon-Zygmund operator Theorem Theorem F. (Bennett, Sharpley, 1988) If f L loc 1 (R) and suppose the euation (2) holds, then ( T f ) (t) CS ( f ) (t), 0 < t <, (9) where C is a constant independent of f and t.

44 Calderon-Zygmund operator Theorem Theorem G. Let f L loc 1 (R) and f satisfies (2). Then the Calderon-Zygmund operator T exists almost everywhere. Furthermore, ( Tf ) (t) CS ( f ) (t), 0 < t <, (10) where C is a constant independent of f and t.

45 Calderon-Zygmund operator The following theorem is the main result of our presentation, in which we get the boundedness of the Calderon-Zygmund operator in the local Morrey-Lorentz spaces M loc p,;λ. Theorem Suppose that f Mp,;λ loc, 1, 0 λ < 1, +λ p λ and the ineuality (2) holds, then the Calderon-Zygmund integral Tf (x) exists almost everywhere. Furthermore, (i) If 1 <, +λ < p < λ, then the Calderon-Zygmund operator T is bounded on the local Morrey-Lorentz space Mp,;λ loc. (ii) If 1 < <, p =, then the operator T is bounded from M loc p,;λ +λ to the weak local Morrey-Lorentz space WMloc p,;λ. (iii) If 1, p = λ, then the operator T 0 is bounded from to BMO. M loc p,;λ

46 Proof. Let 1, 0 λ < 1 and +λ p λ. Since f satisfies (2), by Theorem B the Calderon Zygmund operator Tf exists almost everywhere. (i) Suppose that 1 <, 0 λ < 1, +λ < p < λ and f Mp,;λ loc. From the definition of norm in local Morrey-Lorentz spaces, by using the ineuality (10) and Minkowski s ineuality we get

47 Proof. Tf M loc p,,λ C sup C = sup r λ t 1 p 1 (Tf ) (t) L(0,r) r>0 r λ r>0 ( sup r λ r>0 = I 1 + I 2. ( t 1 p 1 1 t t 0 t 1 p 1 t 1 f (s)ds 0 f (s)ds + t f (s) ds) s L(0,r) + sup r>0 L (0,r) r λ t 1 p 1 t I 1 can be estimated using the same method as in the proof of the boundedness of the maximal operator on M loc p,;λ. f (s) s ds

48 Proof. Let us estimate I 2 : I 2 = C sup r>0 r λ t 1 p 1 t f (s) s ds = C A ( 1 L(0,r) p 1 )g LM,λ (0, ), where g(t) = t 1 p 1 f (t). Since 1 p λ > 0, for β = 1 p 1 the ineuality β > λ 1 holds. By Theorem C we get A ( 1 p 1 )g LM,λ (0, ) C g LM,λ (0, ) = C sup r λ t 1 p 1 f (t) L(0,r) = C f M loc. r>0 p,;λ Therefore we get I 2 C f M loc. Conseuently we obtain the p,;λ boundedness of T in Mp,;λ loc.

49 Proof. (ii) For the limiting case p = +λ, 1 < <, suppose f Mp,;λ loc. From the definition of norm in weak local Morrey-Lorentz spaces and by using the ineuality (10) and Minkowski s ineuality we get Tf WM loc = sup r λ t +λ 1 (Tf ) (t) WL(0,r) +λ,;λ r>0 C sup r>0 + C sup r>0 r λ t λ 1 t 0 r λ λ 1 1+ t t f (s)ds f (s) s WL(0,r) ds = N 1 + N 2. WL(0,r) N 1 can be estimated using the same method as in the proof of the weak boundedness of the maximal operator on M loc p,;λ.

50 Proof. Let us estimate N 2 : N 2 = C sup r>0 r λ t 1+ λ 1 = C A β h WLM,λ (0, ), t f (s) s ds WL(0,r) where β = 1 + λ 1 λ 1 1+ and h(t) = t f (t). From Lemma 2 the operator A β is bounded from the Morrey spaces LM,λ (0, ) to WLM,λ (0, ). Then, N 2 C A β h WLM,λ (0, ) C h LM,λ (0, ) = C sup r λ λ 1 1+ t f (t) L(0,r) = C f M loc. r>0 +λ,;λ Then we obtain the boundedness of the operator T from M loc +λ,;λ to WM loc +λ,;λ.

51 Proof. (iii) For the limiting case p = λ, 1 and 0 λ < 1, suppose f M loc p,;λ. Since the operator T 0 is bounded from L to BMO, then the ineuality T 0 f BMO C f M loc λ,;λ = C f L holds ( Dyn kin, 1991) which proves that Calderon-Zygmund operator T 0 is bounded from M loc to BMO.,;λ λ

52 Theorem Suppose that f Mp,;λ loc, 1, 0 λ < 1, +λ p λ and the ineuality (2) holds, then the maximal Calderon-Zygmund integral T f (x) is finite almost everywhere. Furthermore, (i) If 1 <, +λ < p < λ, then the operator T is bounded in the local Morrey-Lorentz space Mp,;λ loc. (ii) If 1 < <, p =, then the operator T is bounded from Mp,;λ loc +λ to the weak local Morrey- Lorentz space WMloc p,;λ. (iii) If 1, p = λ, then the operator T is bounded from to BMO. M loc p,;λ

53 Proof. Let 1, 0 λ < 1 and +λ p λ. Since f satisfies (2), by Theorem C the maximal Calderon-Zygmund operator T f (x) is finite almost everywhere. The proof of the statements (i) and (ii) of this theorem can be easily obtained from the ineuality (9) and using the same method in the proof the boundedness of the Calderon-Zygmund operator T.

54 Proof. (iii) For the limiting case 1, p = λ and 0 λ < 1, suppose f Mp,;λ loc. Since the operator T is bounded from L to BMO and M loc λ,;λ L, the ineuality T f BMO C f M loc λ,;λ holds (see Bennett, De Vore and Sharpley, 1980) which proves that the operator T is bounded from M loc to BMO.,;λ λ Thus the proof of the theorem is completed.

55 Riesz Potential In this presentation we talk about the boundedness of the Riesz potential I α defined by I α f (x) = R n f (y) dy, 0 < α < n, f Lloc x y n α 1 (R n ), in the local Morrey-Lorentz spaces Mp,;λ loc (Rn ). We will find the necessary and sufficient conditions for the boundedness of I α in the spaces Mp,;λ loc (Rn ). We will apply these results to the boundedness of particular operators such as the fractional maximal operator, fractional Marcinkiewicz operator and fractional powers of some analytic semigroups on Mp,;λ loc (Rn ).

56 Riesz Potential For each measurable function f on (0, ) and each t > 0, the following operator t (S α f )(t) = t α n 1 f (s)ds + s α n 1 f (s)ds 0 t was defined by A.P. Calderon (Calderon 1966). Theorem Theorem H. (Oneil 1963, Sawyer 1990) If the condition ( Sα f ) (1) = 1 0 f (s)ds + s α n 1 f (s)ds <, (11) 1 holds for f L loc 1 (Rn ), then the Riesz potential ( I α f ) (x), x R n exists almost everywhere. Furthermore, the ineuality ( Iα f ) (t) CSα ( f ) (t), 0 < t < (12) is valid, where C is a constant independent of f and t.

57 Riesz Potential We will need the following two Hardy operators to prove our main theorem. Definition Let ϕ be a measurable function on (0, ) and γ and β be real numbers. The weighted Hardy operators with power weights acting on ϕ are defined by H β (γ) ϕ(t) = tγ+β 1 t 0 ϕ(y) y γ dy and ϕ(y) Hβ (γ) ϕ(t) = tγ+β dy. t y γ+1

58 Riesz Potential In the following two lemmas we give the boundedness of the Hardy operators H(γ) α and Hα (γ) on Morrey and weak Morrey spaces. Lemma (N. Samko 2009) Let 0 < λ < 1, 0 < β < 1 λ, 1 r < 1 λ 1 r 1 s = β 1 λ. If γ < 1 r + λ r, then Hβ (γ) is bounded from LM r,λ (0, ) to LM s,λ (0, ), and if γ > λ 1 r, then the operator H β (γ) is bounded from LM r,λ(0, ) to LM s,λ (0, ). Lemma β and (Guliyev, Kucukaslan, Aykol, Serbetci, 2017) Let 0 < λ < 1, 0 < β < 1 λ, 1 r < 1 λ β and 1 r 1 s = β 1 λ. If γ = 1 r + λ r, then H β (γ) is bounded from LM r,λ(0, ) to WLM s,λ (0, ), and if γ = λ 1 r, then the operator H β (γ) is bounded from LM r,λ(0, ) to WLM s,λ (0, ).

59 Riesz Potential The following is a well known theorem about Riesz potential. Theorem Theorem J. Let 0 < α < n, 1 p n α, p < <, 1 r and f L p,r (R n ) satisfies the condition (11). Then the Riesz potential I α f exists almost everywhere. Furthermore, (i) If 1 < p < n α, 1 r s, then the condition 1 p 1 = α n is necessary and sufficient for the boundedness of the operator I α from the Lorentz spaces L p,r to L,s. (ii) If p = 1, then the condition 1 1 = α n is necessary and sufficient for the boundedness of the operator I α from the Lorentz spaces L 1,r to WL.

60 Riesz Potential Theorem Let 0 < α < n, 0 λ < 1, 1 r s, 1, ( ) 1 r r+λ p λ r + α n and f M loc p,r;λ (Rn ) satisfies the condition (11). Then the Riesz potential I α f exists almost everywhere. Furthermore, ( ) r (i) If r+λ < p < λ 1, r + α n then the condition ( ) 1 p 1 = λ 1 r 1 s + α n is necessary and sufficient for the boundedness of the operator I α from the spaces Mp,r;λ loc to Mloc,s;λ. (ii) If p = r r+λ, then the condition 1 1 = α n λ s is necessary and sufficient for the boundedness of the operator I α from the spaces Mp,r;λ loc to WMloc,s;λ.

61 Proof of the theorem If f satisfies (11), then from Theorem H the Riesz potential I α f (x), x R n exists almost everywhere. (i) Sufficiency. Let r r+λ < p < ( λ r + α n ) 1. From the ineuality (12) we get I α f M loc,s,λ = sup t>0 C sup t λ 1 s τ 1 s t>0 C sup t λ 1 s τ 1 s + α τ n 1 t>0 0 + C sup t λ 1 s τ 1 s t>0 t λ 1 s τ 1 s (I α f ) (τ) Ls(0,t) τ (τ αn 1 τ 0 ) f (y)dy + y α n 1 f (y)dy Ls(0,t) τ f (y)dy Ls(0,t) y α n 1 f (y)dy Ls(0,t) = I 1 + I 2.

62 Proof of the theorem We have I 1 = C sup t λ 1 s τ 1 s + α τ n 1 f (y)dy Ls(0,t) = C H β (γ) g LM s,λ (0, ). t>0 0 Consider the Hardy operator H β (γ) such that γ = 1 p 1 r and g(t) = t 1 p 1 r f (t). Since the operator H β (γ) is bounded from the Morrey space LM r,λ (0, ) to LM s,λ (0, ), we get I 1 = C H β (γ) g LM s,λ (0, ) C g LMr,λ (0, ) = C sup t λ 1 r τ p 1 r f (τ) Lr (0,t) = C f M loc. (13) t>0 p,r;λ

63 Proof of the theorem I 2 = C sup t λ 1 s τ 1 s y α n 1 f (y)dy Ls(0,t) = C H β (γ) g LM s,λ (0, ). t>0 τ Consider the Hardy operator H β (γ) such that γ = 1 p 1 r α n and g(t) = t 1 p 1 r f (t). Since the operator H β (γ) is bounded from the Morrey space LM r,λ (0, ) to LM s,λ (0, ) we get I 2 = C H β (γ) g LM s,λ (0, ) C g LMr,λ (0, ) = C sup t λ 1 r τ p 1 r f (τ) Lr (0,t) = C f M loc. (14) t>0 p,r;λ From the ineualities (13) and (14) we obtain the boundedness of the operator I α from Mp,r;λ loc to Mloc,s;λ.

64 Proof of the theorem Necessity. Suppose that the operator I α is bounded from Mp,r;λ loc to Mloc,s;λ ( ), and r r+λ p λ 1. r + α n Define f τ (x) =: f (τx) for τ > 0. Then f Also, and f τ M loc = τ n( 1 p λ r p,r;λ (I α f τ )(x) = τ α (I α f )(τ n x), (I α f τ ) (t) = τ α (I α f ) (tτ n ) I α f τ M loc,s;λ Therefore we get 1 p 1 = λ ( 1 r 1 s τ (t) = f (tτ n ) and ) f M loc. p,r;λ = τ α n( 1 λ s ) I α f M loc,s;λ ) + α n..

65 Proof of the theorem (ii) Sufficiency. r For the limiting case p = r+λ, 1 r s <, suppose f Mloc p,r;λ. By using the ineuality (12) and Minkowski s ineuality we get I α f WM loc,s;λ = sup t λ 1 s τ 1 s (I α f ) (τ) WLs(0,t) t>0 τ (τ αn 1 C sup t λ 1 s τ 1 s t>0 C sup t λ 1 s τ 1 s + α τ n 1 t>0 0 + C sup t λ 1 s τ 1 s t>0 = N 1 + N 2. τ 0 ) f (y)dy + f (y)y α n 1 dy τ f (y)dy WLs(0,t) f (y)y α n 1 dy WLs(0,t)

66 Proof of the theorem We have N 1 = C sup t λ 1 s τ 1 s + α τ n 1 f (y)dy WLs(0,t) t>0 0 = C H β (γ) h WLM s,λ (0, ). Consider the Hardy operator H β λ 1 (γ) such that γ = 1 + r and λ 1 1+ h(t) = t r f (t). From the boundedness of the operator H β (γ) we have N 1 = C H β (γ) h WLM s,λ (0, ) C h LMr,λ (0, ) = C sup t>0 t λ r τ 1+ λ 1 r f (τ) Lr (0,t) = C f M loc. (15) p,r;λ

67 Proof of the theorem Now we consider N 2. N 2 = C sup t λ 1 s τ 1 s t>0 τ f (y)y α n 1 dy WLs(0,t) = C H β (γ) h WL,λ (0, ). Consider the Hardy operator H β λ 1 (γ) such that γ = 1 + r α n and λ 1 1+ h(t) = t r f (t). Since the operator H β (γ) is bounded from the Morrey spaces LM r,λ (0, ) to WLM s,λ (0, ) we get N 2 = C H β (γ) h WLM s,λ (0, ) C h LMr,λ (0, ) = C sup t>0 t λ r τ 1+ λ 1 r f (τ) Lr (0,t) = C f M loc. (16) p,r;λ From the ineualities (15) and (16) we obtain the boundedness of the Riesz potential operator I α from Mp,r;λ loc to WMloc,s;λ.

68 Proof of the theorem Necessity. Suppose that the operator I α is bounded from M loc for p = r r+λ. Define f τ (x) =: f (τx) for τ > 0. p,r;λ to WMloc,s;λ f τ M loc r r+λ,r;λ = τ n f M loc r r+λ,r;λ and I α f τ WM loc = τ α n( 1 λ s ) I α f,s;λ WM loc.,s;λ Therefore we get the euality 1 1 = α n λ s theorem is completed. and the proof of the

69 Some applications Maximal Bochner-Riesz operator Let δ > (n 1)/2, B δ r (f )ˆ(ξ) = (1 r 2 ξ 2 ) δ +ˆf (ξ) and B δ r (x) = r n B δ (x/r) for r > 0. The maximal Bochner-Riesz operator is defined by (see L.Z. Liu and S.Z. Lu (2003) and Y. Liu and D. Chen (2008)) B δ, (f )(x) = sup Br δ (f )(x). r>0 Let H be the space H = {h : h = sup r>0 h(r) < }, then it is clear that B δ, (f )(x) = B δ r (f )(x). By the condition on B δ r (J. Garcia-Cuerva and J.L. Rubio de Francia (1985)), we have and B δ r (x y) Cr n (1 + x y /r) (δ+(n+1)/2) r n, 1 B δ, (f )(x) C sup r>0 r n f (y) dy = CMf (x). B(x,r)

70 Some applications Since the maximal operator M is bounded on the spaces M loc p,;λ, we get the following corollary. Corollary Let 1, 0 λ < 1 and +λ p. (i) If +λ < p <, then the Bochner-Riesz operator Bδ r is bounded on the local Morrey-Lorentz space Mp,;λ loc. (ii) If p = +λ, then the Bochner-Riesz operator Bδ r is bounded from Mp,;λ loc to the weak local Morrey- Lorentz space WMloc p,;λ. (iii) If p = =, then the Bochner-Riesz operator Br δ is bounded on L (R n ).

71 Some applications In this section, we apply the our main theorem to the fractional maximal operator, fractional Marcinkiewicz operator and the fractional powers of some analytic semigroups. Fractional maximal operator For 0 α < n, we define the fractional maximal operator M α f (x) = sup B(x, t) α n 1 f (y) dy, t>0 B(x,t) where B(x, t) is the open ball centered at x of radius t for x R n and B(x, t) is the Lebesgue measure of B(x, t) such that B(x, t) = ω n t n in which ω n denotes the volume of unit ball in R n.

72 Some applications It is well known that the following ineuality holds: M α f (x) ω α 1 n (I α f )(x). From this ineuality we get the following corollary. Corollary Let 0 < α < n, 0 λ < 1, 1 r s, 1, ( ) 1 ( ) r r+λ p λ r + α n and 1 p 1 = λ 1 r 1 s + α n. r ( ) (i) If r+λ < p < λ 1, r + α n then the condition ( ) 1 p 1 = λ 1 r 1 s + α n is necessary and sufficient for the boundedness of the fractional maximal operator M α from the space Mp,r;λ loc to Mloc,s;λ. (ii) If p = r r+λ, then the condition 1 1 = α n λ s necessary and sufficient for the boundedness of the operator M α from the space Mp,r;λ loc to WMloc,s;λ.

73 Some applications Fractional Marcinkiewicz operator Let S n 1 = {x R n : x = 1} be the unit sphere in R n euipped with the Lebesgue measure dσ. Suppose that Ω satisfies the following conditions. (a) Ω is the homogeneous function of degree zero on R n \ {0}, that is, Ω(tx) = Ω(x), for any t > 0, x R n \ {0}. (b) Ω has mean zero on S n 1, that is, S n 1 Ω(x )dσ(x ) = 0. (c) Ω Lip γ (S n 1 ), 0 < γ 1, that is there exists a constant C > 0 such that, Ω(x ) Ω(y ) C x y γ for any x, y S n 1.

74 Some applications In 1958, Stein defined the Marcinkiewicz integral of higher dimension µ Ω as where ( µ Ω,α (f )(x) = F Ω,α,t (f )(x) = 0 x y t F Ω,α,t (f )(x) 2 dt ) 1/2 t 3, Ω(x y) f (y)dy. x y n 1 α By Minkowski ineuality and the conditions on Ω, we get µ Ω,α (f )(x) I α ( f )(x).

75 Some applications Then we have the following corollary. Corollary Let 0 < α < n, 0 λ < 1, 1 r s, 1, ( ) 1 ( ) r r+λ p λ r + α n and 1 p 1 = λ 1 r 1 s + α n. ) 1 ( and 1 p 1 = λ 1 r 1 s ) + α n, then the ( r (i) If r+λ < p < λ r + α n Marcinkiewicz operator µ Ω is bounded from the space M loc M,s;λ loc. (ii) If p = r r+λ and 1 1 = α n λ s, then the operator µ Ω is bounded from the space Mp,r;λ loc to WMloc,s;λ. p,r;λ to

76 Some applications Fractional powers of some analytic semigroups Suppose that L is a linear operator on L 2 which generates and analytic semigoup e tl with the kernel p t (x, y) satisfying a Gaussian upper bound, that is, p t (x, y) c 1 x y 2 t n/2 e c 2 t (17) for x, y R n and all t > 0, where c 1, c 2 > 0 are independent of x, y and t. For 0 < α < n, the fractional powers L α/2 of the operator L are defined by L α/2 f (x) = 1 Γ(α/2) 0 e tl dt f (x) t α/2+1.

77 Some applications Note that if L = is the Laplacian on R n, then L α/2 is the Riesz potential I α. Property (17) is satisfied for large classes of differential operators. Since the semigroup e tl has kernel p t (x, y) which satisfies condition (17), it follows that L α/2 f (x) CI α ( f )(x). Hence we get the following corollary. Corollary Let 0 < α < n, 0 λ < 1, 1 r s, 1 ( ) 1 ( ) r r+λ p λ r + α n and 1 p 1 = λ 1 r 1 s + α n. ) 1 ( and 1 p 1 = λ 1 r 1 s ( r (i) If r+λ < p < λ r + α n operator L α/2 is bounded from the space M loc (ii) If p = ) + α n, then the p,r;λ to Mloc,s;λ. r r+λ and 1 1 = α n λ s, then the operator L α/2 is bounded from the space M loc p,r;λ to WMloc,s;λ.

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