Hardy-Littlewood maximal operator in weighted Lorentz spaces Elona Agora IAM-CONICET Based on joint works with: J. Antezana, M. J. Carro and J. Soria Function Spaces, Differential Operators and Nonlinear Analysis Prague (Czech Republic), 4-9 July, 2016
Classical results
Hardy-Littlewood maximal operator Definition The Hardy-Littlewood maximal operator is defined by: 1 Mf (x) = sup f (y) dy, x I I where I is an interval of the real line and the supremum is considered over all intervals containing x R. I Problem Characterize the weights u L 1 loc (R) so that M : Lp (u) L p (u), that is; find necessary and sufficient conditions on the weight u so that Mf L p (u) f L p (u).
(I) The case of weighted Lebesgue spaces L p (u) ( ) ( p 1 1 1 Recall u A p : sup u(x)dx u (x)dx) 1/(p 1) <. I R I I I I Theorem (Muckenhoupt 1972) For p > 1, the following statements are equivalent: (i) M : L p (u) L p (u); (ii) u A p ; (iii) u(i) u(s) ( ) p ε I, for every interval I and S I and some ε > 0. S
Weak-type Lebesgue space L p, (u) Recall that the space L p, (u) is defined by { } f M(R) : f p L p, (u) = sup λ p u({x R : f (x) > λ}) <. λ>0 It has been also studied the boundedness M : L p (u) L p, (u), equivalently Mf L p, (u) f Lp (u). Theorem (Muckenhoupt 1972) For p > 1, the following statements are equivalent: (i) M : L p (u) L p (u); (ii) M : L p (u) L p, (u); (iii) u A p.
Motivation for introducing the classical Lorentz spaces Note that: f p L = f (x) p dx = p where pλ p 1 {x R : f (x) > λ} dλ = R 0 0 (f (t)) p dt, and f p L p, f (t) = inf{s > 0 : { f > s} t}, = sup λ p {x R : f (x) > λ} = sup t(f ) p (t). λ>0 t>0
Classical Lorentz spaces Definition (Lorentz, 1950) Let p > 0, w L 1 loc (R+ ). The classical Lorentz spaces are: Λ p (w) = {f M(R) : f p Λ p (w) = (f (t)) p w(t)dt < }, Λ p, (w) = {f M(R) : f p Λ p, (w) = sup λ p W( {x : f (x) > λ} ) < }, λ>0 where f (t) = inf{s > 0 : {x R : f (x) > s} t} and W(t) = t 0 w. 0 Examples: (i) If w = 1, we recover Λ p (1) = L p and Λ p, (1) = L p, ; (ii) If w = t q/p 1, we obtain L p,q.
(II) The case of classical Lorentz spaces Λ p (w) Boundedness of M It is known that (Mf ) (t) Pf (t), where Pf (t) = 1 t t 0 f (r)dr = 1 f (t/s) ds s 2 = where E s f (t) = f (t/s). For s [1, ), define E s p Λ p (w) Λ p (w) = By Minkowski inequality we obtain 1 E s f (t) ds s 2, sup E s f p L p (w) = sup W(st) =: W(s). f L p (w) 1 t>0 W(t) M Λp (w) Λ p (w) = sup Mf Λp (w) f Λ p (w) 1 = sup E s f ds Proposition If 1 f L p (w) 1 1 s 2 sup f L p (w) 1 Lp (w) Pf Lp (w) W 1/p (s) ds s 2 <, then M : Λp (w) Λ p (w). 1 W 1/p (s) ds s 2.
The case of classical Lorentz spaces Λ p (w) Lorentz-Shimogaki and Ariño-Muckenhoupt theorem for Λ p (w) Theorem (Lorentz-Shimogaki 1960, Ariño-Muckenhoupt 1990) Let p > 1. Then the following are equivalent: (i) M : Λ p (w) Λ p (w); (ii) M : Λ p (w) Λ p, (w); (iii) 1 W 1/p (s) ds s 2 < ; (iv) There exists s [1, ) so that W(s) < s p ; W(t) ( t ) p ε (v) W(r), t r, for some ε > 0. r Define for some ε > 0. w B p if W(t) ( t ) p ε W(r), t r, r
Weighted Lorentz spaces
Weighted Lorentz spaces Definition (Lorentz, 1950) Let p > 0, u L 1 loc (R), w L1 loc (R+ ). The weighted Lorentz spaces are: Λ p, u where Λ p u(w) = {f M(R) : f p Λ p u(w) = (fu (t)) p w(t)dt < }, (w) = {f M(R) : f Λ p, (w) = sup λ p W(u({x : f (x) > λ})) < }, Examples: u λ>0 fu (t) = inf{s > 0 : u({x R : f (x) > s}) t}, t u(e) = u(x)dx and W(t) = w(s)ds. E (i) If w = 1, we recover Λ p u(1) = L p (u) and Λ p, u (1) = L p, (u); (ii) If u = 1, we obtain Λ p (w) and Λ p, (w). 0 0
Characterization of M : Λ p u(w) Λ p, (w) u Problem I Find necessary and sufficient conditions on u Lloc 1 (R) and w L1 loc (R+ ): M : Λ p u(w) Λ p u(w), equivalently Mf Λ p u (w) C f Λ p u (w). Problem II Find necessary and sufficient conditions on u L 1 loc (R) and w L1 loc (R+ ): M : Λ p u(w) Λu p, (w), equivalently Mf Λ p, u (w) C f Λ p u (w).
Weak-type boundedness of M on the weighted Lorentz spaces Theorem (Carro-Raposo-Soria 2007) Let p > 1. The following statements are equivalent: M : Λ p u(w) Λ p u(w); There exists ε > 0 so that: W(u( j I ( ) j) ) p ε W(u( Ij j S ) max, (1) j ) 1 j J S j for every finite family of disjoint intervals (I j ) J j=1, and every family of measurable sets (S j ) J j=1, so that S j I j, for every j. If w = 1, the above condition is equivalent to A p : If u = 1, it is equivalent to B p : Open u(i) u(s) W(t) ( t ) p ε W(r), t r. r ( ) p ε I. S Characterization of the weak-type boundedness M : Λ p u(w) Λ p, u (w).
Weak-type boundedness of M Theorem (A-Antezana-Carro 2016) Let p > 1, then M : Λ p u(w) Λ p, u (w) M : Λ p u(w) Λ p u(w). Known case w = 1: The implication M : L p (u) L p, (u) M : L p (u) L p (u) is proved using reverse Hölder s inequality: if u A p, then γ > 0 so that ( ) 1 1 u 1+γ 1+γ 1 (t)dt C u(t)dt. I I I I Then, if u A p we have that u A p ε for some ε > 0. Using interpolation M : L p ε (u) L p ε, (u) and M : L L we get the strong type boundedness.
Weak-type boundedness of M Case u = 1: M : Λ p (w) Λ p, (w) M : Λ p (w) Λ p (w). Let s > 1 and 1, if 0 x a; a f (x) = x, if a x sa; 0, if x > sa. Since (Mf ) Pf, by the hypothesis W( {x : Pf (x) > y} ) C 1 y p f p (x)w(x)dx. (2) Note that Pf (as) = 1 + log s (1 + log s). Taking y = we have s s W( (0, as) ) W( {x : Pf (x) > (1+log s)/s} ) C(1+log s) 1 p s p W( (0, a) ). Hence, Thus M : Λ p (w) Λ p (w). 0 W(as) W(a) C(1 + log s)1 p s p. (3)
Weak-type boundedness of M Theorem (A-Antezana-Carro 2015) M : Λ p u(w) Λ p, u (w) M : Λ p u(w) Λ p u(w). General case: Let S I R. Construct f S,I, so that for every λ [ S / I, 1] so that S = λ E λ, where E λ = {x : f S,I (x) > λ}. Then, for s = I S Mf S,I p Λ p, u (w) C f S,I p Λ p u(w) W(u(I)) W(u(S)) C (1 + log s)1 p s p, which implies that M : Λ p u(w) Λ p u(w).
Weak-type boundedness of M If S = S 1 S 2 I R, the function f S,I can be: Several dimensions: We consider the function for any measurable set S R d. χ {MχS>λ}Mχ S,
Possible problem to study Recall that the Riesz transforms are given by R j f (x) = d+1 Γ( 2 ) π d+1 2 lim ε 0 R d \B ε(x) x j y j f (y) dy, x y d+1 for j = 1,..., d, whenever they are well defined. Study the boundedness of the Riesz transforms on weighted Lorentz spaces.
Some references 1. E. Agora, J. Antezana, and M. J. Carro, The complete solution to the weak-type boundedness of Hardy-Littlewood maximal operator on weighted Lorentz spaces, To appear in J. Fourier Anal. Appl. (2016) 2. E. Agora, J. Antezana, M. J. Carro and J. Soria, Lorentz-Shimogaki and Boyd Theorems for weighted Lorentz spaces, J. London Math. Soc. 89 (2014) 321-336. 3. E. Agora, M. J. Carro and J. Soria, Complete characterization of the weak-type boundedness of the Hilbert transform on weighted Lorentz spaces, J. Fourier Anal. Appl. 19 (2013) 712-730. 4. M. J. Carro, J. A. Raposo and J. Soria, Recent developments in the theory of Lorentz Spaces and Weighted Inequalities, Mem. Amer. Math. Soc. 187 (2007) no. 877, xii+128. 5. G. Lorentz, Some new functional spaces, Ann. of Math. 51 (1950) no 2, 37 55.
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