The Hardy-Littlewood maximal operator on graphs Pedro Tradacete - PowerPoint PPT Presentation

The Hardy-Littlewood maximal operator on graphs Pedro Tradacete (UC3M) Joint work with J. Soria (UB) Congreso de J´ ovenes Investigadores RSME 7 - 11 September 2015, Murcia P . Tradacete (UC3M) HL operator on graphs Murcia 2015 1 / 11

Notation Let G = ( V , E ) simple, connected, and finite graph. Shortest path distance d G ( v , v ′ ) = min { k : ∃ ( v j ) k j = 0 , v 0 = v , v k = v ′ , ( v j − 1 , v j ) ∈ E ∀ j ≤ k } . B ( v , r ) ball of center v ∈ V and radius r ≥ 0. ( V , d G , | · | ) metric measure space. Given f : V → R let 1 � M G f ( v ) = sup | f ( w ) | . | B ( v , r ) | r ≥ 0 w ∈ B ( v , r ) (Centered) Hardy-Littlewood maximal function P . Tradacete (UC3M) HL operator on graphs Murcia 2015 2 / 11

Theorem Let G 1 and G 2 be two graphs with V ( G 1 ) = V ( G 2 ) = { 1 , . . . , n } . The following are equivalent: (i) G 1 = G 2 . (ii) For every f : { 1 , . . . , n } → R , M G 1 f = M G 2 f. (iii) For every k ∈ V, M G 1 δ k = M G 2 δ k . In general, it is not true that if G 1 ⊂ G 2 (i.e., V ( G 1 ) = V ( G 2 ) and E ( G 1 ) ⊂ E ( G 2 ) ), then M G 2 f ≤ M G 1 f . For example, if V = { 1 , 2 , 3 , 4 } , G 1 is a linear tree with leafs 1 and 4, G 2 is the 4-cycle C 4 (with a clockwise orientation of V ), then G 1 ⊂ G 2 , but M G 2 δ 4 ( 1 ) = 1 / 3 > 1 / 4 = M G 1 δ 4 ( 1 ) . P . Tradacete (UC3M) HL operator on graphs Murcia 2015 3 / 11

Lemma Let G be graph with n vertices, and T : ℓ p ( G ) → ℓ p ( G ) be a sublinear operator, with 0 < p ≤ 1 . Then, � T � p = max k ∈ V � T δ k � p . K 4 � M K 4 � 1 ∪ ∧ P 4 ⊂ D 4 ⊃ C 4 � M P 4 � 1 > � M D 4 � 1 = � M C 4 � 1 ∪ ∪ ∧ ∧ S 4 L 4 � M S 4 � 1 � M L 4 � 1 P . Tradacete (UC3M) HL operator on graphs Murcia 2015 4 / 11

Proposition � 1 / p � 1 + n − 1 (i) If 0 < p ≤ 1 , then � M K n � p = . n p (ii) If 1 < p < ∞ , then 1 + n − 1 � 1 / p 1 + n − 1 � 1 / p � � ≤ � M K n � p ≤ i . e . � M K n � p ≈ 1 . n p n (iii) For n ≥ 3 , we have � M S n � 1 = n + 1 2 . (iv) For 1 < p < ∞ , then � 1 / p � 1 / p � 1 + n − 1 � n + 5 i . e . � M S n � p ≈ n 1 / p . ≤ � M S n � p ≤ , 2 p 2 (v) For n ≥ 2 we have � n 1 − p − 1  � 1 / p , 0 < p < 1 ,    1 − p � M L n � p ≈   log n , p = 1 .  P . Tradacete (UC3M) HL operator on graphs Murcia 2015 5 / 11

Theorem Let G be a graph with n vertices and 0 < p ≤ 1 . Then, the following optimal estimates hold: 1 + n − 1 � 1 / p 1 + n − 1 � 1 / p � � ≤ � M G � p ≤ . n p 2 p Moreover, 1 + n − 1 � 1 / p � (i) � M G � p = if and only if G = K n ; n p 1 + n − 1 � 1 / p � (ii) � M G � p = if and only if G ∼ S n . 2 p P . Tradacete (UC3M) HL operator on graphs Murcia 2015 6 / 11

Weak p -estimates � 1 / p = max j ∈ V j 1 / p f ∗ � � � f � p , ∞ := sup t � { j ∈ V : f j > t } j . t > 0 � M G f � p , ∞ � M G � p , ∞ = sup . � f � p f Theorem For 0 < p < ∞ , we have � n 1 / p − 1 , if 0 < p ≤ 1 , � M K n � p , ∞ = 1 , if p ≥ 1 . max { n 1 / p / 2 , 1 } ≤ � M S n � p , ∞ ≤ n 1 / p . (In particular, � M S n � p , ∞ ≈ n 1 / p , for every n ≥ 1 and 0 < p < ∞ ) P . Tradacete (UC3M) HL operator on graphs Murcia 2015 7 / 11

Dilation index Definition Given a graph G we define its dilation index as � | B ( x , 3 r ) | � D ( G ) = max | B ( x , r ) | : x ∈ V , r ∈ N , r ≤ diam ( G ) . Example Complete graph: D ( K n ) = 1 . Star graph: D ( S n ) = n 2 . Linear tree: easy to check that D ( L n ) < 3 for all n ∈ N , and lim n →∞ D ( L n ) = 3. For small n : D ( L 3 ) = 3 / 2, D ( L 4 ) = 2, D ( L 5 ) = 2, D ( L 6 ) = 2, D ( L 7 ) = 7 / 3 . . . P . Tradacete (UC3M) HL operator on graphs Murcia 2015 8 / 11

Overlapping index Definition Given a graph G we define its overlapping index as � O ( G ) = min r ∈ N : ∀{ B j } j ∈ J , B j a ball in G , ∃ I ⊂ J , � � � � B j = B i and χ B i ≤ r . j ∈ J i ∈ I i ∈ I Example O ( K n ) = 1 , ∀ n ∈ N ; O ( S n ) = n − 1 , ∀ n ≥ 2 ; � 1 � 1 n ≤ 2 , n ≤ 3 , O ( L n ) = O ( C n ) = 2 n ≥ 3 ; 2 n ≥ 4 . P . Tradacete (UC3M) HL operator on graphs Murcia 2015 9 / 11

Theorem Given a graph G, we have � � � M G � 1 , ∞ ≤ min D ( G ) , O ( G ) . Proposition For the linear graph L n , we have that lim n →∞ � M L n � 1 , ∞ = 2 . Note lim n →∞ � M L n � 1 , ∞ = 2 = �M� 1 , ∞ , where M is the uncentered maximal function in R . Compare to the fact that for the centered Hardy-Littlewood maximal operator M in R and the discrete measures N � � � D = µ = δ a k : a k ∈ R , a k + 1 = a k + H , H fixed, N ∈ N , k = 1 � M µ � 1 , ∞ = 3 sup 2 . � µ � µ ∈D [M. T. Men´ arguez and F . Soria, 1992] P . Tradacete (UC3M) HL operator on graphs Murcia 2015 10 / 11

Thank you for your attention. P . Tradacete (UC3M) HL operator on graphs Murcia 2015 11 / 11