An Upper Bound for the Menchov-Rademacher Operator for Right Triangles
aa r X i v : . [ m a t h . C A ] A ug An Upper Bound for the Menchov-RademacherOperator for Right Triangles
Armen Vagharshakyan
Institute of Mathematics , Armenian National Academy of Sciences
Abstract
We introduce the Menchov-Rademacher operator for right triangles -a sample two-dimensional maximal operator, and prove an upper boundfor its L norm. Keywords:
Maximal operators are one of the important objects in Harmonic Anal-ysis. They usually arise as tools to prove different sorts of pointwise con-vergence results, e.g.: the classical maximal operator is used in Lebesguesdifferentiation theorem, Carleson’s maximal operator is used in Carlesonstheorem on the pointwise convergence of Fourier series, and the Menchov-Rademacher operator is used in theorems on convergence of general or-thogonal series.Whereas much is known for one-dimensional maximal operators, thetopic of higher-dimensional operators is less developed. In this article weare concerned with a specific type of two-dimensional maximal operators -that of the Menchov-Rademacher operator for right triangles - and provean upper bound for its L norm.Proving a corresponding non-trivial lower bound seems to be a muchharder problem. We start by introducing some notations. For a set E ⊂ R d , denote by O ( E ) the family of systems of functions { f ~i } ~i ∈ E ∩ N d that are orthonormal Research was supported by the Science Committee of Armenia, grant 18T-1A081. n a certain set X (whose choice is irrelevant in the context of this paper).That is: f ~i : X → R for ~i ∈ E ∩ N d , Z X f ~i f ~j = 1 if ~i = ~j, and Z X f ~i f ~j = 0 if ~i = ~j. For a family of sets S, denote by ∪ S the union of the sets in S , ∪ S = [ U ∈ S U. For a family of sets S in a vector space R d , a vector a ∈ R d , and a number λ ∈ R, denote by a + λS the following family of sets: a + λS = { a + λU } U ∈ S . We will denote by c i ’s some absolute constants, that is the claim u ( x ) ≤ c i v ( x )means that the inequality holds for some absolute constant c i irrespectiveof x .In this paper the fraction n/b (for n ∈ N ) denotes either ⌊ n/b ⌋ or ⌈ n/b ⌉ . Definition.
Let S be a family of sets in R d . Introduce the generalizedMenchov-Rademacher operator corresponding to the family of sets S asfollows: (cid:16) { f ~i } ~i ∈ ( ∪ S ) ∩ N d , { a i } ~i ∈ ( ∪ S ) ∩ N d (cid:17) → sup I ∈ S X ~i ∈ I ∩ N d a ~i f ~i , where { f ~i } ~i ∈ ( ∪ S ) ∩ N d ∈ O ( ∪ S ) , and { a ~i } ~i ∈ ( ∪ S ) ∩ N d ∈ l (cid:16) ( ∪ S ) ∩ N d (cid:17) . Definition.
Denote by mr ( S ) the sharp upper bound for the L norm ofthe generalized Menchov-Rademacher operator, that is mr ( S ) = sup { f ~i }∈ O ( ∪ S ) sup P ~i ∈ ( ∪ S ) ∩ Nd a ~i ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) sup I ∈ S (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X ~i ∈ I ∩ N d a ~i f ~i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) L ( X ) . Remark.
A motivation to introduce these two definitions above is thatthey generalize the usual notion of the classical Menchov-Rademacher op-erator and its norm. Indeed, take S n to be the family of all intervals in [0 , n ] . Denote α n = mr ( S n )ln( n ) . hen the claim < α = lim sup n → + ∞ α n < + ∞ (1) is the Menchov-Rademacher inequality in its classical setting (see e.g. theoriginal articles of Menchov [1] and Rademacher [2]). Remark.
The best estimate on the constant α from above is provided byW. Bednorz (see [3]) α < , with upper bounds for α obtained earlier being ln 2 / ln 3 by Kounias (see[4]), and / (2 ln 2) by S. Chobanyan (see [5]) . Denote by
T ri a,b,c the triangle with vertices ( a, , ( b, , ( a, c ) . Fur-ther, denote by
T RI n a certain family of right triangles defined by: T RI n = { T ri ,a,b : T ri ,a,b ⊂ T ri ,n,n } . We consider the corresponding Menchov-Rademacher operator for righttriangles (cid:16) { f ~i } ~i ∈ Tri ,n,n ∩ N , { a i } ~i ∈ Tri ,n,n ∩ N (cid:17) → sup I ∈ TRI n X ~i ∈ I ∩ N a ~i f ~i as a model case for maximal operators in two dimensions. We prove thefollowing upper bound: Theorem. mr ( T RI n ) = O (cid:18) n ln (2+ √ (cid:19) . The proof will make us of the following theorem of the Analysis ofAlgorithms first presented by J. Bentley, D. Haken, and B. James (see[6]):
Theorem (Master Theorem) . Let a function T : N → R satisfy T ( n ) ≤ aT ( n/b ) + f ( n ) , and let f ( n ) = O ( n c ) . If c < log b ( a ) , then T ( n ) = O (cid:16) n log b ( a ) (cid:17) . If c > log b ( a ) , then T ( n ) = O ( n c ) . Introduce the following auxiliary families of sets:
REC m,n = { [ x, y ] × [0 , n ] : x, y ∈ [0 , m ] } and HT RI m,n = { T a,b,n : 0 ≤ a ≤ b ≤ m } . tep 1. Let { f ~i } ∈ O ( ∪ REC m,n )and X ~i ∈ ([0 ,m ] × [0 ,n ]) ∩ N a ~i ≤ . Apply the classical Menchov-Rademacher inequality to the orthogonalfamily of functions g i = n X j =1 a i,j f i,j , i = 1 , . . . , m to get (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) sup I ∈ S m (cid:16)P i ∈ I,j ∈ [0 ,n ] ∩ N a i,j f i,j (cid:17)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ln m == (cid:12)(cid:12)(cid:12)(cid:12) sup I ∈ S m (cid:0)P i ∈ I g i (cid:1)(cid:12)(cid:12)(cid:12)(cid:12) ln m ≤ α m , n ∈ N. Therefore, sup n lim sup m →∞ mr ( REC m,n )ln( m ) ≤ α. (2) Step 2.
Let { f ~i } ∈ O ( ∪ HT RI m,n )and X i ∈ ([0 ,m ] × [0 ,n ]) ∩ N a i ≤ . Split the latter sum into two parts as follows: p = X i ∈ ( [0 ,m ] × [ , n ]) ∩ N a i , p > ,p = X i ∈ ( [0 ,m ] × ( n ,n ]) ∩ N a i , p > , so that p + p ≤ . (3)Note that a triangle T a,b,n ∈ HT RI m,n may be split into three parts (atriangle, a RECangle, and another triangle) T a,b,n = T ∪ T ∪ T , by taking its intersection with the RECangles:[0 , m ] × (cid:16) n , n i , [ a, ( a + b ) / × (cid:16) , n i , (( a + b ) / , b ] × (cid:16) , n i . By the triangle inequality: (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) sup I ∈ HTRI m,n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X ~i ∈ I a ~i f ~i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ p · mr (cid:16) HT RI m, n (cid:17) + p · mr (cid:16) REC m, n (cid:17) + p · mr (cid:16) HT RI m, n (cid:17) So that, maximizing over the condition (3), we get mr ( HT RI m,n ) ≤≤ (cid:16) mr (cid:16) HT RI m, n (cid:17) + mr (cid:16) REC m, n (cid:17)(cid:17) + mr (cid:16) HT RI m, n (cid:17) . Consequently, a weaker recursive estimate holds mr ( HT RI m,n ) ≤ √ · mr (cid:16) HT RI m, n (cid:17) + mr (cid:16) REC m, n (cid:17) (4)On the other hand, the initial condition for the recursive estimate (4) is mr ( HT RI m, ) = mr ( S m ) . (5)By arguing along the lines of the Master Theorem, we get mr ( HT RI m ) = O ( √ n ln( m )) , ( m, n ) → ∞ . (6)Indeed, we apply (4) recursively with the initial condition (5) to get mr (cid:0) HT RI m, k (cid:1) ≤ (cid:16) √ (cid:17) k · mr ( S m ) + k − X l =1 ( √ k − − l · mr (cid:0) REC m, l (cid:1) Now, we insert the estimates (1) and (2) to get mr (cid:0) HT RI m, k (cid:1) ≤ (cid:16) √ (cid:17) k + k − X l =1 ( √ k − − l ! · c ln ( m ) ≤≤ (cid:16) √ (cid:17) k · c ln ( m ) , m, k ≥ . We now refer to the estimate mr ( HT RI m,n ) ≤ mr (cid:16) HT RI m, ⌈ log2( n ) ⌉ (cid:17) to claim that the asymptotic bound (6) holds. Step 3.
Let { f ~i } ∈ O ( ∪ T RI n,n )and X ~i ∈ Tri ,n,n a ~i ≤ . Split the triangle T ,n,n into four triangles T , T , T , T as follows: T = T ri , n , n + (0 , n ,T = T ri , n , n + ( n , ,T = T ri , n , n ,T = T ri ,n,n \ (cid:0) T ∪ T ∪ T (cid:1) . enote: p j = X ~i ∈ T j ∩ N a ~i , j = 1 , , , , p j ≥ . The numbers { p j } then satisfy the condition X j =1 p j ≤ . (7)Note that the set T ∈ T RI ,n,n is of either of the following four forms:1. a triangle in T .
2. a union of a triangle in T , a RECangle in REC n , n , and a triangle in HT RI n , n ,
3. a symmetry w.r.t the line y = x of a set described in the case 2 above,4. a union of a triangle in T , a triangle in T , and of a set-theoreticaldifference of two shapes - the square [0 , n ] × [0 , n ] and a triangle in ( n , n ) − T RI n , n . By the triangle inequality: (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) sup I ∈ TRI n,n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X ~i ∈ I a ~i f ~i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤≤ p mr (cid:16) T RI n (cid:17) ++ (cid:18) p mr (cid:16) T RI n (cid:17) + q p + p (cid:16) mr (cid:16) REC n , n (cid:17) + mr (cid:16) HT RI n , n (cid:17)(cid:17)(cid:19) ++ (cid:18) p mr (cid:16) T RI n (cid:17) + q p + p (cid:16) mr (cid:16) REC n , n (cid:17) + mr (cid:16) HT RI n , n (cid:17)(cid:17)(cid:19) ++ (cid:18) ( p + p + p ) mr (cid:16) T RI n (cid:17) + q p + p (cid:19) If we rearrange this estimate by the powers of mr ( T RI n ) , and apply theestimates (2) and (6) obtained for mr ( REC m,n ) and mr ( HT RI m,n ), thenwe get: mr ( T RI n ) ≤ (cid:0) p + p + p + ( p + p + p ) (cid:1) · mr (cid:16) T RI n (cid:17) ++ c √ n ln( n ) · mr (cid:16) T RI n (cid:17) + c n ln ( n ) . We can say that the inequality (7) claims that the vector ( p , p , p , p )belongs to the unit ball. This is why we can estimate a quadratic form ofthe variables p , p , p , p by its largest eigenvalue as follows: p + p + p + ( p + p + p ) ≤ γ = 2 + √ . Consequently, mr ( T RI n ) ≤ √ γ · mr (cid:16) T RI n (cid:17) + c √ n ln( n ) . The proof of our theorem now follows from the Master Theorem. eferences [1] D. Menchoff, “Sur les series de fonctions orthogonales” , Fund. Math.,Vol. 4 (1923), pp. 82105.[2] H. Rademacher, “Einige Satze uber Reihen von allgemeinen Orthogo-nalfunktionen” , Math. Ann., Vol. 87 (1922), Issue 1-2, pp. 112138.[3] W. Bednorz, “A Note on a Mienshov-Rademacher Inequality” , Bull.Acad. Polon. Sci., Vol. 54 (2006), No. 1, pp. 89-93.[4] E. Kounias, “A note on Rademacher’s inequality” , Acta Math. Hun-gar., Vol. 21 (1970), Issue 3-4, pp. 447448.[5] S. Chobanyan, “Some remarks on the MenshovRademacher functional ,Math. Notes, Vol. 59 (1996), Issue 5, pp. 571574.[6] J.L. Bentley, D. Haken, and J.B. Saxe, “A general method for solvingdivide-and-conquer recurrences” , SIGACT News, Vol. 12 (1980), No.3, pp. 36-44., SIGACT News, Vol. 12 (1980), No.3, pp. 36-44.