AA Weighted Transfer Principle in Analysis
Sakin DemirFebruary 25, 2020
Abstract
The power of Calder´on transfer principle is well known when prov-ing strong type and weak type inequalities for certain type of operatorsin ergodic theory. In this article we show that Calder´on’s argument canbe extended to have a transfer principle to be able to prove weightedinequalities for those operators satisfying the condition of Calder´ontransfer principle. We also include some applications of our result.
Mathematics Subject Classifications:
Key Words:
Transfer Principle, Weighted Inequality.
Weighted inequalities has always received great attention in analysis. Theseinequalities has been studied in harmonic analysis extensively. B. Muck-enhoupt [7] first gave a complete characterization for those positive weightfunctions w for which Hardy-Littlewood maximal function maps L p ( w ) toitself for 1 < p < ∞ . R. Hunt et al [6] showed that same characteriza-tion also works for Hilbert transform. It was, of course, quite reasonable toask if there are these types of weighted inequalities in ergodic theory andthis problem has first been studied in E. Atencia and A. De La Torre [1]for ergodic maximal function with discrete parameter and it has first beenstudied in E. Atencia and F. J. Martin-Reyes [2] for ergodic Hilbert trans-form with discrete parameter. Weighted weak type inequalities for ergodicmaximal function and ergodic Hilbert transform with discrete parameter hasfirst been studied by E. Atencia and F. J. Martin-Reyes [3]. When proving1 a r X i v : . [ m a t h . C A ] F e b eighted weak type and strong type inequalities for ergodic maximal func-tion and ergodic Hilbert transform with discrete parameter all authors haveused the method of ergodic rectangles. On the other hand, it is well knownthat classical weak type and strong type inequalities for Hardy-Littlewoodmaximal function and Hilbert transform can be transferred to ergodic theoryby means of Calder´on transfer principle to find the corresponding results forergodic maximal function and ergodic Hilbert transform. It is, of course,quite reasonable to ask if one can have such a transfer principle to be ableto transfer certain weighted inequalities in harmonic analysis to ergodic the-ory. It is the goal of this research to answer this question. We show thatCalder´on’s argument can be modified to develop a transfer principle to beable to prove weighted inequalities for certain operators in ergodic theorysimilar to the classical case. Our result does not only transfer a weightedinequality with A p condition it transfers any weighted inequality as long asthe corresponding operator in harmonic analysis has such an inequality andit also transfers two-weight norm inequalities for the corresponding opera-tor. For example, weighted inequalities for ergodic maximal function andergodic Hilbert transform with continuous parameter become an immediateapplication of our transfer principle to the weighted inequalities for Hardy-Littlewood maximal function and Hilbert transform respectively. We alsoprove by using our result that ergodic square function of the difference ofdyadic ergodic averages with continuous parameter satisfy weak type andstrong type inequalities for those weight functions satisfying A (cid:48) p condition. We will assume that X is a measure space which is totally σ -finite and U t isa one-parameter group of measure-preserving transformations of X . We willalso assume that for every measurable function f on X the function f ( U t x ) ismeasurable in the product of X with the real line. T will denote an operatordefined on the space of locally integrable functions on the real line with thefollowing properties: the values of T are continuous functions on the realline, T is sublinear and commutes with translations, and T is semilocal inthe sense that there exists a positive number (cid:15) such that the support of T f is always contained in an (cid:15) -neighborhood of the support of f .We will associate an operator T (cid:93) on functions on X with such an operator T as follows: 2iven a function f on X let F ( t, x ) = f ( U t x ) . If f is the sum of two functions which are bounded and integrable, respec-tively, then F ( t, x ) is a locally integrable function of t for almost all x andtherefore G ( t, x ) = T ( F ( t, x ))is a well-defined continuous function of t for almost all x . Thus g ( x ) = G (0 , x )has a meaning and we define T (cid:93) f = g ( x ) . Theorem 1.
Let w and v be a non-negative measurable functions on X and T n be a sequence of operators as above and suppose that the operator Sf = sup | T n f | maps L p ( w ) to L p ( v ) for ≤ p ≤ ∞ . Then the same holdsfor the operator S (cid:93) f = sup | T (cid:93)n f | .Also, if Sf satisfies the weighted weak type inequality with respect to w for ≤ p ≤ ∞ so does S (cid:93) f .Proof. We modify the argument of A. P. Calder´on [4] to prove our theorem.Without loss of generality we may assume that the sequence T n is finite, for ifthe theorem is established in this case, the general case follows by a passageto the limit. Under this assumption the operator S has the same propertiesas the operator T above. We note that F ( t, U s x ) = F ( t + s, x ) , which means that for any two given values t , t of t , F ( t , x ) and F ( t , x )are equimeasurable functions of x . On the other hand, due to translationinvariance of S , the function G j ( t, x ) has the same property. In fact we have G ( t, U s x ) = S ( F ( t, U s x )) = S ( F ( t + s, x )) = G ( t + s, x ) . Let now F a ( t, x ) = F j ( t, x ) if | t | < a , F a ( t, x ) = 0 otherwise, and let G a ( t, x ) = S ( F a ( t, x )) . S is positive (i.e., its values are non-negative functions) and sublinear,we have G ( t, x ) = S ( F ) = S ( F a + (cid:15) + ( F − F a + (cid:15) )) ≤ S ( F a + (cid:15) ) + S ( F − F a + (cid:15) )and since F − F a + (cid:15) has support in | t | > a + (cid:15) , and S is semilocal, the last termon the right vanishes for | t | ≤ a for (cid:15) sufficiently large, independently of a .Thus we have G ≤ G a + (cid:15) for | t | ≤ a . Suppose that S satisfies a vector-valuedstrong L p norm inequality. Then since G (0 , x ) and G ( t, x ) are equimeasurablefunctions of x , we have2 (cid:90) X G (0 , x ) p w ( x ) dx = 1 a (cid:90) | t | E and (cid:101) E be the set of points where G (0 , x ) > λ and G a + (cid:15) ( t, x ) > λ ,respectively, and (cid:101) E y the intersection of (cid:101) E with the set { ( t, x ) : x = y } . Thenwe have 2 aw ( E ) ≤ w ( (cid:101) E ) = (cid:90) X w ( (cid:101) E x ) dx. On the other hand, since S satisfies weighted weak type inequality w ( (cid:101) E x ) ≤ C p λ p (cid:90) | F a + (cid:15) ( t, x ) | p dt. By using the above inequalities and the fact that F (0 , x ) and F ( t, x ) areequimeasurable we have aw ( E ) ≤ C p λ p ( a + (cid:15) ) (cid:90) | F (0 , x ) | p dx When we let a tend to ∞ we find the desired result. We say that a non-negative function w ∈ L ( R ) is an A p weight for some1 < p < ∞ ifsup I (cid:20) | I | (cid:90) I w ( x ) dx (cid:21) (cid:20) | I | (cid:90) I [ w ( x )] − / ( p − dx (cid:21) p − ≤ C < ∞ The supremum is taken over all intervals I ⊂ R n ; | I | denotes the measure of I . w is an A ∞ weight if given an interval I there exists δ > (cid:15) > E ⊂ I , | E | < δ · | I | = ⇒ w ( E ) < (1 − (cid:15) ) · w ( I ) . Here w ( E ) = (cid:90) E w ( x ) dx.w ∈ A if I ( w ) = 1 | I | (cid:90) I w ( x ) dx ≤ C ess inf I w I .It is well known and can easily be seen that w ∈ A ∞ implies w ∈ A p if1 < p < ∞ .Let f be a measureable function defined on R and consider the square func-tion S R f ( x ) = (cid:32) ∞ (cid:88) n = −∞ (cid:12)(cid:12)(cid:12)(cid:12) n f ∗ χ [0 , n ] ( x ) − n − f ∗ χ [0 , n − ] ( x ) (cid:12)(cid:12)(cid:12)(cid:12) (cid:33) / . It is proven in A. De La Torre and J. L. Torrea [5] that S R f satisfies theweak type weighted inequality for w ∈ A p with 1 ≤ p < ∞ and it satisfiesthe weighted strong type inequality for w ∈ A p with 1 < p < ∞ .Let now ( X, B , µ ) be an ergodic measure preserving dynamical system and { U t : −∞ < t < ∞} be a one-parameter ergodic measure preserving flow on X .Let us now define the ergodic square function as Sf ( x ) = ∞ (cid:88) n = −∞ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n (cid:90) n f ( U t x ) dt − n − (cid:90) n − f ( U t x ) dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) / for a measureable function f defined on X .We say that w satisfies condition A (cid:48) p if there exists a constant M suchthat for a.e. x ∈ X sup I (cid:20) | I | (cid:90) I w ( U t x ) dt (cid:21) (cid:20) | I | (cid:90) I [ w ( U t x )] − / ( p − dt (cid:21) p − ≤ M The supremum is taken over all intervals I ⊂ R ; | I | denotes the measure of I .Here w ∈ A (cid:48) if 1 | I | (cid:90) I w ( U t x ) dt ≤ C ess inf I w ( U t x )for all intervals I .When we apply our result to the above mentioned weighted weak type andstrong type inequalities for S R f , we see that Sf satisfies the weak typeweighted inequality for w ∈ A (cid:48) p with 1 ≤ p < ∞ and it satisfies the weighted6trong type inequality for w ∈ A (cid:48) p with 1 < p < ∞ .Let f be a measurable function on R , the Hardy-Littlewood maximalfunction M f is defined by
M f ( t ) = sup s s (cid:90) s | f ( t + u ) | du. B. Muckenhoupt [7] has proved that
M f maps L p ( w ) to L p ( w ) for 1 < p < ∞ when w satisfies the A p condition.Consider now the ergodic maximal function with continuous parameter de-fined by M (cid:93) f ( x ) = sup n n (cid:90) n − | f ( U t x ) dt | where f is a measureable function on X .When we apply our result to the above mentioned result of B. Mucken-houpt [7], we see that the ergodic maximal function M (cid:93) f maps L p ( w ) to L p ( w ) for 1 < p < ∞ when w satisfies the A (cid:48) p condition.Similarly when we apply we apply the second part of our result to theweighted weak type inequality of B. Muckenhoupt [7] for the Hardy-Littlewoodmaximal function M f we see that There exists a constant
C > λ > f ∈ L ( X ) (cid:90) { x : M (cid:93) f ( x ) >λ } w ( x ) dµ ≤ Cλ (cid:90) X f ( x ) w ( x ) dµ when w satisfies the A (cid:48) condition.Consider now the Hilber trasnform on R defined by Hf ( x ) = lim (cid:15) → + (cid:90) (cid:15) ≤| t |≤ /(cid:15) f ( t + u ) u du. When we apply our result to the weighted inequalities for Hf given in R.Hunt et al [ ? ] we see that the ergodic Hilbert transform of Cotlar defined as H (cid:93) f ( x ) = lim (cid:15) → + (cid:90) (cid:15) ≤| t |≤ /(cid:15) f ( U t x ) t dt L p ( w ) to L p ( w ) for 1 < p < ∞ when w satisfies the A (cid:48) p condition andThere exists a constant C > λ > f ∈ L ( X ) (cid:90) { x : H (cid:93) f ( x ) >λ } w ( x ) dµ ≤ Cλ (cid:90) X f ( x ) w ( x ) dµ when w satisfies the A (cid:48) condition. References [1] E. Atencia and A. De La Torre,
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