DIMENSION FREE PROPERTIES OF STRONG MUCKENHOUPT AND REVERSE HÖLDER WEIGHTS FOR RADON MEASURES
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1 DIMENSION FEE POPETIES OF STONG MUCKENHOUPT AND EVESE HÖLDE WEIGHTS FO ADON MEASUES O BEZNOSOVA AND A EZNIKOV ABSTACT In this paper we prove self-improvement properties of strong Muckenhoupt and everse Hölder weights with respect to a general adon measure on n We derive our result via a Bellman function argument An important feature of our proof is that it uses only the Bellman function for the one-dimensional problem for Lebesgue measure; with this function in hand, we derive dimension free results for general measures and dimensions Keywords: Bellman function, Dimension free estimates, Muckenhoupt weights, everse Hölder weights Mathematics Subject Classification: Primary: 42B35; Secondary: 43A85 INTODUCTION It is well known that Muckenhoupt weights on a real line with respect to the Lebesgue measure satisfy self-improvement properties in the following sense: for p > q we always have A q A p ; but also for any function w A p, there is an ε > 0 such that w A p ε (we refer to Definition for precise definitions) Besides that, there always exists a q such that w H q These self-improvement properties allow one to prove many important results in harmonic analysis, see, eg, [4] or a more recent paper [5] In [7], the authors considered strong Muckenhoupt classes; in particular, it was proven that for a adon measure µ on n which is absolutely continuous with respect to the Lebesgue measure dx, any weight w A p satisfies a everse Hölder property with an exponent that does not depend on the dimension n For p >, we say that w belongs to the strong Muckenhoupt class with respect to µ, w A p, if there exists a number Q > such that for any rectangular box n with edges parallel to axis, we have w w /(p ) p Q, where ϕ denotes the average of the function ϕ over : ϕ := ϕ(x)dµ(x) µ() For p >, we say that w belongs to the strong everse Hölder class with respect to µ, w Hp, if there exists a constant Q > such that for any rectangular box with edges parallel to axis, we have w p /p Q w We proceed with the following definition Date: February 5, 208
2 2 O BEZNOSOVA AND A EZNIKOV Definition Let µ be a adon measure on n and w be a function which is positive µ-ae For p >, we denote the A p-characteristic of w by [w] p := sup w w /(p ) p and the everse Hölder characteristic of w by [w] Hp := sup w p /p w where both suprema are taken over rectangular boxes with edges parallel to axis If [w] p <, we have w A p and if [w] Hp <, we have w H p In [7] it was proved that if µ is an absolutely continuous adon measure on n and [w] p <, then for some q > we have [w] Hq < C < with an explicit dimension free estimates on q and C It is of a particular importance that we can take q = + 2 p+2 [w] p To prove this result, the authors used a clever version of the Calderón Zygmund decomposition from [6] The aim of this paper is to derive a sharp result from the one-dimensional case for Lebesgue measure (ie, for the classical A p and H p classes on ) In this case the result from [7] can be obtained, for example, by means of a so-called Bellman function; ie, a function of two variables that satisfies certain boundary and concavity conditions in its domain In the one-dimensional case this function is known explicitly, see [0] It has been understood for some time that, for classes of functions like A p, H p or BMO p, when we work with their strong multi-dimensional analogs (eg, A p and H p), the one-dimensional Bellman function should prove the higher-dimensional results with dimension free constants For the Lebesgue measure and the inclusion H p A q, this was done in [] The trick of using the Bellman function for one-dimensional problems was also used in [3], [2] and [9] (in a slightly different setting, the same trick was also used in [8]) In this paper, we present a simple version of this trick for general measures; we prove the result from [7] as well as all other results of self-improving type for strong Muckenhoupt and everse Hölder weights Acknowledgements: we are very grateful to Vasiliy Vasyunin for helpful discussions and suggestions on the presentation of this paper, 2 STATEMENT OF THE MAIN ESULT 2 Properties of Muckenhoupt weights A p For p := /(p ) and every t [0,] define u ± p (t) to be solutions of the equation ( u)( p u) /p = t The function u + p is decreasing and maps [0,] onto [0,]; the function u p is increasing and maps [0,] onto [/p,0] For a fixed Q >, define () s ± p = s ± p (Q) := u ± p (/Q) Our first main result is the following
3 STONG MUCKENHOUPT AND EVESE HÖLDE WEIGHTS 3 Theorem 2 Let µ be a adon measure on n with µ(h) = 0 for every hyperplane H orthogonal to one of the coordinate axis Fix numbers p > and Q > and set p := /(p ) Then for every weight w with [w] p = Q we have and w A q, s p (Q) < q <, w H q, q < /s + p (Q), where s ± p (Q) are defined in () These ranges for q are sharp for n = and µ = dx 22 Properties of everse Hölder weights For p > and every t [0,] we define v ± p (t) to be solutions of the equation ( pv) /p ( v) = t In this case, v + p is a decreasing function that maps [0,] onto [0,/p] and v p is an increasing function that maps [0,] onto [,0] As before for a fixed Q >, we define (2) s ± p = s ± p (Q) := v ± p (/Q) Our second main result concerning everse Hölder weights is the following Theorem 22 Let µ be a adon measure on n with µ(h) = 0 for every hyperplane H orthogonal to one of the coordinate axis Fix numbers p > and Q > Then for every weight w with [w] Hp = Q we have and w A q, s p (Q) < q <, w H q, q < /s + p (Q), where s ± p (Q) are defined in (2) These ranges for q are sharp for n = and µ = dx 3 POOF OF THE MAIN ESULTS We begin with the following Theorem from [0] This theorem ensures the existence of a certain Bellman function for a one-dimensional problem In what follows, by letters without sub-indices (eg, x, x ± ) we denote points in 2 and by letters with sub-indices we denote the corresponding coordinates (eg, x + denotes the first coordinate of x+ ) Theorem 3 (Theorem in [0]) Fix p > and set p := /(p ) Also fix an r (/s p, p ] [,/s + p ) for s ± p (Q) defined in () For every Q > there exists a nonnegative function B Q (x) defined in the domain Ω Q := {x = (x,x 2 ) 2 : x x /p 2 Q} with the following property: B Q (x) is continuous in x and Q, and for any line segment [x,x + ] Ω Q and x = λx + ( λ)x +, λ [0,], we have B Q (x) λb Q (x ) + ( λ)b Q (x + ) Moreover, B(x,x p ) = xr and B Q(x) c(r,q)x r for some positive constant c(r,q) and every x Ω Q To use the concavity property of the function B Q for our proof, we need the following lemma Its proof is given in [0, Lemma4] with an interval instead of the rectangle; however, the proof remains the same in our case
4 4 O BEZNOSOVA AND A EZNIKOV Lemma 32 Let the measure µ be as before Fix two numbers Q > Q > and a rectangular box n with edges parallel to the axis For every coordinate vector e, there exists a hyperplane H normal to e that splits into two rectangular boxes and 2 with the following properties: (i) For i =,2 we have µ( i )/µ() (c, c) for some constant c (0,); (ii) For every weight w with [w] p Q, we have [x,x 2 ] Ω Q and, therefore, where B Q (x,x 2 ) µ( ) µ() B Q (x,x 2) + µ(2 ) µ() B Q (x i 2,x2 2), x = w, x 2 = w p x i = w i, xi 2 = wp i We are ready to prove our main result Proof of Theorem 2 Fix a rectangular box with edges parallel to the axis, and take any Q > Q We first explain how we split into two rectangular boxes Take one of the (n )-dimensional faces of, call it n, that has the largest (n )-area Among all (n 2)-dimensional faces of n, take one of those (call it n 2 ) that have the largest (n 2)-area We proceed like this to get n,, Now take a vector e that is orthogonal to every i, i =,,n We now split according to Lemma 32 Notice that all the corresponding i-dimensional faces of and 2 have smaller i-areas than the corresponding i-dimensional faces of We now take the boxes and 2 and repeat the same procedure If we repeat this M times, we get a family of rectangular boxes = { i,m } i=2 M Denote x = w, x 2 = w p x i,m = w, i,m xi,m 2 = w p i,m Abusing the notation, we also define step-functions x M (t) := 2 M x i,m 2 M i,m(t), xm 2 (t) := x i,m 2 i,m(t) i= i= From the construction of rectangular boxes, we notice that x M (t) w(t) and xm 2 (t) w p (t) as M for µ-ae t Indeed, our splitting procedure (and the fact that we have µ( i )/µ() ( c,c) at every step) guarantees that max diam(i,m ) 0, M, i=,,2 M and we obtain the convergence of x M(t) and xm 2 (t) from the Lebegue differentiation theorem for adon measures Therefore, B Q (x,x 2 ) 2 M µ( i,m ) i= µ() B Q (x i,m,xi,m 2 ) = µ() In the case n = 2, we just take e orthogonal to the longest side of B Q (x M (t),x M 2 (t))dµ(t)
5 By the Fatou lemma, (3) B Q (x,x 2 ) µ() STONG MUCKENHOUPT AND EVESE HÖLDE WEIGHTS 5 lim B Q M (x M (t),x2 M (t))dµ(t) = µ() B Q (w(t),w p (t))dµ(t) = µ() w r (t)dt = w r Since B Q (x,x 2 ) is continuous in Q and the above estimate holds for any Q > Q, we get c(r,q) w r = c(r,q)xr w r If we use this estimate for q = r [,/s + p ), we obtain w Hq If we use this estimate for /(q ) = r (/s p, p ], we obtain w A q for s p (Q) < q < To prove Theorem 22 we need to use a different Bellman function B Q Namely, the following result holds Theorem 33 (Theorem in [0]) Fix p > and r (/s p,] [p,/s + p ) for s ± p defined in (2) For every Q > there exists a non-negative function B Q (x) defined in the domain Ω Q := {x = (x,x 2 ) 2 : x x /p 2 Q} with the following property: B Q (x) is continuous in x and Q, and for any line segment [x,x + ] Ω Q and x = λx + ( λ)x +, λ [0,], we have B Q (x) λb Q (x ) + ( λ)b Q (x + ) Moreover, B(x,x p ) = xr and B Q(x) c(r,q)x r for some positive constant c(r,q) and every x Ω Q We also notice that the analog of Lemma 32 reads the same, and with this in hand, the proof of Theorem 2 is analogous to the proof of Theorem 22; we leave the details to the reader EFEENCES [] M Dindoš and T Wall The sharp A p constant for weights in a reverse-hölder class ev Mat Iberoam, 25(2): , 2009 [2] K Domelevo, S Petermichl, and J Wittwer A linear dimensionless bound for the weighted iesz vector Bull Sci Math, 4(5): , 207 [3] O Dragičević and A Volberg Bellman functions and dimensionless estimates of Littlewood-Paley type J Operator Theory, 56():67 98, 2006 [4] J García-Cuerva and J L ubio de Francia Weighted norm inequalities and related topics, volume 6 of North-Holland Mathematics Studies North-Holland Publishing Co, Amsterdam, 985 Notas de Matemática [Mathematical Notes], 04 [5] T Hytönen and C Pérez Sharp weighted bounds involving A Anal PDE, 6(4):777 88, 203 [6] A A Korenovskyy, A K Lerner, and A M Stokolos On a multidimensional form of F iesz s rising sun lemma Proc Amer Math Soc, 33(5): , 2005 [7] T Luque, C Pérez, and E ela everse Hölder property for strong weights and general measures J Geom Anal, 27():62 82, 207 [8] F Nazarov, A eznikov, S Treil, and A Volberg A Bellman function proof of the L 2 bump conjecture J Anal Math, 2: , 203 [9] S Treil Sharp A 2 estimates of Haar shifts via Bellman function In ecent trends in analysis, volume 6 of Theta Ser Adv Math, pages Theta, Bucharest, 203 [0] V I Vasyunin Mutual estimates for L p -norms and the Bellman function Zap Nauchn Sem S- Peterburg Otdel Mat Inst Steklov (POMI), 355(Issledovaniya po Lineınym Operatoram i Teorii Funktsiı 36):8 38, , 2008
6 6 O BEZNOSOVA AND A EZNIKOV DEPATMENT OF MATHEMATICS, UNIVESITY OF ALABAMA address: ovbeznosova@uaedu DEPATMENT OF MATHEMATICS, FLOIDA STATE UNIVESITY address: reznikov@mathfsuedu
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