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Which Moments of a Logarithmic Derivative Imply Quasiinvariance?

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Which Moments of a Logarithmic Derivative Imply Quasiinvariance? Doc Math J DMV Which Moments of a Logarithmic Derivative Imply Quasi invariance Michael Scheutzow Heinrich v Weizsacker Received June Communicated by Friedrich Gotze Abstract In many sp ecial contexts quasiinvariance of a measure under a oneparameter group of transformations has b een established A remarkable classical general result of AV Skorokhod states that a measure on a Hilb ert space is quasiinvariant in a given direction if it has a logarithmic aj j derivative in this direction such that e is integrable for some a In this note we use the techniques of to extend this result to general oneparameter families of measures and moreover we give a complete char acterization of all functions for which the integrability of j j implies quasiinvariance of If is convex then a necessary and sucient condition is that log xx is not integrable at Mathematics Sub ject Classication A C G Overview The pap er is divided into two parts The rst part do es not mention quasiinvariance at all It treats only onedimensional functions and implicitly onedimensional measures The reason is as follows A measure on R has a logarithmic derivative if and only if has an absolutely continuous Leb esgue density f and is given by 0 f x ae Then the integrability of jj is equivalent to the Leb esgue x f 0 f jf The quasiinvariance of is equivalent to the statement integrability of j f that f x Leb esgueae Therefore in the case of onedimensional measures a function allows a quasiinvariance criterion as indicated in the abstract i for all 0 f jf implies that f is absolutely continuous functions f the integrability of j f strictly p ositive The main result of the rst part Theorem gives necessary and sucient reformulations of this prop erty which are easier to check The most simple R of these reformulations is the divergence of the integrals log xx dx where c is the lower nondecreasing convex envelope of Moreover we give for every with this prop erty explicit lower estimates for the values of f on an interval I in terms Documenta Mathematica Michael Scheutzow Heinrich v Weizsacker R 0 f jf dx Finally we give an j of the length of this interval and of the integral f I example showing that the introduction of the lower convex hull in these results really is necessary The second part of the pap er then proves that the onedimensional situation is typi cal The quasiinvariance criterion works on the real axis if and only if it works for the transp ort of a measure under an arbitrary measurable ow or even more generally for general one parameter families of measures which are dierentiable in the sense of If this criterion applies then one gets even the typical CameronMartin type formula for the RadonNikodymdensities b etween the members of such a family cfeg In the situation of Skorokhods result mentioned in the ax summary we see that the exp onential functions x e a can b e replaced by x x exp but not by exp This shows that Skorokhods exp onential criterion 2 log x log x is not strictly optimal but it gives the optimal p ower of log A class of onedimensional functions Theorem For a measurable function the fol lowing six condi tions are equivalent A Let f R be absolutely continuous such that Z f xj f xdx j f and f Then f x for Lebesgueall x R 0 f xj f x is A Let f R be absolutely continuous such that x j f locally Lebesgue integrable and f Then f x for al l x R B For some a the fol lowing implication holds X X ai z z z e i i i z i i i B The implication holds for al l a C Let be the largest nondecreasing convex function und suppose c Then Z log x dx x c C Similarly lim x and for d in the range of log x Z dx log x d Documenta Mathematica Which Moments of a Logarithmic Derivative In particular the conditions A B hold for if and only if they hold for If p is convex and nondecreasing and some power with p satises one of the conditions then the same is true for Proof Clearly A A P A B Let z and b z Dene f R by i i i s z z i i f s exp z i i i for s z z z z i Note that e f s e i i 0 f and x log f x in this interval Moreover set f s for s b and f z i f s f s for s Then f is absolutely continuous but not strictly p ositive Therefore by assumption A the integral in diverges Hence Z z z 1 i X X i f xdx e z i z z i i z z 1 i1 i i Z b f xj f xdx j f which proves B with a B B Denote by B the statement that B holds with the constant a a P z z Clearly B B if c b We prove B B Supp ose i i b c a a i P and let y y z for j N Then y and hence j j j j j X X ai aj a z e y e e i j z y i j i j From by z B C Let ht log t Dene the number z i i hi x B it follows easily that as x Thus the same holds for Since x is convex and increasing the function h is continuous and decreasing Therefore P for the pro of of it is sucient to prove that the sum z diverges i i P Supp ose on the contrary that z Cho ose y z such that y i i i i i y i c where i x c inf i 1 x x 2z i c i This is p ossible by denition of this inmum c The ane function l x c x i i i z i is since it is negative on and on even the larger function x c x i z z i i is b ounded by Therefore from the denition of we get c i l i z z z i i i We apply B with a and use the summability of the z and hence of the y to i i get X X X i i i y c e e e i i y i i i i Documenta Mathematica Michael Scheutzow Heinrich v Weizsacker i e by construction of the z thus gives Now i z i X X X i i z c e z e i i i z i i i i which is a contradiction C C Both and imply that is continuous nondecreasing and unbounded at innity Therefore there is some c such that is even strictly increasing on c and the assertion follows from lemma b elow applied to log C A Presumably this is the most useful implication We formulate the main part of the pro of as the separate Theorem since it involves only integrals over nite intervals and can b e applied also to functions which do not satisfy the conditions of the theorem In order to deduce our implication from Theorem assume and let x x Then lim log x log x and hence using the x equivalence of and we get Z dy log y 0 f xjf x is lo cally integrable then Now if f is absolutely continuous and x j f 0 f also the function x j xjf x is lo cally integrable and hence b elow gives f a lower b ound for the values of f on any interval s t such that f s The case f t follows by reection In particular f is strictly p ositive which is the assertion of A Finally we prove the last statement Let b e convex and nondecreasing and supp ose p p that satises one of the conditions If p then max and using p criterion B it follows that satises the same condition If p then by p p Jensens inequality is also convex nondecreasing and hence and p p Since log p log the criterion C carries over from to In the pro of we have used the following elementary fact Lemma Let c and let c d be a homeomorphism Then Z Z d x dx dy y x c c d ie both integrals converge at the same time and if they do holds Proof The change of variables y x gives Z Z Z T T T T x x dx dy dxx y x x x c c c d T Since for large T the lefthand side of dominates the righthand side T For the converse inequality assume that the integral on the righthand side of is nite The indenite integral on the lefthand side of is monotone in T so it has Documenta Mathematica Which Moments of a Logarithmic Derivative T a nite or innite limit Therefore by the limit lim exists and it must T T b e b ecause otherwise the integral on the righthand side of would b e innite This implies The following result gives a quantitative version of the implication C A in Theorem Theorem Let R be a convex even function with Let f s t be absolutely continuous such that f s Then min log f xf s sxt Z Z t f x f xdx dx t s f s f x log x s R R 0 y t f x f x dx b e nite Dene F y dx Remark Let I 1 f x log x s I t s which certainly is true for y If the range of F contains the number f s if F y for y then can b e rewritten as I f t f s exp F t s f s This gives a lower estimate of the uctuation of the function f in terms of the integral I and the length of the interval s t ax Remark In the sp ecial case x e there is an elegant more abstract pro of of prop erty A of Theorem see prop That pro of do es not give a lower b ound for the values of f in terms of I but on the other hand it works also in higher dimensions whereas our metho d is strictly onedimensional Proof Both sides of remain unchanged if f is multiplied by some p ositive constant Therefore we may and shall for notational convenience assume f s a ai inf fy s f y e g We also introduce the numbers For a i N let x i a a a a a a z x x z for i N and nally N supfn N x n a 1 i i i i log ai tg We apply Jensens inequality in the second step of the following estimates Documenta Mathematica Michael Scheutzow Heinrich v Weizsacker (a) Z Z N a x t X i f x f x ai e f x dx dx (a) f x f x x s i1 i (a) Z N a x X i f x a ai dx e z i a (a) f x x z i1 i i N a X a a a ai e z ln f x ln f x i i i a z i i N a X a a ai
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