L 1 -smoothing for the Ornstein-Uhlenbeck semigroup

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1 L -smoothing for the Ornstein-Uhlenbeck semigrop K. Ball, F. Barthe, W. Bednorz, K. Oleszkiewicz and P. Wolff September, 00 Abstract Given a probability density, we estimate the rate of decay of the measre of the level sets of its evoltes by the Ornstein-Uhlenbeck semigrop. It is faster than what follows from the preservation of mass and Markov s ineqality. Introdction Let N. For t 0, consider the probability measre µ t = e t δ + +e t δ. We simply write µ for µ = δ + δ ). On the mltiplicative) grop, } N, we consider the semigrop of operators T t ) t 0 defined for fnctions f :, } N R by T t f = f µ N t. In other words, T t fx) = fx y)k t y) dµ N y), where K t y) = N i= + e t y i ). For A,..., N}, we define w A :, } N R by w A y) = N i= y i with the convention w =. This family, known as the Walsh system, forms an orthonormal basis of L ; } N, µ N ). Expanding the prodct in the definition of the kernel K t one readily checks that T t w A = e t carda) w A. The above formlations show that T s T t = T s+t, that T t is self-adjoint in L and preserves positivity and integrals with respect to µ N ). As a conseqence T t is a contraction from L p = L p ; } N, µ N ) into itself: T t f p f p for p. Actally, the hypercontractive estimate of Bonami [] and Beckner [] tells more: if < p < q < + and e t q p, then T t f q f p. Hence the semigrop improves the integrability of fnctions in L p provided p >. A challenging problem is to nderstand the improving effects of T t on fnctions f L. In the paper [5], Talagrand asks the following qestion: for t > 0, is there a fnction ψ t : [, + ) 0, + ) with lim + ψ t ) = +, sch that for every N and every fnction f on, } N with f, and all >, µ N x, T t fx) > }) ψ t )? ) This wold be a strong improvement on the following simple conseqence of Markov s ineqality and the contractivity property: µ N x, T t fx) > }) T tf f. Talagrand actally asks a more specific qestion with ψ t ) = ct) log) and he observes that one cannot expect a faster rate in. Qestion ) is still open; only in some special cases an affirmative answer is known see the last chapter). Its difficlty is essentially de to the lack of convexity of the tail Research of the third, forth and fifth named athors was partially spported by Polish MNiSzW grants PO3A 0 9 and N N

2 condition. Nevertheless, the paper [5] contains a similar reslt for the averaged operator M := 0 T t dt: there exists K sch that for all N and >, µ N x; Mfx) f } ) log log K log The goal of this note is to stdy the analoge of Qestion ) in Gass space. Gassian setting Let n. We work on R n with its canonical Eclidean strctre,, ). Denote by γ n the standard Gassian probability measre on R n : γ n dx) = e x / dx π) n/ Let G be a standard Gassian random vector, with distribtion γ n. Let f : R n R be measrable. Then the Ornstein-Uhlenbeck semigrop U t ) t 0 is defined by U t fx) = Ef e t x + e t G ) = f e t x + e t y ) e y / dy R π) n n/ = e t ) n/ fz)e z e t x) dz e t ) R π) n n/ = e t ) n/ e x / fz)e e t e t ) z et x) dγ n z), R n when f is nonnegative or belongs to L γ n ). The operators U t preserve positivity and mean. They are self-adjoint in L γ n ). By Nelson s hypercontractivity theorem [3], U t is a contraction from L p γ n ) to L q γ n ) provided < p q and p )e t q. It is natral to ask the analoge of Qestion ) for U t : does there exist a fnction ψ t with lim + ψ t ) = + sch that for all n and all nonnegative or γ n -integrable fnction f : R n R, }) γ n x, Ut fx) > f L γ n) ψ t )? ) This ineqality wold actally follow from Talagrand s conjectre on the discrete cbe. f : R n R is continos and bonded, consider the fnction g :, } nk R defined by g ) x, + + x,k x i,j ) i n,j k = f,..., x ) n, + + x n,k. k k Indeed, if By the Central Limit Theorem, when k goes to infinity, the distribtion of g nder µ nk tends to the one of f nder γ n, while the distribtion of T t g nder µ nk tends to the one of U t f nder γ n see e.g. []). This allows to pass from ) for g to ) for f. The above argment ses bondedness and continity. These assmptions can be removed by a classical trncation argment, and sing the semigrop property: U t f = U t/ U t/ f where U t/ f is atomatically continos. We omit the details. To conclde this introdction, let s provide evidence that the fnctions ψ t ) in ) cannot grow faster than log. We will do this for n =, which implies the general case by choosing fnctions depending on only one variable). We have showed that U t fx) = Q t x, z)fz) dγ z), 3) where R Q t x, z) = e t ) exp x z et x) ) ) e t.

3 We are going to choose specific fnctions f 0 with f dγ = for which U t f can be explicitly compted. Note that the previos formla allows to extend the definition of U t to nonnegative) measres ν with ϕ dν < + where ϕt) = e t / / π is the Gassian density. The simplest choice is then to take normalized Dirac masses δ y := ϕy) δ y as test measres. Obviosly ϕ d δ y = and U t δy = Q t, y). Actally, by the semigrop property, Q t, y) = U t/ U t/ δy = U t/ Q t/, y), where x Q t/ x, y) is a nonnegative fnction with nit Gassian integral. Hence, Qt, y); y R } U t/ f; f 0 and } f dγ =. Fix t > 0 and let > e t ) /. Then sing Q t x, y) = Q t y, x) and setting v = e t one readily gets that x; Q t x, y) > } = x, exp y x et y) ) ) e t > v} = e t y e t )y log v) + ; e t y + ) e t )y log v) +. For the particlar choice y = y 0 := e t log v, one gets x; Q t x, y 0 ) > } = log v; e t ) ) log v. Since for 0 < a < b, γ a, b)) b s / a b ds/ / π = e s e a e b / b, we can dedce that π γ x; Qt x, y 0 ) > } ) ) πe t ) log v v. v et ) Combining the above observations yields lim inf log sp γ x; Ut/ fx) > } ) ; f 0 and + } f dγ = > 0. Hence ψ t ) in ) cannot grow faster than log. Using the same one-dimensional test fnctions and similar calclations, one can check that for t > 0, the image by U t of the nit ball B = f L γ n ); f } is not niformly integrable, that is: lim inf sp U t f Utf >c dγ n > 0. c + f B Conseqently U t : L γ n ) L φ γ n ) is not continos when φ is a Yong fnction with lim t + φt)/t = +. Next, we trn to positive reslts. 3 Main reslts In the rest of this section Ba, r) denotes the closed ball of center a and radis r, while Ca, r, r ) = x R n ; r x a r }. We start with an easy inclsion of the pper level-sets of U t f. Lemma. Let f : R n R + be sch that f dγ n =. Then for all t, > 0, } x R n ; U t fx) > B 0, log + n log e t ) ) c. +) Proof. As already explained U t fx) = e t ) n/ e x / fz)e e t e t ) z et x) dγ n z). R n Conseqently U t fx) e t ) n/ e x / f dγ n. Or normalization hypothesis then implies that x; Ut fx) > } x; x > log + n log e t ) }. 3

4 The probability measre of complements of balls appearing in the above lemma can be estimated thanks to the following classical fact. Lemma. For all n N there exists a constant c n sch that for all n it holds γ n B0, ) c ) c n n e /. Actally, when n this is valid for all > 0. Also one may take c = /π. Proof. Polar integration gives that γ n B0, ) c ) = π) n/ n vol n B0, )) + r n e r / dr. For n 4 the map r r n e r /4 is non-decreasing on, ). Ths we may bond the last integral: r n e r /4 re r /4 dr n e /4 re r /4 dr = n e /. Combining the previos statements gives a satisfactory estimate in dimension, which improves on the Markov estimate γ n U t f ) min, /) if f is non-negative with integral. Proposition 3. Let f : R R + be integrable. Then for all t > 0 and v >, f dγ }) γ x; U t fx) > v e t v π log v In higher dimension, the above reasoning gives a weaker estimate than Markov s ineqality. However a more precise approach allows to get a slightly weaker decay for the level sets of U t f. Or main reslt is stated next. It contains a dimensional dependence that we were not able to remove. Theorem 4. Let n and t > 0. Then there exists a constant Kn, t) sch that for all non-negative fnctions f defined on R n with f dγ n =, for all > 0, }) γ n x R n ; U t fx) > log log Kn, t) log Proof. Note that it is enogh to show the ineqality for larger than some nmber 0 n, t) > 0 depending only of n and t. We will jst write that we choose large enogh, bt an explicit vale of 0 n, t) can be obtained from or argment. Let s define R = R, n, t) := log + n log e t ) ) +, R = R, n) := log + n ) log log ). It is clear that for large enogh R > n and also R > R > 0. So by Lemma, γ n B0, R ) c ) c n e R / R n = c n log + n ) log log ) n log ) n c n n + ) log ) n log ) n = c n log 4

5 By Lemma, x; U t fx) > } is a sbset of B0, R ) c. Hence we may write, sing Markov s ineqality on the annls C0, R, R ) and the self-adjointness of U t : γ n x; Ut fx) > } ) γ n x; Ut fx) > } B0, R ) ) + γ n B0, R ) c) ) = γ n x; U t fx) > } B0, R ) c B0, R ) + γ n B0, R ) c) Ut f C0,R,R )dγ n + γ n B0, R ) c) = U t C0,R,R )) f dγ n + γ n B0, R ) c) U t C0,R,R ) + c n log To prove the theorem, it remains to show that U t C0,R,R ) = O log log log ). First note that for any set A R n and all x R n, Therefore U t A x) = E A e t x + e t G ) = P G A e t x ) A e t x ) = γ n. e t e t U t C0,R,R ) = sp a R n γ n Ca, R, R ) ), where R i := R i / e t. The main idea is the above shells can be covered by a thin slab and the complement of a large ball. Set r = r) := log log ), then for large enogh, Lemma yields For an arbitrary point a R n, γ n B0, r) c ) n c n e r / r n n log log ) = c n log ) c log log n. log γ n Ca, R, R ) ) γ n Ca, R, R ) B0, r) ) + γ n B0, r) c ) γ n Ca, R, R ) B0, r) ) + c log log n. log For large enogh, r < R, and the forthcoming Lemma 5 ensres that Ca, R, R ) B0, r) is contained in a strip S of width w := R R r. By the prodct properties of the Gassian measre, γ n S) coincides with the one-dimensional Gassian measre of an interval of length w. Therefore it is not bigger than w/ π w. Hence γ n Ca, R, R ) B0, r) ) R R r log + n ) log log log + n log e t ) = e t e t 4 log log n + 4 e t )) log log n log e t ) e t log + n ) log log log log κn, t), log where the last ineqality is valid for large enogh. The proof of the theorem is therefore complete. Lemma 5. Let 0 < r < ρ < ρ and a, b R n, then the set Ca, ρ, ρ ) Bb, r) is contained in a strip of width at most ρ ρ r. 5

6 Proof. Assme that the intersection is not empty. Then withot loss of generality, a = 0 and b = te with t > 0. Let z be an arbitrary point in the intersection. Obviosly z z ρ. Next, since z Bb, r) and z ρ, one gets r z te = z tz + t ρ tz + t. Hence by the arithmetic mean-geometric mean ineqality z ρ r ) + t t Smmarizing, z [ ρ r, ρ ] R n. ρ r. 4 Prodct fnctions on the discrete cbe Finally, we provide an affirmative answer to the Qestion ) in the case of fnctions with prodct strctre. Proposition 6. Assme that fnctions f, f,..., f N :, } [0, ) satisfy f i dµ = for i =,,..., N. Let f = f f... f N, i.e. fx) = N i= f ix i ). Then for every t > 0 there exists a positive constant c t sch that for all > there is µ N x, T t fx) > }) c t log. Proof. The above reslt is immediately implied by the following ineqality. Proposition 7. [4]) Let b > a > 0. Let X, X,..., X N be independent non-negative random variables sch that EX i = and a X i b a.s. for i =,,..., N. Then for every > we have N P X i > ) C + log ) /, where C is a positive constant which depends only on a and b. i= Indeed, T t f = T t f T t f... T t f N, where T t f i :, } [ e t, + e t ] satisfy T t f i dµ = fi dµ = for i =,,..., N. Ths random variables X, X,..., X N defined on the probability space, } N, µ N ) by X i x) = T t f i x i ) satisfy assmptions of Proposition 7 with a = e t and b = +e t while f = N i= X i. References [] W. Beckner, Ineqalities in Forier analysis. Ann. of Math. ) 0 975), no., [] A. Bonami, Etde des coefficients de Forier des fonctions de L p G). French) Ann. Inst. Forier Grenoble) fasc ). [3] E. Nelson, The free Markoff field. J. Fnctional Analysis 973), 7. [4] K. Oleszkiewicz, On Eaton s property. In preparation) [5] M. Talagrand, A conjectre on convoltion operators, and a non-dnford-pettis operator on L. Israel J. Math ), no.,

7 K. Ball: Department of Mathematics University College London. Gower Street. London WCE 6BT. UNITED KINGDOM. F. Barthe: Institt de Mathématiqes de Tolose CNRS UMR 59). Université Pal Sabatier. 306 Tolose cedex 09. FRANCE. W. Bednorz: Institte of Mathematics. University of Warsaw. Banacha Warszawa. POLAND. K. Oleszkiewicz: Institte of Mathematics. University of Warsaw. Banacha Warszawa. POLAND and Institte of Mathematics. Polish Academy of Sciences. Śniadeckich Warszawa. POLAND. P. Wolff: Department of Mathematics. Case Western Reserve University Eclid Avene Cleveland, OH USA. 7