t 0 Xt sup X t p c p inf t 0

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1 SHARP MAXIMAL L -ESTIMATES FOR MARTINGALES RODRIGO BAÑUELOS AND ADAM OSȨKOWSKI ABSTRACT. Let X be a suermartingale starting from 0 which has only nonnegative jums. For each 0 < < we determine the best constants c, C and c such that su X t C inf t 0 t 0 Xt, su X t c inf t 0 t 0 Xt and su X t c inf t 0 t 0 Xt. The estimates are shown to be shar if X is assumed to be a stoed one-dimensional Brownian motion. The inequalities are deduced from the existence of secial functions, enjoying certain majorization and convexity-tye roerties. Some alications concerning harmonic functions on Euclidean domains are indicated.. INTRODUCTION Suose that Ω, F, P is a comlete robability sace, filtered by F t t 0, a nondecreasing family of sub-σ-algebras of F, such that F 0 contains all the events of robability 0. Assume further that X = X t t 0 is a martingale on this filtration with right-continuous trajectories that have limits from the left. Define the associated suremum, infimum and both-sided suremum rocesses M + = M t + t 0, M = Mt t 0 and M = M t t 0 by the formulas M t + = su X s 0, Mt = inf X s 0 0 s t 0 s t and M t = su X t = Mt M t +, 0 s t where, as usual, a b = max{a, b} and a b = min{a, b}. We will also use the notation X t for X t X t, the jum of X at time t we assume that X 0 = 0 almost surely. The inequalities involving various sizes of X, M +, M and M have layed an imortant role in robability, esecially in the theory of stochastic rocesses and stochastic integration. For instance, recall the classical result of Doob [3]: we have M X, < <, and the constant / is the best ossible, even in the weaker estimates M + X, M X. 200 Mathematics Subject Classification. 60G40, 60G44, 3B05. Key words and hrases. Martingale, Suermartingale, maximal, harmonic function, best constant. R. Bañuelos is suorted in art by NSF Grant # DMS. A. Osȩkowski is suorted in art by NCN grant DEC-202/05/B/ST/0042.

2 2 RODRIGO BAÑUELOS AND ADAM OSȨKOWSKI Here and below, we will use the convention Y = su t 0 E Y t / for any semimartingale Y = Y t t 0. For =, the above L bound does not hold with any finite constant, but we have the following shar LlogL estimate see Gilat [5] and Peskir [6]: for any K >, M K su E X t log + X t + LK, t 0 where LK = + e K K is the best ossible. There are many versions of these results, and we refer the interested reader to the monograh [7] by Peskir and Shiryaev for an overview, resenting the subject from the viewoint of otimal stoing theory. We will be articularly interested in shar L bounds involving X, M +, M and M in the case 0 < <. It is well-known that in general such estimates do not hold with any finite constants unless we assume some additional regularity of the trajectories or the distribution such as continuity, conditional symmetry or nonnegativity. For instance, we have the following result, roved by Shao in [0]. Theorem.. Suose that X is a nonnegative martingale. Then for any 0 < < we have the shar bound M X. / We will be interested in a slightly different class of estimates. Motivated by the study Hardy s H saces for harmonic functions on the uer half-sace R n+ +, Burkholder obtained the following result. Theorem.2 Burkholder []. Suose that X is a martingale with continuous samle aths with X 0 = 0. If Φ is a nondecreasing continuous function on [0, such that Φ0 = 0 and Φβλ γφλ, for some β > γ > and all λ > 0, then. su t 0 EΦM t C su EΦ Mt, t 0 where C deends only on β and γ. In articular, if 0 < <, then.2 M C M. As Burkholder oints out see his Examle 6.3 in [], by stoing Brownian motion at the first time it hits, it follows that.2 does not hold for. Burkholder s roof of. uses good-λ inequalities. Over the years other roofs of.2 have been given, including the recent one in [9] which is written in terms of the functions M + and M. The inequality in [9] is alied to rove a stochastic Gronwall Lemma. The goal of this aer is to obtain the best constant in.2 and its variant roved in [9]. Actually, we will go much further and study a wider class of rocesses: our reasoning will enable us to obtain shar estimates for suermartingales which do not have negative jums. In the formulation of our main results, we will need some additional constants. A straightforward analysis of a derivative shows that there is unique 0 0, for which.3 0 = Comuter simulations show that Now, if 0, 0 ], let.4 α = and C =. On the other hand, if 0,, let α be the unique solution to the equation.5 α / + = α +

3 and set C = + α MARTINGALE INEQUALITIES 3 /. Next, for any 0,, let c = + s s + ds /. Finally, introduce the constant c by s / + c = s + ds if 0 < /2, s / + s + ds if /2 < <. Observe that c = c for 0 < /2; on the other hand, when /2,, the constant c is easily seen to be larger which will also be clear from the reasoning below. We are ready to state the results. The first theorem gives a shar comarison of L norms of a suermartingale and its infimum rocess. Theorem.3. Suose that X is an adated suermartingale with only nonnegative jums, satisfying X 0 = 0 almost surely. Then for any 0 < < we have.6 X C M and the constant C is the best ossible. It is already the best ossible if X is assumed to be a stoed Brownian motion. The second result comares the sizes of the suremum and the infimum rocesses. Theorem.4. Suose that X is an adated suermartingale with only nonnegative jums, satisfying X 0 = 0 almost surely. Then for any 0 < < we have.7 M + c M and the constant c is the best ossible. It is already the best ossible if X is assumed to be a stoed Brownian motion. Our final result is a shar version of the bounds.6 and.7, with two-sided maximal function on the left. This gives the best constant in Burkholder s estimate.2. Here is the recise statement. Theorem.5. Suose that X is an adated suermartingale with only nonnegative jums, satisfying X 0 = 0 almost surely. Then for any 0 < < we have.8 M c M and the constant c is the best ossible. It is already the best ossible if X is assumed to be a stoed Brownian motion. A few words about the aroach and the organization of the aer are in order. Our roofs of.6,.7 and.8 rest on the existence of a certain secial functions, and have their roots in the theory of otimal stoing. We resent them in the next section. In Section 3 we address the otimality of the constants C, c and c. The final section is devoted to the discussion on related results arising in harmonic analysis on Euclidean domains. 2. PROOFS OF.6,.7 AND.8 Throughout this section is a fixed number belonging to 0,. The contents of this section is slit naturally into three arts.

4 4 RODRIGO BAÑUELOS AND ADAM OSȨKOWSKI 2.. Proof of.6. As announced above, the argument deends heavily on the existence of an aroriate secial function. Consider U : R, 0] R defined by the formula Ux, z = α z [ x z ], where α is given by.4 or.5, deending uon the range of. Lemma 2.. The function U has the following roerties. i It is of class C on R, 0. ii For any z < 0 we have 2. U x x, z 0 and U z z, z = 0. iii If x z, then for any d 0 we have 2.2 Ux + d, z = Ux, z + U x x, z. iv If x z, then 2.3 Ux, z x C z. Proof. The first three arts are evident. The only nontrivial statement is the majorization 2.3. By homogeneity, it is enough to show it for z =. Let ϕx = Ux, and ψx = x C for x. The desired bound follows at once from the following four observations: ϕ is increasing, ψ is concave, decreasing on z, 0, and concave, increasing on 0,, ϕα / ϕ ψ. = ψα / and ϕ α / = ψ α /, The first three conditions are clear and follow from straightforward comutations. The latter observation emloys the definition of 0. Indeed, if 0, then +, which is equivalent 2.7. On the other hand, if > 0, then the definitions of α and C guarantee that we actually have equality in 2.7. This finishes the roof. Proof of.6. Let X = X t t 0 be an adated suermartingale starting from 0, which admits only nonnegative jums, and let ε > 0 be a fixed arameter. In view of Lemma 2. i, we may aly Itô s formula to U and the rocess Z ε = X t, Mt ε t 0. We refer the reader to Protter [8] for the general Itô formula used here. As the result of this alication, we get that for each t 0, 2.8 UZ ε t = I 0 + I + I 2 + I I 4,

5 MARTINGALE INEQUALITIES 5 where I 0 = UZ ε 0 = U0, ε, I = I 2 = I 3 = I 4 = 0<s t U x Z ε s dx s, U z Z ε s d M s ε, U xx Z ε s d[x, X] c s, [UZ ε s UZ ε s U x Z ε s X s ]. Note that due to the assumtion on the jums of X, the rocess M is continuous; in articular, this exlains why there is no summand U z Zs ε Ms ε in I 4. Now, let us analyze the behavior of the terms I through I 4 searately. The first of them has nonositive exectation, by the roerties of stochastic integrals. Indeed, if X = N +A is the Doob-Meyer decomosition for X see e.g. Protter [8], then we have I = U x Z ε s dn s + U x Z ε s da s. Now the first term has mean zero, while the second integral is nonositive, because of the first inequality in 2.9 and the fact that A is a nonincreasing rocess. To deal with I 2, we make use of the second condition in 2.. By the aforementioned continuity of M, we see that the rocess M s ε decreases only when X s = M s, i.e., when the coordinates of the variable Z ε s are equal. Then, as we have roved in 2., we have U z Z ε s = 0 and hence the integral I 2 is zero. The term I 3 also vanishes, since for a fixed z, the function x Ux, z is linear. Finally, each summand in I 4 is zero: this is guaranteed by 2.2 and the assumtion that X has only nonnegative jums. Thus, utting all the above facts together and lugging them into 2.8, we obtain or, in view of 2.3, E X t C E M t EUZ ε t U0, ε, ε + α ε. Letting ε 0 gives E X t CE M t, and it remains to take the suremum over t to obtain Proof of.7. Here the reasoning will be more involved. In articular, due to the aearance of the suremum rocess in.7, we are forced to consider secial functions of three variables corresonding to X, M + and M. Introduce U : R [0,, 0] R, given by Ux, y, z = y c z + x z z y/z if y > z, and Ux, y, z = c z + x z z r r + dr r r + dr

6 6 RODRIGO BAÑUELOS AND ADAM OSȨKOWSKI if y z. Let us rove some imortant facts concerning this object; they are gathered in the following statement, which can be regarded as the analogue of Lemma 2.. Lemma 2.2. The function U enjoys the following roerties. i For all z < 0 < y, the function U, y, z : x Ux, y, z is of class C 2 and the artial derivatives U y y, y, z, U z z, y, z exist. ii For all z < x < y we have 2.9 U x x, y, z 0, U y y, y, z = 0 and U z z, y, z 0. iii If z x y, then for any d 0 we have the bound 2.0 Ux + d, x + d y, z Ux, y, z + U x x, y, zd. iv If x z and y 0, then 2. Ux, y, z y c z. Proof. i This is straightforward; we leave the verification to the reader. ii The estimate for U x is evident. The identity U y y, y, z = 0 is also immediate, both for y > z and y z. To show the estimate for U z, note that if y z, then U z z, y, z = c z z by the formula for c. On the other hand, if y > U z z, y, z = c z z r dr = 0, r + z, we easily derive that y/z > c z z r r + dr r dr = 0, r + where the latter equality is again due to the definition of c. iii If x + d y, then both sides are equal, because of the linearity of x Ux, y, z. Therefore, suose that x+d > y and consider the following cases. If x+d z, then also y z and again 2.0 becomes an equality. If y < x + d, then 2.0 reads or x + d c z + x + d z z r x+d/z r + dr c z + x + d z z s F s := s + s r dr 0, r + z < r r + dr, with s = x + d/z. However, the function F vanishes for s =, and its derivative for s > is F s = s s + 2 0,

7 MARTINGALE INEQUALITIES 7 so F is indeed nonositive and 2.0 holds true. The final case we need to consider is when z < y < x + d. Then 2.0 takes the form x + d c z + x + d z z which can be rewritten as x+d/z y c z + x + d z z Gs := s y/z s + y/z s r dr 0, y/z r + r r + dr r r + dr, with s = x+d/z. To see that the latter estimate is valid, we observe that G y/z = 0 and G s = s y/z s rovided s > y/z. This roves the desired bound. iv If x z, then the terms in U involving the aroriate integrals are nonnegative. Therefore, we see that { } 2.2 Ux, y, z max y, z c z y c z. This yields the claim and comletes the roof of the lemma. Proof of.7. Here the reasoning is similar to that aearing in the roof of.6, so we will be brief. Pick an arbitrary adated suermartingale X = X t t 0 starting from 0, which has only nonnegative jums, and let ε > 0. Consider the rocess Z ε = X t, M t + ε, Mt ε t 0. By Lemma 2.2 i, we are allowed to aly Itô s formula to U and this rocess. As the result, we obtain that for t 0, 2.3 UZ ε t = I 0 + I + I 2 + I I 4, where I 0 = UZ ε 0 = U0, ε, ε, I = I 2 = I 3 = I 4 = 0<s t U x Z ε s dx s, U y Zs dm ε s +c ε + U z Zs dm ε s U xx Z ε s d[x, X] c s, [UZ ε s UZ ε s U x Z ε s X s ]. ε, Here M +c s ε s 0 denotes the continuous art of the rocess M + ε. Note that because of the aearance of this rocess in I 2, there is no corresonding term UZ ε s M + s ε in I 4. On the other hand, as in the roof of.6, the rocess M ε is continuous due to the assumtion on the sign of the jums of X. Now, we see that EI 0, by the roerties of stochastic integrals see the roof of.6 for a similar argument. Next, an alication of 2.9 gives that the first integral in I 2

8 8 RODRIGO BAÑUELOS AND ADAM OSȨKOWSKI is zero and the second is nonositive; again, see the analogous reasoning in the roof of.6. The term I 3 vanishes, since U xx is zero. Finally, each summand in I 4 is nonositive: this has been just roved in 2.0 above. Therefore, combining all the above facts, we see that EUZ ε t U0, ε, ε, or, by virtue of 2., E M t + ε c E Mt ε + U0, ε, ε. It remains to let ε 0 and then let t go to infinity. The roof is comlete Proof of.8. Finally, we turn our attention to the bound for the two-sided maximal function. The idea is to roceed exactly in the same manner as in the receding subsection. What roerties should the aroriate secial function have? A careful insection of the above roof shows that it is enough to find U enjoying the conditions of Lemma 2.2, with 2. relaced by Ux, y, z max{y, z} c z. However, the function U introduced in 2.2 does have this roerty when 0, /2]; see the first estimate in 2.2. Consequently, for these values of, the inequality.8 follows at once from the reasoning resented reviously note that c = c rovided 0 < /2. Thus, it remains to establish the desired bound in the range /2, only. Suose that lies in this interval and consider a function U : R [0,, 0] R defined by if y < z, and Ux, y, z = c z + z x z Ux, y, z = y c z + z x z y/z r r + dr r r + dr for remaining x, y, z. For the sake of comleteness, let us list the key roerties of this function in a lemma below. We omit the straightforward roof: analogous argumentation has been already resented in the roof of Lemma 2.2. Lemma 2.3. The function U enjoys the following roerties. i For all z < 0 < y, the function U, y, z is of class C 2 and the artial derivatives U y y, y, z, U z z, y, z exist. ii For all z < x < y we have U x x, y, z 0, U y y, y, z = 0 and U z z, y, z 0. iii If z < x < y, then for any d 0 we have iv If x z, then we have Ux + d, x + d y, z Ux, y, z + U x x, y, zd. Ux, y, z y z c z. Equied with this statement, we obtain the roof of.8 by a word-by-word reetition of the reasoning from 2.2. Since no additional arguments are needed, we omit the details, leaving them to the reader. 3. SHARPNESS For the sake of clarity, we have decided to slit this section into four arts. Throughout, B = B t t 0 denotes a standard, one-dimensional Brownian motion starting from 0.

9 MARTINGALE INEQUALITIES Otimality of C in.6, the case 0, 0 ]. Let β be an arbitrary ositive number smaller than and let δ > 0. Let τ 0 be defined by τ 0 = inf{t : B t {, β}}. Now, define the variable σ and the stoing times τ, τ 2,... by the following inductive rocedure. Suose that n is a given nonnegative integer. If B τn = β + δ n, then ut σ = n and τ n+ = τ n+2 =... = τ n ; on the other hand, if B τn = + δ n, then let τ n+ = inf{t > τ n : B t { + δ n+, β + δ n+ }. To gain some intuition about these random variables, let us look at the behavior of the sequence B τn n 0. The first ste is to wait until the Brownian motion reaches or β. If B τ0 = β, we sto the evolution. If B τ0 =, then we start the second stage and continue until B t reaches + δ or β + δ. If the second case occurs we sto but if B τ = + δ, then we start the third stage and wait until B reaches + δ 2 or β+δ 2. This attern is then reeated. We define the random variable σ to be the number of nontrivial stages which occur before the Brownian motion stos. Using elementary roerties of Brownian motion, we see that 3. Pσ = 0 = β + and, for any nonnegative integer n, 3.2 Pσ > n = β β + n β + δ +. β + + δ Hence, in articular, for any n =, 2,... we have 3.3 Pσ = n = β n β + δ + δ β + β + + δ β + + δ. Consequently, we see that η, the ointwise limit of the sequence τ n n 0, is finite almost surely. Put X = B η t t 0 and let us derive the -th norms of X η and M. By the very construction, X η = β + δ n on the set {σ = n}, so by 3. and 3.3, X η = E X η = β β + + n + δ n β β + δ + δ β + β + + δ β + + δ n= = β β + + β + δ δ β + δ + + δ n β + 2. β + Now observe that n= β + δ + + δ = + δ βδ + δ + β + β + = + + β β + δ + Oδ 2 as δ 0. Since β <, the above exression is less than for small δ. Thus X η is finite. On the other hand, it follows directly from the construction that M +δ n on {σ = n}, that is, we have the ointwise bound βm X η. This gives X X η β M, and since β < was arbitrary, the otimal constant in.6 cannot be smaller than.

10 0 RODRIGO BAÑUELOS AND ADAM OSȨKOWSKI 3.2. Otimality of C in.6, the case 0,. Let β be an arbitrary arameter smaller than α /. Fix a large ositive constant K, an even larger integer N and set δ = K/N. Let τ 0, τ, τ 2,... and σ be as in receding case. The main difference in comarison to the revious construction is that we ut X = B τn t t 0. That is, we terminate the Brownian motion after at most N stages. If n N, then M + δ n and X η = β+δ n on {σ = n}. Furthermore, M = X η = +δ N+ on {σ > N}. Consequently, using 3., 3.2 and 3.3, we derive that M βδ + δ β N β + δ + + δ n=0 β + δn+ β + A similar comutation shows that β + β + δ + β + δ + N n β+δ++δ N βδ + δ β+ = β + β + + δ βδ + δ β + δ β + δ + + δ N +. β + β + β+δ++δ N X η = β βδ + δ β+ β + β + + δ βδ + δ β + δ β + δ + + δ N +. β + β + Now let N go to infinity then δ = K/N converges to 0. The uer bound for M converges to while X η tends to β β e K K/β+ + β β + β β + ek K/β+, β e K K/β+ + β β + β β + ek K/β+. Consequently, the otimal constant in.6 cannot be smaller than β e K K/β+ / β + e K K/β+ e K K/β+ / β + e K K/β+, for any K. Letting K, we easily see that the exression above converges to β + β + β + = + β β +. However, recall that β was an arbitrary ositive constant smaller than α /. Letting β α / and using the definitions of α and C, we see that the exression above converges to C. This roves the sharness of the estimate.6.

11 MARTINGALE INEQUALITIES 3.3. Otimality of c in.7. Here the otimal stoing rocedure will be more comlicated. Let β be a given ositive number smaller than and let δ > 0. Define the stoing times τ 0, τ, τ 2,... and the variable σ with the use of the same formulas as in the receding cases. We will also need an additional stoing time η given as follows: if σ = n and hence B τn = β + δ n, then ut η = inf{t : B t = + δ n }. We easily check that η is a stoing time which is finite almost surely. Put X = B η t t 0 and let us comute the norms M + and M. By the above construction, we see that Mη = + δ n on {σ = n}, so E Mη {σ=n} = + δ n Pσ = n. Therefore, by 3. and 3.3, we have E Mη = + δ n Pσ = n n=0 = β β + + n + δ n β β + δ + δ β + β + δ + β + δ + n= { = β δ [ + δ ] n } β + δ + + β + β + δ + β + = β β + n= + δ + δ β + + δ + βδ β+. Now, if we let δ go to 0, we see that { } 3.4 E Mη β + β + β + β β+ = β2 β β + β. The analysis of EM + η is slightly more comlicated. Suose that σ = n. Then B τn = β + δ n and, using elementary roerties of Brownian motion, we see that for each y > β + δ n, PM + η y σ = n = PB reaches y before it reaches + δ n σ = n = β + + δn y + + δ n and hence the density of M + η, given that σ = n, is equal to Consequently, E [ M + η σ = n ] = gs = β+δ n β + + δn s + + δ n 2, s > β + δn. s β + + δ n s s + + δ n ds = β ++δn 2 β s + 2 ds and hence by 3.3 we obtain, after some straightforward maniulations, EM η + β + δ δ s β + s + 2 ds β + δ + + δ β + = β + δ δ β + β β n= s s + 2 ds + δ + β β+ δ. n

12 2 RODRIGO BAÑUELOS AND ADAM OSȨKOWSKI When δ goes to 0, the latter exression converges to Putting all the above facts together, we see that lim inf δ 0 β s s + 2 ds β β. β EM η + β + E Mη 2 β s s+ ds 2 However, β was an arbitrary ositive number smaller than /. If we let β go to / in the above exression on the right, we see that the otimal constant in.7 cannot be smaller than s / s + 2 ds. This is recisely c, which can be easily verified with the use of integration by arts Otimality of c in.8. If 0 < /2, then the sharness of.8 follows at once from 3.3, since.8 is stronger than.7. Therefore it is enough to study the case /2 < < only. The calculations are very similar to those in the receding section; however, some small but nontrivial changes are required, so we have decided to resent the details. Let τ 0, τ, τ 2,..., σ, η be defined by the same formulas for some fixed β < and ut X = B η t t 0. Now, for a given integer n and 0 < y + δ n, we see that P M η y σ = n =, since M η = + δ n on the set {σ = n}. For y > + δ n we have, as reviously, P M η y σ = n = β + + δn y + + δ n. Consequently, we derive that the conditional -th moment of M η is equal to E[M η σ = n] Therefore, by 3.3, EM η [ β 2 n= [ β = + β + 2 = + δ n + δn β 2 + δ n + +δ n. s β + + δ n s + + δ n 2 ds = β + δ n + β + + δ n s 2 s + 2 ds. + β + s ] βδ + δ n s + 2 ds β + 2 ] s s + 2 ds n β + δ + β + + δ β + δ β + β + + δ β + δ, where the geometric series converges due to the assumtion β <. Letting δ 0, we see that the latter exression converges to [ β β s ] + β + β + β 2 s + 2 ds.

13 MARTINGALE INEQUALITIES 3 Thus, we infer from 3.4 that the uer bound for the ratio EMη /E Mη cannot be smaller than [ β β β+ β 2 + β + ] s s+ ds β β + s s+ ds = 2. 2 β β2 β β+ β It suffices to note that the latter exression converges to c as β. This establishes the desired sharness. 4. HARMONIC FUNCTIONS IN DOMAINS OF R n In [] Burkholder roves an interesting version of inequalities. and.2 for harmonic functions in the uer half-sace R n+ + = {x, y : x R n, y > 0}. We briefly recall his result. If u is harmonic in R n+ +, we let N α u be its non-tangential maximal function defined by N α ux = su{ us, y : s, y Γ α x}, where Γ α x = {s, y : x s < αy} is the cone in R n+ + of aerture α. Setting u = u 0, we define the corresonding one-sided variant of the above object by N α ux = su{ u s, y : s, y Γ α x}. Theorem 4. Burkholder []. Suose u is harmonic in R n+ + satisfying u0, y = oy n, as y. If Φ is as in Theorem.2 then 4. Φ N α ux dx C Φ Nα ux dx, R n R n for some constant C deending on Φ, n and α. In articular, 4.2 N α u C,α,n N α u, 0 < <. It is shown in [,.45] that this inequality fails for. A similar result holds for harmonic functions in the ball of R n with the normalization u0 = 0. Using Theorem.5 and the classical fact that the comosition of a suerharmonic function with a Brownian motion is a suermartingale see Doob [3], we obtain the following robabilistic version of Burkholder s result. Theorem 4.2. Let D R n be a domain an oen connected set. Fix a oint x 0 D and let B = B t t 0 be Brownian motion starting at x 0 and killed uon leaving D. Denote by τ D its exit time from D. Assume further that u is a suerharmonic function in D satisfying the normalization condition ux 0 = 0. Define Mu = su t 0 ub t τd and M u = inf t 0 u B t τd. Then 4.3 Mu c M u, 0 < <, where c is the constant in Theorem.5. This inequality has an interesting alication for harmonic functions in the unit disc D = {z C : z < } in the lane. Suose u is harmonic in D and, as in the uer half-sace, define N α ue iθ and N α ue iθ where this time the suremum is taken over the Stoltz domain given by the interior of the smallest convex set containing the disc {z C : z < α} and the oint e iθ. Here, we assume 0 < α <. It is roved in Burkholder, Gundy and Silverstein [2] that there exists a constant k α deending only on α such that 4.4 m{θ : N α ue iθ > λ} k α P Mu > λ,

14 4 RODRIGO BAÑUELOS AND ADAM OSȨKOWSKI for all λ > 0. Here m denotes the Lebesgue measure on the circle. While the oosite inequality is stated in [2] for harmonic functions, it actually holds for subharmonic functions see also Durrett [4,.37] and we have that there exists a constant K α again deending only on α such that 4.5 P M u > λ K α m{θ : N α ue iθ > λ}. Combining 4.3, 4.4 and 4.5 we obtain Corollary 4.. Let u be a harmonic function in the unit disk D with u0 = 0. Then 4.6 N α u k α K α c N α u, 0 < <, where the constants k α, K α and c are, resectively, those aearing in 4.4, 4.5 and Theorem.5. In articular, the deendence on in the harmonic function inequality 4.6 is the same as in the martingale inequality.8. A similar inequality holds for harmonic functions on the uer half-sace R 2 + satisfying the hyothesis of Theorem 4.. REFERENCES [] D.L. Burkholder, One-sided maximal functions and H, J. Funct. Anal , [2] D.L. Burkholder, R. F. Gundy and M. L. Silverstein, A maximal function characterization of the class H, Trans. Amer. Math. Soc , [3] J. L. Doob, Stochastic rocesses, Wiley, New York, 953. [4] R. Durrett, Brownian Motion and Martingales in Analysis, Wadsworth, Belmont, CA, 984. [5] D. Gilat, The best bound in the LlogL inequality of Hardy and Littlewood and its martingale counterart, Proc. Amer. Math. Soc , [6] G. Peskir, Otimal stoing of the maximum rocess: The maximality rincile, Ann. Probab , [7] G. Peskir and A. Shiryaev, Otimal stoing and free boundary roblems, Lect. Notes in Math. ETH Zurich [8] P. E. Protter. Stochastic integration and differential equations, Sringer-Verlag, Berlin, second edition, [9] M. Scheutzow, A stochastic Gronwall lemma, Infin. Dimens. Anal. Quantum Probab. Relat. To., 6 203, 4 ages. [0] Q.-M. Shao, A comarison theorem on maximal inequalities between negatively associated and indeendent random variables, J. Theoret. Probab , DEPARTMENT OF MATHEMATICS, PURDUE UNIVERSITY, WEST LAFAYETTE, IN 47907, USA address: banuelos@math.urdue.edu DEPARTMENT OF MATHEMATICS, INFORMATICS AND MECHANICS, UNIVERSITY OF WARSAW, BA- NACHA 2, WARSAW, POLAND address: ados@mimuw.edu.l