SHARP INEQUALITIES FOR HILBERT TRANSFORM IN A VECTOR-VALUED SETTING

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1 SHARP INEQUALIIES FOR HILBER RANSFORM IN A VECOR-VALUED SEING ADAM OSȨKOWSKI Abstract. he paper is devoted to the study of the periodic Hilbert transform H in the vector valued setting. Precisely, for any positive integer N we determine the norm of H as an operator from L ; l N ) to Lp ; l N ), p <, and from L p ; l N ) to L ; l N ), for < p. he proof rests on the existence of a certain family of special harmonic functions.. Introduction he motivation for the results obtained in this paper comes from a very natural question about the periodic Hilbert transform and its action on vector-valued functions. Let us start with some historical perspective. Consider the trigonometric polynomial fθ) = a N 2 ak cos kθ b k sin kθ ), θ [, ], k= where a, a,..., a N, b, b 2,..., b N are real coefficients. hen the polynomial conjugate to f or the periodic Hilbert transform of f) is given by Hfθ) = a k sin kθ b k cos kθ). k= A natural question, which interested many mathematicians during the first half of 2th century, can be roughly stated as follows: how does the size of f control the size of its conjugate? Here the sizes of the polynomials can be measured, for instance, in terms of the usual L p -norms with respect to the normalized Haar measure: ) /p f p = fθ) p dθ when p <, and f = esssup θ fθ). By the orthogonality of the trigonometric system, we immediately see that Hf 2 f 2, and this estimate is clearly sharp the equality can be attained for some nontrivial choice of f). Concerning other values of p, Marcel Riesz [23], [24] proved that.) Hf p C p f p, < p <, for some constant C p which depends only on p and not on the coefficients of the polynomial f); furthermore, when p = or p =, then the corresponding estimate does not hold with any finite constant. he optimal value of C p was determined Key words and phrases. Hilbert transform, martingale, best constants. Research supported by the NCN grant DEC-22/5/B/S/42.

2 2 ADAM OSȨKOWSKI almost 5 years later by Pichorides [22] see also Gokhberg and Krupnik [7]): the optimal choice turns out to be max{cot/2p), tan/2p)}, < p <. he above results of Riesz, together with his related works on interpolation, have had a profound influence on the shape of contemporary analysis and have been extended in numerous directions. In particular, ten years after the discovery of.), its validity for vector-valued functions began to be considered. More precisely, what can be said about the constants in.) if we allow the coefficients a i, b j to take values in a certain Banach space B? For example, as Bochner and aylor showed in [], the inequality.) does not hold for any p if f is assumed to take values in l or l. One can also study this question from the perspective of other similar estimates e.g., logarithmic, weak-type, etc.). his problem gained a lot of interest in the literature: see e.g. Aldous [], Bourgain [4], [5], Burkholder [6]-[8], Calderón and Zygmund [], McConnell [9] and Rubio de Francia [25], to mention just a few. It turns out that the good class of spaces, i.e., those in which.) holds true for all < p < with some finite C p, is that of UMD spaces Unconditional for Martingale Differences). We will not recall the definition here; for the detailed description, properties and the interplay between these spaces and the estimate.), the reader is referred to the overview article [9] by Burkholder. he primary goal of this paper is to study a certain version of.), in which the action L p L p is replaced by L L p and/or L p L. Furthermore, we will restrict ourselves to two specific choices of B: l N and l N. Our main contribution is the identification of the corresponding best constants. he result can be stated as follows. heorem.. Fix a positive integer N. i) For any p < and any f L ; l N ) we have /p /4N).2) Hf Lp ;l N ) N 2p2 p log cot s) ds) p f L ;l N ). ii) For any < p and any f L p ; l N ) we have 2 /4N).3) Hf L ;l N ) N 2p log cot s) p ds p ) /p where p denotes the conjugate exponent to p, i.e., p = p/p ). Both inequalities are sharp. f L p ;l N ), he problem of finding sharp or almost sharp versions of various estimates for the Hilbert transform has a long history and has been investigated by many mathematicians. See e.g. Davis [2], Essén [4], Essén, Shea and Stanton [5], [6], Gokhberg and Krupnik [7], Janakiraman [8], Osȩkowski [2], [2], Pichorides [22] and omaszewski [26]. We would like to point out that to the best of our knowledge, the above theorem is the first result in the literature which contains the precise information on the constants in the true Banach-space setting all the results cited above concerned the case of real- or Hilbert-space-valued functions). Of course, it would be most desirable to obtain related results for other value spaces for instance, l N p, l p, or direct sums of such spaces). Unfortunately, this problem seems to be very difficult and not tractable by the methods developed in this paper. On the other hand, we strongly believe that the approach we will introduce below can

3 SHARP INEQUALIIES FOR HILBER RANSFORM IN A VECOR-VALUED SEING 3 be applied to wider classes of Fourier multipliers and/or other types of estimates involving l N and l N -valued functions. As an application, we obtain sharp exponential and LlogL inequalities for H. Here is the precise statement. Corollary.2. i) Suppose that f is a function on with values in a unit ball of l N. hen for any K < /2 we have the sharp bound.4) exp K Hfe iθ ) 4N u 2K/ ) l N dθ u 2 du. cot/4n)) ii) Let Ψs) = s log s s, s. hen for any f : l N with Ψ feiθ ) l N )dθ < and any L > 2/ we have the sharp inequality.5) Hf L ;l N ) L ) Ψ fe iθ ) l dθ 4LN u 2/L) N u 2 du. cot/4n)) A few words about our approach and the organization of the paper are in order. he proof rests on probabilistic methods combined with the existence of a family of certain special harmonic functions on the strip [, ] R. hese special objects are introduced and studied in the next section. Section 3 is devoted to the proofs of the estimates formulated in heorem. and Corollary.2. he final part of the paper addresses the optimality of the constants involved in these bounds. 2. Special functions and their properties Suppose that λ and p are fixed numbers. Let H = R 2 = R [, ) denote the upper halfplane and introduce the harmonic function U λ,p : R, ) R, given by the Poisson integral 2.) U λ,p α, β) = β 2 log s p λ ) α s) 2 β 2 ds, where, as usual, x = max{x, } denotes the positive part of x R. It is easy to check that U satisfies p ) 2.2) lim U 2 λ,pα, β) = log t α,β) t,) λ for t. Next, consider the conformal mapping φz) = ie iz/2, which sends the strip S = [, ] R onto H, and define U λ,p : S R by the formula { Uλ,p φx, y)) if x <, U λ,p x, y) = y p λ) if x =. Clearly, U λ,p is harmonic in the interior of S and, by 2.2), it is continuous on this strip. We study the further properties of U λ,p in the lemma below. We use the notation xx U λ,p, xy U λ,p, etc., for the appropriate second-order partial derivatives of U λ,p. Lemma 2.. he function U λ,p enjoys the following properties. i) U λ,p x, y) = U λ,p x, y) = U λ,p x, y) for all x, y) S. ii) For any x, ) and y R we have yy U λ,p x, y) and xx U λ,p x, y).

4 4 ADAM OSȨKOWSKI iii) For any x [, ] we have U λ,p x, ) U λ,p, ) = 4 /4 ) p 2 ds. iv) For any x, y) S, U λ,p x, y) y p λ). Proof. i) his is an immediate consequence of the following property of U λ,p : for all α R and β >, ) α U λ,p α, β) = U λ,p α, β) = U λ,p α2 β, β. 2 α2 β 2 his can be established by substituting t := t and t := /t in 2.). ii) It suffices to prove the first estimate, then the second follows immediately from the harmonicity of U λ,p. We have, after the substitution t = s expy/2), 2.3) U λ,p x, y) = cos 2 x) 2 log s y p λ ) s sin 2 x)) 2 cos2 2 x) ds. Since for any s R the function y 2 log s y p λ ) is convex, the claim follows. iii) Since xxu 2 λ,p, i) implies that x U λ,p x, ) is nonincreasing on [, ]. Consequently, if x [, ], then 2.3) gives U λ,p x, ) U λ,p, ) = 2 log s p λ ) s 2 ds = 4 2 log s)p λ ) s 2 ds = 4 /4 ) p 2 ds, as desired. iv) Fix y R and apply Jensen s inequality in 2.3), with respect to the convex function t t y p λ). We get 2 cos 2 U λ,p x, y) log s x) p ) s sin 2 x)) 2 cos2 2 x)ds y λ, which is the claim, since the integral inside is equal to simply substitute s := /s). 3. Proofs of.2),.3),.4) and.5) Our reasoning will depend heavily on the theory of continuous-time martingales; let us briefly introduce the necessary notions. Assume that Ω, F, P) is a complete probability space, equipped with F t ) t, a nondecreasing family of sub-σ-fields of F, such that F contains all the events of probability. Let X, Y be two real adapted cádlág martingales, i.e., with right-continuous trajectories that have limits from the left. he symbols [X, X] and [Y, Y ] will stand for the square brackets of X and Y, respectively; see e.g. Dellacherie and Meyer [3] for the definition. Following Bañuelos and Wang [2] and Wang [27], we say that Y is differentially subordinate to X, if the process [X, X] t [Y, Y ] t ) t is nonnegative and nondecreasing as a

5 SHARP INEQUALIIES FOR HILBER RANSFORM IN A VECOR-VALUED SEING 5 function of t. Furthermore, X and Y are said to be orthogonal, if their bracket [X, Y ] defined by the polarization formula [X, Y ] = [X Y, X Y ] [X Y, X Y ])/4) is constant. In our further considerations, the following fact from stochastic analysis will be of importance. Recall that for any real martingale X there exists a unique continuous local martingale part X c of X satisfying [X, X] t = X 2 [X c, X c ] t X s 2 <s t for all t. Here X s = X s X s denotes the jump of X at time s. Furthermore, we have that [X c, X c ] = [X, X] c, the pathwise continuous part of [X, X]. We will require the following statement, which appears as Lemma 2. in [3]. Lemma 3.. If X and Y are semimartingales, then Y is differentially subordinate and orthogonal to X if and only if Y c is differentially subordinate and orthogonal to X c, Y X and Y has continuous paths. Now we are ready to establish the following auxiliary estimate. Lemma 3.2. Suppose that X and Y are orthogonal martingales satisfying the conditions X, Y and such that Y is differentially subordinate to X. hen for any λ and p, 3.) sup t E Y t p λ) 4 he inequality is sharp. /4 ) p 2 ds. Proof. Let t be fixed. Since U λ,p is of class C in the interior of the strip S, we may apply Itô s formula to obtain where 3.2) I = U λ,p X, Y ), I = I 2 = I 3 = t t t I 4 = <s t U λ,p X t, Y t ) = I I I 2 2 I 3 I 4, x U λ,p X s, Y s )dx s t 2 xyu λ,p X s, Y s )d[x c, Y ] s, t xyu 2 λ,p X s, Y s )d[x c, X c ] s y U λ,p X s, Y s )dy s, 2 yyu λ,p X s, Y s )d[y, Y ] s, [ Uλ,p X s, Y s ) U λ,p X s, Y s ) x U λ,p X s, Y s ) X s ]. Note that we have used above the equalities Y s = Y s and Y = Y c, which are due to the continuity of paths of Y. Let us analyse the terms I through I 4 separately: here we will combine Lemma 3. with the properties of U λ,p studied in the previous section. By Lemma 2. iii) we have I 4 /4 2 log cot s ) p λ) ds.

6 6 ADAM OSȨKOWSKI he term I has zero expectation, by the properties of the stochastic integrals. We have I 2 = in view of the orthogonality of X and Y. he differential subordination together with Lemma 2. ii) imply I 3 t 2 xxu λ,p X s, Y s )d[x c, X c ] s t 2 yyu λ,p X s, Y s )d[x c, X c ] s =. Finally, we have that I 4, by the concavity of U λ,p, y) for any fixed y R: see Lemma 2. ii). herefore, by the last part of that lemma, 3.3) E Y t p λ) EU λ,p X t, Y t ) 4 /4 ) p 2 ds and it suffices to take supremum over t to obtain 3.). he optimality of the constant on the right will follow from the sharpness of.2): see the end of Section 4 below. he next step in the analysis is to establish the following statement. Lemma 3.3. Suppose that X and Y are orthogonal martingales satisfying the conditions X, Y and such that Y is differentially subordinate to X. hen for any p and any event A, sup E Y t p A 2p2 t p PA)/4 log cot s) p ds and the number on the right cannot be replaced by a smaller one. Proof. Fix t, an event A and let ) p 2 PA) λ = log cot. 4 Next, consider the splitting A = A A, where A = A { Y t p λ} and A = A { Y t p < λ}. By the previous lemma, we have E Y t p λ) A E Y t p λ) 4 = 4 /4 PA)/4 = 2p2 p ) p 2 ds ) p 2 ds PA)/4 log cot s) p ds λpa). Furthermore, we obviously have E Y t p λ) A. Adding the latter two estimates gives or, equivalently, E Y t p λ) A 2p2 p E Y t p A 2p2 p PA)/4 PA)/4 log cot s) p ds λpa) log cot s) p ds. It remains to take the supremum over t to get the desired estimate. sharpness is deferred to the end of Section 4. Its

7 SHARP INEQUALIIES FOR HILBER RANSFORM IN A VECOR-VALUED SEING 7 he next result can be regarded as a probabilistic version of.2) and is of independent interest. heorem 3.4. Suppose that X = X, X 2,..., X N ) is a bounded martingale taking values in l N and let Y = Y, Y 2,..., Y N ) be an l N -valued martingale such that for each j, Y j and X j are orthogonal and Y j is differentially subordinate to X j. hen we have the sharp inequality sup Y t l N p N /p t 2 p2 p /4N) /p log cot s) ds) p sup X t l N. t Proof. By homogeneity, we may assume that X l N ; then each coordinate X j is a martingale taking values in the interval [, ]. Suppose that t is fixed. hen there are pairwise disjoint events A, A 2,..., A N such that Y t p l N = and hence, by the previous lemma, E Y t p l N = j= E Y j t p Aj j= j= Y j t p Aj. 2 p2 p PAj)/4 log cot s) p ds. However, the function u u log cot s)p ds is concave: indeed, the integrand is nonincreasing. Consequently, by Jensen s inequality, we obtain E Y t p l N N 2p2 p /4N) log cot s) p ds. It remains to take the supremum over t to get the claim. he sharpness will be made clear at the end of Section 4. Now let us deduce our main result for the Hilbert transform. Proof of.2). his follows from a well-known and straightforward argument. Let B = B t ) t be a planar Brownian motion starting from and let τ denote the first moment B hits the unit circle. Let f be a fixed function on the unit circle, taking values in the unit ball of l N. Denote by u and v the harmonic extensions of f and Hf to the unit disc. hen the processes X, Y given by X t = ub τ t ), Y t = vb τ t ) for t ) are orthogonal martingales and Y is differentially subordinate to X more precisely, these properties hold for each pair X j and Y j of coordinates). his follows at once from the identities [X j, X j ] t = u 2 ξ) [Y j, Y j ] t = [X j, Y j ] t = τ t τ t τ t u j B s ) 2 ds, v j B s ) 2 ds = τ t u j B s ) v j B s )ds =, u j B s ) 2 ds, where the latter equality is due to Cauchy-Riemann equations. But u takes values in the unit ball of l N, since so does f. Consequently, the martingale X also has

8 8 ADAM OSȨKOWSKI this boundedness property. herefore, by.2), we may write Hf L p ;l N ) sup Yt l N p t N /p N /p 2 p2 p 2 p2 p /4N) /4N) /p log cot s) ds) p sup Xt l N t log cot s) p ds) /p f L ;l N ), which gives the first estimate. o obtain.3), we use duality argument: clearly, we have Hf L ;l N ) = Hf j dθ = sup Hf j g j dθ, j= j= where the supremum is taken over all functions g = g, g 2,..., g N ) on satisfying g L ;l N ). Now, for any such g, we may write j= Hf j g j dθ = f j Hg j dθ j= Hg L p ;l N ) f L p ;l N ) 2 /4N) N 2p log cot s) p ds p ) /p f L p ;l N ), where in the last passage we have applied.2) to g. his proves the claim. Finally, we are ready to establish the inequalities of Corollary.2. Proof of.4) and.5). o show the exponential bound, observe that.2) implies, for any n, Hf n L n ;l N ) 4N /4N) ) n 2 log cot s ds. Divide both sides by n!, sum over all n add to both sides to get exp K Hfe iθ ) 4N /4N) ) 2K ) l N dθ exp log cot s ds, which is.4) use the substitution u = cot s). o show the LlogL estimate, we use duality, as in the above proof of.3). We will need an auxiliary bound 3.4) xy Ψx) e y valid for all nonnegative x, y we leave the straightforward proof to the reader). We have Hf L ;l N ) = sup f j e iθ )Hg j e iθ )dθ : g L ;l N ) j= { } L sup fe iθ Hge iθ ) ) l N L dθ : g L ;l N ) l N

9 SHARP INEQUALIIES FOR HILBER RANSFORM IN A VECOR-VALUED SEING 9 and, by 3.4) and.4) applied to K = /L), L fe iθ Hge iθ ) ) l N L dθ l N L ) Ψ fe iθ ) l dθ L exp N L ) Ψ fe iθ ) l dθ 4LN N his completes the proof. cot/4n)) Hge iθ ) L l N ) u 2/L) u 2 du. dθ 4. Sharpness In this section, we show that the constants appearing in.2),.3),.4),.5) as well as in the estimates of Lemmas 3.2, 3.3 and heorem 3.4 cannot be improved. First let us focus on the estimate.2). Note that the mapping Gz) = 2i/) log[iz )/z i)] is conformal, satisfies G) = and sends the unit disc onto the strip [, ] R. We easily derive that w = Re G and its conjugate Hw = Im G admit the formulas we iθ ) = { θ /2} { θ >/2}, Hwe iθ ) = 2 log sin θ cos θ for θ [, ]. Now, given a positive integer N, consider the set { } A = θ [, ] : θ /2 < /2N) or θ /2 < /2N), which is a sum of two intervals of length /N, with centers at /2 and /2. he function Hw satisfies the symmetry condition for all θ [, ], and therefore Hwe iθ ) p dθ = 2 A Hwe iθ ) = Hwe i θ) ) = Hwe iθ ) = 2p p = 2p p = 2p2 p /2/2N) /2 /2/2N) /2 /2N) /4N) Hwe iθ ) p dθ p log sin θ cos θ dθ p log cos θ sin θ dθ log cot s) p ds. Now consider the pairwise disjoint subsets A = A, A 2 = A /N, A 3 = A /N,..., A N = A N )/N of [, ] here, as usual, A u = {x u : x A} and we identify x and y if their difference is a multiple of ). Composing the function G with an appropriate rotation z e iϕ z, we see that for any j {, 2,..., N}, there is a function w j : {, }, such that Hw Aj j e iθ ) p dθ = 2p2 p /4N) log cot s) p ds.

10 ADAM OSȨKOWSKI herefore, if we take w = w, w 2,..., w N ), we see that w L ;l N ) = and /p Hw L p l N ) Hw j p dθ A j = j= N 2p2 p /4N) log cot s) p ds) /p. herefore, the inequality.2) is sharp. his also proves that the constants involved in the estimates of Lemmas 3.2, 3.3 and heorem 3.4 are also the best possible: indeed, if any of them could be improved, then it would be possible to decrease the constant in.2). Finally, since the above extremal example is the same for each p, we immediately obtain the sharpness of the exponential bound.4). It remains to show that the constants in.3) and.5) are the best as well. his can be deduced from the sharpness of.2) and.4) with the use of duality argument. We will only present the details for the first estimate, and leave the analysis of.5) to the reader. So, assume that for a given < p, the optimal constant in.3) equals β p. Fix f L ; l N ) and observe that Hf L p ;l N ) = sup Hf j g j dθ, j= where the supremum is taken over all g = g, g 2,..., g N ) on satisfying g L p ;l N ). Now, for any such g we may write Hf j g j dθ = f j Hg j dθ j= j= f L ;l N ) Hg L ;l N ) β p f L ;l N ). Here we have used the assumed fact that.3) holds with the constant β p. Consequently, Hf L p ;l N ) β p f L ;l N ) and by the sharpness of.2), we get β p his is precisely the claim. N 2p 2 p /4N) log cot s) p ds Acknowledgment ) /p he above results were obtained when the author was visiting Purdue University, West Lafayette, USA. References [] D. J. Aldous, Unconditional bases and martingales in L pf ), Math. Proc. Cambridge Phil. Soc ), [2] R. Bañuelos and G. Wang, Sharp inequalities for martingales with applications to the Beurling-Ahlfors and Riesz transformations, Duke Math. J ), [3] R. Bañuelos and G. Wang, Davis s inequality for orthogonal martingales under differential subordination, Michigan Math. J. 47 2), [4] J. Bourgain, Some remarks on Banach spaces in which martingale difference sequences are unconditional, Ark. Mat ),

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