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BMO Functions and Carleson Measures with Values in Uniformly Convex Spaces

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BMO Functions and Carleson Measures with Values in Uniformly Convex Spaces Canad. J. Math. Vol. 62 (4), 2010 pp. 827–844 doi:10.4153/CJM-2010-043-6 °c Canadian Mathematical Society 2010 BMO Functions and Carleson Measures with Values in Uniformly Convex Spaces Caiheng Ouyang and Quanhua Xu Abstract. This paper studies the relationship between vector-valued BMO functions and the Carleson measures defined by their gradients. Let dA and dm denote Lebesgue measures on the unit disc D and the unit circle T, respectively. For 1 < q < ∞ and a Banach space B, we prove that there exists a positive constant c such that q−1 q q q sup (1 − |z|) k∇ f (z)k Pz0 (z)dA(z) ≤ c sup k f (z) − f (z0)k Pz0 (z)dm(z) z0∈D ZD z0∈D ZT holds for all trigonometric polynomials f with coefficients in B if and only if B admits an equivalent norm which is q-uniformly convex, where 2 1 − |z0| P (z) = . z0 2 |1 − z¯0z| The validity of the converse inequality is equivalent to the existence of an equivalent q-uniformly smooth norm. 1 Introduction Let T be the unit circle of the complex plane equipped with normalized Haar measure dm. Recall that an integrable function f on T is of bounded mean oscillation (BMO) if 1 k f k∗ = sup | f − fI|dm < ∞, I |I| ZI T = −1 where the supremum runs over all arcs of and fI |I| I f dm is the mean of f over I. Let BMO(T) denote the space of BMO functions onRT. The means over arcs in this definition can be replaced by the averages of f against the Poisson kernel Pz0 for the unit disc D: 2 = 1 − |z0| T Pz0 (z) 2 , z0 ∈ D, z ∈ . |1 − z¯0z| Then 2 2 k f k∗ ≈ sup | f (z) − f (z0)| Pz0 (z)dm(z), z0∈D ZT Received by the editors January 17, 2008. Published electronically May 20, 2010. C. Ouyang is partially supported by the National Natural Science Foundation of China (No. 10971219). Q. Xu is partially supported by ANR 06-BLAN-0015. AMS subject classification: 46E40, 42B25, 46B20. Keywords: BMO, Carleson measures, Lusin type, Lusin cotype, uniformly convex spaces, uniformly smooth spaces. 827 Downloaded from https://www.cambridge.org/core. 02 Oct 2021 at 06:41:55, subject to the Cambridge Core terms of use. 828 C. Ouyang and Q. Xu with universal equivalence constants. Here, as well as in the sequel, we denote also by f its Poisson integral in D: = f (z0) f (z)Pz0 (z)dm(z), z0 ∈ D. ZT On the other hand, it is well known that BMO functions can be characterized by Carleson measures. A positive measure µ on D is called a Carleson measure if 2 = 1 − |z0| kµkC sup 2 dµ(z) < ∞. z0∈D ZD |1 − z¯0z| Let f ∈ L1(T). Then f ∈ BMO(T) if and only if |∇ f (z)|2(1−|z|2)dA(z) is a Carleson measure, where dA(z) denotes Lebesgue measure on D. In this case, we have 2 2 2 2 (1 − |z| )(1 − |z0| ) (1.1) k f k∗ ≈ sup |∇ f (z)| 2 dA(z). z0∈D ZD |1 − z¯0z| We refer to [6] for all these results. This paper concerns the vector-valued version of (1.1). More precisely, we are interested in characterizing Banach spaces B for which one of the two inequalities in (1.1) holds for B-valued functions f . Given a Banach space B, let Lp(T; B) denote the usual Lp-space of Bochner p-integrable functions on T with values in B. The space BMO(T; B) of B-valued functions on T is defined in the same way as in the scalar case just by replacing the absolute value of C by the norm of B. Then the vector-valued analogue of (1.1) is the following: 2 2 −1 2 2 (1 − |z| )(1 − |z0| ) 2 (1.2) c1 k f k∗ ≤ sup k∇ f (z)k 2 dA(z) ≤ c2k f k∗ z0∈D ZD |1 − z¯0z| for all f ∈ BMO(T; B), where c1, c2 are two positive constants (depending on B), and where ∂ f ∂ f k∇ f (z)k = (z) + (z) , z = x + iy. ° ∂x ° ° ∂y ° ° ° ° ° It is part of the folklore that (1.2)° holds° if° and only° if B is isomorphic to a Hilbert space (see [2]). We include a proof of this result at the end of the paper for the convenience of the reader. However, if one considers the validity of only one of the two inequalities in (1.2), the matter becomes much subtler and the corresponding class of Banach spaces is much larger. The following theorem solves this problem. Theorem 1.1 Let B be a Banach space. (i) There exists a positive constant c such that 2 2 2 (1 − |z| )(1 − |z0| ) 2 sup k∇ f (z)k 2 dA(z) ≤ ck f k∗ z0∈D ZD |1 − z¯0z| holds for all trigonometric polynomials f with coefficients in B if and only if B admits an equivalent norm that is 2-uniformly convex. Downloaded from https://www.cambridge.org/core. 02 Oct 2021 at 06:41:55, subject to the Cambridge Core terms of use. BMO Functions and Carleson Measures with Values in Uniformly Convex Spaces 829 (ii) There exists a positive constant c such that 2 2 2 (1 − |z| )(1 − |z0| ) −1 2 sup k∇ f (z)k 2 dA(z) ≥ c k f k∗ z0∈D ZD |1 − z¯0z| holds for all trigonometric polynomials f with coefficients in B if and only if B admits an equivalent norm that is 2-uniformly smooth. We refer to the next section for the definition of uniform convexity (smoothness). This theorem is intimately related to the main result of [15], where the vector-valued Littlewood–Paley theory is studied. Given f ∈ L1(T; B), define the Littlewood–Paley g-function 1 2 G( f )(z) = (1 − r)k∇ f (rz)k2dr, z ∈ T. Z ¡ ¢ 0 The following fact is again well known. The equivalence kG( f )kL2(T) ≈ k f − f (0)kL2(T;B) holds uniformly for all B-valued trigonometric polynomials f if and only if B is iso- morphic to a Hilbert space. However, the two one-sided inequalities are related to uniform convexity (smoothness). More precisely, we have the following result from [15]. Theorem 1.2 Let B be a Banach space. (i) B has an equivalent 2-uniformly convex norm if and only if for some p ∈ (1, ∞) (or equivalently, for every p ∈ (1, ∞)) there exists a positive constant c such that (1.3) kG( f )kLp(T) ≤ ck f kLp(T;B) holds for all B-valued trigonometric polynomials f . (ii) B has an equivalent 2-uniformly smooth norm if and only if for some p ∈ (1, ∞) (or equivalently, for every p ∈ (1, ∞)) there exists a positive constant c such that (1.4) k f − f (0)kLp(T;B) ≤ ckG( f )kLp(T) holds for all B-valued trigonometric polynomials f . According to [15], the spaces satisfying (1.3) (resp. (1.4)) are said to be of Lusin cotype 2 (resp. Lusin type 2). The name Lusin refers to the fact that the Littlewood– Paley g-function can be replaced by the Lusin area function. At this stage, let us also recall that by Pisier’s renorming theorem [10], B has an equivalent 2-uniformly convex (resp. smooth) norm if and only if B is of martingale cotype (resp. type) 2. The value p = ∞ is, of course, not allowed in Theorem 1.2. At the time of the writing of [15], the second author guessed that a correct version of Theorem 1.2 for p = ∞ should be Theorem 1.1, but could not confirm this. Our proof of Theorem 1.1 relies heavily on Theorem 1.2 and Calderon–Zygmund´ singular integral theory. In fact, we will work in the more general setting of a Euclidean space Rn instead of T. On the other hand, the power 2 in k∇ f k2 no longer plays any special role in the vector-valued setting. We will consider the analogue of Theorem 1.1 for k∇ f kq with 1 < q < ∞. The corresponding result is stated separately in Theorems 4.1 and 5.1 below, which correspond to the end point p = ∞ of the results of [9, 15]. Downloaded from https://www.cambridge.org/core. 02 Oct 2021 at 06:41:55, subject to the Cambridge Core terms of use. 830 C. Ouyang and Q. Xu 2 Preliminaries Our references for harmonic analysis are [5, 6, 14]. All results quoted in this paper without explicit reference can be found there. However, one sometimes needs to adapt arguments in the scalar case to the vector-valued setting. Let (Ω, µ) be a measure space and B a Banach space. For 1 ≤ p ≤ ∞, we denote by Lp(Ω, µ; B) the usual Lp-space of Bochner (or strongly) measurable functions on p Ω with values in B. The norm of L (Ω, µ; B) is denoted by k · kp. The n-dimensional Rn 1 Rn Euclidean space is equipped with Lebesgue measure, and Lloc( ; B) denotes the space of locally integrable functions on Rn with values in B. Recall the Poisson kernel on Rn: Γ( n+1 ) t P (x) = 2 , x ∈ Rn, t > 0. t π(n+1)/2 (t2 + |x|2)(n+1)/2 1 Rn Let f ∈ Lloc( ; B) such that 1 k f (x)k n+1 dx < ∞.
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