Let ϕ : R n × [0, ∞) → [0, ∞) be a growth function such that ϕ(x, ·) is nondecreasing, ϕ(x, 0) = 0, ϕ(x, t) > 0 when t > 0, lim t→∞ ϕ(x, t) = ∞, and ϕ(·, t) is a Muckenhoupt A ∞ (R n ) weight uniformly in t. In this paper, the authors establish the Lusin area function and the molecular characterizations of the Musielak-Orlicz Hardy space H ϕ (R n ) introduced by Luong Dang Ky via the grand maximal function. As an application, the authors obtain the ϕ-Carleson measure characterization of the Musielak-Orlicz BMO-type space BMO ϕ (R n ), which was proved to be the dual space of H ϕ (R n )[1] and Orlicz in [24], which is widely used in various branches of analysis (see, for example, [25,26] and their references). Moreover, as a development of the theory of Orlicz spaces, Orlicz-Hardy spaces and their dual spaces were studied by Strömberg [30] and Janson [14] on R n and, quite recently, Orlicz-Hardy spaces associated with divergence form elliptic operators by Jiang and Yang [15].Furthermore, the classical BMO space (the space of functions with bounded mean oscillation), originally introduced by John and Nirenberg [16], plays an important role in the study of partial differential equations and harmonic analysis. In particular, Fefferman and Stein [9] proved that BMO(R n ) is the dual space of H 1 (R n ) and also obtained the Carleson measure characterization of BMO(R n ). Moreover, the generalized BMO-type space BMO ρ (R n ) was studied in [30,14,13] and it was proved therein to be the dual space of the Orlicz-Hardy space H Φ (R n ), where the function Φ : [0, ∞) → [0, ∞) satisfies the following assumptions:). Here and in what follows, Φ −1 denotes the inverse function of Φ. Observe that Φ may not be convex and hence may not be an Orlicz function in the classical sense. Meanwhile, the Carleson measure characterization of BMO ρ (R n ) was obtained in [13]. Recently, a new Musielak-Orlicz Hardy space H ϕ (R n ) was introduced by Ky [17], via the grand maximal function, which includes both the Orlicz-Hardy space in [30,14] and the weighted Hardy space H p ω (R n ) with p ∈ (0, 1] and ω ∈ A ∞ (R n ) in [11,31]. Here and in what follows, ϕ : R n × [0, ∞) → [0, ∞) is a growth function such that ϕ(x, ·), for any fixed x ∈ R n , satisfies (1.1) with uniformly upper type 1 and lower type p for some p ∈ (0, 1] (see Section 2 for the definitions of uniformly upper or lower types), and ϕ(·, t) is a Muckenhoupt A ∞ (R n ) weight uniformly in t, and A q (R n ) with q ∈ [1, ∞] denotes the class of Muckenhoupt weights (see, for example, [12] for their definitions and properties). In [17], Ky first established the atomic characterization of H ϕ (R n ) and further introduced the Musielak-Orlicz BMO-type space BMO ϕ (R n ), which was proved to be the dual space of H ϕ (R n ). Furthermore, some interesting applications of these spaces were also presented in [2,4,5,17,18,19]. Moreover, the local Musielak-Orlicz Hardy space, h ϕ (R n ), and its dual space, bmo ϕ (R n ), were studied in [33] and some applications of h ϕ (R n ) and bmo ϕ (R n...

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In this paper, the Orlicz addition of measures is proposed and an interpretation of the f -divergence is provided based on a linear Orlicz addition of two measures. Fundamental inequalities, such as, a dual functional Orlicz-Brunn-Minkowski inequality, are established. We also investigate an optimization problem for the f -divergence and establish functional affine isoperimetric inequalities for the dual functional Orlicz affine and geominimal surface areas of measures. Index Termsaffine isoperimetric inequality, affine surface area, Brunn-Minkowski theory, dual Brunn-Minkowski theory, the f -divergence, geominimal surface area, optimization problem for the f -divergence.with equality if and only if p = q almost everywhere with respect to Q. When f is strictly concave with f (1) = 0, one gets similar results with "≥" replaced by "≤". From this viewpoint, the f -divergence can be used to distinguish two measures. Without doubt, the f -divergence plays fundamental roles in, such as, image analysis, information September 11, 2018 DRAFT 2 theory, pattern matching and statistical learning (see [5], [11], [24], [27], [34]), where the measure of difference between measures is required. Moreover, in general, the f -divergence is arguably better than the L p distance. Recent development in convex geometry has witnessed the strong connections between the f -divergence and convex geometry. For instance, it has been proved that the L p affine surface area [7], [32], [37], a central notion in convex geometry, is related to the Renyi entropy [38]; while the general affine surface area [28], [30] is associated to the f -divergence [9]. Note that these affine surface areas are valuations; and valuations are the key ingredients for the Dehn's solution of Hilbert third problem. Moreover, under certain conditions (such as, semicontinuity), it has been proved that these affine surface areas can be used to uniquely characterize all valuations which remain unchanged under linear transforms with determinant ±1 (see e.g. [23], [29], [30]). On the other hand, as showed in the Subsection V-A, the affine and geominimal surface areas (see e.g. [32], [35], [40], [41]) can be translated to an optimization problem for the f -divergence. This observation leads us to investigate the dual functional Orlicz affine and geominimal surface areas for measures, which are invariant under linear transforms with determinant ±1. The Brunn-Minkowsi inequality is arguably one of the most important inequalities in convex geometry. It can be used to prove, for instance, the celebrated Minkowski's and isoperimetric inequalities. (Note that the isoperimetric problem has a history over 1000 years). See the excellent survey [18] for more details. On the other hand, the dual Brunn-Minkowski inequality and dual Minkowski inequality are crucial for the solutions of the famous Busemann-Petty problem (see e.g., [17], [22], [31], [43]). The Brunn-Minkowsi inequality and its dual have been extended to the Orlicz theory in [19], [20], [39], [44].This paper is dedicated to...