Doctor of philosophy in mathematics: Some problems in pluripotential theory

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The dissertation is written on the basis of the paper published in Annales Polonici Mathematici, the paper published in Acta Mathematica Vietnamica and the paper published in Results in Mathematics.

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VIETNAM ACADEMY OF SCIENCE AND TECHNOLOGY INSTITUTE OF MATHEMATICS DO THAI DUONG SOME PROBLEMS IN PLURIPOTENTIAL THEORY DISSERTATION SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY IN MATHEMATICS HANOI - 2021
VIETNAM ACADEMY OF SCIENCE AND TECHNOLOGY INSTITUTE OF MATHEMATICS DO THAI DUONG SOME PROBLEMS IN PLURIPOTENTIAL THEORY Speciality: Mathematical Analysis Speciality code: 9460102 (62 46 01 02) DISSERTATION SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY IN MATHEMATICS Supervisor: Prof. Dr.Sc. PHAM HOANG HIEP Prof. Dr.Sc. DINH TIEN CUONG HANOI - 2021
Confirmation This dissertation was written on the basis of my research works carried out at Institute of Mathematics, Vietnam Academy of Science and Technology, under the supervision of Prof. Dr.Sc. Pham Hoang Hiep and Prof. Dr.Sc. Dinh Tien Cuong. All the presented results have never been published by others. January 3, 2021 The author Do Thai Duong i
Acknowledgments First of all, I am deeply grateful to my academic advisors, Professor Pham Hoang Hiep and Professor Dinh Tien Cuong, for their invaluable help and support. I am sincerely grateful to IMU (The International Mathematical Union), FIMU (Friends of the IMU) and TWAS (The World Academy of Sciences) for supporting my PhD studies through the IMU Breakout Graduate Fellowship. The wonderful research environment of the Institute of Mathematics, Vietnam Academy of Science and Technology, and the excellence of its staff have helped me to complete this work within the schedule. I would like to thank my colleagues for their efficient help during the years of my PhD studies. Especially, I would like to express my special appreciation to Do Hoang Son for his valuable comments and suggestions on my research results. I also would like to thank the participants of the weekly seminar at Department of Mathematical Analysis for many useful conversations. Furthermore, I am sincerely grateful to Prof. Le Tuan Hoa, Prof. Phung Ho Hai, Prof. Nguyen Minh Tri, Prof. Le Mau Hai, Prof. Nguyen Quang Dieu, Prof. Nguyen Viet Dung, Prof. Doan Thai Son for their guidance and constant encouragement. Valuable remarks and suggestions of the Professors from the Department-level PhD Dissertation Evaluation Committee and from the two anonymous indepen- dent referees are gratefully acknowledged. Finally, I would like to thank my family for their endless love and unconditional support. ii
Contents Table of Notations v Introduction x Chapter 1. A comparison theorem for subharmonic functions 1 1.1 Some basic properties of subharmonic functions . . . . . . . . . . 1 1.2 Some basic properties of Hausdorff measure . . . . . . . . . . . . . 5 1.3 An extension of the mean value theorem . . . . . . . . . . . . . . 8 1.4 A comparison theorem for subharmonic functions . . . . . . . . . . 13 1.5 Other versions of main results . . . . . . . . . . . . . . . . . . . . 16 Chapter 2. Complex Monge-Ampère equation in strictly pseudo- convex domains 18 2.1 Some properties of plurisubharmonic functions . . . . . . . . . . . 19 2.2 Domain of Monge-Ampère operator and notions of Cegrell classes . 21 2.3 Some basic properties of relative capacity . . . . . . . . . . . . . . 25 2.4 Dirichlet problem for the Monge-Ampère equation is strictly pseu- doconvex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.5 A remark on the class E . . . . . . . . . . . . . . . . . . . . . . . 31 Chapter 3. Decay near boundary of volume of sublevel sets of plurisub- harmonic functions 36 3.1 Some properties of the class F . . . . . . . . . . . . . . . . . . . . 37 3.2 An integral theorem for the class F . . . . . . . . . . . . . . . . . 39 3.3 Some necessary conditions for membership of the class F . . . . . 42 3.4 A sufficient condition for membership of the class F . . . . . . . . 46 iii
List of Author’s Related Papers 50 References 51 iv
Table of Notations N the set of positive integers. R the set of real numbers. C the set of complex numbers. Rn the real vector space of dimension n. Cn the complex vector space of dimension n. Bn the unit ball in Rn . B2n the unit ball in Cn . ∂ Bn the unit sphere in Rn . ∂ B2n the unit sphere in Cn . B(x, r) the open ball of center x and radius r in real vector space or complex vector space. B(x, r) the closed ball of center x and radius r in real vector space or complex vector space. ∂ B(x, r) the sphere of center x and radius r in real vector space or complex vector space. Vn the Lebesgue measure on Rn . V2n the Lebesgue measure on Cn . σ the surface measure (in any dimension) on a surface. ∅ the empty set. ∥x∥ the norm of a vector x. A(Ω) the set of analytic functions on Ω. C(Ω) the set of continuous functions on Ω. C k (Ω) the set of k−times differentiable functions with derivatives of order k are continuous on Ω. C0k (Ω) the set of k−times differentiable functions with derivatives of order k are continuous and compact support on Ω. C ∞ (Ω) the set of smooth functions on Ω. v
C0∞ (Ω) the set of smooth functions with compact support on Ω. E0 (Ω), E(Ω), F(Ω), N (Ω) Cegrell’s classes on Ω. H(Ω) the set of harmonic functions on Ω. USC(Ω) the set of upper semicontinuous functions on Ω. ∞ L (Ω) the set of bounded functions on Ω. ∞ Lloc (Ω) the set of locally bounded functions on Ω. p L (Ω) the set of p-th power integrable functions on Ω. Lploc (Ω) the set of locally p-th power integrable functions on Ω. SH(Ω) the set of subharmonic functions on Ω. PSH(Ω) the set of plurisubharmonic functions on Ω. − PSH (Ω) the set of negative plurisubharmonic functions on Ω. MPSH(Ω) the set of maximal plurisubharmonic functions on Ω. OX,z the space of germs of holomorphic functions at a point z ∈ X. u∗v the convolution of u and v . vi
Introduction It has been known since the 19th century that gravity and electrostatic forces are two fundamental forces of nature. They were believed to be derived from the use of functions so-called “potentials”, that satisfied Laplace’s equation. The term “(classical) potential theory” arose to describe a linear theory associated to the Laplacian. This theory focused on harmonic functions, subharmonic functions, the Dirichlet problem, harmonic measure, Green’s functions, potentials and capacity in several real variables. The potential theory in two dimensional space, which is always considered as the potential theory in the complex plane, has attracted considerable interest since it is closely related to complex analysis. In particular, there is a connection between Laplace’s equation and analytic functions. While the real and imaginary parts of analytic functions of a complex variable satisfy the Laplace’s equation in two dimensions, the solution to Laplace’s equation is the real part of an analytic function. In general, some techniques of complex analysis, particularly conformal mapping, can be used to simplify proofs of some results in the potential theory, while some theorems in the potential theory have analogies and applications in complex analysis. In the 20th century, pluripotential theory was developed as the complex mul- tivariate analogue of the classical potential theory in the complex plane. This theory is highly non-linear and associated to complex Monge-Ampère operators. The basic objects are plurisubharmonic functions of several complex variables that were defined in 1942 by Kiyoshi Oka and Pierre Lelong. This class is the natural counterpart of the class of subharmonic functions of one complex variable. The plurisubharmonic functions are also considered as subharmonic functions on sev- eral real variables which are invariant with respect to all biholomorphic coordinate systems. In this dissertation, we study some specific problems in the pluripotential theory and the potential theory. vii
In Chapter 1, motivated by the fact that two subharmonic functions which agree almost everywhere on a domain with respect to Lebesgue measure must coincide everywhere on that domain, we are interested in the following problem. Problem 1. Whether we can conclude that two subharmonic functions on a do- main of Rn which agree almost everywhere on a m−dimensional submanifold with respect to m-dimensional Hausdorff measure must coincide everywhere on that submanifold? Chapter 1 is devoted to answer Problem 1 completely. For this purpose, we prove two main theorems with similar assumptions. They concern restrictions of sub- harmonic functions in Ω to a Borel subset K ⊂ Ω which, together with a measure µ, is subject to some technical assumptions. These allow K to have co-dimension one (and a little more, but not two), with µ being more or less a corresponding Hausdorff measure. The first main result (Theorem 1.3.3) is an extension of the mean value theorem. It states that the mean value theorem in an infinitesimal form still holds when restricted to K , and with respect to µ. The second main re- sult (Theorem 1.4.1) is a comparison theorem for subharmonic functions. It states that a comparison between an upper semicontinuous function and a subharmonic function which holds almost everywhere (with respect to µ) on K actually holds at every point of K . By these theorems, we prove that Problem 1 has a positive answer in the case of hypersurfaces. We also provide a counterexample (Example 1.4.4) in the case of subspaces of higher co-dimension. In addition, we apply the main theorems to Ahlfors-David regular sets to obtain some consequences, and prove other versions of the main results in terms of measure densities. In Chapter 2, we study the Dirichlet problem for the complex Monge-Ampère equation. We are interested in the following problem. Problem 2. Find conditions for µ such that the solution u of Dirichlet problem for complex Monge-Ampère equation is continuous outside an analytic set but u may not be continuous in Ω. This problem arises from the fact that there are some plurisubharmonic functions which are not continuous in the whole domain, though they are continuous outside an analytic set. For example, u(z) = −(− log ∥z∥)1/2 is not continuous in the whole unit ball B2n , but it is continuous in B2n \{0}. In studying this problem, we prove a sufficient condition (Theorem 2.4.8) which relaxes assumptions of a well- known result of Kolodziej (Theorem B in [26]) to some technical assumptions. These assumptions naturally lead to the following problem. viii
Problem 3. Find conditions for α such that v = −(− log(|f1 |λ1 + ... + |fm |λm ))α belongs to the domain of Monge-Ampère operator, where f1 , ..., fm are analytic functions. Note that if 0 < α < n1 , then v ∈ E (see [7], [11]). Our result (Proposition 2.5.1) is a necessary and sufficient condition where we further assume some conditions of f1 , ..., fm and the non-singularity of their zero-sets. Chapter 3 is devoted to study the behavior near boundary of the functions from class F in a strictly pseudoconvex domain Ω. In particular, we estimate the sublevel set of plurisubharmonic functions near the boundary of a domain. The study in this chapter is motivated by two sources. Firstly, we are inspired from [8] where the author gave the characterization of the class F in terms of the capacity of the sublevel set. Secondly, for a bounded hyperconvex domain Ω, although for every u ∈ F (Ω) we have lim sup u(z) = 0, z→∂Ω being in F(Ω) does not necessarily ensure that limz→∂Ω u(z) = 0. We are inter- ested in the following problem. Problem 4. Find necessary and sufficient conditions of the volume of the sub- level sets near certain boundary points of functions from the class F in strictly pseudoconvex domains. In studying this problem, we give a necessary condition (Theorem 3.3.1) for upper bound of their volume and its consequences. When the domain is the unit ball, we prove a necessary condition (Theorem 3.3.3) and a sufficient condition (Theorem 3.4.1) for membership of the class F . They both concern decay near boundary of volume of the sublevel sets. The dissertation is written on the basis of the paper [19] published in Annales Polonici Mathematici, the paper [20] published in Acta Mathematica Vietnamica and the paper [21] published in Results in Mathematics. The results of this dissertation were presented at - The weekly seminar of the Department of Mathematical Analysis, Institute of Mathematics, Vietnam Academy of Science and Technology; ix
- Analysis session of “The 9th Vietnam Mathematical Congress” (August 14-18, 2018, Telecommunications University, Nha Trang, Vietnam); - International Conference “Differential Equations and Dynamical Systems” (September 5-6, 2019, Hanoi Pedagogical University 2, Xuan Hoa, Phuc Yen, Vinh Phuc, Vietnam); - “Vietnam-USA Joint Mathematical Meeting” (June 10-13, 2019, Quy Nhon, Vietnam) (a poster presentation). x
Chapter 1 A comparison theorem for subharmonic functions The present chapter, which is written on the basis of the paper “Thai Duong Do, A comparison theorem for subharmonic functions, Results Math. 74 (2019), Paper No. 176, 13pp”, devotes to study Problem 1 mentioned at the Introduction. This chapter is organized as follows. - In Sections 1.1 and 1.2, we recall some basic properties of subharmonic func- tions and some basic properties of Hausdorff measure which will be used in the sequel. - In Section 1.3, we prove our first main result “An extension of the mean value theorem” and its corollaries. - In Section 1.4, we prove our second main result “A comparison theorem for subharmonic functions” by using the first main result as main tool. We also establish its corollaries and construct a counterexample which answer the Problem 1 completely. - In Section 1.5, we prove other versions of the main results in terms of measure densities. Throughout this chapter, we always assume that Ω is a domain of Rn (n ≥ 2). 1.1 Some basic properties of subharmonic functions In this section, we recall definition and basic properties of subharmonic func- tions. For more details, the reader is referred to [25, 32, 22]. 1
Definition 1.1.1. Let u : Ω −→ R be a C 2 -function. The function u is said to be harmonic in Ω if it satisfies the Laplace equation: ∑ n ∂ 2u △u = ≡ 0 in Ω. j=1 ∂x2j We denote by H(Ω) the family of all harmonic functions in Ω. Definition 1.1.2. Let u : Ω −→ [−∞, +∞) be an upper semicontinuous function which is not identically −∞. The function u is said to be subharmonic if for every relatively compact open subset U of Ω and every function φ ∈ H(U ) ∩ C(U ¯ ), the following implication is true: u ≤ φ on ∂U =⇒ u ≤ φ on U . We denote by SH(Ω) the family of all subharmonic functions in Ω. The following elementary result, which is an immediate consequence of Defini- tion 1.1.2, helps us to generate further examples of subharmonic functions. Theorem 1.1.3. Let u, v ∈ SH(Ω). Then: 1, max{u, v} ∈ SH(Ω); 2, αu + βv ∈ SH(Ω) for all α, β ≥ 0. The following theorem gives us a characterization of subharmonicity in terms of integral means. Theorem 1.1.4. Let u : Ω −→ [−∞, +∞) be an upper semicontinuous function which is not identically −∞. Then the following conditions are equivalent: 1, u ∈ SH(Ω); 2, if B(a, R) ⊂ Ω, then ∫ 1 u(a) ≤ u(x)dσ(x); σ(∂ B(0, 1))Rn−1 ∂ B(a,R) 3, if B(a, R) ⊂ Ω, then ∫ 1 u(a) ≤ u(x)dVn (x). Vn (B(0, 1))Rn B(a,R) The local integrability of subharmonic functions is an immediate consequence of Theorem 1.1.4. 2
Corollary 1.1.5. Let u ∈ SH(Ω). Then u ∈ L1loc (Ω). Theorem 1.1.4 implies also that we can sometimes glue subharmonic functions together to give a new subharmonic function. Corollary 1.1.6. Let Ω be a domain in Rn , ω be a non-empty proper open subset of Ω, and let u ∈ SH(Ω), v ∈ SH(ω). Suppose that lim sup v(y) ≤ u(y), x→y for each y ∈ ∂ω ∩ Ω. Then the formula  max{u, v} in ω, w= u in Ω\ω defines a subharmonic function in Ω. The next theorem follows by the semi-continuity of subharmonic function and Theorem 1.1.4. It is the motivation of this chapter as mentioned in the beginning. Theorem 1.1.7. Let u, v ∈ SH(Ω). If u = v almost everywhere in Ω then u = v in Ω. The following theorem, known as Riesz Decomposition theorem, will be used as a major technical tool in proving an extension of the mean value theorem. Here, we identify ∆u with a positive measure, see Theorem 1.1.12 below. Theorem 1.1.8. Let u ∈ SH(Ω) and U be a relatively compact open subset of Ω. Then we can decompose u as ∫ −1 u(x) = ( ) g(|x − w|)dν(w) + φ(x), max{1, n − 2}σ ∂ B(0, 1) U on U , where ν = ∆u|U , φ ∈ H(U ) and the kernel g : (0, +∞) → R is defined by  − log r (n = 2) g(r) = . (1.1) r2−n (n > 2) Next, we recall the notation of convolution and its application on approximating a subharmonic function by a family of smooth subharmonic functions. Definition 1.1.9. Let u, v ∈ L1 (Rn ), then the convolution u ∗ v is defined by the formula ∫ (u ∗ v)(x) = u(x − y)v(y)dVn (y). Rn 3
Moreover, the convolution u∗v is also well-defined if u ∈ L1loc (Rn ) and v ∈ L1 (Rn ) has compact support. We define χ : R → R by the formula  e−1/t (t > 0) χ(t) = 0 (t ≤ 0). And we define ρ : Rn → R as follows ρ(x) = Kχ(1 − ∥x∥2 ), where ( ∫ )−1 K= χ(1 − ∥x∥ )dλ(x) 2 . B(0,1) Obviously, ρ ∈ C ∞ (Rn ), suppρ = B(0, 1) and ∫ ρ(x)dVn (x) = 1. Rn For ϵ > 0, we define 1 (x) ρϵ (x) = ρ . ϵn ϵ The next theorem is known as the main approximation theorem for subharmonic functions. Theorem 1.1.10. Let Ω be a domain in Rn and let u ∈ SH(Ω). If ϵ > 0 such that Ωϵ := {x ∈ Ω : dist(x, ∂Ω) > ϵ} ̸= ∅, then u ∗ ρϵ ∈ C ∞ ∩ SH(Ωϵ ). Moreover, u ∗ ρϵ monotonically decreases with decreasing ϵ and lim u ∗ ρϵ (x) = u(x) ϵ→0 for each x ∈ Ω. We end this section with the characterization of subharmonicity. The following theorem gives us a characterization of smooth subharmonic functions. Theorem 1.1.11. Let u ∈ C 2 (Ω). Then u ∈ SH(Ω) if and only if ∆u ≥ 0 in Ω. Since subharmonic functions are locally integrable, their Laplacians can be eval- uated in the sense of distributions. The next theorem shows that we can apply the characterization of subharmonicity described in Theorem 1.1.11 in a much wider context. 4
Theorem 1.1.12. Let u ∈ SH(Ω). Then ∆u ≥ 0 in the sense of distributions, i.e. ∫ u(x)∆φ(x)dVn (x) ≥ 0 Ω for any non-negative test function φ ∈ C0∞ (Ω). Conversely, if v ∈ L1loc (Ω) is such that ∆v ≥ 0 in Ω in the sense of distributions, then the function u = lim(v ∗ ρϵ ) ϵ→0 is well-defined, subharmonic in Ω, and equal to v almost everywhere in Ω. 1.2 Some basic properties of Hausdorff measure In this section, we recall definitions and basic properties of Hausdorff measure, the Ahlfors-David regular set, the upper and lower densities of a Radon mea- sure. For more details, the reader is referred to [17, 29, 30, 36]. First, we recall Carathéodory’s construction. Definition 1.2.1. Let X be a metric space, F be a family of subsets of X and φ be a non-negative function on F . Assume that 1, for every δ > 0, there is {Ei }i=1,2,... ⊂ F such that X = ∪∞ i=1 Ei and d(Ei ) ≤ δ , where d(E) is the diameter of E : d(E) = sup |x − y|; x,y∈E 2, for every δ > 0, there is E ∈ F such that φ(E) ≤ δ and d(E) ≤ δ . For 0 < δ ≤ ∞ and U ⊂ X , we define {∑ ∪ } ψδ (U ) = inf φ(Ei ) : U ⊂ Ei , d(Ei ) ≤ δ, Ei ∈ F , i i ψ(U ) = lim ψδ (U ). δ↓0 Theorem 1.2.2. 1, ψ is a Borel measure. 2, If the members of F are Borel sets then ψ is Borel regular. Now let X be separable and 0 ≤ k < ∞. We choose F = {U ⊂ X} and φ(U ) = d(U )k with the interpretations 00 = 1 and d(∅)k = 0. The resulting measure ψ is called the k−dimensional Hausdorff measure. Definition 1.2.3. Let U ⊂ Rn , we define: {∑ ∪ } Hδk (U ) = inf d(Ei ) : U ⊂ k Ei , d(Ei ) ≤ δ . i i 5
The k -dimensional Hausdorff measure of U , denoted by H k (U ), is defined by H k (U ) = lim Hδk (U ). δ↓0 Example 1.2.4. • For k = 0, H 0 is the counting measure. • For k = m, where m is an integer and 1 ≤ m < n, let U be an m−dimensional surface in Rn , then the restriction H m |U gives a constant multiple of the sur- face measure on U . • For k = n, n 2n H = Vn . Vn (B(0, 1)) In particular, H n (B(a, R)) = (2R)n for a ∈ Rn and 0 < R < ∞. • For k > n, since H k (Rn ) = 0, H k in Rn is uninteresting. The k -dimensional Hausdorff measure is a Borel measure, but usually it is not a Radon measure since it need not be locally finite. For example, if k < n then every non-empty open set in Rn has infinite H k measure. The next elementary result, as a direct consequence of Definition 1.2.3, shows that Hausdorff measures behave nicely under translations and dilations in Rn . Theorem 1.2.5. For U ⊂ Rn , a ∈ Rn and 0 < α < ∞, we have 1, H k (U + a) = H k (U ) where U + a = {x + a : x ∈ U }, 2, H k (αU ) = αk H k (U ) where αU = {αx : x ∈ U }. Each of the measures H k can be compared with the others as follows. Theorem 1.2.6. For 0 ≤ k < l < ∞ and U ⊂ Rn , we have: 1, if H k (U ) < ∞ then H l (U ) = 0; 2, if H l (U ) > 0 then H k (U ) = ∞. According to Theorem 1.2.6, we may define the Hausdorff dimension as follows. Definition 1.2.7. Let U ⊂ Rn . Then the Hausdorff dimension of U is dim U = sup{k : H k (U ) > 0} = sup{k : H k (U ) = ∞} = inf{k : H k (U ) < ∞} = inf{k : H k (U ) = 0}. 6
In other words, dim U is the unique number (maybe ∞) for which 1, if k < dim U then H k (U ) = ∞; 2, if k > dim U then H k (U ) = 0. Example 1.2.8. • dim Rn = n. • dim C = log 2/ log 3 where C is the Cantor ternary set. Notice that, at the borderline case when k = dim U , all three cases H k (U ) = 0, 0 < H k (U ) < ∞, H k (U ) = ∞ are possible and we cannot have any general non- trivial information about the value H k (U ). In fact, for a given subset U of Rn , there may be no value k for which U has positive and finite H k measure. Therefore, the values of the Hausdorff measures often do not give much extra information. But, by replacing φ(U ) = d(U )k by some other function of the diameter, we can construct the Generalized Hausdorff measures which measure the given set in a more delicate manner. Definition 1.2.9. Let h : [0, +∞) → [0, +∞) be a gauge, i.e. h is non- decreasing, right-continuous and equal to 0 only at 0. For U ⊂ Rn , we define the Generalized Hausdorff h-measure as follows: {∑ ∪ } Λh (U ) = lim inf h(d(Ei )) : U ⊂ Ei , d(Ei ) ≤ δ . δ↓0 i i The next definitions are about Ahlfors-David regular and densities of Radon measures. Definition 1.2.10. A subset U of Rn is said to be Ahlfors-David regular with dimension k if it is closed and if there is a constant C0 > 0 such that C0−1 Rk ≤ H k (U ∩ B(x, R)) ≤ C0 Rk , for all x ∈ U and 0 < R < d(U ). Definition 1.2.11. Let 0 ≤ k < ∞ and let η be a Radon measure on Rn . The upper and lower k -densities of η at x ∈ Rn are defined by ( ) ∗k η B(x, r) Θ (η, x) = lim sup , r↓0 (2r)k ( ) η B(x, r) Θk∗ (η, x) = lim inf . r↓0 (2r)k 7
If they agree, their common value Θk (η, x) = Θ∗k (η, x) = Θk∗ (η, x) is called the k−density of η at x. Information on upper k−densities can be used to compare η with H k . Theorem 1.2.12. Let η be a Radon measure on Rn , U ⊂ Rn , and 0 < α < ∞. 1, If Θ∗k (η, x) ≤ α for x ∈ U , then η(U ) ≤ 2k αH k (U ). 2, If Θ∗k (η, x) ≥ α for x ∈ U , then η(U ) ≥ αH k (U ). 1.3 An extension of the mean value theorem In this section, we consider an extension of the mean value theorem and its corollaries. We first introduce a family of admissible functions as follow. { H= h : (0, +∞) → (0, +∞) : ∃ M > 0, c > 4 such that ∫cϵ } h(r) h(ϵ) dr ≤ M , for ϵ small enough . rn−1 ϵn−2 0 Remark 1.3.1. The family H contains many functions such as rk , rk | log r| where k > n − 2. If h1 , h2 ∈ H then h1 + h2 ∈ H and a · h1 ∈ H for every positive number a. Here is one simple way of interpreting the admissible functions h. For a function F : (0, +∞) → (0, +∞), we could consider the following property: 1 ∫cϵ cϵ F (r)dr ∃c > 4 such that lim sup 0 < ∞. (∗) ϵ→0+ F (ϵ) This can be viewed as a very weak asymptotic one-dimensional mean value property as we are comparing integral means with the value of the integrated function inside the interval of integration. A function h belongs to H if and only if F (r) = h(r)/rn−1 satisfies (∗). The following classical result will be used in the sequel. 8