Parameter depending almost monotonic functions and their applications to dimensions in metric measure spaces

In connection with application to various problems of operator theory, we study almost monotonic functions w(x, r) depending on a parameter x which runs a metric measure space X , and the so called index numbers m(w,x), M(w, x) of such functions, and consider some generalized Zygmund, Bary, Lozinskii and Stechkin conditions. The main results contain necessary and sufficient conditions, in terms of lower and upper bounds of indices m(w, x) and M(w, x) , for the uniform belongness of functions w(·, r) to Zygmund-BaryStechkin classes. We give also applications to local dimensions in metric measure spaces and characterization of some integral inequalities involving radial weights and measures of balls in such spaces.


Introduction
Last decades there was observed an increasing interest to the study of function spaces whose characteristics may vary from point to point.A well known typical example is the generalized Lebesgue space L p(•) with variable exponent, see for instance the surveying papers [2], [11], [25] on harmonic analysis in such spaces.Another example is a generalized Hölder space with a given dominant of continuity modulus, which may vary from point to point.Hölder spaces of variable order λ(x), and more general variable generalized Hölder spaces, were studied in [9], [16], [28], [29], [30], [32], [31].The generalized Hölder classes were there defined as spaces of functions whose continuity modulus at a point x has a dominant (characteristic) w(x, r), where r ∈ [0, ], 0 < < ∞ and x runs a certain set X ⊆ R n (typically an interval in R 1 , or either a domain or a unit sphere in R n ).This characteristic belongs to this or other Zygmund-Bary-Stechkin class Φ α β in the variable r ∈ R 1 + for every x ∈ R n and one needs to have some properties of these functions uniform with respect to x.
But the main reason for this study lies in some problems in the theory of metric measure spaces with variable dimensions.As is known, the onedimensional Muckenhoupt condition (or the multi-dimensional one in the case of radial weights), includes integral constructions similar to those which are involved in the Zygmund integral condition.When such Muckenhoupt conditions are used within the frameworks of metric measure spaces X where μB(x, r) depends on the point x, one arrives at a similar problem of dependence of a function in the Zygmund-Bary-Stechkin class Φ α β on a parameter x running a metric measure space, see [12].These facts led us to the study of Zygmund-Bary-Stechkin functions w(x, r) depending on a parameter x belonging in general to an arbitrary metric measure space X , which is undertaken in this paper.
The paper is organized as follows.In Section 2 we give necessary preliminaries.In Subsection (2.1), we deal with the so called indices m(w, x) and M (w, x) of functions w(x, r) almost monotonic in r ∈ R 1 + .We refer to papers [10], [18], [17], [21], [22], [20], [19], [24], [23] for properties of such indices in the case w = w(r) and their usage in the study of the Fredholmness of singular integral operators in weighted generalized Hölder spaces (see also [15] for Fredholmness in the case of usual Hölder spaces).In Subsection (2.1), we consider the index numbers m(w, x), M(w, x) of Zygmund-Bary-Stechkin functions in dependence on a point x.We study, in particular, their lower and upper bounds when x runs X .In Subsections (2.3) and (2.4), we deal with some generalized Zygmund, Bary, Lozinskii and Stechkin conditions.
Section 3 contains results on the characterization of the uniform belongness of functions to Zygmund-Bary-Stechkin classes in terms of lower and upper bounds of indices m(w, x) and M (w, x).In the proofs we follow some ideas developed earlier in [1] and [10].
In Section 4 we consider applications to some questions in measure metric spaces.We suggest a new approach to define local lower and upper dimensions.In terms of some bounds of these local dimensions we give a new characterization of the validity of some Zygmund-type integral inequalities involving measures μB(x, r) of balls.Such inequalities arise in Muckehhoupt conditions for radial monotonic weights.

2.1.
Indices m(w), m * (w), m * * (w) and M (w), M * (w), M * * (w) of functions w ∈ W depending on parameter.Let X be a metric space with a positive measure μ (not necessarily satisfying the doubling condition) and Ω a bounded open set in X with = diam Ω, 0 < < ∞.
We will deal with functions w(x, r) defined on Ω × [0, ] which are almost increasing in variable r , and we will be interested in properties related to this almost monotonicity, uniform with respect to x.
We recall that a non-negative function f on [0, ] is said to be almost increasing (or almost decreasing ) if there exists a constant 4) for any fixed x ∈ X the function w(x, r) is almost increasing in r with the uniform coefficient 1 ≤ C w < ∞ not depending on x: In future for brevity we say that a function w(x, r) is uniformly almost increasing in r meaning that property (4) holds with C w not depending on x.When it is admitted that w(x, r) is almost increasing at every point x ∈ Ω with C w which may depend on x, we say that w(x, r) is pointwise almost increasing in r .Similar notions for almost decreasing functions are defined.
Whenever necessary we always assume that w(x, r) is defined as (2.3) w(x, r) = w(x, ) for r ≥ .
We also introduce the wider class will be referred to as the lower and upper Matuszewska-Orlicz indices of a function w ∈ W with respect to r .We refer to [14], p. 20, [18], [10] for the Matuszewska-Orlicz indices m(w), M(w) in the case where w = w(r).The function involved in (2.5) and (2.6), is submultiplicative.Recall that a nonnegative everywhere finite function f (t) defined on (0, ∞), is called submultiplicative, if f (t 1 t 2 ) ≤ f (t 1 )f (t 2 ) for all t 1 , t 1 ∈ (0, ∞).Following [13], Ch.II, formula (1.16) we call the function W (x, r) a dilation of w(x, r) (note that in [13] such a dilation was defined with sup h>0 instead of lim h→0 ).
The lower index may be also written in terms of the dilation of w(x, r): We will mainly deal with the numbers m(w), M(w) and in the sequel give sufficient conditions on w for the coincidences m(w) = m * (w) = m * * (w) and M (w) = M * (w) = M * * (w).
Remark 2.4.The number m(w) and m * (w) may be also represented as (2.17) Proof.Indeed, we transform m * (w) as follows:  w(x,h) is obviously submultiplicative.Hence by Lemma 2.3 we arrive at (2.18).The arguments for m(w) are similar.

On the coincidence m(w)
= m * (w) = m * * (w).In Theorem 3.1 characterizing the generalized Bary-Stechkin class, we use only the numbers m(w) and m * (w).However, in applications of these numbers to measuring lower and upper dimensions of metric measure spaces (see Section 4) the usage of the number m * * (w) seems to be more natural as the lower bound for local lower dimensions.So there arises an interest for sufficient conditions on w(x, r) for the coincidence m(w) = m * (w) = m * * (w).
We introduce the following definition. Proof.
Of course, conditions (2.19)-(2.20) of Definition 2.5 are nothing else but just a reformulation of the fact that m(w) = m * (w) = m * * (w).So we are interested in some easy to check sufficient conditions for w to satisfy condition (A).
The conditions of the next two lemmas are aimed , in particular, to the case where w(x, r) = μB(x, r) may be a measure of balls in metric measure spaces.
Lemma 2.7.Let w(x, r) have a form Proof.The proof is direct.Indeed, From the uniform decay condition on ψ(x, r) we easily obtain that Hence the formulas m(w, x) = M (w, x) = ψ(x, 0) follow.
To check the validity of (2.19), in view of (2.22) it is sufficient to verify that for small r , which holds by (2.21).The validity of (2.20) is obvious since the The following lemma provides sufficient conditions when m(w, x) may be different from M (w, x), which is important in those applications where lower and upper lower dimensions in measure metric spaces may be different.

Lemma 2.8. Let w(x, r) have a form
. Then where Φ(r) = lim h→0 ϕ(rh) ϕ(h) is the dilation of the function ϕ.Hence the formulas (2.23) follow.By direct verification it is also easy to check that w(x, r) satisfies condition (A).Then m(w) = m * (w) = m * * (w) by Lemma 2.6.
We refer to [20], where there are given various non-trivial examples of functions ϕ(r) with non-coinciding local indices m(ϕ), M(ϕ).where c = c(w) > 0 does not depend on r ∈ (0, ] and x ∈ Ω.The class Φ of functions w = w(r) was introduced in [1], where conditions (2.24) were imposed on monotonous functions in W ; we deal with almost monotonous functions.In the case where functions w do not depend on x, the following statement characterizing the class Φ in terms of the indices m(w) and M (w), was proved in [18], p. 125.

Theorem 2.10. A function w(r) ∈ W([0, ]) is in the Zygmund-Bary-Stechkin class Φ if and only if
and for w ∈ Φ and any ε > 0 there exist constants c 1 = c 1 (w, ε) > 0 and Besides this, condition m(w) > 0 is equivalent to the first inequality in (2.24), while condition M (w) < 1 is equivalent to the second one.
We will obtain a similar statement for the general case treated in this paper and for the generalized Zygmund-Bary-Stechkin class Φ β γ .Let β ≥ 0, γ > 0 .The following classes Φ β γ , considered in [27] and [26], p. 253, generalize the class and by Z γ the class of functions w(x, r) ∈ W satisfying the condition where A = A(w) > 0 does not depend on r ∈ (0, ] and x ∈ Ω.We denote In the sequel we refer to the above conditions as (Z β )-and (Z γ )conditions.
The class Φ β γ is nonempty if and only if β < γ , see Corollary 3.4 below.Similarly to (2.25), we shall show that the condition β < m(w) ≤ M (w) < γ, with numbers m(w) and M (w) introduced in (2.11) is a characterization of the "uniform" class Φ β γ (Ω × [0, ]), see Theorem 3.5.2.4.Generalized Bary, Lozinskii and Stechkin conditions.Let β, γ ∈ R 1 .For functions w ∈ W(Ω × [0, ]) we consider the following well known conditions (see [1], where such conditions were treated for β = 0 and in the case of increasing functions w = w(r) belonging to W([0, ])): 1) Bary type conditions: where A = A(w) > 0 does not depend on n ∈ N + and x ∈ Ω ,  Proof.We may assume that β ≥ 0 , the case of negative β being reduced to the case of positive β by passing to the function w 1 (x, t) = t a w(x, t) with a > −β .I) According to (2.12) we have Similarly, by (2.11) we obtain For part II) we prove the following chain We suppose that β > 0 , modifications for the case β = 0 are easy: power functions should be replaced by the logarithmic function under the corresponding integration.We take = 1 without loosing generality.
The implication (Z β ) =⇒ (sL β ).Given (Z β ), that is, we shall show that condition (sL β ), that is, the condition for all ξ ∈ (0, 1), r ∈ (0, 1) such that ξ r ≤ 1 2 ("ersatz" of the almost monotonicity).Indeed, from (3.3) it follows that which yields (3.5) since ξ r ≤ 1 2 .Now the key moment is that we repeat the same idea once more.For all ξ > 2η from (3.3) we obtain We choose now a relation between η and ξ in the following way: 2 and from (3.6) we obtain w(x,η) for η sufficiently small (0 < η < 1 2 e −2AM ) and all x.Therefore, w(x, Cη) ≥ 2C β w(x, η).Hence lim inf The implication (sL β ) =⇒ (S β ).Let (sL β ) be valid: there exists a C > 1 such that We shall show that the function w(x,r) r β+δ with every is uniformly a.i. in r .Let 0 < ε 1 < ν − 1 .From (3.7) it follows that for every such ε there exists an r 1 not depending on x such that We choose δ = δ(C, ε, ε 1 ) = ln (ν−ε1) ln C > 0 and show that w(x) x β+δ is almost increasing under this choice of δ .With this δ , inequality (3.9) takes the form Now, for arbitrary 0 < ρ < r ≤ r 1 we choose an integer N by the condition . Then by (3.10) we get Since w(x, r) is a.i., we obtain (3.11) Thus w(x,r) r β+δ is almost increasing in r on [0, r 1 ].Since any positive function bounded from below is a.i, from condition (2.1) it follows that w(x,r) r β+δ is almost increasing on [0, ].This fact has been proved for any δ such that (3.12) with an arbitrarily small ε 2 .This means that one may take any δ satisfying (3.12) with an arbitrary C > 1 such that the inequality Since ε > 0 is arbitrarily small, the right hand side in (3.13) may be arbitrarily close to m(w) in case m(w) < ∞.If m(w) = ∞, then δ may be taken arbitrarily large.In both cases one may take an arbitrary δ < m(w).
The implication (P β ) =⇒ (B β ).We have for any choice of the integer p.Since the function w(x, r) is almost increasing in r , we then get By condition (P β ), for any θ ∈ (0, 1) we can choose an integer p such that w x, , that is, (B β ) holds.

III).
This part was already proved under the passage (sL β ) =⇒ (S β ), see (3.8).The proof of Theorem 3.2 is symmetrical to that of Theorem 3.1 and thereby is omitted.Remark 3.3.Statement II) of Theorems 3.1 and 3.2 was proved in [1] in the case when β = 0 and functions w = w(r) were monotonous.A modification of the proof from [1] adjusted for the case of β ≥ 0 and almost increasing functions w(r) was given in [10].For the case of functions w(x, r) depending on a parameter x belonging to a metric measure space, we followed mainly the arguments of the proof in [10], with modifications everywhere, where the uniformness of various estimates with respect to x was needed.Theorems 3.1 and 3.2 for almost increasing functions w = w(r) were earlier proved for β = 0 and γ = 1 in [18].
To get at inequalities (3.15), it suffices to observe that the functions w(x, r) r m(w)−ε and w(x, r) r M(w)+ε are uniformly almost increasing and decreasing, respectively, for any ε > 0 according to statement III) of Theorems 3.1 and 3.2, and any uniformly almost increasing or almost decreasing function is uniformly bounded from above or from below, respectively.
The following theorem characterizes the conditions (S β ) and (S γ ) in terms of the indices m(w) and M (w).Theorem 3.6.For any function w ∈ Z β its lower index bound m(w) may be calculated by the formula while for any w ∈ Z γ its upper index bound M (w) is calculated by the formula and a non-negative measure μ, and let B(x, r) = {y ∈ X : d(x, y) < r} .We refer to [8], [4] for basics on metric measure spaces.In the particular case where for every point x ∈ X there exists a positive number s = s(x) such that for every positive ε > 0 , where the constants C 1 > 0, C 2 > 0 in general depend on x and ε , then the space X may be said to have a local dimension at a point x calculated by the formula In the general case, the lower and upper local dimensions may be attributed to a point x.We refer to [3] (p. 25), [6], [7], where such local dimensions were introduced and/or used.Formulas (4.4) reflect the case where instead of (4.3) one has for arbitrary ε > 0 (with C 1 and C 2 in general depending not only on x, but on ε as well).
Comparison with property (3.15) of almost monotonic functions allows us to suggest new formulas for local lower and upper dimensions of metric measure spaces, based the technique of the lower and upper indices adjusted for nonnegative monotonic (or almost monotonic) functions and presented in Section 2-3.Namely, the lower and upper local dimensions at a point x may be introduced as lower and upper indices m(μB(x, r)), M (μB(x, r)) of the measure of the ball B(x, r).In the case when X is bounded we introduce the local lower and upper dimensions in the form According to (2.8) and Lemma 2.3, we can also write the dimensions dim X (x) and dim X (x) in the form An advantage of the usage of these local dimensions in comparison with dimensions (4.4) is in the fact that just in terms of the dimensions dim X (x) and dim X (x) it is possible to characterize some integral inequalities involving the measures μB(x, r), which appear when one considers the Muckenhoupt type A p -condition on metric measure spaces, see Theorem 4.2.
The following statement is obvious.In applications to weighted estimations of maximal, singular and potential operators on metric measure spaces the following more general integral inequalities for radial weights w[d(x, x 0 )] and measures μB(x, r) are of importance, where u, v are almost increasing functions and 0 < h ≤ < ∞. (For simplicity, we do not take functions u and v depending on the parameter x).The next theorem provides sufficient conditions for their validity in terms of the indices of the weight and the lower dimension of the space.

Theorem 4.3. Let u, v ∈ W([0, ]). Under assumptions i)-iii), the conditions
are sufficient for inequalities (4.16)-(4.17) to hold, respectively.Proof.It suffices to apply Theorem 3.1 and make use of the following properties of the index numbers 4.4.The case of unbounded metric measure spaces.When X is unbounded, conditions (4.18) with = ∞ do not guarantee the validity of inequalities (4.16)-(4.17):an information about the behavior of functions u, v and the measures μB(x, r) as r → ∞, is also needed.Characteristic of the measures μB(x, r), required for this goal, may be introduced in terms similar to the above introduced dimensions: where The numbers dim X (∞), dim X (∞) as introduced in (4.20)-(4.21)do not depend on x ∈ X , which is proved in the following lemma.Note that they are not limits of dimensions dim X (x) and dim X (x) or dimensions dim X (x), dim X (x) as x → ∞: simple examples show that these limits may not exist, while dim X (∞), dim X (∞) always exist as finite or infinite number, and when those limits exist they do not necessarily coincide with dim X (∞), dim X (∞).ln M ∞ (x, r) ln r do not depend on x.In the case where the measure μ satisfies also assumption ii) and d(x, y) is a distance, that is, a = 1 in (4.2), then even the function M ∞ (x, r) does not depend on x.
Proof.Let x, y be arbitrary points of X and let t > ad(x, y).By the triangle inequality (4. by the arbitrariness of x and y .Similarly, the coincidence of the limits as r → 0 is proved. In the case where a = 1 and the measures μB(x, r) are continuous in r , the last statement of the lemma follows from (4.22).
We dwell on an extension of Theorem 4.3 to unbounded spaces X for the case of inequality (4.17).To this end, we introduce one more characteristic of the space X at infinity: For many "good" metric measure spaces one has for instance, for spaces with the measures of balls μB(x, r) behaving as r → ∞ similarly to functions w(x, r) in Lemmas 2.7 and 2.8 as r → 0. We also introduce the indices for functions w on an infinite interval, similarly to (2.5)-(2.8),(2.10); for simplicity, in this case we take functions w not depending on a parameter x ∈ X : The following theorem provides sufficient conditions for the validity of inequality (4.17), in terms of the indices (they are necessary in the case where v(r) is a power function).M (v) < dim(X) and M ∞ (v) < dim X (∞), are sufficient for the validity of (4.17).Proof.By Theorem 4.3 and the first condition in (4.26), inequality (4.17) holds for any finite interval 0 < r < N .To guarantee that the constant C in (4.17) does not depend on N , we have to use properties of v(r) and μB(x, r) at infinity.We may consider the values r ≥ 2 .It suffices to prove that  ln r + M ∞ (v) < 0, is sufficient for the validity of (4.28).It is easy to see that the last inequality is nothing else but the inequality M ∞ (v) < dim X (∞), which holds by the assumption.
) ess inf x∈Ω w(x, r) := d 0 (r) > 0 for every r > 0, where ess inf x∈Ω w(x, r) is considered with respect to the measure μ on X ;

.
as r → ∞.With the change of variables t → 1 t , ρ = 1 r , inequality (4By Theorem 3.2, the validity of the latter inequality with the uniform constant C not depending on x and r , is equivalent to the numerical inequality M (w) < 0 for the index (2.11) of the function w(x, r) = μB(x,