On the Rate of Convergence of (‖f‖p‖f‖∞)p\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\big (\frac{\Vert f\Vert _{p}}{\Vert f\Vert _{\infty }}\big )^{p}$$\end{document} as p→∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p\rightarrow \infty $$\end{document}

Let (X,E,μ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(X,{\mathcal {E}},\mu )$$\end{document} be a measure space and let f:X→R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f:X\rightarrow \mathbb R$$\end{document} be a measurable function such that ‖f‖p<∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Vert f\Vert _{p}<\infty $$\end{document} for all p≥1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p\ge 1$$\end{document} and ‖f‖∞>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Vert f\Vert _{\infty }>0$$\end{document}. In this paper, we describe the rate of convergence of (‖f‖p‖f‖∞)p\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\frac{\Vert f\Vert _{p}}{\Vert f\Vert _{\infty }})^{p}$$\end{document} as p→∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p\rightarrow \infty $$\end{document}.


Statement of Results
Let (X, E, μ) be a measure space and let f : X → R be a measurable function such that f p < ∞ for all p ≥ 1 and f ∞ > 0. There are many results describing the limiting behaviour of f p as p → ∞. For example, it is wellknown that f p f ∞ → 1 as p → ∞; see [2, p. 201] for a proof of this and some related results. However, other limiting behaviours may also be of interest. For example, in the study of the regularity of solutions to the Navier-Stokes equation, it is sometimes of interest to know the limiting behaviour of the p'th power of f p f ∞ , i.e. it is of interest to know how ( f p f ∞ ) p behaves for large values of p; see, for example, [5,Equation (38)] for a more detailed discussion of this. We first note that it is not difficult to show that ( f p f ∞ ) p converges as p → ∞. More precisely, if we let i.e. E f is the extremum set of f , then and so It follows immediately from this and the Dominated Convergence theorem that (1.4) Finally, (1.2) follows immediately from (1.3) and (1.4).
We will now show that it is, in fact, possible to compute the rate of convergence in (1.2). This is the main result in this note and the statement of Theorem 1.1 below. (1.5) If we put The proof of Theorem 1.1 is given in Sect. 2.
Remark. Theorem 1.1 shows that for each ε > 0, there is a number p ε ≥ 1, such that Loosely speaking, this says that Δ f (p) behaves roughly like p −a f for large values of p, i.e. the rate at which ( Remark. Let (X, E, μ) be a measure space. In Theorem 1.1 we assume that the function f : X → R satisfies the following two conditions: (1) f p < ∞ for all p ≥ 1 and (2) f ∞ > 0. We will now briefly discuss what happens if these conditions are not satisfied.
Regarding condition (2). Of course, if condition (2) is not satisfied, i.e., if f ∞ = 0, then f is the null function, whence f p = f ∞ = 0 for all p ≥ 1, and it follows from this that the ratio In particular, we conclude that the ratio f p f ∞ is undefined for all p ≥ 1, and the problem of computing the rate of convergence of Regarding condition (1). It is not difficult to see that the conclusion in Theorem 1.1 remains valid even if condition (1) is replaced by the following slightly weaker condition: there is a real number p 0 ≥ 1 such that f p < ∞ for all p ≥ p 0 . If this condition is not satisfied, i.e., if there is no number = ∞ for all sufficient large real numbers p, and the problem of computing the rate of convergence of ∞ for all sufficiently large real numbers p. In particular, we conclude that if f ∞ = ∞, then the ratio f p f ∞ is undefined for all sufficiently large real numbers p, and the problem of computing the rate of convergence of In several important and natural cases the numbers a f and a f can be computed explicitly. This is the content of the next corollary.
In particular, if E f = {x 0 } consists of just one element and there are polynomials P and Q with deg P = deg Q = N and a positive number δ > 0 such that Proof. In this case, it is not difficult to see that a f = log λ(B(x, r)) log r , provided the limit exists. The detailed study of the local dimensions of measures is known as multifractal analysis and has received enormous interest during the past 20 years; the reader is refereed to the texts by Falconer [1] or Pesin [6] for a more thorough discussion of this. It is now generally believed by experts that local dimensions provide important information about the geometric properties of measures.
We will now describe the relation between the statement in Theorem 1.1 and local dimensions of measures. Let (X, E, μ) be a measure space and let f : X → R be a measurable function such that f p < ∞ for all p ≥ 1 and f ∞ > 0. Define the function Φ f : X → R by and let μ f denote the distribution of Φ f , i.e. μ f is the Borel probability measure on R defined by for Borel subsets B of R. It is clear that if r > 0, then B(1, r)) = μ f (B(1, r)) , and the statement in Theorem 1.1, therefore, says that In particular, if the local dimension dim loc (μ f ; 1) of μ f at 1 exists, then

Proof of Theorem 1.1
We first prove two auxiliary results that will be used in the proof of Theorem 1.1, namely Lemmas 2.2 and 2.4. Lemma 2.2 provides an alternative expressing for the p'th moment of a measure. This expression will allow us to bound Δ f (p) by an integral of the form Proof. It now follows from Lemma 2.1 that (2.1) Introducing the substitution u = t 1 p into the integral in (2.1), it now follows that and f (p) → 1 as p → ∞ and |g(p)| ≤ (1 − δ) p for all p ≥ 1.