GEORGIAN MATHEMATICAL JOURNAL: Vol. 3, No. 1, 1996, 49-52

ON THE UNIQUENESS OF MAXIMAL FUNCTIONS

L. EPHREMIDZE

Abstract. The uniqueness theorem for the one-sided maximal operator has been proved.

Let L be the class of real 2π-periodic integrable functions and let M be the one-sided maximal operator

M(f)(x) = sup_{b>x} \frac{1}{b − x} ∫_{x}^{b} fdm, f ∈ L, x ∈ ℝ

(m denotes the Lebesgue measure on the line ℝ).

In this paper we shall prove the following uniqueness

Theorem 1. Let f, g ∈ L and M(f) = M(g). Then f = g a.e. on ℝ.

Sets of the type {x ∈ ℝ : M(f)(x) > t} = {x ∈ ℝ : M(g)(x) > t} will be briefly denoted by (M > t). Obviously (M > t)_{t∈ℝ} is a class of bundled open sets continuous from the right, i.e.,

∪_{t>τ}(M > t) = (M > τ).

Let

t_{0} = inf{M(f) : x ∈ ℝ} = inf{M(g)(x) : x ∈ ℝ}.

For an arbitrary integrable function f if t = \frac{1}{2π} ∫_{0}^{2π} fdm, then M(f) ≥ t on the whole line and M(f)(x_{0}) = t for x_{0} being the point of maximum of the function x ↦ ∫_{0}^{x} fdm − tx. Thus we can conclude that

∫_{0}^{2π}fdm = ∫_{0}^{2π}gdm = t_{0}.

49

1991 Mathematics Subject Classification. 42B25.

Key words and phrases. Maximal function, uniqueness theorem.

1072-947X/96/0100-0049$09.50/0 \begin{matrix}c \\ ◯ \end{matrix} 1996 Plenum Publishing Corporation

Because of the Lebesgue differentiation theorem f, g ≤ t_{0} a.e. on ℝ\(M > t_{0}). On the other hand, applying the Riesz rising sun lemma (see [1]), we have

∫_{(M>t_{0})} fdm = ∫_{(M>t_{0})} gdm = t_{0} · m(M > t_{0})  (1)

(see also [2], p. 58). Consequently f = g = t_{0} a.e. on ℝ\(M > t_{0}) and to prove the theorem it suffices to show the validity of

Lemma 1. Let (a, b) be a (finite) connected component of (M > t_{0}). Then

∫_{x}^{b} fdm = ∫_{x}^{b} gdm  (2)

for each x ∈ (a, b).

Proof. Assume x fixed and let t_{x} = M(f)(x) = M(g)(x). For each t ∈ [t_{0}, t_{x}) suppose (a_{t}, b_{t}) to be the connected component of (M > t) which contains x and assume that b_{t} = x whenever t = t_{x} (note that b_{t_{0}} = b, by assumption). Obviously

∪_{t>τ}(a_{t}, b_{t}) ⊂ (a_{τ} , b_{τ})

and it is easy to show that t ↦ b_{t} is a non-increasing function on [t_{0}, t_{x}] continuous from the right.

Let D be the set of points of discontinuity of this function and let

D_{c} = {t : b_{τ} = b_{t} for some τ > t}.

If t ∈ [t_{0}, t_{x})\(D ∪ D_{c}) and b_{t} is a Lebesgue point of both functions f and g, then

f(b_{t}), g(b_{t}) ≤ t

(since b_{t} ∉ (M > t)). On the other hand, for each τ ∈ (t, t_{x}) we have

\frac{1}{b_{t} − b_{τ}} ∫_{b_{τ}}^{b_{t}} fdm, \frac{1}{b_{t} − b_{τ}} ∫_{b_{τ}}^{b_{t}} gdm > t

(since (a_{t}, b_{t}) is a connected component of (M > t) and b_{τ} ∈ (a_{t}, b_{t}); see Lemma 1 in [3]). Hence we can conclude that

f(b_{t}) = g(b_{t}) = t.

For t ∈ D let

b′_{t} = lim_{τ→t−} b_{τ} .

Then

\frac{1}{b′_{t} − b_{t}} ∫_{b_{t}}^{b′_{t}} fdm, \frac{1}{b′_{t} − b_{t}} ∫_{b_{t}}^{b′_{t}} gdm ≤ t

(since b_{t} ∉ (M > t)) and for each τ ∈ [t_{0}, t) we have

\frac{1}{b_{τ} − b_{t}} ∫_{b_{t}}^{b_{τ}} fdm, \frac{1}{b_{τ} − b_{t}} ∫_{b_{t}}^{b_{τ}} gdm > τ

(since (a_{τ} , b_{τ}) is a connected component of (M > τ) and b_{t} ∈ (a_{τ} , b_{τ})). Hence, letting τ converge to t from the left, we get

∫_{b_{t}}^{b′_{t}} fdm = ∫_{b_{t}}^{b′_{t}} gdm = t(b′_{t} − b_{t}).

Since [x, b] = A_{1} ∪ A_{2} ∪ A_{3}, where

A_{1} = {b_{t} : t ∈ [t_{0}, t_{x}]\(D ∪ D_{c})},
A_{2} = ∪_{t∈D}[b_{t}, b′_{t}],
A_{3} = {b_{t} : t ∈ D_{c}},

and since f = g a.e. on A_{1},

∫_{A_{2}} fdm = ∫_{A_{2}} gdm

and A_{3} is a denumerable set, we can conclude that (2) holds.

Note that the lemma remains true if f and g are locally integrable functions on ℝ. Hence if we use the balancing ergodic equality (see [4]) instead of the equality (1), then we get the uniqueness theorem for the ergodic maximal operator.

Theorem 2. Let (T_{λ})_{λ≥0} be an ergodic semiflow of measure-preserving transformations on a finite measure space (X, 𝕊, μ) and let M be the ergodic maximal operator

M(f)(x) = sup_{a>0} \frac{1}{a} ∫_{0}^{a} f(T_{λ}x)dλ, f ∈ L(X).

Then M(f) = M(g) implies that f = g a.e. (in the sense of measure μ) on X.

References

1. F. Riesz, Sur un théoreme de maximum de MM. Hardy et Littlewood. J. London Math. Soc. 7(1932), 10–13.

2. M. de Guzmán, Differentiation of integrals in R^{n}. Springer-Verlag, Berlin, 1975.

3. L. Ephremidze, On a relationship between the integrabilities of various maximal functions. Georgian Math. J. 2(1995), No. 1, 9–20.

4. B. Marcus and K. Petersen, Balancing ergodic averages. Lecture Notes in Math. 729(1979), No. 4, 126–143.

(Received 06.10.1994)

Author’s address: A. Razmadze Mathematical Institute Georgian Academy of Sciences 1, Z. Rukhazde St., Tbilisi 380093 Republic of Georgia