BEST CONSTANTS FOR UNCENTERED MAXIMAL FUNCTIONS

Loukas Grafakos^{∗} and Stephen Montgomery-Smith^{∗}

University of Missouri, Columbia

Abstract. We precisely evaluate the operator norm of the uncentered Hardy-Littlewood maximal function on L^{p}(ℝ^{1}). Consequently, we compute the operator norm of the “strong” maximal function on L^{p}(ℝ^{n}), and we observe that the operator norm of the uncentered Hardy-Littlewood maximal function over balls on L^{p}(ℝ^{n}) grows exponentially as n → ∞.

For a locally integrable function f on ℝ^{n}, let

(M_{n}f)(x) = sup_{B∋x} \frac{1}{|B|} ∫_{B} |f(y)| dy,

where the supremum is taken over all closed balls B that contain the point x. M_{n}f is called the uncentered Hardy-Littlewood maximal function of f on ℝ^{n}. In this paper we find the precise value of the operator norm of M_{1} on L^{p}(ℝ^{1}). It turns out that this operator norm is the solution of an equation. Our main result is the following:

Theorem. For 1 < p < ∞, the operator norm of M_{1} : L^{p}(ℝ^{1}) → L^{p}(ℝ^{1}) is the unique positive solution of the equation

(1)  (p − 1) x^{p} − p x^{p−1} − 1 = 0.

In order to prove our Theorem, we fix a nonnegative f and we introduce the left and right maximal functions:

(M_{L}f)(x) = sup_{a<x} \frac{1}{x − a} ∫_{a}^{x} f(t) dt  and  (M_{R}f)(x) = sup_{b>x} \frac{1}{b − x} ∫_{x}^{b} f(t) dt.

For the proof of the next result, known popularly as the “sunrise lemma”, we refer the reader to Lemma (21.75) (i), Ch VI in [HS].

Typeset by AMS-TEX

∗Research partially supported by the NSF.

Lemma 1. Let f ≥ 0 be in L^{1}(ℝ^{1}). For each λ > 0, let C_{λ} = {x : (M_{L}f)(x) > λ} and D_{λ} = {x : (M_{R}f)(x) > λ}. Then

(2)  λ|C_{λ}| = ∫_{C_{λ}} f dt  and  λ|D_{λ}| = ∫_{D_{λ}} f dt.

Now we are ready to prove the main lemma that leads to our Theorem. This next result may be viewed as the “correct” weak type estimate for the maximal function M_{1}.

Lemma 2. Let f ≥ 0 be in L^{1}(ℝ^{1}). For each λ > 0, let A_{λ} = {x : (M_{1}f)(x) > λ} and B_{λ} = {x : f(x) > λ}. Then

(3)  λ(|A_{λ}| + |B_{λ}|) ≤ ∫_{A_{λ}} f dt + ∫_{B_{λ}} f dt.

To prove (3), first note that

(4)  sup(M_{L}, M_{R}) = M_{1}.

For, clearly sup(M_{L}, M_{R}) ≤ M_{1}. On the other hand, it is easy to see that for each real number x, (M_{1}f)(x) is bounded by a convex combination of (M_{L}f)(x) and (M_{R}f)(x).

Now we add the two equalities in (2). Then using the fact that A_{λ} = C_{λ} ∪ D_{λ} which follows from (4), we obtain

(5)  λ(|A_{λ}| + |C_{λ} ∩ D_{λ}|) = ∫_{A_{λ}} f dt + ∫_{C_{λ}∩D_{λ}} f dt.

Clearly B_{λ} − (C_{λ} ∩ D_{λ}) is a set of measure zero, and f ≤ λ on (C_{λ} ∩ D_{λ}) − B_{λ}. Therefore

(6)  ∫_{(C_{λ}∩D_{λ})−B_{λ}} f dt ≤ λ|(C_{λ} ∩ D_{λ}) − B_{λ}|.

Equations (5) and (6) now imply equation (3), as required.

To prove the inequality in our Theorem, we require the following fact.

Lemma 3. Let f and g be nonnegative functions on ℝ^{1}. Then if p > 1, we have

∫_{0}^{∞} λ^{p−2} ∫_{g(t)>λ} f(t) dt dλ = \frac{1}{p − 1} ∫_{ℝ^{1}} fg^{p−1} dt,

and if p > 0, we have

∫_{0}^{∞} λ^{p−1}|\{g > λ\}| dλ = \frac{1}{p} ∫_{ℝ^{1}} g^{p} dt.

The first equality is easily proved, since by Fubini’s theorem, the left hand side is

∫_{−∞}^{∞} f(t) ∫_{0}^{g(t)} λ^{p−2} dλ dt,

which is readily seen to equal the right hand side. The second equality is the special case of the first when f = 1.

We now continue the proof of our Theorem. Multiplying (3) by λ^{(p−2)}, integrating λ from 0 to ∞, and applying Lemma 3, we obtain

\frac{1}{p}∥M_{1}f∥_{p}^{p} + \frac{1}{p}∥f∥_{p}^{p} ≤ \frac{1}{p − 1}∥f∥_{p}^{p} + \frac{1}{p − 1} ∫_{ℝ^{1}} f(x)[(M_{1}f)(x)]^{p−1} dx,

that is,

(p − 1)∥M_{1}f∥_{p}^{p} − ∫_{ℝ^{1}} f(x)[(M_{1}f)(x)]^{p−1} dx − ∥f∥_{p}^{p} ≤ 0.

Applying Hölder’s inequality with exponents p and p/(p−1) to the second term, we obtain

(7)  (p − 1)(\frac{∥M_{1}f∥_{p}}{∥f∥_{p}} )^{p} − p(\frac{∥M_{1}f∥_{p}}{∥f∥_{p}} )^{p−1} − 1 ≤ 0,

from which we conclude that \frac{∥M_{1}f∥_{p}}{∥f∥_{p}} ≤ c_{p}, where c_{p} is the unique positive solution of the (1).

To show that c_{p} is in fact the operator norm of M_{1} on L^{p}(ℝ^{1}), we construct an example. Note that equality in (3) is satisfied when f is even symmetrically decreasing and equality in (7) is satisfied when M_{1}f is a multiple of f. We are therefore led to the following example. Let f_{ε,N}(t) = |t|^{− \frac{1}{p}} χ_{ε,N}(|t|), where χ_{ε,N} is the characteristic function of the interval [ε, N]. It can be easily seen that

(8)  lim_{ε→0} lim_{N→∞} \frac{∥M_{1}f∥_{p}}{∥f∥_{p}} = M_{1}(f_{0})(1),

where f_{0}(t) = |t|^{− \frac{1}{p}} ∈ L_{loc}^{1}. An easy calculation gives that

(9)  M_{1}(f_{0})(1) = \frac{p}{p − 1} \frac{γ^{\frac{1}{p′}} + 1}{γ + 1},

where γ is the unique positive solution of the equation

(10)  \frac{p}{p − 1} \frac{γ^{\frac{1}{p′}} + 1}{γ + 1} = γ^{− \frac{1}{p}} .

Using (9) and (10), it is a matter of simple arithmetic to now show that M_{1}(f_{0})(1) is the unique positive root of equation (1). This completes the proof of our Theorem.

Before we conclude, we would like to make some remarks. Denote by x = (x_{1}, …, x_{n}) points in ℝ^{n}. For a locally integrable function f on ℝ^{n}, define

(N_{n}f)(x) = sup_{\begin{matrix}a_{1}<x_{1} \\ b_{1}>x_{1} \end{matrix}} ⋯ sup_{\begin{matrix}a_{n}<x_{n} \\ b_{n}>x_{n} \end{matrix}} \frac{1}{(b_{1} − a_{1}) ⋯ (b_{n} − a_{n})} ∫_{a_{1}}^{b_{1}} ⋯ ∫_{a_{n}}^{b_{n}} f(y_{1}, …, y_{n}) dy_{n} ⋯ dy_{1}.

N_{n} is called the “strong” maximal function on ℝ^{n}. Clearly N_{1} = M_{1}. Observe that

N_{n} ≤ M_{1}^{(1)} ∘ ⋯ ∘ M_{1}^{(n)},

where M_{1}^{(j)} denotes the maximal operator M_{1} applied to the x_{j} coordinate. This shows that the operator norm of N_{n} on L^{p}(ℝ^{n}) is less than or equal to c_{p}^{n}. By considering the function

g(x) = ∏_{j=1}^{n} f_{ϵ,N}(x_{j}),

where f_{ϵ,N} is as above, we obtain the converse inequality. We have therefore proved the following:

Corollary. For 1 < p < ∞, the operator norm of N_{n} : L^{p}(ℝ^{n}) → L^{p}(ℝ^{n}) is c_{p}^{n}, where c_{p} is the unique positive solution of equation (1).

One can show that \frac{p}{p−1} < c_{p} < \frac{2p}{p−1}. This implies that the operator norm of N_{n} on L^{p}(ℝ^{n}) grows exponentially with n, as n → ∞. Next, we observe that the same is true for the uncentered maximal function M_{n}. There are several ways to see this. One way is by considering the sequence of functions

h_{ϵ,N}(x) = |x|^{− \frac{n}{p}} χ_{ϵ,N}(|x|).

Let U_{n} be the open unit ball in ℝ^{n}. For x ∈ ℝ^{n}, let B_{x} = \frac{x}{2} + \frac{|x|}{2} \overline{U_{n}}. Then x ∈ B_{x} and

(11) (M_{n}(h_{ε,N}))(x) ≥ \frac{1}{|B_{x}|} ∫_{B_{x}} |y|^{− \frac{n}{p}} χ_{ε,N}(|y|) dy = \frac{1}{|U_{n}|}(\frac{2}{|x|})^{n} ∫_{B_{x}} |y|^{− \frac{n}{p}} χ_{ε,N}(|y|) dy.

Therefore for 1 < p < ∞ and for all ε, N > 0 we have

\frac{∥M_{n}(h_{ε,N})∥_{L^{p}}}{∥h_{ε,N}∥_{L^{p}}} ≥\frac{2^{n}}{∥h_{ε,N}∥_{L^{p}}|U_{n}|}\{ ∫_{r=0}^{+∞} ∫_{S^{n−1}} [\frac{1}{r^{n}} ∫_{B_{rϕ}} |y|^{− \frac{n}{p}} χ_{ϵ,N}(|y|)dy]^{p}dϕ r^{n} \frac{dr}{r} \}^{\frac{1}{p}}

(12)  =\frac{2^{n}}{∥h_{ϵ,N}∥_{L^{p}}|U_{n}|}\{ ∫_{r=0}^{+∞} ∫_{S^{n−1}} [\frac{1}{r^{n}} ∫_{t=0}^{r} ∫_{S_{ϕ}( \frac{t}{r})} t^{− \frac{n}{p}} χ_{ϵ,N}(t)t^{n} \frac{dt}{t}dθ]^{p}dϕ r^{n} \frac{dr}{r} \}^{\frac{1}{p}} ,

where S_{ϕ}(t) = \{θ ∈ S^{n−1} : |tθ − \frac{ϕ}{2}| ≤ \frac{1}{2}\}. By a change of variables (12) is equal to

\frac{2^{n}}{∥h_{ε,N}∥_{L^{p}}|U_{n}|}\{ ∫_{S^{n−1}} ∫_{r=0}^{+∞}[ ∫_{t=0}^{1} ∫_{S_{ϕ}(t)} χ_{ε,N}(rt) t^{\frac{n}{p′}} \frac{dt}{t}dθ]^{p} \frac{dr}{r} dϕ\}^{\frac{1}{p}}

(13)  =\frac{2^{n}}{|U_{n}|}\{ ∫_{S^{n−1}} [\frac{∫_{r=0}^{∞} |(K_{ϕ} ∗ χ_{ε,N})(r)|^{p} \frac{dr}{r}}{∫_{r=0}^{∞} χ_{ε,N}^{p}(r)\frac{dr}{r}} ]\frac{dϕ}{ω_{n−1}} \}^{\frac{1}{p}} ,

where K_{ϕ}(t) = t^{n/p′}χ_{[0,1]}(t) ∫_{S_{ϕ}(t)} |θ|_{B}^{−n/p}dθ, ω_{n−1} = |S^{n−1}| = \frac{(n−1)π^{\frac{n−1}{2}}}{Γ( \frac{n+1}{2} )} , and ∗ denotes

convolution on the multiplicative group G = (ℝ^{+}, \frac{dt}{t}). If K ≥ 0 on G, the sequence of functions χ_{ϵ,N} gives equality in the convolution inequality ∥g ∗ K∥_{L^{p}(G)} ≤ ∥K∥_{L^{1}(G)}∥g∥_{L^{p}(G)} as ε → 0 and N → ∞. Therefore, the expression inside brackets in (13) converges to ∥K_{ϕ}∥_{L^{1}(G)}^{p} as ϵ → 0 and N → ∞, and we obtain the estimate

lim_{\begin{matrix}ϵ→0 \\ N→∞ \end{matrix}} \frac{∥M_{n}(h_{ε,N})∥_{L^{p}}}{∥h_{ε,N}∥_{L^{p}}} ≥ \frac{2^{n}}{|U_{n}|}\{ ∫_{S^{n−1}} [ ∫_{0}^{1}t^{\frac{n}{p′}} ∫_{S_{ϕ}(t)} dθ \frac{dt}{t}]^{p} \frac{dϕ}{ω_{n−1}} \}^{\frac{1}{p}} = \frac{n2^{n}}{ω_{n−1}} ∫_{0}^{1}t^{\frac{n}{p′}} ∫_{\begin{matrix}S^{n−1} \\ θ_{1}≥t \end{matrix}} dθ \frac{dt}{t}

= 2^{n}p′ \frac{ω_{n−2}}{ω_{n−1}} ∫_{0}^{1}s^{\frac{n}{p′}} (1 − s^{2})^{\frac{n−3}{2}} ds = 2^{n−1}p′ \frac{ω_{n−2}}{ω_{n−1}} B(\frac{n}{2p′} − \frac{1}{2}, \frac{n−3}{2}). (14)

Stirling’s formula gives that expression (14) is asymptotic to \{\frac{4( \frac{1}{p′})^{\frac{1}{p′}}}{( \frac{1}{p′} +1)^{( \frac{1}{p′} +1)}} \}^{\frac{n}{2}} as n → ∞

and since the number inside the braces above is bigger than 1 when 1 < p < ∞, we also deduce exponential growth for the operator norm of M_{n} on L^{p}(ℝ^{n}), as n → ∞.

These remarks should be compared to the fact that for 1 < p < ∞, the operator norm of the Hardy-Littlewood maximal function on L^{p}(ℝ^{n}) is bounded above by some constant A_{p} independent of the dimension n (see [S] and [SS]).

References

[CG] M. Christ and L. Grafakos, Best constants for two non-convolution inequalities, Proc. Amer. Math. Soc. 123 (1995), 1687–1693.

[HS] Hewitt and Strömberg, Real and Abstract Analysis, Springer-Verlag, New York, NY, 1965.

[S] E.M. Stein, Some results in harmonic analysis in ℝ^{n}, for n → ∞, Bull. Amer. Math. Soc. 9 (1983), 71–73.

[SS] E.M. Stein and J.O. Strömberg, Behavior of maximal functions in ℝ^{n}, for large n, Ark. Math. 21 (1983), 259–269.

[Z] A. Zygmund, Trigonometric Series, Cambridge Univ. Press, Cambridge, UK, 1959.

Department of Mathematics, University of Missouri, Columbia, MO 65211

   E-mail address: loukas@msindy3.cs.missouri.edu, stephen@mont.cs.missouri.edu