ON THE ADJOINT OF THE MAXIMAL FUNCTION

Alberto de la Torre

1. Introduction

Let A_{p}, p > 1, denote the class of functions satisfying Muckenhoupt’s condition, i.e.,

∫_{Q} w (∫_{Q} w^{1−q})^{p−1} ≤ C|Q|^{p}, p + q = pq.

It is clear that w ∈ A_{p} if and only if w^{1−q} ∈ A_{q}. This means that if Mf is the usual maximal operator Mf(x) = sup_{x∈Q} \frac{1}{|Q|} ∫_{Q} |f|, then [Mu]

(1)  ∫ (Mf)^{p}w ≤ C ∫ |f|^{p}w ∀f ∈ L_{p}(w),

if and only if

(2)  ∫ (Mf)^{q}w^{1−q} ≤ C ∫ |f|^{q}w^{1−q} ∀f ∈ L_{q}(w^{1−q}),

but this is quite surprising because the equivalence of (1) and (2) is natural only for self-adjoint operators.

If instead of A_{p} we consider the classes A_{p}^{+}, i.e., the functions w for which there exists C so that for any a < b < c the following holds:

∫_{a}^{b} w (∫_{b}^{c} w^{1−q})^{p−1} ≤ C(c − a)^{p},

then we see that now w ∈ A_{p}^{+} if and only if w^{1−q} ∈ A_{q}^{−} (the definition of A_{p}^{−} is the same as A_{p}^{+} except that we reverse the orientation). Since we know that these classes characterize the weights for which the one-sided maximal operators M^{+}f(x) = sup_{h>0} \frac{1}{h} ∫_{x}^{x+h} f and M^{−}f(x) = sup_{h>0} \frac{1}{h} ∫_{x−h}^{x} f are bounded [Sa], now we have that

Research supported by DGICYT Grant (pb94 1496-413) and Junta de Andalucía

(3)  ∫ (M^{+}f)^{p}w ≤ C ∫ |f|^{p}w ∀f ∈ L_{p}(w),

if and only if

(4)  ∫ (M^{−}f)^{q}w^{1−q} ≤ C ∫ |f|^{q}w^{1−q} ∀f ∈ L_{q}(w^{1−q}).

This means that M^{−} behaves as the “adjoint” of M^{+}. Our aim is to prove that, although certainly the operator M is not “self-adjoint”, it controls the adjoints of the linearizations of M. We will also prove that for the onesided operator M^{+}, what happens is that the adjoints of the linearizations of M^{+} are controlled by M^{−}. This will explain why the maximal operator behaves as a self-adjoint operator, while M^{+} behaves as the adjoint of M^{−}. Throughout the paper C will denote an absolute constant that may change from line to line. By A_{∞} we will denote the union of the A_{p} weights, p > 1.

2. Linearizations of the maximal operator

Let ϕ be a smooth non-negative function, radial, decreasing, with support on {x ∈ ℝ^{n}: |x| ≤ 1} and normalized so that ∫ ϕ = 1. It is well known [SW] that then the maximal operator

M_{ϕ}f(x) = sup_{ε>0} \frac{1}{ε^{n}} ∫ |ϕ(\frac{x − y}{ε} )f(y)| dy = sup_{ε>0} |f| ∗ ϕ_{ε}(x),

is pointwise equivalent to Mf. It is also clear that we can replace the sup in this definition by sup_{ε_{i}}, where ε_{i} runs over a countable family.

Now to each partition P = (A_{i}), of ℝ^{n} we can associate a linear operator T_{P} f(x) = ∑_{i} χ_{A_{i}}(x)(f ∗ ϕ_{ε_{i}})(x). It is clear that

|T_{P} f(x)| ≤ ∑_{i} χ_{A_{i}}(x)|f ∗ ϕ_{ε_{i}}(x)| ≤ ∑_{i} χ_{A_{i}}M_{ϕ}f(x) = M_{ϕ}f(x).

Conversely, for any f ≥ 0 there exists a partition P = (A_{i}) of ℝ^{N} such that M_{ϕ}f(x) ≤ 2(f ∗ ϕ_{ε_{i}})(x) for any x ∈ A_{i}, and then

M_{ϕ}f = ∑_{i} χ_{A_{i}}M_{ϕ}f ≤ 2 ∑_{i} χ_{A_{i}}(f ∗ ϕ_{ε_{i}}) = 2T_{P} f.

From these observations we obtain that (1) is equivalent to

(5)  ∫ (T_{P} |f|)^{p}w ≤ C ∫ |f|^{p}w ∀f ∈ L^{p}(w),

where C is a constant independent of the particular linearization.

Let T_{P}^{∗} be the adjoint of T_{P}, i.e.,

T_{P}^{∗}g(x) = ∑_{i}(gχ_{A_{i}}) ∗ ϕ_{ε_{i}}(x).

If we can control T_{P}^{∗} by M in the sense that there exists C, independent of the partition P, such that

(6)  ∫ |T_{P}^{∗}g|^{r}w ≤ C ∫ (Mg)^{r}w, r > 0, w ∈ A_{∞},

then we will have that (1) implies

(7)  ∫ |T_{P}^{∗}g|^{p}w ≤ C ∫ |g|^{p}w,

By duality this is equivalent to

∫ |T_{P} f|^{q}w^{1−q} ≤ C ∫ |f|^{q}w^{1−q},

where C does not depend on P. Since this is equivalent to (2), we will have shown that the equivalence between (1) and (2) is due to the fact that, although the maximal operator is not self-adjoint, it controls the adjoints of all its linearizations. To prove (6) we will make use of the sharp maximal function, defined as follows:

f^{♯}(x) = sup_{x∈B} \frac{1}{|B|} ∫_{B} |f − m_{B}f|,

where m_{B}f = \frac{1}{|B|} ∫_{B} f. It is easy to see that in this definition one can replace m_{B}f by any number C_{B} depending only on the ball B. The main fact about f^{♯}, due to C. Fefferman and E. Stein, is that for weights in A_{∞} the following inequality holds,

(8)  ∫ (Mf)^{r}w ≤ C ∫ f^{♯}^{r}w, r > 0,

and this clearly implies

(9)  ∫ |f|^{r}w ≤ C ∫ f^{♯}^{r}w.

We will control the adjoint of the linearizations of M_{ϕ} in the following precise sense. Theorem 1. Let T_{P}^{∗} be the adjoint of some linearization of M_{ϕ}, then there exists C independent of the particular linearization such that

(T_{P}^{∗}|g|)^{♯}(x) ≤ CMg(x)

for any x and g.

This result, together with (9), implies (6) as we wanted.

Proof. Let g ≥ 0. It is enough to prove that for any fixed x and any ball that contains x, B = B(x_{B}, r), there exists C_{B} so that

\frac{1}{|B|} ∫_{B} |T_{P}^{∗}g(z) − C_{B}| dz ≤ CMg(x).

Let us write g = g_{1} + g_{2} with g_{1} = gχ_{3B}, where 3B denotes a ball with center x_{B} and radius 3r. Let us choose C_{B} = T_{P}^{∗}g_{2}(x_{B}), then

\frac{1}{|B|} ∫_{B} |T_{P}^{∗}g(z) − C_{B}| dz

≤ \frac{1}{|B|} ∫_{B} |T_{P}^{∗}g_{1}(z)| + \frac{1}{|B|} ∫_{B} |T_{P}^{∗}g_{2}(z) − T_{P}^{∗}g_{2}(x_{B})|

= I_{1} + I_{2}.

For I_{1} we simply use the fact that the operator T_{P}^{∗} is obviously bounded in L^{1} and obtain,

I_{1} ≤ \frac{1}{|B|} ∫ g_{1} = \frac{1}{|B|} ∫_{3B} g ≤ CMg(x).

For I_{2} we will prove that there exists C such that for any z ∈ B

|T_{P}^{∗}g_{2}(z) − T_{P}^{∗}g_{2}(x_{B})| ≤ CMg(x).

Indeed,

(10) |T_{P}^{∗}g_{2}(z) − T_{P}^{∗}g_{2}(x_{B})| = |∑_{i}(g_{2}χ_{A_{i}}) ∗ ϕ_{ε_{i}}(z) − (g_{2}χ_{A_{i}}) ∗ ϕ_{ε_{i}}(x_{B})|

≤ ∑_{i} ∫_{y∈ℝ^{n}\3B} g_{2}(y)χ_{A_{i}}(y)|ϕ_{ε_{i}}(z − y) − ϕ_{ε_{i}}(x_{B} − y)| dy.

Now our assumptions on the support of ϕ imply, first of all, that only those ε_{i} that are greater then r give a non zero contribution to this last sum. Furthermore, for each y ∈ ℝ^{n} \3B one has that ϕ_{ε_{i}}(z −y) = ϕ_{ε_{i}}(x_{B} −y) = 0 unless ε_{i} ≥ \frac{1}{4}|y − x|, since otherwise for any z ∈ B

ε_{i} < \frac{1}{4}|y − x| ≤ \frac{1}{4}(|y − z| + |z − x|) ≤ \frac{1}{4}(|y − z| + 2r) ≤ \frac{1}{4}|y − z| + \frac{ε_{i}}{2} ,

and this implies |y − z|/ε_{i} > 2. If we now use that ϕ is smooth we can dominate our last term in (10) by

C ∑_{i} ∫ χ_{A_{i}}(y)g_{2}(y)\frac{|z − x_{B}|}{|y − x|^{n+1}} dy ≤ rC ∫_{ℝ^{n}\3B} g(y)\frac{1}{|y − x|^{n+1}} dy.

This can be dominated by g ∗ h(x) where h(y) = rCχ_{ℝ^{n}\B(0,2r)}(y)|y|^{−(n+1)}, and this is dominated by CMg(x) because h is radial, decreasing and integrable. □

Remark. Observe that, in particular, we have proved that the adjoints of the linearizations of the maximal operator map L_{∞} into BMO, i.e., near L^{1} they behave as a nice convolution operator, but near L^{∞} they behave as singular integrals.

3. One-sided operators

The ingredients of the proof in the previous case are the sharp maximal inequality for A_{∞} weights in (8), and Theorem 1. For the one-sided case one needs to consider A_{∞}^{+} weights. This class, and the corresponding A_{∞}^{−}, were introduced and studied in [MPT]. The lateral version of the sharp maximal function was introduced in [MT], where a substitute of (8) was proved, i.e., for any w ∈ A_{∞}^{+},

∫ (M^{+}f)^{r}w ≤ C ∫ (f^{♯+})^{r}w,

where the lateral sharp function is defined as

f^{♯+}(x) = sup_{h>0} \frac{1}{h} ∫_{x}^{x+h} |f − a_{I}|,

where a_{I} is a constant that depends only on the interval I = (x, x + h). (Actually in [MT], a smaller sharp function is used, but for our purpose this is enough.) Since the operator M^{−} is equivalent to

M_{ϕ}^{−}f(x) = sup_{ε>0} \frac{1}{ε^{n}} ∫ ϕ(\frac{x − y}{ε} )f(y) dy = sup_{ε>0} f ∗ ϕ_{ε}(x)

with ϕ supported in [0, 1) integrable and decreasing, it is enough to prove a lateral version of Theorem 1 but with respect to the adjoint of the linearizations of this operator. Observe that any linearization of M_{ϕ}^{−} is of the form T_{P}^{−}f(x) = ∑_{i} χ_{A_{i}}(x)(f ∗ ϕ_{ε_{i}})(x), but due to the lack of symmetry in ϕ, now the adjoint is of the form

T_{P}^{∗}g(x) = ∑_{i}(gχ_{A_{i}}) ∗ ϕ_{ε_{i}}(x),

where ϕ is increasing and supported on (−1, 0].

Theorem 2. Let T_{P}^{∗} be the adjoint of some linearization of M_{ϕ}^{−}, then there exists C independent of the particular linearization such that

(T_{P}^{∗}|g|)^{♯+}(x) ≤ CM^{+}g(x)

for any x and g.

Proof. The proof runs along the same lines as the proof of Theorem 1. For fixed x and h > 0 we consider the interval I = [x, x + h], and write g = g_{1}+g_{2}+g_{3}, where g_{1} = gχ_{(−∞,x)}, g_{2} = gχ_{[x,x+3h)}, and choose a_{I} = T_{P}^{∗}g_{3}(x). The fact that ϕ is supported on (−1, 0] implies that T_{P}^{∗}g_{1}(z) = 0, for z ∈ I, and then

\frac{1}{h} ∫_{I} |T_{P}^{∗}g(z)−a_{I}| dz ≤ \frac{1}{h} ∫_{I} |T_{P}^{∗}g_{2}(z)|+ \frac{1}{h} ∫_{I} |T_{P}^{∗}g_{3}(z)−T_{P}^{∗}g_{3}(x)| = I_{1} +I_{2}.

For I_{1} we use again that our operator is bounded in L^{1} and obtain I_{1} ≤ \frac{1}{h} ∫_{x}^{x+3h} g ≤ CM^{+}g(x). For I_{2} the same reasoning as before gives

|T_{P}^{∗}g_{3}(z) − T_{P}^{∗}g_{3}(x)| = | ∑_{i}(g_{3}χ_{A_{i}}) ∗ ϕ_{ε_{i}}(z) − (g_{3}χ_{A_{i}}) ∗ ϕ_{ε_{i}}(x_{B})|

≤ ∑_{i} ∫_{y>x+3h} g(y)χ_{A_{i}}(y)|ϕ_{ε_{i}}(z − y) − ϕ_{ε_{i}}(x_{−}y)| dy,

≤ C ∑_{i} ∫ χ_{A_{i}}(y)g_{3}(y)\frac{(z − x)}{(y − x)^{2}} dy

≤ hC ∫_{y>x+3h} g(y)\frac{1}{(y − x)^{2}} dy ≤ CM^{+}g(x).

□

References

[MPT] F. J. Martín-Reyes, L. Pick and A. de la Torre, A_{∞}^{+} condition, Canad. J. Math. 45 (1993), 1231–1244.

[MT] F. J. Martín-Reyes and A. de la Torre, One-sided BMO spaces, J. London Math. Soc. 49 (1994), 529–542.

[Mu] B. Muckenhoupt, Weighted norm inequalities for the Hardy maximal function, Trans. Amer. Math. Soc. 165 (1972), 207–226..

[Sa] E. Sawyer, Weighted inequalities for the one sided Hardy-Littlewood maximal functions, Trans. Amer. Math. Soc. 297 (1986), 53–61.

[SW] E. Stein and G. Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton University Press, 1971.