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![ydunyan@ucas.ac.cn](http://html.rhhz.net/ZGKXYDXXB/images/REemail.gif)
中国科学院大学数学科学学院, 北京 100049
摘要: 研究迭代的非中心型哈代-李特伍德极大函数和迭代的中心型哈代-李特伍德极大函数。证明迭代极大函数的极限是极大算子的一个不动点。作为不动点理论的一个应用,最终得到,对于非中心型哈代-李特伍德极大算子,这个不动点处处为‖f‖∞。对于中心型哈代-李特伍德极大算子,仅在n=1,2时有相同的结果。
关键词: 哈代-李特伍德极大函数不动点迭代的哈代-李特伍德极大函数
Define the centered Hardy-Littlewood maximal function by
${M_c}f\left( x \right) = \mathop {\sup }\limits_{r > 0} \frac{1}{{\left| {B\left( {x,r} \right)} \right|}}\int_{B\left( {x,r} \right)} {\left| {f\left( y \right)} \right|{\rm{d}}y} ,$ | (1) |
$Mf\left( x \right) = \mathop {\sup }\limits_{B \ni x} \frac{1}{{\left| B \right|}}\int_B {\left| {f\left( y \right)} \right|{\rm{d}}y} ,$ | (2) |
It is clear that
${M_c}f\left( x \right) \le Mf\left( x \right) \le {2^n}{M_c}f\left( x \right)$ | (3) |
Let M be the non-centered Hardy-Littlewood maximal function defined by (2). Define the iterated non-centered Hardy-Littlewood maximal function denoted by Mk+1 as follows:
${M^{k + 1}}f\left( x \right): = M\left( {{M^k}f} \right)\left( x \right),$ | (4) |
In the same way, we can set
$M_c^{k + 1}f\left( x \right): = {M_c}\left( {M_c^kf} \right)\left( x \right).$ | (5) |
Wei et al[5] studied the limit of Mkf and obtained Theorem A as follows.
Theorem A??For any f∈L∞(
$\mathop {\lim }\limits_{k \to \infty } {M^k}f\left( x \right) = {\left\| f \right\|_\infty }$ | (6) |
For Mc, we want to know whether it has the same properties as M. Unexpectedly, the limit of Mckf is essentially different from the limit of Mkf.
Now we formulate our main results as follows.
Theorem B??If f∈Lloc1(
$\mathop {\lim }\limits_{k \to \infty } M_c^kf\left( x \right) = {\left\| f \right\|_\infty }$ | (7) |
Theorem C??Let f∈Lloc1(
$\mathop {\lim }\limits_{k \to \infty } {M^k}f\left( x \right) = {\left\| f \right\|_\infty },$ |
We remark that the range of function f in Theorem C is wider than that in Theorem A. Furthermore in this paper we will use some novel ideas to prove Theorem C.
1 Fixed point of Hardy-Littlewood maximal operatorTo prove our main theorems, we first provide some definitions and lemmas which will be used in the follows. Some lemmas can be found in classic literatures and here we omit their proofs.
Definition 1.1??A function F is called a fixed point of a operator T, if
$TF\left( x \right) = F\left( x \right)$ | (8) |
Obviously if F is a fixed point of the operator T, then we have
$\mathop {\lim }\limits_{k \to \infty } {T^k}F\left( x \right) = F\left( x \right).$ |
${M_c}f\left( x \right) \ge \left| {f\left( x \right)} \right|$ |
$Mf\left( x \right) \ge \left| {f\left( x \right)} \right|.$ |
Lemma 1.1??For x∈
${M^{k + 1}}f\left( x \right) \ge {M^k}f\left( x \right)$ | (9) |
$M_c^{k + 1}f\left( x \right) \ge M_c^kf\left( x \right)$ | (10) |
Proof??Set E={x:x is not the Lesbegue point of |f|}. It follows from the Lesbegue differentiation theorem that m(E)=0. Actually we merely need to prove
${M^2}f\left( x \right) \ge Mf\left( x \right)$ |
We conclude that
$\begin{array}{l}Mf\left( x \right) = \mathop {\sup }\limits_{B \ni x} \frac{1}{{\left| B \right|}}\int_B {\left| {f\left( y \right)} \right|{\rm{d}}y} \\ = \mathop {\sup }\limits_{B \ni x} \frac{1}{{\left| B \right|}}\int_{B\backslash E} {\left| {f\left( y \right)} \right|{\rm{d}}y} \\ = \mathop {\sup }\limits_{B \ni x} \frac{1}{{\left| B \right|}}\int_{B\backslash E} {\left\{ {\mathop {\lim }\limits_{k \to 0} \frac{1}{{\left| {B\left( {y,r} \right)} \right|}}\int_{B\left( {y,r} \right)} {\left| {f\left( u \right)} \right|{\rm{d}}u} } \right\}{\rm{d}}y} \\ \le \mathop {\sup }\limits_{B \ni x} \frac{1}{{\left| B \right|}}\int_{B\backslash E} {Mf\left( y \right){\rm{d}}y} = \mathop {\sup }\limits_{B \ni x} \frac{1}{{\left| B \right|}}\int_B {Mf\left( y \right){\rm{d}}y} \\ = {M^2}f\left( x \right).\end{array}$ | (11) |
By Lemma 1.1, since Mkf monotonously increases, the limit of Mkf(x) exists for all x∈
Lemma 1.2??If f∈Lloc1(
Proof??Set ?∈Cc∞(
$\int_{{\mathbb{R}^n}} {\phi \left( x \right){\text{d}}x} = 1.$ |
${f_t} = f * {\phi _t},$ |
${f_t} \in {C^\infty }\left( {{\mathbb{R}^n}} \right) \cap L_{{\text{loc}}}^1\left( {{\mathbb{R}^n}} \right).$ |
${{\chi }_{r}}=\frac{1}{\left| {{B}_{r}} \right|}{{\chi }_{{{B}_{r}}}}.$ |
${M_c}\left( f \right)\left( x \right) = \mathop {\sup }\limits_{r > 0} {\chi _r} * \left| f \right|\left( x \right).$ | (12) |
$\begin{array}{*{20}{c}} {{\chi _r} * \left| {{\phi _t} * f} \right|\left( x \right) = \int_{{\mathbb{R}^n}} {{\phi _t}\left( y \right)\left( {{\chi _r} * {\tau _y}f} \right)\left( x \right){\text{d}}y} } \\ { \leqslant \int_{{\mathbb{R}^n}} {{\phi _t}\left( y \right){M_c}\left( {{\tau _y}f} \right)\left( x \right){\text{d}}y} = {\phi _t} * {M_c}\left( f \right)\left( x \right)} \\ { = {f_t}\left( x \right).} \end{array}$ | (13) |
${M_c}\left( {{f_t}} \right)\left( x \right) \ge {f_t}\left( x \right).$ |
Using the similar method, we can easily prove that ft is a fixed point of M if f is a fixed point of M.
Lemma 1.3??There is a non-constant fixed point of Mc in Lloc1(
There is a non-constant fixed point of M in Lloc1(
Lemma 1.3 is due to that for a smooth function, every point in
Lemma 1.4??If f is a non-constant and smooth function, and f is a fixed point of M, then, in any closed ball, the minimum value of f is gotten only in the sphere.
Lemma 1.4 has the same proof as the proof of extremism principle of harmonic function. For the details please see Ref.[6].
Lemma 1.5??Suppose that f is a fixed point of M. If f∈Lloc1(
Proof. Since f is a fixed point of M, it follows from Lemma 1.2 that there exists ft∈
Suppose that B is a ball in
If ft is not a constant, then, by Lemma 1.4, there is at least one point x∈?B such that ft(y) > ft(x) holds for all y∈B°.
Note that ft is a fixed point of M. Thus we have that
$\begin{array}{*{20}{c}}{{f_t}\left( x \right) < \frac{1}{{m\left( {{B^ \circ }} \right)}}\int_{{B^ \circ }} {\left| {{f_t}\left( y \right)} \right|{\rm{d}}y} }\\={\frac{1}{{m\left( B \right)}}\int_B {\left| {{f_t}\left( y \right)} \right|{\rm{d}}y} \le M{f_t}\left( x \right) = {f_t}\left( x \right).}\end{array}$ | (14) |
Next we will prove that f(x)=C for all x∈
Choose a radial nonnegative function ?∈Cc∞(
∫
For each t > 0, we have ft(x)=C.
Set BR={x∈
$\begin{gathered} C={{f}_{t}}\left( x \right)=\int_{{{\mathbb{R}}^{n}}}{\left( f{{\chi }_{{{B}_{R}}}}+f{{\chi }_{{{\mathbb{R}}^{n}}\backslash {{B}_{R}}}} \right)\left( x-y \right){{\phi }_{t}}\left( y \right)\text{d}y} \hfill \\ = f{\chi _{{B_R}}} * {\phi _t}\left( x \right) + f{\chi _{{\mathbb{R}^n}\backslash {B_R}}} * {\phi _t}\left( x \right). \hfill \\ \end{gathered} $ | (15) |
$\mathop {\lim }\limits_{t \to 0} f{\chi _{{B_R}}} * {\phi _t}\left( x \right) + \mathop {\lim }\limits_{t \to 0} f{\chi _{{\mathbb{R}^n}\backslash {B_R}}} * {\phi _t}\left( x \right) = C.$ | (16) |
$\mathop {\lim }\limits_{t \to 0} f{\chi _{{B_R}}} * {\phi _t}\left( x \right) = f{\chi _{{B_R}}}\left( x \right)$ | (17) |
${\text{supp}}f{\chi _{{\mathbb{R}^n}\backslash {B_R}}} * {\phi _t} \subset \left\{ {x \in {\mathbb{R}^n}:\left| x \right| \geqslant R - t} \right\}.$ | (18) |
$f{\chi _{{B_R}}}\left( x \right) + \mathop {\lim }\limits_{t \to 0} f{\chi _{{\mathbb{R}^n}\backslash {B_R}}} * {\phi _t}\left( y \right) = C$ |
$f{\chi _{{B_R}}}\left( x \right) = C$ |
$f\left( x \right) = C$ |
$Mf\left( x \right) = f\left( x \right).$ |
$f\left( x \right) = C$ |
If f?Lloc1(
$\int_B {\left| {f\left( x \right)} \right|{\rm{d}}x} = \infty .$ |
We remark that Mc has essential difference with M with respect to the fixed point. We all know that when n≥3, the function f(x)=|x|2-n is a harmonic function in
Korry[7] obtained the following lemma 1.6.
Lemma 1.6??For the Mc, if f∈Lloc1(
2 Main theoremsIn the section, for any local integral function, the limit of the iterated Hardy-Littlewood maximal function is a fixed point of Hardy-Littlewood maximal operator.
Theorem 2.1??Write
$\mathop {\lim }\limits_{k \to \infty } {M^k}f\left( x \right) = F\left( x \right).$ |
$MF\left( x \right) = F\left( x \right).$ |
In the same way, write
$\mathop {\lim }\limits_{k \to \infty } M_c^kf\left( x \right) = {F_c}\left( x \right),$ |
Proof??We only prove the first part of Theorem 2.1. It follows that
$\begin{array}{*{20}{c}}{MF\left( x \right) = M\mathop {\lim }\limits_{k \to \infty } {M^k}f\left( x \right) = }\\{\mathop {\sup }\limits_{B \ni x} \frac{1}{{m\left( B \right)}}\int_B {\mathop {\lim }\limits_{k \to \infty } {M^k}f\left( y \right){\rm{d}}y} .}\end{array}$ | (19) |
$\begin{array}{l}MF\left( x \right) - \varepsilon \le \frac{1}{{m\left( {{B_\varepsilon }} \right)}}\int_{{B_\varepsilon }} {\mathop {\lim }\limits_{k \to \infty } {M^k}f\left( y \right){\rm{d}}y} \\\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \le \mathop {\lim }\limits_{k \to \infty } \frac{1}{{m\left( {{B_\varepsilon }} \right)}}\int_{{B_\varepsilon }} {{M^k}f\left( y \right){\rm{d}}y} \\\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \le \mathop {\lim }\limits_{k \to \infty } {M^{k + 1}}f\left( x \right) = F\left( x \right)\\\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \le MF\left( x \right).\end{array}$ | (20) |
$MF\left( x \right) = F\left( x \right).$ | (21) |
$\mathop {\lim }\limits_{k \to \infty } {M^k}f\left( x \right) = F\left( x \right).$ |
Proof??Since F(x)=C, it implies from the definition of Hardy-Littlewood maximal function that C≤‖f‖∞. By the definition of essential supremum of function, associate to an arbitrary ε > 0, a set E?
$\left| {f\left( x \right)} \right| > {\left\| f \right\|_\infty } - \varepsilon ,$ |
$C = F\left( x \right) \ge Mf\left( x \right) \ge \left| {f\left( x \right)} \right| > {\left\| f \right\|_\infty } - \varepsilon .$ |
$C \ge {\left\| f \right\|_\infty }.$ |
Next we will prove our main theorems.
The proof of Theorem B
Proof??It follows from Theorem 2.1 that
${F_c}\left( x \right) = \mathop {\lim }\limits_{k \to \infty } M_c^kf\left( x \right)$ | (22) |
By Lemm 1.6, if Fc∈Lloc1(
Since Fc(x)=C, it implies from the definition of center Hardy-Littlewood maximal function that C≤‖f‖∞.
By the definition of essential supremum of function, associate to an arbitrary ε > 0 a set E?
$\left| {f\left( x \right)} \right| > {\left\| f \right\|_\infty } - \varepsilon ,$ |
$C = F\left( x \right) \ge {M_c}f\left( x \right) \ge \left| {f\left( x \right)} \right| > {\left\| f \right\|_\infty } - \varepsilon .$ |
$C \ge {\left\| f \right\|_\infty }.$ |
If Fc?Lloc1(
The proof of Theorem C
Proof??It follows from Theorem 2.1 that
$F\left( x \right) = \mathop {\lim }\limits_{k \to \infty } {M^k}f\left( x \right)$ | (23) |
By Lemma 1.5, if F∈Lloc1(
It follows from Lemma 2.1 that F=‖f‖∞.
If F?Lloc1(
$\int_{B\left( {0,R} \right)} {\left| {F\left( x \right)} \right|{\rm{d}}x} = \infty .$ |
$F\left( x \right) = MF\left( x \right) = \infty .$ |
References
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