GEORGIAN MATHEMATICAL JOURNAL: Vol. 3, No. 1, 1996, 69-80

LITTLEWOOD–PALEY OPERATORS ON THE GENERALIZED LIPSCHITZ SPACES

SHANZHEN LU, CHANGMEI TAN, AND KOZO YABUTA

Abstract. Littlewood–Paley operators defined on a new kind of generalized Lipschitz spaces E_{0}^{α,p} are studied. It is proved that the image of a function under the action of these operators is either equal to infinity almost everywhere or is in E_{0}^{α,p}, where −n < α < 1 and 1 < p < ∞.

1. Introduction

For x ∈ ℝ^{n}, y > 0, the Poisson kernel is P (x, y) = c_{n}y(y^{2} +|x|^{2})^{−(n+1)/2}. Denote the Poisson integral of f by

f(x, y) = ∫_{ℝ^{n}} f(z)P (x − z, y) dz.

We have (see [1])

|∇f(x, y)| ≤ c_{n} ∫_{ℝ^{n}} |f(z)| (y + |x − z|)^{−(n+1)}dz.  (1)

Let us now consider the following two kinds of Littlewood–Paley functions:

S(f)(x) = ( ∫∫_{Γ(x)} y^{1−n}|∇f(z, y)|^{2}dz dy)^{1/2}

and

g_{λ}^{∗}(f)(x) = \{ ∫∫_{ℝ_{+}^{n+1}} (\frac{y}{y + |x − z|})^{λn}y^{1−n}|∇f(z, y)|^{2}dz dy\}^{1/2}.

1991 Mathematics Subject Classification. 42B25. Key words and phrases. Littlewood–Paley functions, generalized Lipschitz spaces,

Poisson integral, Poisson kernel.

69

1072-947X/96/0100-0069$09.50/0 \begin{matrix}c \\ ◯ \end{matrix} 1996 Plenum Publishing Corporation

The generalized Lipschitz space E^{α,p} consists of functions f which are locally integrable and satisfy the following condition: there exists a constant C such that for any cube Q

∫_{Q} |f(x) − f_{Q}|^{p}dx ≤ C|Q|^{1+ \frac{αp}{n}} ,  (2)

where f_{Q} = \frac{1}{Q} ∫_{Q} f(x)dx. Denote the norm of f in E^{α,p} by

∥f∥_{α,p} = inf {C^{1/p} : C satisfies (2)}.

Recently, Qiu [2] has obtained the following result.

Theorem A. Let 1 < p < ∞, −n/p ≤ α < 1/2, α ≠ 0, and λ > max(1, 2/p). If f ∈ E^{α,p} and T f is S(f) or g_{λ}^{∗}(f), then either T f(x) = ∞ a.e. or T f(x) < ∞ a.e., and there exists a constant C independent of f such that

∥T f∥_{α,p} ≤ C∥f∥_{α,p}.

We notice that the range of α in Theorem A seems somewhat rough. It is natural to consider whether the conclusion of the above theorem holds for −n < α < 1. The last named author of this paper proved that the conclusion of Theorem A holds for −n/p < α < 1 (see [3]). In this paper, with the aid of the idea in [4], we shall introduce a variant of E^{α,p}, E_{0}^{α,p}, and prove that the conclusion of Theorem A holds for E_{0}^{α,p} with −n < α < 1. Let us first definie E_{0}^{α,p}.

Definition. A locally integrable function f is called a generalized Lipschitz function of central type if there exists a constant C such that (2) holds for any cube Q centered at the origin. Moreover, the space consisting of all generalized Lipschitz functions of central type is denoted by E_{0}^{α,p}. We call E_{0}^{α,p} the generalized Lipschitz space of central type.

It is easy to see that E^{α,p} ⊂ E_{0}^{α,p} and E_{0}^{α,p} is just the bounded mean oscillation space of central type, BMO_{0} in [4]. Let us now formulate our results.

Theorem 1. Let 1 < p < ∞ and −n < α < 1. If f ∈ E_{0}^{α,p}, then either S(f)(x) = ∞ a.e. or S(f)(x) < ∞ a.e., and there exists a constant C independent of f such that

∥S(f)∥_{α,p} ≤ C∥f∥_{α,p}.

Theorem 2. Let 1 < p < ∞, −n < α < 1, and λ > max(1, 2/p) + 2/n. If f ∈ E_{0}^{α,p}, then either g_{λ}^{∗}(f)(x) = ∞ a.e. or g_{λ}^{∗}(f)(x) < ∞ a.e., and there exists a constant C = C(n, α, p, λ) such that

∥g_{λ}^{∗}(f)∥_{α,p} ≤ C∥f∥_{α,p}.

2. Some Lemmas

Lemma 1. Let 1 < p < ∞, −n < α < 1, α ≠ 0, and 0 < d, α < d. If f ∈ E_{0}^{α,p} and Q is a cube centered at the origin with the edge length r, then there exists a constant C = C(n, p, α, d) such that for any y > 0

∫_{ℝ^{n}} \frac{|f(x) − f_{Q}|}{y^{n+d} + |x|^{n+d}} dx ≤ Cy^{−d}(y^{α} + r^{α})∥f∥_{α,p}.  (3)

See [1] and [2] for its proof.

Lemma 1′. Let 1 < p < ∞ and d > 0. If f ∈ BMO_{0} = E_{0}^{0,p} and Q is a cube centered at the origin with the edge length r, then there exists a constant C = C(n, p, d) such that for any y > 0,

∫_{ℝ^{n}} \frac{|f(x) − f_{Q}|}{y^{n+d} + |x|^{n+d}} dx ≤ Cy^{−d}(1 + | log_{2} \frac{y}{r} |)∥f∥_{0,p}.

Proof. By the known result in [5] we have

∫_{ℝ^{n}} \frac{|f(x) − f_{Q}|}{r^{n+d} + |x|^{n+d}} dx ≤ Cr^{−d}∥f∥_{0,p}.

Let R be the cube centered at the origin with the edge length y. Then

∫_{ℝ^{n}} \frac{|f(x) − f_{Q}|}{y^{n+d} + |x|^{n+d}} dx ≤ ∫_{ℝ^{n}} \frac{|f(x) − f_{R}|}{y^{n+d} + |x|^{n+d}} dx

+|f_{R} − f_{Q}| ∫_{ℝ^{n}} \frac{dx}{y^{n+d} + |x|^{n+d}} dx ≤ Cy^{−d}∥f∥_{0,p} + Cy^{−d}|f_{R} − f_{Q}|.

Thus it remains to prove

|f_{R} − f_{Q}| ≤ C(1 + | log_{2} \frac{y}{r} |)∥f∥_{0,p}.

Let y > r, and let k satisfy 2^{k} ≤ y < 2^{k+1}r. Then k ≤ log_{2} \frac{y}{r} and

|f_{R} − f_{Q}| ≤ |f_{R} − f_{Q_{k}}| + ∑_{j=1}^{k} |f_{Q_{j}} − f_{Q_{j−1}}|

≤ 2^{n}(\frac{1}{|R|} ∫_{R} |f(x) − f_{R}|^{p}dx)^{1/p} + ∑_{j=1}^{k} 2^{n}∥f∥_{0,p}

≤ 2^{n}(1 + k)∥f∥_{0,p}

≤ C(1 + log_{2} \frac{y}{r} )∥f∥_{0,p},

where Q_{k} is the concentric extension of Q by 2^{k} times.

When y < r, by exchanging y and r, we shall get the same estimate as above with log_{2} \frac{r}{y} = | log_{2} \frac{y}{r} |.

Let χ_{E} be the characteristic function of E. For a cube Q in ℝ^{n} and d > 0 let dQ be the concentric extension of Q by d times.

Lemma 2. Suppose that 1 < p < ∞, −n < α < 1, and f ∈ E_{0}^{α,p}. Let Q be a cube centered at the origin with the edge length r, and h_{Q}(x) = [f(x) − f_{Q}]χ_{Q^{c}} (x). If there is x′ ∈ dQ such that S(h_{Q})(x′) < ∞, where d = (8\sqrt{n})^{−1}, then there exists a constant C = C(n, α, p) such that

S(h_{Q})(x) < ∞, ∀x ∈ dQ

and

|S(h_{Q})(x) − S(h_{Q})(x′)| < Cr^{α}∥f∥_{α,p}, ∀x ∈ dQ.

Proof. Let us first consider the case of α ≠ 0. Fix x ∈ dQ. Set

Γ^{−}(x) = {(z, y) ∈ Γ(x) : y ≤ dr}

and

Γ^{+}(x) = {(z, y) ∈ Γ(x) : y > dr}.

Then

S(h_{Q})(x) ≤ S^{−} + S^{+}, x ∈ dQ,

where

S^{−} = ( ∫∫_{Γ^{−}(x)} y^{1−n}|∇h_{Q}(z, y)|^{2}dz dy)^{1/2}

and

S^{+} = ( ∫∫_{Γ^{+}(x)} y^{1−n}|∇h_{Q}(z, y)|^{2}dz dy)^{1/2}.

LITTLEWOOD–PALEY OPERATORS 73

Estimating S^{−} as in [2], we have

S^{−} ≤ Cr^{α}∥f∥_{α,p}.  (4)

For S^{+} we have

S^{+} = ( ∫∫_{Γ^{+}(0)} y^{1−n}|∇h_{Q}(x + z, y)|^{2}dz dy)^{1/2}

≤ ( ∫∫_{Γ^{+}(0)} y^{1−n}|∇h_{Q}(x′ + z, y)|^{2}dz dy)^{1/2}

+ ( ∫∫_{Γ^{+}(0)} y^{1−n}|∇h_{Q}(x + z, y) − ∇h_{Q}(x′ + z, y)|^{2}dz dy)^{1/2}

≤ S(h_{Q})(x′) + { ∫∫_{Γ^{+}(0)} y^{1−n}

× ( ∫_{Q^{c}} |∇P (x + z − t, y)∇P (x′ + z + t, y)| |f(t) − f_{Q}|dt)^{2}dz dy}^{1/2}.

Note that

|∇P (x, y) − ∇P (x′, y)| = ( ∑_{j=1}^{n+1} |\frac{∂}{∂x_{j}} p(x, y) − \frac{∂}{∂x_{j}} P (x′, y))^{1/2},

where \frac{∂}{∂x_{n+1}} = \frac{∂}{∂y}. By the mean value theorem we have

|\frac{∂}{∂x_{j}} p(x, y) − \frac{∂}{∂x_{j}} P (x′, y)|

= |∇\frac{∂}{∂x_{j}} P (x, y)|_{x+θ_{j}(x−x′)}|x − x′|, 0 < θ_{j} < 1,

where

|∇\frac{∂}{∂x_{j}} P (x, y)| ≤ \frac{C}{(y + |x|)^{n+2}} .

Thus

|∇P (x + z − t, y) − ∇P (x′ + z − t, y)|
≤ C|x − x′|{ ∑_{j=1}^{n+1} (y + |x + z − t + θ_{j}(x − x′)|)^{−2(n+2)}}^{1/2}.  (5)

Since (x, y) ∈ Γ^{+}(0), x, x′ ∈ dQ, and t ∉ Q, we have |t| > r/2, |z| < y, |x| < r/16 < |t|/8, and |x − x′| < r/8 < |t|/4. Thus,

|t| ≤ |x + z − t + θ_{j}(x − x′)| + |x| + |z| + |x − x′|
≤ |x + z − t + θ_{j}(x − x′)| + |t|/8 + y + |t|/4

and

\frac{5}{16} (y + |t|) ≤ |x + z − t + θ_{j}(x − x′)| + y,

where 1 ≤ j ≤ n| + 1. Therefore

|∇P (x + z − t, y) − ∇P (x′ + z − t, y)| ≤ \frac{Cr}{(y + |t|)^{n+2}} .  (6)

Using (6) and (3), we obtain

S^{+} ≤ S(h_{Q})(x′) + C\{ ∫∫_{Γ^{+}(0)} y^{1−n}[ ∫_{Q^{c}} \frac{r|f(t) − f_{Q}|}{(y + |t|)^{n+2}}dt]^{2}dzdy\}^{1/2}

≤ S(h_{Q})(x′) + C{ ∫∫_{Γ^{+}(0)} y^{1−n}r^{2}[y^{−2}(y^{α} + r^{α})∥f∥_{α,p}]^{2}dzdy}^{1/2}

≤ S(h_{Q})(x′) + Cr∥f∥_{α,p}{ ∫_{dr}^{∞} ∫_{|z|<y} y^{1−n}y^{−4}(y^{2α} + r^{2α})dzdy}^{1/2}

≤ S(h_{Q})(x′) + Cr^{α}∥f∥_{α,p}.  (7)

Combining (4) with (7) we have

S(h_{Q})(x) ≤ S(h_{Q})(x′) + Cr^{α}∥f∥_{α,p}.

Thus S(h_{Q})(x) < ∞. Exchanging x and x′, we obtain

|S(h_{Q})(x) − S(h_{Q})(x′)| ≤ Cr^{α}∥f∥_{α,p}.

Hence the proof of Lemma 2 is complete for the case of α ≠ 0.

When α = 0, by using Lemma 1′ instead of Lemma 1 we obtain

S^{+} ≤ S(h_{Q})(x′) + C\{ ∫∫_{Γ^{+}(0)} y^{1−n}[ ∫_{Q^{c}} \frac{r|f(t) − f_{Q}|}{(y + |t|)^{n+2}}dt]^{2}dzdy\}^{1/2}

≤ S(h_{Q})(x′) + C\{ ∫∫_{Γ^{+}(0)} y^{1−n}r^{2}[y^{−2}(1 + | log_{2} \frac{y}{r} |)∥f∥_{0,p}]^{2}dzdy\}^{1/2}

≤ S(h_{Q})(x′) + Cr∥f∥_{0,p}\{ ∫_{dr}^{∞} ∫_{|z|<y} y^{−3−n}(1 + | log_{2} \frac{y}{r} |)^{2}dzdy\}^{1/2}

≤ S(h_{Q})(x′) + Cr^{α}∥f∥_{0,p}{ ∫_{1}^{∞}u^{−3}(1 + | log_{2} u|)^{2}du}^{1/2}

≤ S(h_{Q})(x′) + Cr^{α}∥f∥_{0,p}.  (8)

Now, it is easy to see that the conclusion of the lemma for α = 0 follows from (8) and (4) with α = 0.

Lemma 3. Under the hypothesis of Lemma 2, if there is x′ ∈ dQ such that g_{λ}^{∗}(h_{Q})(x′) < ∞, where λ > max(1, 2/p) + 2/n, then there exists a constant C = C(n, α, λ, p) such that g_{λ}^{∗}(h_{Q})(x) < ∞ and

|g_{λ}^{∗}(h_{Q})(x) − g_{λ}^{∗}(x′) ≤ Cr^{α}∥f∥_{α,p}, ∀x ∈ dQ.

Proof. We only consider the case of α ≠ 0. As in Lemma 2, the proof in the case α = 0 is similar. Let

J_{k} = {(z, y) ∈ ℝ_{+}^{n+1} : |z| < 2^{k−2}r, 0 < y < 2^{k−2}r}, k ≥ 0.

For fixed x ∈ dQ we have

g_{λ}^{∗}(h_{Q})(x) ≤ G^{−} + G^{+},

where

G^{−} = ( ∫∫_{J_{0}} (\frac{y}{y + |z|})^{λn}y^{1−n}|∇h_{Q}(x + z, y)|^{2}dz dy)^{1/2}

and

G^{+} = ( ∫∫_{ℝ_{+}^{n+1}\J_{0}} (\frac{y}{y + |z|})^{λn}y^{1−n}|∇h_{Q}(x + z, y)|^{2}dz dy)^{1/2}.

Note that if (z, y) ∈ J_{0}, x ∈ dQ, and t ≠ Q, then |z| < r/4, |x| < r/16, and |t| > r/2. Thus

|t| ≤ |t − x − z| + |x| + |z| ≤ |x + z − t| + \frac{5}{8}|t|

and

\frac{1}{16} (r + |t|) ≤ |x + z − t| + y.

By (1) and Lemma 1 we get

G^{−} ≤ C\{ ∫∫_{J_{0}} (\frac{y}{y + |z|})^{λn}y^{1−n}[ ∫_{Q^{c}} \frac{r|f(t) − f_{Q}|}{(y + |x + z − t|)^{n+1}} dt]^{2}dzdy\}^{1/2}

≤ C\{ ∫∫_{J_{0}} (\frac{y}{y + |z|})^{λn}y^{1−n}[ ∫_{Q^{c}} \frac{r|f(t) − f_{Q}|}{(r + |t|)^{n+1}}dt\}^{2}dzdy\}^{1/2}

≤ C\{ ∫_{dr}^{∞} ∫_{|z|<y} (\frac{y}{y + |z|})^{λn}y^{1−n}[r^{α−1}∥f∥_{α,p}]^{2}dzdy\}^{1/2}

≤ Cr^{α−1}∥f∥_{α,p}( ∫_{0}^{r} y^{1−n}r^{n}dy)^{1/2}

≤ Cr^{α}∥f∥_{α,p}.

To estimate C^{+} we observe that

G^{+} ≤ \{ ∫∫_{ℝ_{+}^{n+1}\J_{0}} (\frac{y}{y + |z|})^{λn}y^{1−n}|∇h_{Q}(x′ + z, y)|^{2}dzdy\}^{1/2}

+ \{ ∫∫_{ℝ_{+}^{n+1}\J_{0}} (\frac{y}{y + |z|})^{λn}y^{1−n}|∇h_{Q}(x + z, y) −

− ∇h_{Q}(x′ + z, y)|^{2}dzdy}^{1/2}

≤ g_{λ}^{∗}(h_{Q})(x′) + D,

where

D = \{ ∫∫_{ℝ_{+}^{n+1}\J_{0}} (\frac{y}{y + |z|})^{λn}y^{1−n}|∇h_{Q}(x + z, y) −

− ∇h_{Q}(x′ + z, y)|^{2}dzdy}^{1/2}

≤ C{ ∑_{k=1}^{∞}(2^{k}r)^{−λn} ∫∫_{J_{k}\J_{k−1}} y^{λn+1−n}

× [ ∫_{Q^{c}} |∇P (x + z − t, y) − ∇P (x′ + z − t, y)| |f(t) − f_{Q}|dt]^{2}dzdy}^{1/2}

≤ C{ ∑_{k=1}^{∞}(2^{k}r)^{−λn}(A_{k} + B_{k})}^{1/2}.

Here

A_{k} = ∫∫_{J_{k}\J_{k−1}} y^{λn+1−n}

× [ ∫_{Q_{k+1}^{c}} |∇P (x + z − t, y) − ∇P (x′ + z − t, y)| |f(t) − f_{Q}|dt]^{2}dzdy,

B_{k} = ∫∫_{J_{k}\J_{k−1}} y^{λn+1−n}

× [ ∫_{Q_{k+1}\Q} |∇P (x + z − t, y) − ∇P (x′ + z − t, y)| |f(t) − f_{Q}|dt]^{2}dzdy,

and Q_{k+1} = 2^{k+1}Q. Without loss of generality we may assume that max(1, 2/p) + 2/n < λ < 3 + 2/n. By the easy inequality (see [3])

|∇P (x, y) − ∇P (x′, y)|

≤ C|x − x′|(\frac{1}{(y + |x|)^{n+2}} + \frac{1}{(y + |x′|)^{n+2}} ), ∀x, x′ ∈ ℝ^{n}, y > 0,

together with the Minkowski inequality for integrals, we have

B_{k} ≤ Cr^{2} ∫∫_{J_{k}\J_{k−1}} y^{λn+1−n}\{ ∫_{Q_{k+1}\Q} |f(t) − f_{Q}|(\frac{1}{(y + |x + z − t|)^{n+2}}

+ \frac{1}{(y + |x′ + z − t|)^{n+2}} )dt\}^{2}dz dy

≤ Cr^{2} ∫_{0}^{∞}∫_{ℝ^{n}} y^{λn+1−n}\{ ∫_{Q_{k+1}} |f(t) − f_{Q}|(\frac{1}{(y + |x + z − t|)^{n+2}}

+ \frac{1}{(y + |x′ + z − t|)^{n+2}} )dt\}^{2}dz dy

≤ Cr^{2} ∫_{ℝ^{n}}[ ∫_{Q_{k+1}} |f(t)−f_{Q}|( ∫_{0}^{∞}\frac{y^{λn+1−n}}{(y+|x + z − t|)^{2(n+2)}} dy)^{1/2}dt]^{2}dz

+ Cr^{2} ∫_{ℝ^{n}}[ ∫_{Q_{k+1}} |f(t)−f_{Q}|( ∫_{0}^{∞}\frac{y^{λn+1−n}}{(y+|x′ + z − t|)^{2(n+2)}} dy)^{1/2}dt]^{2}dz

=Cr^{2} ∫_{ℝ^{n}}[ ∫_{Q_{k+1}} \frac{|f(t)−f_{Q}|}{|z+x−t|^{(3n−λn+2)/2}} ( ∫_{0}^{∞}\frac{y^{λn+1−n}}{(1+y)^{2(n+2)}} dy)^{1/2}dt]^{2}dz

+Cr^{2} ∫_{ℝ^{n}}[ ∫_{Q_{k+1}} \frac{|f(t) − f_{Q}|}{|z+x′−t|^{(3n−λn+2)/2}} ( ∫_{0}^{∞}\frac{y^{λn+1−n}}{(1+y)^{2(n+2)}} dy)^{1/2}dt]^{2}dz

= Cr^{2} ∫_{ℝ^{n}} ( ∫_{Q_{k+1}} \frac{|f(t) − f_{Q}|}{|u − t|^{n−[(λn−1)−2]/2}} dt)^{2}du.

Using the Hardy–Littlewood–Sobolev theorem on fractional integration with γ = [(λ − 1)n − 2]/2, q = 2, and 1/s = 1/q + γ/n = λ/2 − 1/n (see [6]), we obtain

B_{k} ≤ Cr^{2}( ∫_{Q_{k+1}} |f(z) − f_{Q}|^{s}dz)^{2/s}.

Since λ ≥ 2/p + 2/n, then p ≥ s. Thus

B_{k} ≤ Cr^{2}( ∫_{Q_{k+1}} |f(z) − f_{Q}|^{p}dz)^{2/p}|Q_{k+1}|^{2(1/s−1/p)}

≤ Cr^{2}{( ∫_{Q_{k+1}} |f(z) − f_{Q}|^{p}dz)^{1/p}

+ |Q_{k+1}|^{1/p}|f_{Q_{k+1}} − f_{Q}|}^{2}|Q_{k+1}|^{2(1/s−1/p)}

≤ Cr^{2}{|Q_{k+1}|^{1/p+α/n}∥f∥_{α,p}

+ |Q_{k+1}|^{1/p}(2^{k}r)^{α}∥f∥_{α,p}}^{2}|Q_{k+1}|^{2(1/s+1/p)}

≤ Cr^{2}(2^{k}r)^{2α}(2^{k}r)^{λn−2}∥f∥_{α,p}

≤ C(2^{k}r)^{λn}(2^{2k(α−1)}r^{2α}∥f∥_{α,p}.

To estimate A_{k} we observe that if (z, y) ∈ J_{k}\J_{k−1} and t ∉ Q_{k+1}, then |t| > 2^{k}r, k ≥ 1, and |z| < 2^{k−2}r < |t|/4. Thus,

|t| ≤ |x + z − t + θ_{j}(x − x′)| + |x| + |z| + |x − x′|

≤ |x + z − t + θ(x − x′)| + \frac{5}{16}|t|.

By Lemma 1 we have

A_{k} ≤ C ∫∫_{J_{k}\J_{k+1}} y^{λn+1−n}[ ∫_{Q_{k+1}^{c}} \frac{r|f(t) − f_{Q}|}{(2^{k}r + |t|)^{n+2}} dt]^{2}dz dy

≤ Cr^{2} ∫∫_{J_{k}\J_{k+1}} y^{λn+1−n}{(2^{k}r)^{−2}[(2^{k}r)^{α} + r^{α}]∥f∥_{α,p}}^{2}dz dy

≤ Cr^{2}(2^{k}r)^{−4+2α}∥f∥_{α,p} ∫_{0}^{2^{k}r} ∫_{|z|<2^{k}r} y^{λn+1−n}dz dy

≤ Cr^{2α}(2^{k}r)^{λn}2^{2k(α−1)}∥f∥_{α,p}.

Combining the estimate of A_{k} with that of B_{k}, we obtain

D ≤ C{ ∑_{k=1}^{∞}(2^{k}r)^{−λn}(2^{k}r)^{λn}r^{2α}2^{2k(α−1)}∥f∥_{α,p}}^{1/2} ≤ Cr^{α}∥f∥_{α,p}.

Therefore

g_{λ}^{∗}(h_{Q})(x) ≤ g_{λ}^{∗}(h_{Q})(x′) + Cr^{α}∥f∥_{α,p}.

As in the last part of the proof of Lemma 2, we have

|g_{λ}^{(}h_{Q})(x) − g_{λ}^{∗}(h_{Q})(x′) ≤ Cr^{α}∥f∥_{α,p}.

3. The Proofs of the Theorems

Let T be one of the Littlewood–Paley functions as in Section 1. Suppose that T f(x) ≠ ∞ a.e. Then |E| =^{Δ} |{x : T f(x) < ∞}| > 0. Thus there is a cube Q′ centered at the origin such that |Q′ ∩ E| > 0. Set Q = (1/d)Q′ (then Q′ = dQ). We write f as

f(x) = f_{Q} + [f(x) − f_{Q}]χ_{Q}(x) + [f(x) − f_{Q}]χ_{Q^{c}} (x)

=^{Δ} f_{Q} + g_{Q}(x) + h_{Q}(x).

Since

T f(x) ≤ T g_{Q}(x) + T h_{Q}(x)  (9)

and

T h_{Q}(x) ≤ T f(x) + T g_{Q}(x),  (10)

it is easy to see that the inequality

∥g_{Q}∥_{p} = ( ∫_{Q} |f(t) − f_{Q}|^{p}dt)^{1/p} ≤ C|Q|^{1/p+α/n}∥f∥_{α,p}  (11)

implies that g_{Q} ∈ L^{p}. Then it follows from the L^{p}-boundedness of the Littlewood–Paley operator that T g_{Q}(x) < ∞ a.e.. Since |Q′ ∩ E| > 0, there is x′ ∈ Q′ ∩ E ⊂ dQ such that T f(x′) < ∞ and T g_{Q}(x′) < ∞. By (10) and Lemmas 2 and 3, we have T h_{Q}(x′) < ∞ and

T h_{Q}(x) < ∞, ∀x ∈ dQ = Q′.

By (9) we obtain

T f(x) < ∞ a.e. x ∈ Q′.

Finally, let the edge length of Q′ tend to ∞; we have T f(x) < ∞ a.e., x ∈ ℝ^{n}.

Let Q′ be a cube centered at the origin, and Q = (1/d)Q′. Choose x′ ∈ dQ so that T h_{Q}(x′) < ∞. Then it follows from (11) and Lemmas 2 and 3 that

( ∫_{Q′} |T f(x) − T h_{Q}(x′)|^{p}dx)^{1/p}

≤ ( ∫_{Q′} |T g_{Q}(x)|^{p}dx)^{1/p} + ( ∫_{Q′} |T h_{Q}(x) − T h_{Q}(x′)|^{p}dx)^{1/p}

≤ C∥g_{Q}∥_{p} + C|Q′|^{1/p}r^{α}∥f∥_{α,p}

≤ C|Q′|^{1/p+α/n}∥f∥_{α,p}.

This completes the proof of the theorems.

Acknowledgement

The work is supported by the NSFC (Tian Yuan). Shanzhen Lu would like to thank Kozo Yabuta for inviting him to visit Nara during his sabbatiacal in Japan.

References

1. D. S. Kurtz, Littlewood–Paley operators on BMO. Proc. Amer. Math. Soc. 99(1987), 657–666.

2. S. G. Qiu, Boundedness of Littlewood–Paley operators and Marcinkiewicz integral on E^{α,p}. J. Math. Res. Exposition 12(1992), 41–50.

3. K. Yabuta, Boundedness of Littlewood–Paley operators. Preprint.

4. S. Z. Lu and D. C. Yang, The Littlewood–Paley function and φ-transform characterizations of a new Hardy space HK_{2} associated with the Herz space. Studia Math. 101(1992), 285–298.

5. F. B. Fabes, R. L. Janson, and U. Neri, Spaces of harmonic functions representable by Poisson integrals of functions in BMO and L_{p,λ}. Indiana Univ. Math. J. 25(1976), 159–170.

6. E. M. Stein, Singular integrals and differentiability properties of functions. Princeton University Press, Princeton, 1970.

(Received 04.05.1994)

Authors’ addresses:

Shanzhen Lu and Changmei Tan Department of Mathematics Beijing Normal University Beijing 100875, P. R. China Kozo Yabuta Department of Mathematics Nara Women’s University Nara 630, Japan