Computational Methods and Function Theory Volume 5 (2005), No. 2, 431–444
Generalized Lipschitz Functions Javad Mashreg...

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Computational Methods and Function Theory Volume 5 (2005), No. 2, 431–444

Generalized Lipschitz Functions Javad Mashreghi

(Communicated by Matti Vuorinen) Abstract. In this note the class Lipα(t) of continuous functions is introduced. The definition is arranged so that for the constant function α(t) ≡ α, the class Lipα(t) is nothing but the classical Lipschitz space Lipα . Then, to justify that our set of axioms for α(t) are properly chosen, some celebrated theorems of Privalov, Titchmarsh, Hardy and Littlewood about Lipα functions are shown to be also valid for Lipα(t) functions. Keywords. Lipschitz functions, test function, associated test function, Hilbert transform, Hardy classes. 2000 MSC. Primary 26A16; Secondary 32A40.

1. Introduction Let f : R → C be a continuous function. The modulus of continuity of f is ωf (t) :=

sup

|f (x + y) − f (x)|,

t > 0.

x∈R, |y|≤t

The function ωf is used to define different classes of continuous functions. For example, f is called Lipα , for some α with 0 < α ≤ 1, if ωf (t) = O(tα )

as t → 0+ .

In this note, our main goal is to replace tα by tα(t) , where α(t) is a function defined for small values of t and properly tends to α as t → 0+ . To be more precise, we assume that α(t) is a real continuous function defined in a right neighborhood Received October 21, 2004, in revised form October 18, 2005. This work was supported by two research grants from NSERC (Canada) and FQRNT (Qu´ebec). c 2005 Heldermann Verlag ISSN 1617-9447/$ 2.50

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of zero, say (0, t0 ), and, as t → 0+ , we have (1.1) (1.2) (1.3)

α ∈ R,

α(t) = α + O(1), tα(t)−β+1 τ α(τ )−β dτ = + O(tα(t)−β+1 ), α + 1 − β 0 Z t0 tα(t)−β+1 τ α(τ )−β dτ = + O(tα(t)−β+1 ), β−α−1 t Z

t

β < α + 1, β > α + 1.

Let us call α(t) a test function. The space of all continuous functions f : R → C satisfying ωf (t) = O(tα(t) ) as t → 0+ will be denoted by Lipα(t) . Clearly the classical space Lipα is a special case corresponding to the test function α(t) ≡ α. Given a test function α(t), let 1 α ˆ (t) = α + log log t

Z

t0

τ

α(τ )−α−1

dτ

.

t

In Section 2, we show that α ˆ (t) is also a test function. We call it the test function associated with α(t). The conditions (1.2) and (1.3) deal with cases β < α + 1 and β > α + 1. The associated test function is introduced to deal with the troublesome case β = α + 1. In Section 3, we provide a simple sufficient condition for a continuously differentiable function to be a test function. As a consequence we see that (1.4)

log2 1t log3 1t logn+1 1t α(t) = α − α1 − α2 − · · · − αn , log 1t log 1t log 1t

where α, α1 , α2 , . . . , αn ∈ R and n times

z }| { logn := log log · · · log, is a test function. For this test function, we have α α α 1 1 1 2 1 n α(t) α t = t log log2 · · · logn . t t t In Section 4, we generalize three classical theorems: a Theorem of Privalov about the Hilbert transform of bounded Lipα functions with 0 < α < 1, a Theorem of Titchmarsh about the Hilbert transform of bounded Lip1 functions, and a Theorem of Hardy-Littlewood about the boundary behavior of functions in the disc algebra. We show that these theorems remain valid if in the statement of theorem one replaces Lipα by Lipα(t) . Therefore, we hope that this section will convince our readers that the set of axioms (1.1), (1.2) and (1.3) are properly chosen.

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2. The associated test function The associated test function is defined so that Z t0 α(t) ˆ α τ α(τ )−α−1 dτ. t =t t

Here we show that α ˆ (t) is really a test function. Lemma 2.1. Let α(t) be a test function and let Z t0 1 α(τ )−α−1 α ˆ (t) = α + τ dτ . log log t t Then α ˆ (t) is a test function. Proof. Clearly, α ˆ (t) is a continuous function on (0, t0 ). Our first task is to show that limt→0+ α ˆ (t) = α. For any ε > 0, by (1.3), we have Z t0 Z t0 α(t)−α ˆ α(τ )−α−1 t = τ dτ = τ α(τ )−α−1−ε τ ε dτ t t α(t)−α−ε Z t0 t ε α(t)−α−ε α(τ )−α−1−ε ε ≥t + O(t ) . τ dτ = t ε t ˆ Hence lim inf t→0+ tα(t)−α(t) ≥ 1/ε. When ε tends to zero we get

(2.1)

ˆ lim tα(t)−α(t) = ∞.

t→0+

Let λ ∈ R and let ˆ ϕλ (t) = tα(t)−α+λ .

Then, we have ˆ ϕ0λ (t) = tλ−1−α+α(t) (λtα(t)−α(t) − 1). If λ ≤ 0, then ϕ0λ < 0 and thus ϕλ is strictly decreasing. On the other hand, if λ > 0, by (2.1), ϕ0λ (t) > 0 for small values of t and thus ϕλ is strictly increasing.

Fix ε > 0. For small enough τ , say 0 < τ < τ0 < min{1, t0 }, we have α − ε ≤ α(τ ) ≤ α + ε. Hence τ ε−1 ≤ τ α(τ )−α−1 ≤ τ −ε−1 . For 0 < t < τ0 , we thus get Z τ0 Z τ0 Z τ0 τ0ε tε t−ε τ0−ε ε−1 α(τ )−α−1 − = τ dτ ≤ τ dτ ≤ τ −ε−1 dτ = − . ε ε ε ε t t t Therefore, for 0 < t < τ0 , −ε Z τ0 ε 1 t τ0−ε 1 1 τ0 tε α(τ )−α−1 log − ≤ log τ dτ ≤ log − , log t ε ε log t log t ε ε t

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which implies Z t0 1 α(τ )−α−1 τ dτ log −ε ≤ lim inf t→0+ log t t Z t0 1 α(τ )−α−1 ≤ lim sup log τ dτ ≤ 0. t→0+ log t t Now we, let ε → 0 to get 1 lim+ log t→0 log t

Z

t0

τ

α(τ )−α−1

dτ

= 0.

t

Hence, lim α ˆ (t) = α.

t→0+

Let β < α + 1. Then, for (1.2), Z t Z t Z t ˆ tα(t)−β+1 α−β α(τ ˆ )−α α(t)−α ˆ α(τ ˆ )−β . τ τ dτ ≥ t τ α−β dτ = τ dτ = α−β+1 0 0 0 On the other hand, for 0 < ε < 1 + α − β, we have Z t Z t Z t α(τ ˆ )−β α−β−ε α(τ ˆ )−α+ε α(t)−α+ε ˆ τ dτ = τ τ dτ ≤ t τ α−β−ε dτ 0

0

0 α(t)−β+1 ˆ

=

t . α−β+1−ε

Hence t

Z

ˆ )−β τ α(τ dτ =

0

ˆ tα(t)−β+1 ˆ + O(tα(t)−β+1 ). α−β+1

Finally, let β > α + 1. Then, for (1.3), Z t0 Z t0 Z α(τ ˆ )−β α−β α(τ ˆ )−α α(t)−α ˆ τ dτ = τ τ dτ ≤ t t

t

t0

τ α−β dτ ≤

t

ˆ tα(t)−β+1 . β−α−1

On the other hand, for any ε > 0, we have Z τ0 Z τ0 Z τ0 α(τ ˆ )−β α−β−ε α(τ ˆ )−α+ε α(t)−α+ε ˆ τ dτ = τ τ dτ ≥ t τ α−β−ε dτ t t t β+ε−α−1 ! α(t)−β+1 ˆ t t = 1− . β−α−1+ε τ0 Hence Z

t0

τ t

α(τ ˆ )−β

ˆ tα(t)−β+1 ˆ dτ = + O(tα(t)−β+1 ). β−α−1

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Using similar calculations, we can show that Z t0 1 α(τ )−α−1 α(t) + log τ dτ log t t is also a test function. But, we will not use this result in what follows. The following simple result will be used several times in what follows. Let us mention that f (t) g(t), as t → 0+ , means that there are constants c, C > 0 such that cg(t) ≤ f (t) ≤ Cg(t) in a right neighborhood of zero. Corollary 2.2. Let α(t) be a test function with limt→0+ α(t) = α. Let β, γ, C be real constants such that C > 0, γ > 0 and β − 1 < α. Then, α(t)+1−β−γ if α < γ + β − 1, Z t0 α(τ )−β t τ α(t)−α ˆ dτ t as t → 0+ . if α = γ + β − 1, γ + Ctγ τ 0 1 if α > γ + β − 1, Proof. We decompose the integral over two intervals (0, t) and (t, t0 ) and then we use (1.2) and (1.3). Hence Z t Z t0 α(τ )−β Z t0 α(τ )−β τ τ + dτ = dτ γ γ τ + Ct τ γ + Ctγ t 0 0 Z t0 α(τ )−β Z t α(τ )−β τ τ dτ + dτ γ t τγ t 0 tα(τ )−β+1 + tα(τ )−β−γ+1 . tγ The first estimate is true since α > β − 1, and the second one holds since α < β + γ − 1. Similarly, by (1.2) and (2.1), we have Z t Z t0 α(τ )−α−1+γ Z t0 α(τ )−α−1+γ τ τ + dτ = dτ γ γ τ + Ct τ γ + Ctγ t 0 0 Z t α(τ )−α−1+γ Z t0 α(τ )−α−1+γ τ τ dτ + dτ γ t τγ t 0 ˆ ˆ tα(t)−α + tα(t)−α tα(t)−α .

The last case is proved similarly.

3. A sufficient condition In this section we give a simple and sufficient criterion verifying that the function α(t) defined in (1.4) is a test function.

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Theorem 3.1. Let α ∈ R and let α(t) be a real continuously differentiable function defined on (0, t0 ) with limt→0+ α(t) = α. Suppose that lim α0 (t) t log t = 0.

t→0+

Then α(t) is a test function. Proof. Since d(tα(t)−γ ) = (α(t) − γ + α0 (t) t log t)tα(t)−γ−1 , dt then α(t) − γ + α0 (t) t log t → α − γ

as t → 0+ .

Hence, for β < α + 1, we have Z t Z t α(τ )−β τ dτ = τ α(τ )−α τ α−β dτ 0 0 τ =t τ α(τ )−β+1 = α − β + 1 τ =0 Z t 1 (α(τ ) − α + α0 (τ ) τ log τ )τ α(τ )−β dτ − α−β+1 0 Z t α(t)−β+1 t = τ α(τ )−β dτ. + O(1) α−β+1 0 Therefore, t

Z

τ α(τ )−β dτ =

(1 + O(1)) 0

tα(t)−β+1 . α−β+1

Thus (1.2) holds. For β > α + 1, we have Z t0 Z t0 α(τ )−β τ dτ = τ α(τ )−α τ α−β dτ t t τ =t τ α(τ )−β+1 0 = α − β + 1 τ =t Z t0 − (α(τ ) − α + α0 (τ ) τ log τ )τ α(τ )−β dτ t Z t0 tα(t)−β+1 = O(1) − + O(1) τ α(τ )−β dτ. α−β+1 t Hence Z (1 + O(1)) t

Thus (1.3) holds.

t0

τ α(τ )−β dτ =

tα(t)−β+1 . β−α−1

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Corollary 3.2. Let α, α1 , α2 , . . . , αn be real constants, and let log2 1t log3 1t logn+1 1t α(t) = α − α1 − α2 − · · · − αn . log 1t log 1t log 1t Then α(t) is a test function. Moreover, ( tα ˆ 1+α1 α αn tα(t) α log2 1t 2 · · · logn 1t t log 1t

if α1 < −1, if α1 > −1.

If α1 = α2 = . . . = αk−1 = −1, then ( tα ˆ 1+αk α αn tα(t) α t logk 1t logk+1 1t k+1 · · · logn 1t

if αk < −1, if αk > −1.

Finally, if α1 = α2 = . . . = αn = −1, then ˆ tα(t) tα logn+1 1t .

4. Applications In this section, we generalize a theorem of Privalov and a theorem of Titchmarsh about the Hilbert transform of bounded Lipα functions, and a theorem of HardyLittlewood about the boundary behavior of functions in the disc algebra. 4.1. Hilbert transform on R. Let u : R → R be in L1 (dt/(1 + t2 )). Then the integral Z t i ∞ 1 + u(t) dt π −∞ z − t 1 + t2 converges uniformly on compact subsets of the upper half plane and thus it defines an analytic function there. Hence its real and imaginary parts Z 1 ∞ Im z u(t) dt, U (z) := π −∞ |z − t|2 Z ∞ 1 t Re z − t e (z) := (4.1) U + u(t) dt π −∞ |z − t|2 1 + t2 are a pair of harmonic conjugates in the upper half plane. According to the well known approximate identity property of the Poisson kernel Im z/π|z − t|2 , the non-tangential limit of U exists at almost every x ∈ R and is equal to u(x). e also exists almost everywhere on R. Moreover, the non-tangential limit of U This fact is a deep and fundamental result in the theory of functions. This limit, wherever it exists, is denoted by u˜ and is called the Hilbert transform of u; u˜ can also be found by the singular integral Z 1 t 1 + u(t) dt. (4.2) u˜(x) = lim ε→0 π |x−t|>ε x − t 1 + t2

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This subject has been comprehensively studied in [1] and [5]. It is also briefly discussed in [4] and [6]. There are two celebrated results about the Hilbert transform of bounded Lipschitz functions. Theorem (Privalov [7]). If u is a bounded Lipα , 0 < α < 1, function, then u˜(x) exists for all x ∈ R, and besides u˜ is also Lipα . Theorem (Titchmarsh [8]). If u is a bounded Lip1 function, then u˜(x) exists for all x ∈ R, and besides |˜ u(x + t) − u˜(x)| ≤ const t log 1t for all x ∈ R and for small values of t > 0. ˆ If α(t) ≡ 1, then, by Corollary 3.2, tα(t) t log(1/t). Therefore, Titchmarsh’s Theorem can be rephrased in the following way: If u is a bounded Lip1 function, then u˜(x) exists for all x ∈ R, and besides it is Lipˆ1 . Now, we may generalize these two theorems in the following way.

Theorem 4.1. Let α(t) be a test function with limt→0+ α(t) ≤ 1. Let u be a bounded Lipα(t) function on R. Then u˜(x) exists for all x ∈ R, and furthermore ( Lipα(t) if limt→0+ α(t) < 1, u˜ ∈ Lipα(t) if limt→0+ α(t) = 1. ˆ Proof. To estimate |˜ u(x+t)− u˜(x)|, instead of taking the real line as our straight path to go from x to x + t, we go from x up to x + it, then to x + t + it and finally down to x + t: e (x + it)| |˜ u(x) − u˜(x + t)| ≤ |˜ u(x) − U (4.3)

e (x + it) − U e (x + t + it)| +|U e (x + t + it) − u˜(x + t)|. +|U

Hence we proceed to study each term of the right side. By (4.1) and (4.2), we have Z 1 τ 1 e (x + it)| = lim |˜ u(x) − U + u(τ ) dτ ε→0 π x−τ 1 + τ2 |x−τ |>ε Z 1 ∞ x−τ τ − + u(τ ) dτ 2 2 2 π −∞ (x − τ ) + t 1+τ Z 1 1 x − τ = lim − u(τ ) dτ ε→0 π |x−τ |>ε x−τ (x − τ )2 + t2 Z t2 ∞ u(x − τ ) − u(x + τ ) = lim dτ ε→0 π ε τ (τ 2 + t2 ) Z t2 ∞ |u(x + τ ) − u(x − τ )| ≤ dτ. π 0 τ (τ 2 + t2 )

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Since u is bounded and Lipα(t) , there is a constant C such that |u(x + τ ) − u(x − τ )| ≤ C(2τ )α(2τ ) for all values of x ∈ R, small values of τ , say 0 < τ < 1. Hence, by Corollary 2.2, we have Z Z t2 1 C(2τ )α(2τ ) t2 ∞ 2kuk∞ e |˜ u(x) − U (x + it)| ≤ dτ + dτ π 0 τ (τ 2 + t2 ) π 1 τ3 Z kuk∞ 2 4Ct2 2 τ α(τ )−1 t dτ + ≤ 2 2 π π 0 τ + 4t ≤ C 0 tα(t) + C 00 t2 . Therefore, for any x ∈ R and for any test function α(t) with limt→0+ α(t) ≤ 1, we have (4.4)

e (x + it)| = O(tα(t) ). |˜ u(x) − U

Incidentally, the preceding calculation also shows that u˜(x) exists for all x ∈ R. By the Mean Value Theorem, ∂U e e (x + it) − U e (x + t + it)| ≤ t sup |U (s + it) , s∈R ∂x where Z e ∂U 1 ∞ (s + it) = ∂x π −∞ Z 1 ∞ = π −∞ Z 1 ∞ = π −∞ Z 1 ∞ = π −∞

∂ x−τ τ u(τ ) dτ + ∂x (x − τ )2 + t2 1 + τ 2 x=s t2 − (s − τ )2 u(τ ) dτ (t2 + (s − τ )2 )2 t2 − τ 2 u(s + τ ) dτ (t2 + τ 2 )2 t2 − τ 2 (u(s + τ ) − u(s)) dτ. (t2 + τ 2 )2

Hence Z t ∞ |u(s + τ ) − u(s)| e e |U (x + it) − U (x + t + it)| ≤ sup dτ t2 + τ 2 s∈R π −∞ Z 1 α(τ ) Z τ t ∞ 2kuk∞ ≤ Ct dτ + dτ. 2 2 π 1 τ2 0 t +τ The asymptotic behavior of Z 0

1

τ α(τ ) dτ t2 + τ 2

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depends on limt→0+ α(t). According to Corollary 2.2, we have ( α(t) e (x + it) − U e (x + t + it)| = O(t ) if limt→0+ α(t) < 1, (4.5) |U ˆ O(tα(t) ) if limt→0+ α(t) = 1. Finally, (4.3), (4.4) and (4.5) give the required result. In the light of Corollary 3.2 and Theorem 4.1, we get the following special result. Corollary 4.2. Let α1 , α2 , . . . , αn ∈ R. Let u be a bounded function on R such that α α αn |u(x + t) − u(x)| ≤ Ct log 1t 1 log2 1t 2 · · · logn 1t for all x ∈ R and for all t ∈ (0, t0 ), where t0 is a small enough positive constant. Then, u˜(x) exists for all values of x ∈ R and besides, for all x ∈ R and for all t ∈ (0, t0 ), we have ( Ct if α1 < −1, |˜ u(x + t) − u˜(x)| ≤ 1 1+α1 1 αn 1 α2 Ct log t · · · logn t if α1 > −1; log2 t if α1 = α2 = . . . = αk−1 = −1, then ( Ct 1+αk α αn |˜ u(x + t) − u˜(x)| ≤ logk+1 1t k+1 · · · logn 1t Ct logk 1t

if αk < −1, if αk > −1;

finally, if α1 = α2 = . . . = αn = −1, then |˜ u(x + t) − u˜(x)| ≤ Ct logn+1 1t . Now, we give an example to show that Corollary 4.2 gives a sharp result. Let ( 1 if |t| ≤ 1e , |t| log |t| u(t) = 1 if |t| ≥ 1e , e and let α(t) = 1 −

log2 1t , log 1t

1 0 0, and let f be ¯ and is Lipα (t) on T if and only if analytic on D. Then f is continuous on D 0 iθ α(1−r)−1 − f (re ) = O((1 − r) ) as r → 1 .

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Proof. Suppose that f 0 (reiθ ) = O((1 − r)α(1−r)−1 ) as r → 1− . According to (1.1), this assumption ensures that the limit Z R lim− f 0 (reiθ )eiθ dr R→1

ρ

exists and is finite at all points eiθ ∈ T. Hence, the radial limit f (eiθ ) = lim f (reiθ ) r→1

exists at all points of eiθ ∈ T, and iθ

Z

iθ

f (e ) − f (ρe ) =

1

f 0 (reiθ )eiθ dr.

ρ

Thus, by (1.2), iθ

Z

iθ

1 0

|f (e ) − f (ρe )| ≤

|f (re )| dr ≤ C ρZ

(4.6)

Z

iθ

1

(1 − r)α(1−r)−1 dr

ρ

1−ρ

τ α(τ )−1 dτ ≤ C(1 − ρ)α(1−ρ) .

=C 0

Moreover, we have (cf. Figure 1) iϕ

iθ

Z

f (e ) − f (e ) =

f 0 (ζ) dζ

Γ

for all θ and ϕ with 0 < ϕ − θ < 1 and ρ = 1 − (ϕ − θ). eiϕ ρeiϕ

Γ eiθ ρeiθ

Figure 1.

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Therefore, by (1.2), iϕ

Z

1 0

Z

ϕ

|f (e ) − f (e )| ≤

iθ

0

Z

1

|f (re )| dr + |f (ρe )| ρdt + |f 0 (reiϕ )| dr θ ρ Z 1 (1 − r)α(1−r)−1 dr + C|ϕ − θ|(1 − ρ)α(1−ρ)−1 ≤ 2C Zρ |ϕ−θ| τ α(τ )−1 dτ + C|ϕ − θ|α(|ϕ−θ|) = 2C

iθ

it

ρ

(4.7)

0

≤ C|ϕ − θ|α(|ϕ−θ|) . ¯ and is Lipα(t) on T. Hence, by (4.6) and (4.7), f is continuous on D ¯ and is Lipα(t) on T. Hence, the Poisson Now, suppose that f is continuous on D representation formula Z π 1 − r2 iθ it dt f (re ) = f (e ) , reiθ ∈ D, 2 − 2r cos(θ − t) 1 + r 2π −π is valid. Taking the derivative of both sides with respect to θ gives Z π −2r(1 − r2 ) sin(θ − t) dt iθ 0 iθ ire f (re ) = f (eit ) 2 2 2π −π (1 + r − 2r cos(θ − t)) Z π 2 2r(1 − r ) sin t i(θ+t) dt f (e ) = 2 2 2π −π (1 + r − 2r cos t) Z π 2r(1 − r2 ) sin t dt (f (ei(θ+t) ) − f (eiθ )) . = 2 2 2π −π (1 + r − 2r cos t) Thus, by Corollary 2.2, 2(1 − r) |f (re )| ≤ π 0

π

Z

iθ

−π π

Z ≤ C(1 − r)

0

|t|ωf (|t|) dt ((1 − r)2 + 4r( πt )2 )2 tα(t)+1 dt 2 ((1 − r)2 + 2 πt 2 )2

α(1−r)−1

≤ C(1 − r)

as r → 1− .

The boundary values of a function in the Hardy space H p (D) is far away from being continuous. Nevertheless, for f ∈ H p (D), the function Ωf (t) = kft − f kp , where ft (z) = f (eit z), represents a continuous function [5, p. 9]. In the same papers, Hardy and Littlewood proved the following result. Theorem (Hardy-Littlewood [2, 3]). Let f ∈ H p (D). Then, the the function Ωf is Lip α, 0 < α ≤ 1, if and only if kfr0 kp = O((1 − r)α−1 ) as r → 1− . Similarly, this theorem can also be generalized as follows.

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Theorem 4.4. Let α(t) be a test function and let f ∈ H p (D). Then, Ωf is Lipα(t) if and only if kfr0 kp = O((1 − r)α(1−r)−1 ) as r → 1− .

5. Further generalization In the definition of α(t), instead of (1.2) and (1.3), we may assume that Z t τ α(τ )−β dτ = O(tα(t)−β+1 ), β < α + 1, 0 Z t0 τ α(τ )−β dτ = O(tα(t)−β+1 ), β > α + 1. t

Then, instead of Corollary 2.2, we would α(t)+1−β−γ ) Z t0 α(τ )−β O(t τ ˆ dτ = O(tα(t)−α ) τ γ + Ctγ 0 O(1)

have if α < γ + β − 1, if α = γ + β − 1, if α > γ + β − 1,

as t → 0+ .

Nevertheless, theorems like 4.1, 4.3 and 4.4 are still valid. One of the main reasons we did not pick this more generalized definition was that we were interested in the test function α α αn tα(t) = tα log 1t 1 log2 1t 2 · · · logn 1t , for which (1.2) and (1.3) hold. Acknowledgement. The author thanks the referee for many valuable remarks and suggestions.

References 1. J. Garnett, Bounded Analytic Functions, Academic Press, New York, 1981. 2. G. H. Hardy and J. E. Littlewood, Some properties of fractional integrals I, Math. Z. 27 (1928), 565–606. 3. , Some properties of fractional integrals II, Math. Z. 34 (1932), 403–439. 4. V. Havin and J. Mashreghi, Admissible majorants for model subspaces of H 2 , Part I and Part II, Canad. J. Math. 55 (2003), 1231–1263 and 1264–1301. 5. P. Koosis, Introduction to Hp Spaces, Second Edition, Cambridge Tracts in Mathematics 115, Cambridge University Press, Cambridge, 1998. 6. J. Mashreghi, Hilbert transform of log |f |. Proc. Amer. Math. Soc. 130 (2002), 683–688. 7. I. I. Privalov, Int´egrale de Cauchy, Bulletin de l’Universit´e `a Saratov, 1918. 8. E. C. Titchmarsh, Introduction to the Theory of Fourier Integrals, third edition, Chelsea Publishing Co., New York, 1986. Javad Mashreghi E-mail: [email protected] Address: D´epartement de math´ematiques et de statistique, Universit´e Laval, Qu´ebec, QC, G1K 7P4, Canada.

Generalized Lipschitz Functions Javad Mashreghi

(Communicated by Matti Vuorinen) Abstract. In this note the class Lipα(t) of continuous functions is introduced. The definition is arranged so that for the constant function α(t) ≡ α, the class Lipα(t) is nothing but the classical Lipschitz space Lipα . Then, to justify that our set of axioms for α(t) are properly chosen, some celebrated theorems of Privalov, Titchmarsh, Hardy and Littlewood about Lipα functions are shown to be also valid for Lipα(t) functions. Keywords. Lipschitz functions, test function, associated test function, Hilbert transform, Hardy classes. 2000 MSC. Primary 26A16; Secondary 32A40.

1. Introduction Let f : R → C be a continuous function. The modulus of continuity of f is ωf (t) :=

sup

|f (x + y) − f (x)|,

t > 0.

x∈R, |y|≤t

The function ωf is used to define different classes of continuous functions. For example, f is called Lipα , for some α with 0 < α ≤ 1, if ωf (t) = O(tα )

as t → 0+ .

In this note, our main goal is to replace tα by tα(t) , where α(t) is a function defined for small values of t and properly tends to α as t → 0+ . To be more precise, we assume that α(t) is a real continuous function defined in a right neighborhood Received October 21, 2004, in revised form October 18, 2005. This work was supported by two research grants from NSERC (Canada) and FQRNT (Qu´ebec). c 2005 Heldermann Verlag ISSN 1617-9447/$ 2.50

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of zero, say (0, t0 ), and, as t → 0+ , we have (1.1) (1.2) (1.3)

α ∈ R,

α(t) = α + O(1), tα(t)−β+1 τ α(τ )−β dτ = + O(tα(t)−β+1 ), α + 1 − β 0 Z t0 tα(t)−β+1 τ α(τ )−β dτ = + O(tα(t)−β+1 ), β−α−1 t Z

t

β < α + 1, β > α + 1.

Let us call α(t) a test function. The space of all continuous functions f : R → C satisfying ωf (t) = O(tα(t) ) as t → 0+ will be denoted by Lipα(t) . Clearly the classical space Lipα is a special case corresponding to the test function α(t) ≡ α. Given a test function α(t), let 1 α ˆ (t) = α + log log t

Z

t0

τ

α(τ )−α−1

dτ

.

t

In Section 2, we show that α ˆ (t) is also a test function. We call it the test function associated with α(t). The conditions (1.2) and (1.3) deal with cases β < α + 1 and β > α + 1. The associated test function is introduced to deal with the troublesome case β = α + 1. In Section 3, we provide a simple sufficient condition for a continuously differentiable function to be a test function. As a consequence we see that (1.4)

log2 1t log3 1t logn+1 1t α(t) = α − α1 − α2 − · · · − αn , log 1t log 1t log 1t

where α, α1 , α2 , . . . , αn ∈ R and n times

z }| { logn := log log · · · log, is a test function. For this test function, we have α α α 1 1 1 2 1 n α(t) α t = t log log2 · · · logn . t t t In Section 4, we generalize three classical theorems: a Theorem of Privalov about the Hilbert transform of bounded Lipα functions with 0 < α < 1, a Theorem of Titchmarsh about the Hilbert transform of bounded Lip1 functions, and a Theorem of Hardy-Littlewood about the boundary behavior of functions in the disc algebra. We show that these theorems remain valid if in the statement of theorem one replaces Lipα by Lipα(t) . Therefore, we hope that this section will convince our readers that the set of axioms (1.1), (1.2) and (1.3) are properly chosen.

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2. The associated test function The associated test function is defined so that Z t0 α(t) ˆ α τ α(τ )−α−1 dτ. t =t t

Here we show that α ˆ (t) is really a test function. Lemma 2.1. Let α(t) be a test function and let Z t0 1 α(τ )−α−1 α ˆ (t) = α + τ dτ . log log t t Then α ˆ (t) is a test function. Proof. Clearly, α ˆ (t) is a continuous function on (0, t0 ). Our first task is to show that limt→0+ α ˆ (t) = α. For any ε > 0, by (1.3), we have Z t0 Z t0 α(t)−α ˆ α(τ )−α−1 t = τ dτ = τ α(τ )−α−1−ε τ ε dτ t t α(t)−α−ε Z t0 t ε α(t)−α−ε α(τ )−α−1−ε ε ≥t + O(t ) . τ dτ = t ε t ˆ Hence lim inf t→0+ tα(t)−α(t) ≥ 1/ε. When ε tends to zero we get

(2.1)

ˆ lim tα(t)−α(t) = ∞.

t→0+

Let λ ∈ R and let ˆ ϕλ (t) = tα(t)−α+λ .

Then, we have ˆ ϕ0λ (t) = tλ−1−α+α(t) (λtα(t)−α(t) − 1). If λ ≤ 0, then ϕ0λ < 0 and thus ϕλ is strictly decreasing. On the other hand, if λ > 0, by (2.1), ϕ0λ (t) > 0 for small values of t and thus ϕλ is strictly increasing.

Fix ε > 0. For small enough τ , say 0 < τ < τ0 < min{1, t0 }, we have α − ε ≤ α(τ ) ≤ α + ε. Hence τ ε−1 ≤ τ α(τ )−α−1 ≤ τ −ε−1 . For 0 < t < τ0 , we thus get Z τ0 Z τ0 Z τ0 τ0ε tε t−ε τ0−ε ε−1 α(τ )−α−1 − = τ dτ ≤ τ dτ ≤ τ −ε−1 dτ = − . ε ε ε ε t t t Therefore, for 0 < t < τ0 , −ε Z τ0 ε 1 t τ0−ε 1 1 τ0 tε α(τ )−α−1 log − ≤ log τ dτ ≤ log − , log t ε ε log t log t ε ε t

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which implies Z t0 1 α(τ )−α−1 τ dτ log −ε ≤ lim inf t→0+ log t t Z t0 1 α(τ )−α−1 ≤ lim sup log τ dτ ≤ 0. t→0+ log t t Now we, let ε → 0 to get 1 lim+ log t→0 log t

Z

t0

τ

α(τ )−α−1

dτ

= 0.

t

Hence, lim α ˆ (t) = α.

t→0+

Let β < α + 1. Then, for (1.2), Z t Z t Z t ˆ tα(t)−β+1 α−β α(τ ˆ )−α α(t)−α ˆ α(τ ˆ )−β . τ τ dτ ≥ t τ α−β dτ = τ dτ = α−β+1 0 0 0 On the other hand, for 0 < ε < 1 + α − β, we have Z t Z t Z t α(τ ˆ )−β α−β−ε α(τ ˆ )−α+ε α(t)−α+ε ˆ τ dτ = τ τ dτ ≤ t τ α−β−ε dτ 0

0

0 α(t)−β+1 ˆ

=

t . α−β+1−ε

Hence t

Z

ˆ )−β τ α(τ dτ =

0

ˆ tα(t)−β+1 ˆ + O(tα(t)−β+1 ). α−β+1

Finally, let β > α + 1. Then, for (1.3), Z t0 Z t0 Z α(τ ˆ )−β α−β α(τ ˆ )−α α(t)−α ˆ τ dτ = τ τ dτ ≤ t t

t

t0

τ α−β dτ ≤

t

ˆ tα(t)−β+1 . β−α−1

On the other hand, for any ε > 0, we have Z τ0 Z τ0 Z τ0 α(τ ˆ )−β α−β−ε α(τ ˆ )−α+ε α(t)−α+ε ˆ τ dτ = τ τ dτ ≥ t τ α−β−ε dτ t t t β+ε−α−1 ! α(t)−β+1 ˆ t t = 1− . β−α−1+ε τ0 Hence Z

t0

τ t

α(τ ˆ )−β

ˆ tα(t)−β+1 ˆ dτ = + O(tα(t)−β+1 ). β−α−1

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Using similar calculations, we can show that Z t0 1 α(τ )−α−1 α(t) + log τ dτ log t t is also a test function. But, we will not use this result in what follows. The following simple result will be used several times in what follows. Let us mention that f (t) g(t), as t → 0+ , means that there are constants c, C > 0 such that cg(t) ≤ f (t) ≤ Cg(t) in a right neighborhood of zero. Corollary 2.2. Let α(t) be a test function with limt→0+ α(t) = α. Let β, γ, C be real constants such that C > 0, γ > 0 and β − 1 < α. Then, α(t)+1−β−γ if α < γ + β − 1, Z t0 α(τ )−β t τ α(t)−α ˆ dτ t as t → 0+ . if α = γ + β − 1, γ + Ctγ τ 0 1 if α > γ + β − 1, Proof. We decompose the integral over two intervals (0, t) and (t, t0 ) and then we use (1.2) and (1.3). Hence Z t Z t0 α(τ )−β Z t0 α(τ )−β τ τ + dτ = dτ γ γ τ + Ct τ γ + Ctγ t 0 0 Z t0 α(τ )−β Z t α(τ )−β τ τ dτ + dτ γ t τγ t 0 tα(τ )−β+1 + tα(τ )−β−γ+1 . tγ The first estimate is true since α > β − 1, and the second one holds since α < β + γ − 1. Similarly, by (1.2) and (2.1), we have Z t Z t0 α(τ )−α−1+γ Z t0 α(τ )−α−1+γ τ τ + dτ = dτ γ γ τ + Ct τ γ + Ctγ t 0 0 Z t α(τ )−α−1+γ Z t0 α(τ )−α−1+γ τ τ dτ + dτ γ t τγ t 0 ˆ ˆ tα(t)−α + tα(t)−α tα(t)−α .

The last case is proved similarly.

3. A sufficient condition In this section we give a simple and sufficient criterion verifying that the function α(t) defined in (1.4) is a test function.

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Theorem 3.1. Let α ∈ R and let α(t) be a real continuously differentiable function defined on (0, t0 ) with limt→0+ α(t) = α. Suppose that lim α0 (t) t log t = 0.

t→0+

Then α(t) is a test function. Proof. Since d(tα(t)−γ ) = (α(t) − γ + α0 (t) t log t)tα(t)−γ−1 , dt then α(t) − γ + α0 (t) t log t → α − γ

as t → 0+ .

Hence, for β < α + 1, we have Z t Z t α(τ )−β τ dτ = τ α(τ )−α τ α−β dτ 0 0 τ =t τ α(τ )−β+1 = α − β + 1 τ =0 Z t 1 (α(τ ) − α + α0 (τ ) τ log τ )τ α(τ )−β dτ − α−β+1 0 Z t α(t)−β+1 t = τ α(τ )−β dτ. + O(1) α−β+1 0 Therefore, t

Z

τ α(τ )−β dτ =

(1 + O(1)) 0

tα(t)−β+1 . α−β+1

Thus (1.2) holds. For β > α + 1, we have Z t0 Z t0 α(τ )−β τ dτ = τ α(τ )−α τ α−β dτ t t τ =t τ α(τ )−β+1 0 = α − β + 1 τ =t Z t0 − (α(τ ) − α + α0 (τ ) τ log τ )τ α(τ )−β dτ t Z t0 tα(t)−β+1 = O(1) − + O(1) τ α(τ )−β dτ. α−β+1 t Hence Z (1 + O(1)) t

Thus (1.3) holds.

t0

τ α(τ )−β dτ =

tα(t)−β+1 . β−α−1

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Corollary 3.2. Let α, α1 , α2 , . . . , αn be real constants, and let log2 1t log3 1t logn+1 1t α(t) = α − α1 − α2 − · · · − αn . log 1t log 1t log 1t Then α(t) is a test function. Moreover, ( tα ˆ 1+α1 α αn tα(t) α log2 1t 2 · · · logn 1t t log 1t

if α1 < −1, if α1 > −1.

If α1 = α2 = . . . = αk−1 = −1, then ( tα ˆ 1+αk α αn tα(t) α t logk 1t logk+1 1t k+1 · · · logn 1t

if αk < −1, if αk > −1.

Finally, if α1 = α2 = . . . = αn = −1, then ˆ tα(t) tα logn+1 1t .

4. Applications In this section, we generalize a theorem of Privalov and a theorem of Titchmarsh about the Hilbert transform of bounded Lipα functions, and a theorem of HardyLittlewood about the boundary behavior of functions in the disc algebra. 4.1. Hilbert transform on R. Let u : R → R be in L1 (dt/(1 + t2 )). Then the integral Z t i ∞ 1 + u(t) dt π −∞ z − t 1 + t2 converges uniformly on compact subsets of the upper half plane and thus it defines an analytic function there. Hence its real and imaginary parts Z 1 ∞ Im z u(t) dt, U (z) := π −∞ |z − t|2 Z ∞ 1 t Re z − t e (z) := (4.1) U + u(t) dt π −∞ |z − t|2 1 + t2 are a pair of harmonic conjugates in the upper half plane. According to the well known approximate identity property of the Poisson kernel Im z/π|z − t|2 , the non-tangential limit of U exists at almost every x ∈ R and is equal to u(x). e also exists almost everywhere on R. Moreover, the non-tangential limit of U This fact is a deep and fundamental result in the theory of functions. This limit, wherever it exists, is denoted by u˜ and is called the Hilbert transform of u; u˜ can also be found by the singular integral Z 1 t 1 + u(t) dt. (4.2) u˜(x) = lim ε→0 π |x−t|>ε x − t 1 + t2

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This subject has been comprehensively studied in [1] and [5]. It is also briefly discussed in [4] and [6]. There are two celebrated results about the Hilbert transform of bounded Lipschitz functions. Theorem (Privalov [7]). If u is a bounded Lipα , 0 < α < 1, function, then u˜(x) exists for all x ∈ R, and besides u˜ is also Lipα . Theorem (Titchmarsh [8]). If u is a bounded Lip1 function, then u˜(x) exists for all x ∈ R, and besides |˜ u(x + t) − u˜(x)| ≤ const t log 1t for all x ∈ R and for small values of t > 0. ˆ If α(t) ≡ 1, then, by Corollary 3.2, tα(t) t log(1/t). Therefore, Titchmarsh’s Theorem can be rephrased in the following way: If u is a bounded Lip1 function, then u˜(x) exists for all x ∈ R, and besides it is Lipˆ1 . Now, we may generalize these two theorems in the following way.

Theorem 4.1. Let α(t) be a test function with limt→0+ α(t) ≤ 1. Let u be a bounded Lipα(t) function on R. Then u˜(x) exists for all x ∈ R, and furthermore ( Lipα(t) if limt→0+ α(t) < 1, u˜ ∈ Lipα(t) if limt→0+ α(t) = 1. ˆ Proof. To estimate |˜ u(x+t)− u˜(x)|, instead of taking the real line as our straight path to go from x to x + t, we go from x up to x + it, then to x + t + it and finally down to x + t: e (x + it)| |˜ u(x) − u˜(x + t)| ≤ |˜ u(x) − U (4.3)

e (x + it) − U e (x + t + it)| +|U e (x + t + it) − u˜(x + t)|. +|U

Hence we proceed to study each term of the right side. By (4.1) and (4.2), we have Z 1 τ 1 e (x + it)| = lim |˜ u(x) − U + u(τ ) dτ ε→0 π x−τ 1 + τ2 |x−τ |>ε Z 1 ∞ x−τ τ − + u(τ ) dτ 2 2 2 π −∞ (x − τ ) + t 1+τ Z 1 1 x − τ = lim − u(τ ) dτ ε→0 π |x−τ |>ε x−τ (x − τ )2 + t2 Z t2 ∞ u(x − τ ) − u(x + τ ) = lim dτ ε→0 π ε τ (τ 2 + t2 ) Z t2 ∞ |u(x + τ ) − u(x − τ )| ≤ dτ. π 0 τ (τ 2 + t2 )

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Since u is bounded and Lipα(t) , there is a constant C such that |u(x + τ ) − u(x − τ )| ≤ C(2τ )α(2τ ) for all values of x ∈ R, small values of τ , say 0 < τ < 1. Hence, by Corollary 2.2, we have Z Z t2 1 C(2τ )α(2τ ) t2 ∞ 2kuk∞ e |˜ u(x) − U (x + it)| ≤ dτ + dτ π 0 τ (τ 2 + t2 ) π 1 τ3 Z kuk∞ 2 4Ct2 2 τ α(τ )−1 t dτ + ≤ 2 2 π π 0 τ + 4t ≤ C 0 tα(t) + C 00 t2 . Therefore, for any x ∈ R and for any test function α(t) with limt→0+ α(t) ≤ 1, we have (4.4)

e (x + it)| = O(tα(t) ). |˜ u(x) − U

Incidentally, the preceding calculation also shows that u˜(x) exists for all x ∈ R. By the Mean Value Theorem, ∂U e e (x + it) − U e (x + t + it)| ≤ t sup |U (s + it) , s∈R ∂x where Z e ∂U 1 ∞ (s + it) = ∂x π −∞ Z 1 ∞ = π −∞ Z 1 ∞ = π −∞ Z 1 ∞ = π −∞

∂ x−τ τ u(τ ) dτ + ∂x (x − τ )2 + t2 1 + τ 2 x=s t2 − (s − τ )2 u(τ ) dτ (t2 + (s − τ )2 )2 t2 − τ 2 u(s + τ ) dτ (t2 + τ 2 )2 t2 − τ 2 (u(s + τ ) − u(s)) dτ. (t2 + τ 2 )2

Hence Z t ∞ |u(s + τ ) − u(s)| e e |U (x + it) − U (x + t + it)| ≤ sup dτ t2 + τ 2 s∈R π −∞ Z 1 α(τ ) Z τ t ∞ 2kuk∞ ≤ Ct dτ + dτ. 2 2 π 1 τ2 0 t +τ The asymptotic behavior of Z 0

1

τ α(τ ) dτ t2 + τ 2

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depends on limt→0+ α(t). According to Corollary 2.2, we have ( α(t) e (x + it) − U e (x + t + it)| = O(t ) if limt→0+ α(t) < 1, (4.5) |U ˆ O(tα(t) ) if limt→0+ α(t) = 1. Finally, (4.3), (4.4) and (4.5) give the required result. In the light of Corollary 3.2 and Theorem 4.1, we get the following special result. Corollary 4.2. Let α1 , α2 , . . . , αn ∈ R. Let u be a bounded function on R such that α α αn |u(x + t) − u(x)| ≤ Ct log 1t 1 log2 1t 2 · · · logn 1t for all x ∈ R and for all t ∈ (0, t0 ), where t0 is a small enough positive constant. Then, u˜(x) exists for all values of x ∈ R and besides, for all x ∈ R and for all t ∈ (0, t0 ), we have ( Ct if α1 < −1, |˜ u(x + t) − u˜(x)| ≤ 1 1+α1 1 αn 1 α2 Ct log t · · · logn t if α1 > −1; log2 t if α1 = α2 = . . . = αk−1 = −1, then ( Ct 1+αk α αn |˜ u(x + t) − u˜(x)| ≤ logk+1 1t k+1 · · · logn 1t Ct logk 1t

if αk < −1, if αk > −1;

finally, if α1 = α2 = . . . = αn = −1, then |˜ u(x + t) − u˜(x)| ≤ Ct logn+1 1t . Now, we give an example to show that Corollary 4.2 gives a sharp result. Let ( 1 if |t| ≤ 1e , |t| log |t| u(t) = 1 if |t| ≥ 1e , e and let α(t) = 1 −

log2 1t , log 1t

1 0 0, and let f be ¯ and is Lipα (t) on T if and only if analytic on D. Then f is continuous on D 0 iθ α(1−r)−1 − f (re ) = O((1 − r) ) as r → 1 .

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Proof. Suppose that f 0 (reiθ ) = O((1 − r)α(1−r)−1 ) as r → 1− . According to (1.1), this assumption ensures that the limit Z R lim− f 0 (reiθ )eiθ dr R→1

ρ

exists and is finite at all points eiθ ∈ T. Hence, the radial limit f (eiθ ) = lim f (reiθ ) r→1

exists at all points of eiθ ∈ T, and iθ

Z

iθ

f (e ) − f (ρe ) =

1

f 0 (reiθ )eiθ dr.

ρ

Thus, by (1.2), iθ

Z

iθ

1 0

|f (e ) − f (ρe )| ≤

|f (re )| dr ≤ C ρZ

(4.6)

Z

iθ

1

(1 − r)α(1−r)−1 dr

ρ

1−ρ

τ α(τ )−1 dτ ≤ C(1 − ρ)α(1−ρ) .

=C 0

Moreover, we have (cf. Figure 1) iϕ

iθ

Z

f (e ) − f (e ) =

f 0 (ζ) dζ

Γ

for all θ and ϕ with 0 < ϕ − θ < 1 and ρ = 1 − (ϕ − θ). eiϕ ρeiϕ

Γ eiθ ρeiθ

Figure 1.

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Therefore, by (1.2), iϕ

Z

1 0

Z

ϕ

|f (e ) − f (e )| ≤

iθ

0

Z

1

|f (re )| dr + |f (ρe )| ρdt + |f 0 (reiϕ )| dr θ ρ Z 1 (1 − r)α(1−r)−1 dr + C|ϕ − θ|(1 − ρ)α(1−ρ)−1 ≤ 2C Zρ |ϕ−θ| τ α(τ )−1 dτ + C|ϕ − θ|α(|ϕ−θ|) = 2C

iθ

it

ρ

(4.7)

0

≤ C|ϕ − θ|α(|ϕ−θ|) . ¯ and is Lipα(t) on T. Hence, by (4.6) and (4.7), f is continuous on D ¯ and is Lipα(t) on T. Hence, the Poisson Now, suppose that f is continuous on D representation formula Z π 1 − r2 iθ it dt f (re ) = f (e ) , reiθ ∈ D, 2 − 2r cos(θ − t) 1 + r 2π −π is valid. Taking the derivative of both sides with respect to θ gives Z π −2r(1 − r2 ) sin(θ − t) dt iθ 0 iθ ire f (re ) = f (eit ) 2 2 2π −π (1 + r − 2r cos(θ − t)) Z π 2 2r(1 − r ) sin t i(θ+t) dt f (e ) = 2 2 2π −π (1 + r − 2r cos t) Z π 2r(1 − r2 ) sin t dt (f (ei(θ+t) ) − f (eiθ )) . = 2 2 2π −π (1 + r − 2r cos t) Thus, by Corollary 2.2, 2(1 − r) |f (re )| ≤ π 0

π

Z

iθ

−π π

Z ≤ C(1 − r)

0

|t|ωf (|t|) dt ((1 − r)2 + 4r( πt )2 )2 tα(t)+1 dt 2 ((1 − r)2 + 2 πt 2 )2

α(1−r)−1

≤ C(1 − r)

as r → 1− .

The boundary values of a function in the Hardy space H p (D) is far away from being continuous. Nevertheless, for f ∈ H p (D), the function Ωf (t) = kft − f kp , where ft (z) = f (eit z), represents a continuous function [5, p. 9]. In the same papers, Hardy and Littlewood proved the following result. Theorem (Hardy-Littlewood [2, 3]). Let f ∈ H p (D). Then, the the function Ωf is Lip α, 0 < α ≤ 1, if and only if kfr0 kp = O((1 − r)α−1 ) as r → 1− . Similarly, this theorem can also be generalized as follows.

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Theorem 4.4. Let α(t) be a test function and let f ∈ H p (D). Then, Ωf is Lipα(t) if and only if kfr0 kp = O((1 − r)α(1−r)−1 ) as r → 1− .

5. Further generalization In the definition of α(t), instead of (1.2) and (1.3), we may assume that Z t τ α(τ )−β dτ = O(tα(t)−β+1 ), β < α + 1, 0 Z t0 τ α(τ )−β dτ = O(tα(t)−β+1 ), β > α + 1. t

Then, instead of Corollary 2.2, we would α(t)+1−β−γ ) Z t0 α(τ )−β O(t τ ˆ dτ = O(tα(t)−α ) τ γ + Ctγ 0 O(1)

have if α < γ + β − 1, if α = γ + β − 1, if α > γ + β − 1,

as t → 0+ .

Nevertheless, theorems like 4.1, 4.3 and 4.4 are still valid. One of the main reasons we did not pick this more generalized definition was that we were interested in the test function α α αn tα(t) = tα log 1t 1 log2 1t 2 · · · logn 1t , for which (1.2) and (1.3) hold. Acknowledgement. The author thanks the referee for many valuable remarks and suggestions.

References 1. J. Garnett, Bounded Analytic Functions, Academic Press, New York, 1981. 2. G. H. Hardy and J. E. Littlewood, Some properties of fractional integrals I, Math. Z. 27 (1928), 565–606. 3. , Some properties of fractional integrals II, Math. Z. 34 (1932), 403–439. 4. V. Havin and J. Mashreghi, Admissible majorants for model subspaces of H 2 , Part I and Part II, Canad. J. Math. 55 (2003), 1231–1263 and 1264–1301. 5. P. Koosis, Introduction to Hp Spaces, Second Edition, Cambridge Tracts in Mathematics 115, Cambridge University Press, Cambridge, 1998. 6. J. Mashreghi, Hilbert transform of log |f |. Proc. Amer. Math. Soc. 130 (2002), 683–688. 7. I. I. Privalov, Int´egrale de Cauchy, Bulletin de l’Universit´e `a Saratov, 1918. 8. E. C. Titchmarsh, Introduction to the Theory of Fourier Integrals, third edition, Chelsea Publishing Co., New York, 1986. Javad Mashreghi E-mail: [email protected] Address: D´epartement de math´ematiques et de statistique, Universit´e Laval, Qu´ebec, QC, G1K 7P4, Canada.

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