PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE Nouvelle série, tome 96 (110) (2014), 181–192 DOI: 10.2298/PIM1410181L

WEIGHTED MARKOV–BERNSTEIN INEQUALITIES FOR ENTIRE FUNCTIONS OF EXPONENTIAL TYPE

Doron S. Lubinsky

Abstract. We prove weighted Markov–Bernstein inequalities of the form

∫_{−∞}^{∞} |f′(x)|^{p}w(x) dx ⩽ C(σ + 1)^{p} ∫_{−∞}^{∞} |f(x)|^{p}w(x) dx

Here w satisfies certain doubling type properties, f is an entire function of exponential type ⩽ σ, p > 0, and C is independent of f and σ. For example, w(x) = (1 + x^{2})^{α} satisfies the conditions for any α ∈ ℝ. Classical doubling inequalities of Mastroianni and Totik inspired this result.

1. Introduction

The classical Markov–Bernstein inequality for the unit circle asserts that for polynomials P of degree ⩽ n, and 0 < p ⩽ ∞,

(1.1)  ∥P′∥_{L_{p} (Γ)} ⩽ n∥P ∥_{L_{p} (Γ)} .

Here Γ is the unit circle, and if p < ∞,

∥P ∥_{L_{p} (Γ)} = ( ∫_{−π}^{π} |P (e^{iθ})|^{p}dθ) ^{1/p}.

Of course, it was proved earlier for 1 ⩽ p ⩽ ∞, and later for 0 < p < 1 by Arestov [1]. There is a close cousin for entire functions f of exponential type ⩽ σ, and 0 < p ⩽ ∞:

(1.2)  ∥f′∥_{L_{p} (ℝ)} ⩽ σ∥f∥_{L_{p} (ℝ)} .

It too was earlier proved for 1 ⩽ p ⩽ ∞, and later for 0 < p < 1. See [15]. In fact, these inequalities are equivalent, and can be derived from each other–as follows, for example, from the methods of [10] where there is a similar equivalence between Marcinkiewicz–Zygmund and Plancherel–Polya inequalities. These are yet more illustrations of the classical link between approximation theory for polynomials and that for entire functions of exponential type, amply explored in the memoir of Ganzburg [8], and in the books of Timan [17], and Trigub and Belinsky [18], for example.

2010 Mathematics Subject Classification: 42C05.

Key words and phrases: Entire functions of exponential type, Bernstein inequalities.

Research supported by NSF grant DMS1001182 and US-Israel BSF grant 2008399.

In honor of the retirement of Giuseppe Mastroianni.

There is a vast literature on Markov–Bernstein inequalities, both for polynomials [5,12,14], and entire functions of exponential type. For the latter, there are Szegő type inequalities, and sharp inequalities for various subclasses of entire functions with special properties–see [4,6,16]. In another direction, weighted Bernstein inequalities involving inner functions, and model spaces have been investigated by Baranov [2, 3].

For polynomials, one of the most beautiful results involves doubling weights, and is due to Mastroianni and Totik [13]. Recall the setting: let W : [−π, π] → [0, ∞) be measurable. Extend W as a 2π periodic function to the real line. We say that W is doubling if there is a constant L (called a doubling constant for W) such that for all intervals I, we have ∫_{2I} W ⩽ L ∫_{I} W. Here 2I is the interval with the same center as I, but with twice the length. A typical doubling weight is

W (t) = h(t) ∏_{j=1}^{k} |t − β_{j} |^{γ_{j}} ,

where h is bounded above and below by positive constants, and all {β_{j} } are distinct and lie in [−π, π], while all γ_{j} > −1. An immediate consequence of Theorem 4.1 in [13, p. 45] is that for 1 ⩽ p < ∞,

∫_{−π}^{π} |P′(e^{iθ})|^{p} W (θ) dθ ⩽ Cn^{p} ∫_{−π}^{π} |P (e^{iθ})|^{p} W (θ) dθ,

valid for n ⩾ 1 and all polynomials P of degree ⩽ n. This was extended to 0 < p < 1 by Erdelyi [7]. The constant C depends only on p and the doubling constant L, not on the particular W.

In this paper, inspired by the results of Mastroanni, Totik, and Erdelyi, we prove weighted Markov–Bernstein inequalities. Our most general result follows.

Theorem 1.1. Let σ, p > 0, r ∈ (0, 1], and let w : ℝ → [0, ∞) be a measurable function satisfying the following:

(I) The one-sided doubling condition about 0: there exists L > 1, such that for |a| ⩾ r,

(1.3)  | ∫_{a}^{2a} w| ⩽ L| ∫_{a/2}^{a} w|.

(II) The growth condition about integers: there exist B, β ⩾ 1 such that for k ⩾ 0 and −1 ⩽ j ⩽ max\{2k + 1, \frac{1}{r}\},

(1.4)  ∫_{jr}^{(j+1)r} w ⩽ B(1 + r|j − k|)^{β} ∫_{kr}^{(k+1)r} w.

Assume also the analogous condition for k < 0. For t ∈ ℝ, let

(1.5)  w_{r}(t) = \frac{1}{2r} ∫_{t−r}^{t+r} w(s) ds.

Then for entire functions f of exponential type ⩽ σ, we have

(1.6)  ∫_{−∞}^{∞} |f′(t)|^{p}w_{r}(t) dt ⩽ C(σ + 1)^{p} ∫_{−∞}^{∞} |f(t)|^{p}w_{r}(t) dt,

provided the right-hand side is finite. Here C depends on B, β, p and L, but is independent of σ, r, f, and the particular w.

Corollary 1.2. Let p > 0. Assume that all the conditions of Theorem 1.1 hold for some r_{0} ∈ (0, 1), and all r ∈ (0, r_{0}), with L, B and β independent of r. Then for σ > 0, and entire functions f of exponential type ⩽ σ, we have

(1.7)  ∫_{−∞}^{∞} |f′(t)|^{p}w(t) dt ⩽ C(σ + 1)^{p} ∫_{−∞}^{∞} |f(t)|^{p}w(t) dt,

provided the right-hand side is finite. Here C depends on B, β, p and L, but is independent of σ, f, and the particular w.

Corollary 1.3. Let σ, p > 0, and let w : ℝ → (0, ∞) be a measurable function satisfying the following: for some M ⩾ 1, we have for for t ∈ ℝ ∖ {0} and both \frac{1}{2} ⩽ \frac{s}{t} ⩽ 2 and |s − t| ⩽ 2,

(1.8)  \frac{1}{M} ⩽ \frac{w(s)}{w(t)} ⩽ M.

Then for entire functions f of exponential type ⩽ σ, we have

(1.9)  ∫_{−∞}^{∞} |f′(t)|^{p}w(t) dt ⩽ C(σ + 1)^{p} ∫_{−∞}^{∞} |f(t)|^{p}w(t) dt,

provided the right-hand side is finite. Here C depends on M, but is independent of σ, w and f.

Corollary 1.4. Let σ, p > 0, and α ∈ ℝ. Then for entire functions f of exponential type ⩽ σ, we have

(1.10)  ∫_{−∞}^{∞} |f′(t)|^{p}(1 + t^{2})^{α} dt ⩽ C(σ + 1)^{p} ∫_{−∞}^{∞} |f(t)|^{p}(1 + t^{2})^{α} dt,

provided the right-hand side is finite. Here C is independent of σ and f.

To the best of our knowledge, even the inequalities in Corollary 1.4 are new. Almost all existing inequalities in the literature are unweighted, though they involve sharp constants as in (1.2). We note that if 1 = λ_{1} < λ_{2} < ⋯, and f(x) = ∑ _{j=1}^{m} c_{j} λ_{j}^{−ix} , we used orthogonal Dirichlet polynomials in [11] to prove

( ∫_{−∞}^{∞} \frac{|f′(x)|^{2}}{1 + x^{2}} dx) ^{1/2} ⩽ \{logλ_{m} + (logλ_{m})^{1/2} \} ( ∫_{−∞}^{∞} \frac{|f(x)|^{2}}{1 + x^{2}} dx) ^{1/2} .

Here one cannot replace logλ_{m} +(logλ_{m})^{1/2} by any factor smaller than logλ_{m} +C_{1} for some C_{1} > 0. This inequality reflects the fact that f is entire of type ⩽ logλ_{m}.

It is noteworthy that if one allows the weight to depend on the exponential type of the function, then it suffices to prove results for entire functions of exponential type 1. Indeed, suppose that for some weight w and all entire functions f of exponential type at most 1; we have

∫_{−∞}^{∞} |f′(t)|^{p}w(t) dt ⩽ C_{1} ∫_{−∞}^{∞} |f(t)|^{p}w(t) dt.

If now f is entire of exponential type ⩽ σ, and we apply this last inequality to g(t) = f(t/σ), which does have type ⩽ 1, and then make a substitution t = σs, we obtain for all entire functions f of exponential type ⩽ σ,

∫_{−∞}^{∞} |f′(s)|^{p}w(σs) ds ⩽ C_{1}σ^{p} ∫_{−∞}^{∞} |f(s)|^{p}w(σs) ds.

However, the goal of this paper is estimates in which the weight does not depend on σ.

We prove the results in Section 2. Throughout C, C_{1}, C_{2}, … denote positive constants independent of f, σ, r. The same symbol does not necessarily denote the same constant in different occurrences.

2. Proofs of the results

Throughout, we let S(t) = sinπt/πt denote the sinc kernel. We will use the bounds |S(t)| ⩽ min{1, 1/π|t|}. We begin by applying (1.2) to

(2.1)  g(t) = f(t)[S( \frac{t}{ℓ}) + iS( \frac{t}{ℓ} + \frac{1}{2}) ]^{ℓ},

where ℓ is a fixed positive integer. This yields:

Lemma 2.1. Let ℓ ⩾ 1, p > 0, and f be entire of exponential type ⩽ σ. Then

(2.2)  ∫_{−∞}^{∞} |f′(t)|^{p}(1 + |t|)^{−ℓp} dt ⩽ C(σ + 1)^{p} ∫_{−∞}^{∞} |f(t)|^{p}(1 + |t|)^{−ℓp} dt,

where C is independent of f and σ.

Proof. Let h(t) = S ( \frac{t}{ℓ}) + iS ( \frac{t}{ℓ} + \frac{1}{2}) , so that g of (2.1) satisfies g = fh^{ℓ}. First note that for real t,

(2.3)  |h(t)| ⩽ min\{ 2, \frac{ℓ}{π|t|} + \frac{2ℓ}{π(|t| + ℓ/2)} \} ⩽ C(1 + |t|)^{−1} ,

where C depends only on ℓ. By (1.2), and some simple calculations, also,

(2.4)  |h′(t)| ⩽ C(1 + |t|)^{−1} ,

where again C depends only on ℓ. In the other direction, we see that

|h(t)|^{2} = ( \frac{sinπ \frac{t}{ℓ}}{π \frac{t}{ℓ}} ) ^{2} + ( \frac{cosπ \frac{t}{ℓ}}{π ( \frac{t}{ℓ} + \frac{1}{2})} ) ^{2}

⩾ \frac{(sinπ \frac{t}{ℓ})^{2} + (cosπ \frac{t}{ℓ})^{2}}{(π (| \frac{t}{ℓ} | + \frac{1}{2}) )^{2}} ⩾ C(1 + |t|)^{−2} . (2.5)

Then, recalling (2.1),

|f′(t)h(t)^{ℓ}| = |g′(t) − f(t)ℓh(t)^{ℓ−1} h′(t)|

⩽ |g′(t)| + C|f(t)|(1 + |t|)^{−ℓ} , (2.6)

by (2.3) and (2.4). Now g is entire of exponential type ⩽ σ + 1, and (2.3) shows that

∫_{−∞}^{∞} |g(t)|^{p}dt ⩽ C ∫_{−∞}^{∞} |f(t)|^{p}(1 + |t|)^{−ℓp} dt < ∞,

so applying (1.2) to g gives

∫_{−∞}^{∞} |g′(t)|^{p}dt ⩽ (σ + 1)^{p} ∫_{−∞}^{∞} |g(t)|^{p}dt ⩽ C(σ + 1)^{p} ∫_{−∞}^{∞} |f(t)|^{p}(1 + |t|)^{−ℓp} dt.

Together with (2.6), and (2.5), this yields

∫_{−∞}^{∞} |f′(t)|^{p}(1 + |t|)^{−ℓp} dt ⩽ C ∫_{−∞}^{∞} |f′(t)h(t)^{ℓ}|^{p} dt
⩽ C ∫_{−∞}^{∞} (|g′(t)|^{p} + C|f(t)|^{p}(1 + |t|)^{−ℓp} ) dt
⩽ C{(σ + 1)^{p} + 1} ∫_{−∞}^{∞} |f(t)|^{p}(1 + |t|)^{−ℓp} dt.

So we have the result. □

From this we deduce:

Lemma 2.2. Let σ, p > 0, ℓ ⩾ 1, and let w : ℝ → [0, ∞) be a measurable function. Let

(2.7)  H(t) = ∫_{−∞}^{∞} \frac{w(x)}{(1 + |x − t|)^{ℓp}} dx, t ∈ ℝ,

and assume that this is finite for t ∈ ℝ. Then for entire functions f of exponential type ⩽ σ for which the right-hand side is finite,

(2.8)  ∫_{−∞}^{∞} |f′(t)|^{p}H(t) dt ⩽ C(σ + 1)^{p} ∫_{−∞}^{∞} |f(t)|^{p}H(t) dt,

where C depends only on ℓ and p. In particular, it is independent of f, σ, w, H.

Proof. For a given x, and f, apply Lemma 2.1 to the function f(· + x), so that we are translating the variable. Making a substitution s = t + x yields

∫_{−∞}^{∞} |f′(s)|^{p} \frac{ds}{(1 + |s − x|)^{ℓp}} ⩽ C(σ + 1)^{p} ∫_{−∞}^{∞} |f(s)|^{p} \frac{ds}{(1 + |s − x|)^{ℓp}} .

Now multiply by w(x) and integrate over all real x, and then interchange the integrals. The convergence of the right-hand side in (2.8), and the nonnegativity of the integrand justifies the interchange of integrals. □

Our final lemma before proving Theorem 1.1 involves upper and lower bounds on w_{r}.

Lemma 2.3. Assume the hypotheses of Theorem 1.1. Then for some C_{1}, C_{2} > 0 that depend on L, B, β, ∫_{−1}^{0} w, ∫_{0}^{1} w, but not on r, t, nor on the particular w,

(2.9)  C_{2}(1 + |t|)^{−β} ⩽ w_{r}(t) ⩽ C_{1}(1 + |t|)^{log_{2} L}.

Proof. We first establish the lower bound. Let us assume first that t ⩾ 0 and choose j_{0} ⩾ 0 such that j_{0}r ⩽ t < (j_{0} + 1)r. Note that then

j_{0}r ⩾ t − r and (j_{0} + 1)r ⩽ t + r;
(2.10)  (j_{0} − 1)r ⩽ t − r and (j_{0} + 2)r ⩾ t + r.

Then, using (1.4),

∫_{t−r}^{t+r} w ⩾ ∫_{j_{0} r}^{(j_{0} +1)r} w ⩾ B^{−1} (1 + j_{0}r)^{−β} ∫_{0}^{r} w
⩾ B^{−1} (1 + t)^{−β} ∫_{0}^{r} w ⩾ B^{−1} (1 + t)^{−β} (1 + B(1 + 2^{β}))^{−1} ∫_{−r}^{r} w,

again by (1.4). Thus for t ⩾ 0, and some C depending only on B, β,

(2.11)  w_{r}(t) ⩾ C(1 + t)^{−β} w_{r}(0).

Next, using (1.4),

∫_{0}^{1}w ⩽ ∑_{j=0}^{[1/r]} ∫_{jr}^{(j+1)r} w ⩽ B ( ∫_{0}^{r} w) ∑_{j=0}^{[1/r]}(1 + jr)^{β} ⩽ B ( ∫_{0}^{r} w) ∫_{0}^{[1/r]+1} (1 + sr)^{β}ds

= B ( \frac{1}{r} ∫_{0}^{r} w) ∫_{0}^{r[1/r]+r} (1 + y)^{β}dy ⩽ B ( \frac{1}{r} ∫_{0}^{r} w) ∫_{0}^{2}(1 + y)^{β}dy.

A similar estimate holds for ∫_{−1}^{0} w, so for some C depending only on B, β,

(2.12)  ∫_{−1}^{1} w ⩽ Cw_{r}(0).

Together with (2.11), this establishes the lower bound for t ⩾ 0, and of course t < 0 is similar. We turn to the upper bound. Again, we assume t ⩾ 0, and that j_{0} is as above. We see using (2.10), and then (1.4), that

∫_{t−r}^{t+r} w ⩽ ∫_{(j_{0} −1)r}^{(j_{0} +2)r} w ⩽ (1 + 2B2^{β}) ∫_{j_{0} r}^{(j_{0} +1)r} w.

We continue this using (1.4), as

⩽ (1 + 2^{β+1} B)\frac{1}{[1/r] + 1} ∑_{k=j_{0}}^{j_{0} +[1/r]} B(1 + |j_{0} − k|r)^{β} ∫_{kr}^{(k+1)r} w

⩽ (1 + 2^{β+1} B)B2^{β}r ∫_{j_{0} r}^{(j_{0} +[1/r]+1)r} w.

Thus we have shown that

w_{r}(t) ⩽ C ∫_{j_{0} r}^{j_{0} r+2} w,

where C is independent of r, t, but depends on B and β. We continue this using (2.10), and then (1.3), as

⩽ C [∫_{0}^{1} w + ∫_{1}^{t+2} w] ⩽ C[∫_{0}^{1} w + ∑_{0⩽k⩽log_{2} (t+2)} ∫_{2^{k}}^{2^{k+1}} w]

⩽ C ( ∫_{0}^{1} w) [1 + ∑_{0⩽k⩽log_{2} (t+2)} L^{k+1} ] ⩽ C ( ∫_{0}^{1} w) L^{log_{2} (t+2)}

= C ( ∫_{0}^{1} w) (t + 2)^{log_{2} L}.

This gives the upper bound for t ⩾ 0, and the case t < 0 is similar. □

Proof of Theorem 1.1. Choose ℓ so large that

(2.13)  log_{2} L + β − ℓp ⩽ −2.

Note that this choice does not depend on w. Let H be as in Lemma 2.2. We estimate H above and below. Let us assume first that t ⩾ 0 and choose j_{0} ⩾ 0 such that j_{0}r ⩽ t < (j_{0} + 1)r, so that (2.10) holds. Split

H(t) = ( ∫_{−∞}^{0} + ∫_{0}^{max \{(2j_{0} +1)r,1\}} + ∫_{max \{(2j_{0} +1)r,1\}}^{∞} ) \frac{w(s)}{(1 + |s − t|)^{ℓp}} ds

= I_{1} + I_{2} + I_{3}. (2.14)

We start with the central integral I_{2} as it will contribute to both our upper and lower bounds. We use our growth condition (1.4) as well as that fact that for s ∈ [jr, (j + 1)r], we have |s − t| ⩾ |j − j_{0}|r − r ⩾ \frac{1}{2}|j − j_{0}|r if |j − j_{0}| ⩾ 2. If |j − j_{0}| ⩽ 2, observe that |j − j_{0}|r ⩽ 2. Thus

(2.15)  I_{2} ⩽ ∑_{j=0}^{max \{2j_{0} +1,[1/r]\}} ∫_{jr}^{(j+1)r} \frac{w(s)}{(1 + |s − t|)^{ℓp}} ds

⩽ ∑_{j=0}^{max \{2j_{0} +1,[1/r]\}} \frac{1}{( \frac{1}{4}(1 + |j − j_{0}|r))^{ℓp}} ∫_{jr}^{(j+1)r} w(s) ds

⩽ 4^{ℓp}B( ∫_{j_{0} r}^{(j_{0} +1)r} w(s) ds) ∑_{j=0}^{max \{2j_{0} +1,[1/r]\}} \frac{1}{(1 + |j − j_{0}|r)^{ℓp−β}}

⩽ 4^{ℓp}B ( ∫_{t−r}^{t+r} w(s) ds) ∑_{k=−∞}^{∞} \frac{1}{(1 + |k|r)^{ℓp−β}}

⩽ 4^{ℓp+3} Brw_{r}(t) ∫_{−∞}^{∞} \frac{1}{(1 + |s|r)^{ℓp−β}} ds ⩽ C_{1}w_{r}(t).

Here C_{1} depends on B, β, ℓ, p but is independent of r and w. We have also used (2.13) and L ⩾ 1 to ensure the convergence of the integral in the second last line. Note that we could not simply use the upper bound in Lemma 2.3 for w_{r}, as we need the last right-hand side of (2.15) to involve w_{r}(t). In the other direction, we see from (1.4) that

(2.16)  I_{2} ⩾ ∑_{j=j_{0}}^{max \{2j_{0} +1,[1/r]−1\}} \frac{1}{(1 + |j − j_{0}|r + r)^{ℓp}} ∫_{jr}^{(j+1)r} w(s) ds

⩾ B^{−1} ( ∫_{j_{0} r}^{(j_{0} +1)r} w(s) ds) ∑_{j=j_{0}}^{max \{2j_{0} +1,[1/r]−1\}} \frac{1}{(2 + |j − j_{0}|r)^{ℓp+β}}

⩾ B^{−1} ( ∫_{j_{0} r}^{(j_{0} +1)r} w(s) ds) ∑_{k=0}^{max \{j_{0} +1,[1/r]−1−j_{0} \}}  \frac{1}{(2 + kr)^{ℓp+β}} .

Here, using our growth condition (1.4), and then (2.10),

(1 + 2B2^{β}) ∫_{j_{0} r}^{(j_{0} +1)r} w(s) ds ⩾ ∫_{(j_{0} −1)r}^{(j_{0} +2)r} w(s)ds ⩾ ∫_{t−r}^{t+r} w(s) ds = 2rw_{r}(t),

and

∑_{k=0}^{max \{j_{0} +1,[1/r]−1−j_{0} \}}  \frac{1}{(2 + kr)^{ℓp+β}} ⩾ ∫_{0}^{max \{j_{0} +2,[1/r]−j_{0} \}}  \frac{1}{(2 + tr)^{ℓp+β}} dt

= \frac{1}{r} ∫_{0}^{max \{(j_{0} +1)r,r[1/r]−j_{0} r\}} \frac{1}{(2 + s)^{ℓp+β}} ds

⩾ \frac{1}{r} ∫_{0}^{1/2} \frac{1}{(2 + s)^{ℓp+β}} ds,

since if (j_{0} + 1)r ⩽ \frac{1}{2}, then r[1/r] − j_{0}r ⩾ 1 − r − j_{0}r ⩾ \frac{1}{2}. Substituting the last two inequalities in (2.16), and using (2.15), we have shown that for t ⩾ 0,

(2.17)  C_{1}w_{r}(t) ⩾ I_{2} ⩾ C_{2}w_{r}(t),

where C_{1} and C_{2} depend on ℓ, p, β, B, but not on r or the particular w. Next, our doubling condition (1.3) gives

(2.18) I_{1} ⩽ ∑_{j=0}^{∞} ∫_{−2^{j+1}}^{−2^{j}}  \frac{w(s)}{(1 + |s| + t)^{ℓp}} ds + \frac{1}{(1 + t)^{ℓp}} ∫_{−1}^{0} w

⩽ ∑_{j=0}^{∞} \frac{1}{(1 + 2^{j} + t)^{ℓp}} ∫_{−2^{j+1}}^{−2^{j}} w(s) ds + \frac{1}{(1 + t)^{ℓp}} ∫_{−1}^{0} w

⩽ ∑_{j=0}^{∞} \frac{L^{j+1}}{(1 + 2^{j} + t)^{ℓp}} ∫_{−1}^{0} w + \frac{1}{(1 + t)^{ℓp}} ∫_{−1}^{0} w

⩽ ( \frac{1}{(1 + t)^{ℓp}} ∑_{0⩽j⩽log_{2} (1+t)}L^{j+1} + L ∑_{j>log_{2} (1+t)} ( \frac{L}{2^{ℓp}})^{j} + \frac{1}{(1 + t)^{ℓp}} )∫_{−1}^{0} w

⩽ C( \frac{1}{(1 + t)^{ℓp}} L^{log_{2} (1+t)} + ( \frac{L}{2^{ℓp}})^{log_{2} (1+t)} ) ∫_{−1}^{0} w

⩽ C ( ∫_{−1}^{0} w) (1 + t)^{log_{2} L−ℓp} ,

by (2.13). Here C depends only on p, ℓ, L. Next, let N = log_{2} max{[(2j_{0} + 1)r], 1}, and let j ⩾ N, and s ∈ [2^{j} , 2^{j+1} ]. We claim that

(2.19)  1 + |s − t| ⩾ \frac{1}{3}2^{j} .

If first j_{0} = 0, then N = 1 and t < r, so 1 + |s − t| ⩾ 1 + 2^{j} − 1 = 2^{j}. If j_{0} ⩾ 1, then (j_{0} + 1)r ⩽ \frac{2}{3}(2j_{0} + 1)r ⩽ \frac{2}{3}2^{N}, so |s − t| ⩾ 2^{j} − (j_{0} + 1)r ⩾ 2^{j} − \frac{2}{3}2^{N} ⩾ \frac{1}{3}2^{j}. Thus we have (2.19). Then our doubling hypothesis (1.3) gives

I_{3} ⩽ ∑_{j=N}^{∞} ∫_{2^{j}}^{2^{j+1}} \frac{w(s)}{(1 + |s − t|)^{ℓp}} ds ⩽ ∑_{j=N}^{∞} \frac{1}{(3^{−1} 2^{j})^{ℓp}} ∫_{2^{j}}^{2^{j+1}} w(s) ds

⩽ ∑_{j=N}^{∞} \frac{1}{(3^{−1} 2^{j})^{ℓp}}L^{j+1} ∫_{0}^{1} w ⩽ 3^{ℓp}L( ∫_{0}^{1} w) ∑_{j=N}^{∞} ( \frac{L}{2^{ℓp}})^{j}

⩽ (2)3^{ℓp}L( ∫_{0}^{1} w) ( \frac{L}{2^{ℓp}})^{N}

⩽ C( ∫_{0}^{1} w) ( max{[(2j_{0} + 1)r], 1})^{log_{2} L−ℓp}

⩽ C( ∫_{0}^{1} w) (1 + t)^{log_{2} L−ℓp} .

In the third last line, we used L/2^{ℓp} ⩽ 1/4, as follows from (2.13). In the last line, we used (2.10). Together with (2.14), (2.17), and (2.18), we have shown that for t ⩾ 0,

C_{2}w_{r}(t) ⩽ H(t) ⩽ C_{1}( w_{r}(t) + ( ∫_{−1}^{1} w) (1 + t)^{log_{2} L−ℓp} ) .

Next, from (2.11) and (2.12), we can continue this as

C_{2}w_{r}(t) ⩽ H(t) ⩽ C_{1}w_{r}(t)(1 + (1 + t)^{log_{2} L−ℓp+β} ) ⩽ C_{3}w_{r}(t),

by (2.13). The case t < 0 is similar. Now the result follows from Lemma 2.2. □

We note that at least for p ⩾ 1, one can use the Markov–Bernstein inequalities in Theorem 1.1 to prove that there exists δ_{0} ∈ (0, 1) such that for σ > 0, and nonidentically vanishing entire functions f of exponential type ⩽ σ, we have

\frac{1}{2} ⩽ ∫_{−∞}^{∞} |f(t)|^{p}w_{δ_{0} /(σ+1)} (t)dt/ ∫_{−∞}^{∞} |f(t)|^{p}w(t) dt ⩽ \frac{3}{2}.

This gives one way to prove Corollary 1.2. However, we use a different method below.

Proof of Corollary 1.2. First note that Lemma 2.3 and our hypotheses imply that for some C > 1,

(2.20)  C^{−1} (1 + |t|)^{−β} ⩽ w_{r}(t) ⩽ C(1 + |t|)^{log_{2} L}, r ∈ (0, r_{0}) and t ∈ ℝ.

Here C is independent of r and t. Let σ > 0 and f be entire of type ⩽ σ, with the integral in the right-hand side of (1.7) finite. Let us choose k such that kp ⩾ β + log_{2} L + 2, and choose ε > 0, and set g(t) = f(t)S(εt)^{k}. By Lebesgue’s differentiation theorem, we have for a.e. t ∈ ℝ,

lim_{r→0+} w_{r}(t)|g(t)|^{p} = w(t)|g(t)|^{p}.

Next, (2.20) shows that for r ∈ (0, r_{0}) and all real t

w_{r}(t)|g(t)|^{p} ⩽ C(1 + |t|)^{log_{2} L}|f(t)|^{p} min{1, 1/πε|t|}^{kp}
⩽ C_{1}(1 + |t|)^{log_{2} L−kp} |f(t)|^{p}
⩽ C_{1}(1 + |t|)^{log_{2} L−kp+β} w(t)|f(t)|^{p} ⩽ C_{1}C_{2}w(t)|f(t)|^{p},

by Lemma 2.3 and our choice of k. Here C_{1} and C_{2} are independent of r, f but depend on ε and w. Since C_{1}C_{2}w(t)|f(t)|^{p} is independent of r and integrable by (1.7), Lebesgue’s dominated convergence theorem gives

lim_{r→0+} ∫_{−∞}^{∞} w_{r}(t)|g(t)|^{p}dt = ∫_{−∞}^{∞} w(t)|g(t)|^{p}dt.

Next, for each given R > 0, as g′ is bounded in each finite interval, and w_{r} is bounded independently of r,

lim_{r→0+} ∫_{−R}^{R} w_{r}(t)|g′(t)|^{p}dt = ∫_{−R}^{R} w(t)|g′(t)|^{p}dt.

Then as g has exponential type ⩽ σ + kεπ, Theorem 1.1 and the last two limits yield

∫_{−R}^{R} w(t)| \frac{d}{dt}(f(t)S(εt))|^{p}dt
⩽ C(σ + kεπ + 1)^{p} ∫_{−∞}^{∞} w(t)|f(t)S(εt)|^{p}dt
⩽ C(σ + kεπ + 1)^{p} ∫_{−∞}^{∞} w(t)|f(t)|^{p}dt,

recall that |S| ⩽ 1. Here C is independent of ε, σ, f, R. We can now let ε → 0+, and use the fact that S(εt) converges uniformly for t in compact subsets of ℂ to S(0) = 1. A similar statement then holds for the derivatives. We deduce that

∫_{−R}^{R} w(t)|f′(t)|^{p}dt ⩽ C(σ + 1)^{p} ∫_{−∞}^{∞} w(t)|f(t)|^{p}dt.

Finally, let R → ∞. □

Proof. Proof of Corollary 1.3 We choose r = 1 in Theorem 1.1. Our condition (1.8) shows that for some C > 1 and all t ∈ ℝ,

(2.21)  M^{−1} ⩽ w_{1}(t)/w(t) ⩽ M.

That condition also gives for a ⩾ 0,

∫_{a}^{2a} w ⩽ M2aw(a) ⩽ 4M^{2} ∫_{a/2}^{a} w

and similarly,

∫_{−2a}^{−a} w ⩽ 4M^{2} ∫_{−a/2}^{0} w.

So we can choose L = 4M^{2} in (1.3). Next, let k ⩾ 0 and −1 ⩽ j ⩽ max\{2k + 1, \frac{1}{r}\} = 2k + 1. We have to show that (1.4) holds for the given j, k and with r = 1. Firstly if j = −1 or 0, (1.8) gives

(2.22)  ∫_{j}^{j+1} w ⩽ M^{2} ∫_{1}^{2} w.

So now let us consider 1 ⩽ j ⩽ 2k + 1. Let us first suppose that j ⩽ k, and choose 0 ⩽ n ⩽ log_{2} k such that

\frac{k}{2^{n+1}} ⩽ j ⩽ \frac{k}{2^{n}} .

Then by repeated use of (1.8),

(2.23)  ∫_{j}^{j+1} w ⩽ Mw(j) ⩽ M^{2}w( \frac{k}{2^{n}} ) ⩽ M^{n+2} w(k) ⩽ M^{n+3} ∫_{k}^{k+2} w.

Here

M^{n} = 2^{n log_{2} M} ⩽ (k/j)^{log_{2} M} = (1 + (k − j)/j)^{log_{2} M} ⩽ (1 + |k − j|)^{log_{2} M} .

Combined with (2.22) and (2.23), we have shown that for −1 ⩽ j ⩽ k,

∫_{j}^{j+1} w ⩽ M^{5}(1 + |k − j|)^{log_{2} M} ∫_{k}^{k+1} w.

Next, if k < j ⩽ 2k + 1,

∫_{j}^{j+1} w ⩽ M^{2}w(k) ⩽ M^{3} ∫_{k}^{k+1} w ⩽ M^{3}(1 + |k − j|)^{log_{2} M} ∫_{k}^{k+1} w.

In summary, we have established (1.4) with B = M^{5} and β = log_{2} M. Then, recalling(2.21), Theorem 1.1 gives the result. □

Proof of Corollary 1.4. It is easy to see that w(x) = (1 + x^{2})^{α} satisfies (1.8), with, for example, M = 17^{|α|} . □

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School of Mathematics Georgia Institute of Technology Atlanta, GA 30332-0160 USA

 lubinsky@math.gatech.edu