Uniform Lorentz norm estimates for convolution operators

Abstract

Uniform endpoint Lorentz norm improving estimates for convolution operators with affine arclength measure supported on simple plane curves are established. The estimates hold for a wide class of simple curves, and the condition is stated in terms of averages of the square of the affine arclength weight, extending previously known results.

MSC:44A35, 42B35.

1 Introduction

Let $\varphi :(a,b)\to \mathbb{R}$ be a $C^2$ function such that ${\varphi }^{″}(t)\ge 0$ for all $t\in (a,b)$. In this paper, we consider the convolution operator given by

$$\mathcal{T}f(x_1,x_2)={\int }_a^{b}f(x_1-t,x_2-\varphi (t))\omega (t)dt$$

(1.1)

for $f\in C_0^{\mathrm{\infty }}({\mathbb{R}}^2)$. Here and in what follows, we denote $\omega (t):={({\varphi }^{″}(t))}^{1/3}$. Curves of the form $(t,\varphi (t))$ are said to be simple according to Drury and Marshall [1]. The measure $\omega (t)dt$ supported on the curve $(t,\varphi (t))$ is known as the affine arclength measure, which is based on the affine arclength parameter as in [2], and was introduced by Drury and Marshall [1] in dealing with the Fourier restriction problem related to curves, and later by Drury [3] in studying convolution operators with measures supported on curves. We refer interested readers to [2–4] for the relevance of affine geometry in this subject. One big benefit of using the affine arclength measure in place of the Euclidean arclength measure $\sqrt{1+{\varphi }^{\prime }{(t)}^2}dt$ has been its effect of mitigating degeneracies and it is believed that various uniform sharp estimates hold for a wide class of curves.

As is well known, the typeset $\mathcal{S}=\{(p^{-1},q^{-1}):\mathcal{T}\text{ is bounded from }{L}^p({\mathbb{R}}^2)\text{ to }{L}^q({\mathbb{R}}^2)\}$ of is contained in the convex hull of $\{(0,0),(1,1),(2/3,1/3)\}$ and uniform estimates in a, b, and ϕ are expected only for $(1/p,1/q)=(2/3,1/3)$. Many conditions to guarantee optimal uniform $L^{3/2}$-$L^3$ estimates have been known so far. See [3, 5–12] for example. Among other things, the author proved the following.

Theorem 1.1 (Choi [12])

Let J be an open interval in ℝ, and $\varphi :J\to \mathbb{R}$ be a $C^2$ function such that ${\varphi }^{″}\ge 0$. Suppose that there exists a positive constant A such that

$${\omega }^{1/2}(t_1){\omega }^{1/2}(t_2)\le {(\frac{A}{t_2-t_1}{\int }_{t_1}^{t_2}{\omega }^3(t)dt)}^{1/3}$$

holds whenever $t_1<t_2$ and $[t_1,t_2]\subset J$. Let be the operator defined as in (1.1). Then there exists a constant C that depends only on A such that

$${\parallel \mathcal{T}f\parallel }_{L^3({\mathbb{R}}^2)}\le C{\parallel f\parallel }_{L^{3/2}({\mathbb{R}}^2)}$$

holds uniformly in $f\in C_0^{\mathrm{\infty }}({\mathbb{R}}^2)$.

Under somewhat stronger assumptions on $\varphi (t)$ or $\omega (t)$, the endpoint Lebesgue norm estimate aforementioned can be improved to optimal Lorentz norm estimates, namely from $L^{3/2}({\mathbb{R}}^2)$ into $L^{3,3/2}({\mathbb{R}}^2)$ and $L^{3/2,3}({\mathbb{R}}^2)$ into $L^3({\mathbb{R}}^2)$. We refer interested readers to [6, 8, 10, 11] for known sufficient conditions for optimal and nearly optimal Lorentz norm estimates. Most importantly, Oberlin established the following uniform optimal Lorentz norm improving estimates.

Theorem 1.2 (Oberlin [11])

Let J be an open interval. Suppose that $\omega (t)$ is monotone increasing and that there exists a positive constant A such that

$$\sqrt{\omega (t_1)\omega (t_2)}\le A\omega ((t_1+t_2)/2)$$

(1.2)

holds whenever $t_1<t_2$ and $[t_1,t_2]\subset J$. Then the operator given by (1.1) satisfies

$$\begin{array}{c}{\parallel \mathcal{T}f\parallel }_{L^{3,3/2}({\mathbb{R}}^2)}\le C{\parallel f\parallel }_{L^{3/2}({\mathbb{R}}^2)},\hfill \\ {\parallel \mathcal{T}f\parallel }_{L^3({\mathbb{R}}^2)}\le C{\parallel f\parallel }_{L^{3/2,3}({\mathbb{R}}^2)}\hfill \end{array}$$

for all $f\in C_0^{\mathrm{\infty }}({\mathbb{R}}^2)$, where C is a constant depending only on A.

For the proof of the optimality, see [13] by Stovall along with [8] by Bak et al. It is interesting to ask if the condition in Theorem 1.2 can be relaxed to cover more general curves. Based on an ingenious argument of Oberlin in [11], the author aims to establish a uniform optimal Lorentz norm improving estimate under a condition on averages of the square of $\omega (t)$. The average condition is a slightly stronger version of that in Theorem 1.1, and yet covers most simple plane curves studied up to now including those in Theorem 1.2.

This paper is organized as follows: in the following section, conditions on $\omega (t)$ are introduced and the main theorem is stated. The last section is devoted to the proof of the main theorem. As usual, absolute constants may grow from line to line.

2 Statement of the main theorem

Before we state our main result, we introduce certain conditions on functions defined on intervals.

Definition 2.1 Let $0<p<\mathrm{\infty }$. For an interval $J_1$ in ℝ, a locally $L^p$ function $\mathrm{\Phi }:J_1\to {\mathbb{R}}^{+}$, and a positive real number A, we let

$$\begin{array}{rcl}{\mathfrak{G}}_p(\mathrm{\Phi },A)& :=& \{F:J_1\to {\mathbb{R}}^{+}|\sqrt{F(t_1)F(t_2)}\le A{(\frac{1}{t_2-t_1}{\int }_{t_1}^{t_2}{\mathrm{\Phi }}^p(t)dt)}^{1/p}\\ \text{whenever }{t}_1<t_2\text{ and }[t_1,t_2]\subset J_1\}\end{array}$$

and we let

$${\mathcal{E}}_p(A):=\{\mathrm{\Phi }:J\to {\mathbb{R}}^{+}\mid J\text{ is an interval and }\mathrm{\Phi }\in {\mathfrak{G}}_p(\mathrm{\Phi },A)\}.$$

An interesting subclass of ${\mathcal{E}}_p(2^{1/p}A)$, $0<p<\mathrm{\infty }$, was introduced by Bak et al. [14] in studying Fourier restriction estimates related to degenerate curves.

Definition 2.2 For an interval J and a positive real number A, a function $\mathrm{\Phi }:J\to {\mathbb{R}}^{+}$ is said to be a member of $\tilde{\mathcal{E}}(A)$ if

• Φ is monotone; and

• whenever $t_1<t_2$ and $[t_1,t_2]\subset J$,

  $$\sqrt{\mathrm{\Phi }(t_1)\mathrm{\Phi }(t_2)}\le A\mathrm{\Phi }((t_1+t_2)/2)$$

holds.

The condition (1.2) can be rewritten as $\omega \in \tilde{\mathcal{E}}(A)$.

Remark 2.3 It seems appropriate to mention some properties of ${\mathcal{E}}_p(A)$ and $\tilde{\mathcal{E}}(A)$ mentioned above.

• 1. It is a simple matter to check:

• $\tilde{\mathcal{E}}(A)\subset {\mathcal{E}}_p(2^{1/p}A)$ for all $p\in (0,\mathrm{\infty })$;

• $\mathrm{\Phi }\in {\mathcal{E}}_p(A)$ if and only if ${\mathrm{\Phi }}^p\in {\mathcal{E}}_1(A^p)$;

• $\mathrm{\Phi }\in {\mathcal{E}}_p(A)$ implies $\lambda \mathrm{\Phi }\in {\mathcal{E}}_p(A)$ for all $\lambda >0$; and

• $\mathrm{\Phi }\in {\mathcal{E}}_p(A)$ implies $\mathrm{\Phi }(a\cdot +b)\in {\mathcal{E}}_p(A)$ for all $(a,b)\in \mathbb{R}\setminus \{0\}\times \mathbb{R}$.

• 2. If $0<p_1<p_2<\mathrm{\infty }$, $\mathrm{\Phi }:J\to {\mathbb{R}}^{+}\in {\mathcal{E}}_{p_1}(A)$, and $\mathrm{\Phi }\in L_{\mathrm{loc}}^{p_2}(J)$, then $\mathrm{\Phi }\in {\mathcal{E}}_{p_2}(A)$ by Hölder’s inequality.

• 3. The class $\tilde{\mathcal{E}}(1)$ is essentially the class of logarithmically concave functions, which already encompasses many useful examples. Simplest examples are the exponential function and $\mathrm{\Phi }(t)=t^{\alpha }$, $t>0$, for $\alpha \ge 0$. More interesting example is the function $\mathrm{\Phi }(t)=e^{-1/t}$, $t>0$, which models a curve ‘flat’ at the origin. A hierarchy of flatter functions that belong to $\tilde{\mathcal{E}}(1)$ was constructed by Bak et al. [14].

• 4. For a polynomial $p(t)$ of degree N, $|p(t)|$ belongs to $\tilde{\mathcal{E}}(2^{N/2})$ after (possibly) decomposing the real line into at most $3^{N/2}$ intervals.

• 5. Nevertheless, there are functions that belong to ${\mathcal{E}}_p(A)$ but do not belong to $\tilde{\mathcal{E}}(A^{\prime })$ for any $A^{\prime }>0$. Two examples of curves that our result covers that are not covered in [11] can be constructed with the aid of the examples given below.

Example 2.4 Consider ${\mathrm{\Phi }}_{\beta }(t)=t^{-\beta }$, $t>0$, for $\beta \ge 2$. Then, for given $0<t_1<t_2<\mathrm{\infty }$, we have by a change of variable

$$\begin{array}{rcl}\frac{1}{t_2-t_1}{\int }_{t_1}^{t_2}{\mathrm{\Phi }}_{\beta }(t)dt& =& \frac{1}{(\lambda -1)t_1^{\beta }}{\int }_1^{\lambda }{t}^{-\beta }dt\\ =& \frac{1}{(\lambda -1)t_1^{\beta }}{\int }_{{\lambda }^{-1}}^1{t}^{\beta -2}dt,\end{array}$$

where $\lambda :=t_2/t_1>1$. Since $t^{\beta -2}$ is logarithmically concave, we see

$$\begin{array}{rcl}\frac{1}{t_2-t_1}{\int }_{t_1}^{t_2}{\mathrm{\Phi }}_{\beta }(t)dt& \ge & \frac{1}{2}\frac{1-{\lambda }^{-1}}{(\lambda -1)t_1^{\beta }}{\lambda }^{(-\beta +2)/2}\\ =& \frac{1}{2}\frac{1}{t_1^{\beta }{\lambda }^{\beta /2}}=\frac{1}{2}\sqrt{{\mathrm{\Phi }}_{\beta }(t_1){\mathrm{\Phi }}_{\beta }(t_2)},\end{array}$$

which implies ${\mathrm{\Phi }}_{\beta }\in {\mathcal{E}}_1(2)$. In view of Remark 2.3, given $\beta >0$, ${\mathrm{\Phi }}_{\beta }\in {\mathcal{E}}_p(2^{1/p})$ if $p\ge 2/\beta$. One can easily see ${\mathrm{\Phi }}_{\beta }\notin \tilde{\mathcal{E}}(A^{\prime })$ for any $A^{\prime }>0$ and $\beta >0$.

Example 2.5 Consider $\mathrm{\Phi }:(0,\mathrm{\infty })\to {\mathbb{R}}^{+}$ given by $\mathrm{\Phi }(t)={(2t)}^{1/2}{e}^{t^2}$. Then we have $\sqrt{\mathrm{\Phi }(t)\mathrm{\Phi }(1)}\sim t^{1/4}{e}^{t^2/2}$ and $\mathrm{\Phi }((t+1)/2)=O(t^{1/2}{e}^{t^2/3})$ as $t\to \mathrm{\infty }$, which clearly implies $\mathrm{\Phi }\notin \tilde{\mathcal{E}}(A)$ for all $A>0$. On the other hand, $\mathrm{\Phi }\in {\mathcal{E}}_2(1)$ by the following.

Proposition 2.6 Let $\psi :J\to \mathbb{R}$. Suppose that ${\psi }^{\prime }\in {\mathcal{E}}_1(A)$ for some $A>0$. Then the function Φ given by $\mathrm{\Phi }(t)={({\psi }^{\prime })}^{1/p}(t)exp(\psi (t))$ belongs to ${\mathcal{E}}_p(A^{1/p})$ for $0<p<\mathrm{\infty }$.

Proof Let $t_1<t_2$. Since ${\psi }^{\prime }\in {\mathcal{E}}_1(A)$, we have

$$\psi (t_2)-\psi (t_1)={\int }_{t_1}^{t_2}{\psi }^{\prime }(t)dt\ge A^{-1}(t_2-t_1)\sqrt{{\psi }^{\prime }(t_1){\psi }^{\prime }(t_2)}>0$$

by the fundamental theorem of calculus and the assumption on ${\psi }^{\prime }(t)$. A change of variable gives

$$\begin{array}{rcl}\frac{1}{t_2-t_1}{\int }_{t_1}^{t_2}{\mathrm{\Phi }}^p(t)dt& =& \frac{1}{t_2-t_1}{\int }_{t_1}^{t_2}{e}^{p\psi (t)}{\psi }^{\prime }(t)dt\\ =& \frac{1}{p(t_2-t_1)}{\int }_{p\psi (t_1)}^{p\psi (t_2)}{e}^{t}dt\\ =& \frac{\psi (t_2)-\psi (t_1)}{t_2-t_1}\times \frac{e^{p\psi (t_2)}-e^{p\psi (t_1)}}{p(\psi (t_2)-\psi (t_1))}.\end{array}$$

From

$$\begin{array}{rcl}\frac{e^b-e^a}{b-a}& =& e^{(b+a)/2}\times \frac{e^{(b-a)/2}-e^{-(b-a)/2}}{2\times (b-a)/2}\\ =& e^{(b+a)/2}\times \frac{sinh((b-a)/2)}{(b-a)/2}\ge e^{(b+a)/2}\end{array}$$

for all $a<b$, we see

$$\frac{e^{p\psi (t_2)}-e^{p\psi (t_1)}}{p(\psi (t_2)-\psi (t_1))}\ge e^{p(\psi (t_1)+\psi (t_2))/2}.$$

Altogether, we obtain

$$\frac{1}{t_2-t_1}{\int }_{t_1}^{t_2}{\mathrm{\Phi }}^p(t)dt\ge A^{-1}{e}^{p(\psi (t_1)+\psi (t_2))/2}\sqrt{{\psi }^{\prime }(t_1){\psi }^{\prime }(t_2)}=A^{-1}{(\mathrm{\Phi }(t_1)\mathrm{\Phi }(t_2))}^{p/2}.$$

By taking the p th root we obtain the desired estimate. □

We are now ready to state the main theorem of this paper.

Theorem 2.7 Let $-\mathrm{\infty }\le a<b\le \mathrm{\infty }$, and let $\varphi :(a,b)\to \mathbb{R}$ be a $C^2$ function such that ${\varphi }^{″}\ge 0$ on the interval. Suppose that there exists a positive constant A such that $\omega \in {\mathcal{E}}_2(A)$, i.e.

$${\omega }^{1/2}(t_1){\omega }^{1/2}(t_2)\le A{(\frac{1}{t_2-t_1}{\int }_{t_1}^{t_2}{\omega }^2(t)dt)}^{1/2}$$

holds whenever $a<t_1<t_2<b$. Let be the operator given by (1.1). Then there exists a constant C that depends only on A such that

$${\parallel \mathcal{T}f\parallel }_{L^{3,3/2}({\mathbb{R}}^2)}\le C{\parallel f\parallel }_{L^{3/2}({\mathbb{R}}^2)},$$

(2.1)

$${\parallel \mathcal{T}f\parallel }_{L^3({\mathbb{R}}^2)}\le C{\parallel f\parallel }_{L^{3/2,3}({\mathbb{R}}^2)}$$

(2.2)

holds uniformly in $f\in C_0^{\mathrm{\infty }}({\mathbb{R}}^2)$.

Remark 2.8 Some remarks are in order.

• In view of Remark 2.3, Proposition 2.6, Example 2.4 and Example 2.5, the condition $\omega \in \tilde{\mathcal{E}}(A)$ is strictly stronger than the condition $\omega \in {\mathcal{E}}_2(\sqrt{2}A)$ in Theorem 2.7, and therefore our result improves Theorem 1.2.

• An explicit example is also available. Consider $\varphi (t)=t^{-1/2}exp(t^2)$ defined for $t\in (c,\mathrm{\infty })$, where c is a large constant. A simple calculation shows $\omega (t)\sim t^{1/2}exp(t^2/3)$. By Proposition 2.6, $\omega \in {\mathcal{E}}_2(A)$ for some $A>0$. Thus, the corresponding operator satisfies endpoint Lorentz estimates (2.1) and (2.2) by Theorem 2.7, but Theorem 1.2 is not directly applicable.

• It is not known whether $\omega \in {\mathcal{E}}_2(A)$ in Theorem 2.7 can be further relaxed to $\omega \in {\mathcal{E}}_p(A)$ for some $p>2$. More generally, one can ask for the optimal p such that $\omega \in {\mathcal{E}}_p(A)$ guarantees the boundedness of from $L^{\frac{3}{2},q}({\mathbb{R}}^2)$ to $L^{3,r}({\mathbb{R}}^2)$ for given $q\le r$.

3 Proof of the main theorem

Before we prove the theorem, we begin with a couple of lemmas.

Lemma 3.1 Let J be an interval in ℝ, and let $\omega :J\to {\mathbb{R}}_{+}$ be a continuous function such that $\omega \in {\mathcal{E}}_2(A)$ for some $A>0$, i.e.

$${\omega }^{1/2}(t_1){\omega }^{1/2}(t_2)\le A{(\frac{1}{t_2-t_1}{\int }_{t_1}^{t_2}{\omega }^2(t)dt)}^{1/2}$$

holds whenever $t_1<t_2$ and $[t_1,t_2]\subset J$. Then the following holds:

$$\omega {(t_1)}^{1/3}\omega {(t_2)}^{1/3}\omega {\left(t^{\ast }\right)}^{1/3}\le 6^{1/3}A{(\frac{1}{t_2-t_1}{\int }_{t_1}^{t_2}{\omega }^3(t)dt)}^{1/3}$$

(3.1)

whenever $t_1<t_2$ and $t^{\ast }\in [t_1,t_2]\subset J$.

Proof of Lemma 3.1 Let $t^{\ast }\in [t_1,t_2]\subset J$. From

$${\omega }^{1/2}(t_1){\omega }^{1/2}(t_2)\le A{(\frac{1}{t_2-t_1}{\int }_{t_1}^{t_2}{\omega }^2(t)dt)}^{1/2},$$

we obtain

$$\begin{array}{c}{\omega }^{1/2}(t_1){\omega }^{1/2}(t_2){\omega }^{1/2}\left(t^{\ast }\right)\hfill \\ \le A{(\frac{1}{t_2-t_1}{\int }_{t_1}^{t_2}{\omega }^2(t)\omega \left(t^{\ast }\right)dt)}^{1/2}\hfill \\ \le A^{3/2}{(\frac{1}{t_2-t_1}{\int }_{t_1}^{t_2}\omega (t)|\frac{1}{t^{\ast }-t}{\int }_t^{t^{\ast }}{\omega }^2(s)ds\left|dt\right)}^{1/2}\hfill \\ \le A^{3/2}{(\frac{1}{t_2-t_1}{\int }_{t_1}^{t_2}{\omega }^3(t)dt)}^{1/6}\hfill \\ \times {\left(\frac{1}{t_2-t_1}{\int }_{t_1}^{t_2}\right|\frac{1}{t^{\ast }-t}{\int }_t^{t^{\ast }}{\omega }^2(s)ds{|}^{3/2}dt)}^{1/3}\hfill \end{array}$$

by hypothesis and Hölder’s inequality. Applying Hardy’s inequality twice gives us

$${\left({\int }_{t_1}^{t_2}\right|\frac{1}{t^{\ast }-t}{\int }_t^{t^{\ast }}{\omega }^2(s)ds{|}^{3/2}dt)}^{2/3}\le 6{({\int }_{t_1}^{t_2}{\omega }^3(t)dt)}^{2/3},$$

and so we obtain

$${\omega }^{1/2}(t_1){\omega }^{1/2}(t_2){\omega }^{1/2}\left(t^{\ast }\right)\le \frac{6^{1/2}{A}^{3/2}}{{(t_2-t_1)}^{1/2}}{({\int }_{t_1}^{t_2}{\omega }^3(t)dt)}^{1/2}.$$

By taking the $2/3$th power, we obtain the desired estimate. □

The following lemma, which is nearly a triviality, generalizes a version of Lemma 2.2 in [11].

Lemma 3.2 Suppose F is nonnegative and continuous on some interval $[a,b]$. For $t\in [a,b]$, we let $\tilde{F}(t):={max}_{[t,b]}F$, and for $\rho >0$, we let

$$E_{\rho }=\{t\in [a,b]:\tilde{F}(t)(b-t)\le \rho \}.$$

Then we have

$${\int }_{E_{\rho }}F(t)dt\le \rho .$$

Proof of Lemma 3.2 Observe that the function $t\mapsto \tilde{F}(t)(b-t)$ is a monotone decreasing function. Let $\rho >0$ be given. Since $b\in E_{\rho }$, $E_{\rho }$ is nonempty. Let $t_{\ast }:=inf{E}_{\rho }$. Then we have $\tilde{F}(t_{\ast })(b-t_{\ast })\le \rho$. From this, we obtain

$$\begin{array}{rcl}{\int }_{E_{\rho }}F(t)dt& =& {\int }_{t_{\ast }}^{b}F(t)dt\\ \le & \tilde{F}(t_{\ast })(b-t_{\ast })=\rho ,\end{array}$$

which finishes the proof. □

Proof of Theorem 2.7 It suffices to prove (2.1) by duality. We may further assume, without loss of generality, $-\mathrm{\infty }<a<b<\mathrm{\infty }$, since a uniform estimate independent of a and b will allow us a suitable limiting argument. For a measurable subset E of either ℝ or ${\mathbb{R}}^2$, we denote the Lebesgue measure and the characteristic function of E by $|E|$ and , respectively. We also write $\gamma (t)=(t,\varphi (t))$.

By a well-known interpolation argument [7, 8], it suffices to show that

holds for measurable sets $E\subset {\mathbb{R}}^2$. In view of the simple identities

where $\overline{\gamma }(t):=(t,\overline{\varphi }(t))$, $\overline{\varphi }(t):=\varphi (a+b-t)$, $\overline{\omega }(t):={({\overline{\varphi }}^{″}(t))}^{1/3}=\omega (a+b-t)\in {\mathcal{E}}_2(A)$, and $\overline{E}:=\{(x_1,x_2):(-x_1,x_2)\in E\}$, it is enough to establish that

(3.2)

holds for measurable sets $E\subset {\mathbb{R}}^2$. To do this, we let $\mathrm{\Delta }:=\{(t_1,t_2):a<t_1<t_2<b\}$. The mapping $\mathrm{\Phi }:\mathrm{\Delta }\to {\mathbb{R}}^2$ given by $\mathrm{\Phi }(t_1,t_2)=\gamma (t_2)-\gamma (t_1)$ is one-to-one and the absolute value of the Jacobian determinant $J(t_1,t_2)$ of Φ is given by

$$J(t_1,t_2)={\varphi }^{\prime }(t_2)-{\varphi }^{\prime }(t_1).$$

Given measurable $\mathrm{\Omega }\subset \mathrm{\Delta }$ and $t_2\in (a,b)$, we apply Lemma 3.2 with

to obtain

where $\tilde{\omega }(t_1;t_2):={max}_{[t_1,t_2]}\omega$. From this, we get

and so

Multiplying by $\omega (t_2)$ and integrating with respect to $t_2$ provides us with

Notice that for $a<t_1<t_2<b$, there exists $t_{\ast }\in [t_1,t_2]$ such that $\tilde{\omega }(t_1;t_2)=\omega (t_{\ast })$. By Lemma 3.1, we have

$$\begin{array}{rcl}\omega (t_1)\omega (t_2)\tilde{\omega }(t_1;t_2)(t_2-t_1)& =& \omega (t_1)\omega (t_2)\omega (t_{\ast })(t_2-t_1)\\ \le & 6A^3{\int }_{t_1}^{t_2}{\omega }^3(t)dt\\ =& 6A^3{\int }_{t_1}^{t_2}{\varphi }^{″}(t)dt\\ =& 6A^3({\varphi }^{\prime }(t_2)-{\varphi }^{\prime }(t_1))\\ =& 6A^{3}J(t_1,t_2),\end{array}$$

which further implies

Letting $\mathrm{\Omega }=\{(t_1,t_2)\in \mathrm{\Delta }:\gamma (t_1)-\gamma (t_2)\in E\}$ and making a change of variables, we obtain the desired estimate (3.2). □