Interpolation on Lebesgue Spaces

ca.classical-analysis, Expository, fa.functional-analysis Add comments

Dec 172011

This is the set of notes I wrote up for the talk I gave at the student analysis seminar on December 1 and December 8.

* * * * *

Let $T:X \to Y$ be an operator between vector spaces satisfying some linearity condition. We fix Banach subspaces $A_0$ and $A_1$ fo $X$ and $B_0$ and $B_1$ of $Y$ . If the restrictions $T|_{A_0}$ and $T|_{A_1}$ are operators into $B_0$ and $B_1$ , respectively, satisfying some norm estimates, then an interpolation theorem tells us that we can often find “intermediate” Banach spaces $(A_t,B_t)$ between $(A_0,B_0)$ and $(A_1,B_1)$ such that $T|_{A_t}$ , for each $t \in [0,1]$ , is a bounded operator into $B_t$ with its norm depending canonically on the norm estimates at the endpoints.

Contents

Lecture 1 – Strong Endpoint Estimates: The theorems of Riesz-Thorin and Stein. If a linear map is bounded as an operator from to and from to , respectively, the the Riesz-Thorin interpolation theorem establishes the boundedness of as an operator from to , where and depend on the exponents and the parameter . As an immediate application, we shall prove the Hausdorff-Young inequality for the Fourier transform and the generalized Young’s inequality. We also mention the variable-parameter version of Riesz-Throin, proven by E. Stein in his 1955 thesis.
Lecture 2 – Weak Endpoint Estimates: The theorems of Marcinkiewicz and Fefferman-Stein. What if an operator just fails to be bounded at the endpoints? In the second part of the talk, we shall consider the weak-type operators, which are bounded as an operator from a Lebesgue space to a weak Lebesgue space. The prototypical examples of such operators are the Hardy-Littlewood maximal function in the theory of differentiation and the Hilbert transform in the theory of singular integrals. We also mention an extension of Stein’s interpolation theorem by C. Fefferman, which allows the endpoint-operator to be bounded from a Lebesgue space to a Hardy space.

Lecture 1. Strong Endpoint Estimates: The theorems of Riesz-Throin and Stein

The first theorem on our list is the interpolation theorem of M. Riesz and Thorin, concerning operators on Lebesgue spaces. The theorem depends on complex-analytic methods for its proof, whence the operators must take complex scalars. The general theorems of its kind, pioneered by Calderón, are known as complex interpolation.

- 1. The Fourier transform -

To motivate the Riesz-Thorin interpolation theorem, we briefly review the $L^1$ and $L^2$ theory of the Fourier transform on Euclidean spaces. For each $f \in L^1(\mathbb{R}^d)$ , we define the Fourier transform $\hat{f}:\mathbb{R}^d \to \mathbb{C}$ by setting

$\displaystyle \hat{f}(\xi) = \int_{\mathbb{R}^d} f(x) e^{-2 \pi i x \cdot \xi} \, dx,$

where $x \cdot \xi$ is the standard dot product of $x$ and $\xi$ . We immediately see that

$\displaystyle \|\hat{f}\|_\infty \leq \|f\|_1,$

which shows that the Fourier transform is a bounded operator from $L^1(\mathbb{R}^d)$ to $L^\infty(\mathbb{R}^d)$ . Straightforward computation yields the following basic properties of the Fourier transform—here we let $f \in L^1(\mathbb{R}^d)$ and $\tau \in \mathbb{R}^d$ :

If $f_\tau(x) = f(x-\tau)$ , then $\widehat{f_\tau} (x) = e^{-2 \pi i \tau \cdot \xi} \hat{f}(\xi)$ .
If $e_\tau(x) = e^{2 \pi i x \cdot \tau}$ , then $\widehat{e_\tau f}(\xi) = \hat{f}(\xi - \tau)$ .
If $\tilde{f}(x) = \overline{f(-x)}$ , then $\widehat{\tilde{f}} = \overline{\hat{f}}$ .

A natural setting for the study of the Fourier transform is the Schwartz space $\mathbb{S}(\mathbb{R}^d)$ , which is the collection of smooth functions on $\mathbb{R}^d$ that are rapidly decreasing. The formal definition is stated succinctly with the aid of multi-indices, which are vectors in $\mathbb{R}^d$ with nonnegative integer components. Each multi-index $\alpha$ comes with the following set of shorthand notations:

$\displaystyle D^\alpha = \frac{\partial^{\alpha_1}}{\partial x_1^{\alpha_1}} \cdots \frac{\partial^{\alpha_1}}{\partial x_d^{\alpha_d}};$

$\displaystyle x^\alpha = x_1^{\alpha_1} \cdots x_d^{\alpha_d};$

$|\alpha| = \alpha_1 + \cdots + \alpha_d.$

We are now ready to state the

Definition. The Schwartz space $\mathscr{S}(\mathbb{R}^d)$ on $\mathbb{R}^d$ is the collection of $\mathscr{C}^\infty$ -functions $\varphi:\mathbb{R}^d \to \mathbb{C}$ such that

$\displaystyle \sup_{x \in \mathbb{R}^d} |x^\alpha D^\beta \varphi(x)| < \infty$

for every pair of multi-indices $\alpha$ and $\beta$ .

We remark that $\mathscr{S}(\mathbb{R}^d)$ contains $\mathscr{C}_0^\infty(\mathbb{R}^d)$ , hence there are plenty of Schwartz functions. In fact, $\mathscr{S}(\mathbb{R}^d)$ is dense in $L^p(\mathbb{R}^d)$ for $1 \leq p < \infty$ . Furthermore, the Fourier transform of a Schwartz function is again a Schwartz function.

An important result in the $L^1$ theory of the Fourier transform is the Fourier inversion theorem. We shall only need a simple case of the theorem: see pp.5-17 of [SW71] for a discussion of the general version.

Theorem 1 (Fourier inversion theorem). If $f \in L^1(\mathbb{R}^d)$ and $\hat{f} \in L^1(\mathbb{R}^d)$ , then

$\displaystyle f(x) = \int_{\mathbb{R}^d} \hat{f}(\xi) e^{2 \pi i \xi \cdot x} \, d\xi$

for almost every $x \in \mathbb{R}^d$ .

Proof. We first note that the Fourier transform of the Gaussican $\Gamma(x) = e^{-\pi |x|^2}$ is again the Gaussian, whence Theorem 1 holds for $\Gamma$ . Setting

$\displaystyle \Gamma_{\varepsilon}(x) = e^{-\pi \varepsilon^2 |x|^2}$

for each $\varepsilon > 0$ , we see that

$\displaystyle \widehat{\Gamma_\varepsilon}(\xi) = \varepsilon^{-d} e^{-\pi \varepsilon^{-2} |\xi|^2},$

whereby the theorem also holds for all $\Gamma_\varepsilon$ .

We also observe that a simple application of the Fubini-Tonelli theorem yields the following multiplication formula for $f,g \in L^1(\mathbb{R}^d)$ :

$\displaystyle \int_{\mathbb{R}^d} \hat{f}(t) g(t) \, dt = \int_{\mathbb{R}^d} f(t) \hat{g}(t) \, dt.$

Lastly, we will need the following standard fact, whose proof can be found, for example, in page 243 of [Fol99] or page 63 of [Ste70].

Lemma 2 (Approximation to the identity).Let $\varphi \in L^1(\mathbb{R}^d)$ and set

$\displaystyle \varphi_\varepsilon(x) = \varepsilon^{-d} \varphi(\varepsilon^{-1} x)$

for each $\varepsilon > 0$ . If $\int_{\mathbb{R}^d} \varphi_\varepsilon(x) \, dx = 0$ for each $\varepsilon >0$ , then

$\displaystyle \lim_{\varepsilon \to 0} \|f * \varphi_\varepsilon - f\|_p = 0$

for every $f \in L^p(\mathbb{R}^d)$ with a fixed $1 \leq p < \infty$ .

We are now ready to prove Theorem 1. Consider the following modification of the inversion formula:

$\displaystyle I_\varepsilon (x) = \int_{\mathbb{R}^d} \hat{f}(\xi) e^{- \pi \varepsilon^2 |xi|^2} e^{2 \pi i \xi \cdot x} \, d\xi$ .

Since $\hat{f} \in L^1(\mathbb{R}^d)$ , the dominated convergence theorem implies that

$\displaystyle \lim_{\varepsilon \to 0} I_\varepsilon(x) = \int_{\mathbb{R}^d} \hat{f}(\xi) e^{2 \pi i \xi \cdot x} \, d \xi$ .

We set $g_\varepsilon(\xi) = e^{-\pi \varepsilon^2 |\xi|^2} e^{2 \pi i \xi \cdot x}$ and note that

$\displaystyle I_\varepsilon (x) = \int \hat{f}(t) g(t) \, dt = \int f(t) \hat{g}(t) \, dt,$

by the multiplication formula. Since $g(\xi) = \Gamma_\varepsilon(\xi) e^{2 \pi i \xi \cdot x}$ , we see that

$\displaystyle \hat{g}(t) = \widehat{\Gamma_\varepsilon}(t-s) = \Gamma^\varepsilon(s-t),$

where $\Gamma^\varepsilon(t) = \varepsilon^{-d} \Gamma(\varepsilon^{-1} t)$ . Observing that

$\displaystyle I_\varepsilon = f * \Gamma^\varepsilon,$

we conclude from Lemma 2 that

$\displaystyle \lim_{\varepsilon \to 0} \|I_\varepsilon - f\|_1 = 0.$

The theorem now follows. $\square$

Note that the inversion theorem can be reformulated as

$\displaystyle \widehat{\hat{f}}(x) = f(-x).$

If $\varphi,\phi \in \mathcal{S}(\mathbb{R}^d)$ , then the inversion theorem implies that

$\displaystyle \int_{\mathbb{R}^d} \varphi(t) \bar{\phi}(t) \, dt = \int_{\mathbb{R}^d} \widehat{\hat{\varphi}}(-t) \bar{\phi}(t) \, dt = \int_{\mathbb{R}^d} \widehat{\hat{\varphi}}(-t) \overline{\phi(-t)} \, dt.$

Therefore,

$\displaystyle \int_{\mathbb{R}^d} \hat{\varphi} \overline{\hat{\phi}} = \int_{\mathbb{R}^d} \widehat{\hat{\varphi}} \tilde{\phi},$

whence by the multiplication formula we obtain

$\displaystyle \int_{\mathbb{R}^d} \varphi \overline{\phi} = \int_{\mathbb{R}^d} \hat{\varphi} \widehat{\tilde{\phi}} = \int_{\mathbb{R}^d} \hat{\varphi} \overline{\hat{\phi}}.$

This implies that the Fourier transform restricted to $\mathcal{S}(\mathbb{R}^d)$ is an isometry in the $L^2$ -norm. Since the Schwartz space $\mathcal{S}(\mathbb{R}^d)$ is dense in $L^2(\mathbb{R}^d)$ , there exists a unique norm-preserving extension of the Fourier transform onto all of $L^2(\mathbb{R}^d)$ . We concede that

$\displaystyle \|\hat{f}\|_2 = \|f\|_2.$

- 2. The Riesz-Thorin interpolation theorem -

We have thus seen that the Fourier transform $\mathscr{F}$ is a bounded linear operator from $L^1(\mathbb{R}^d)$ to $L^\infty(\mathbb{R}^d)$ and from $L^2(\mathbb{R}^d)$ to $L^2(\mathbb{R}^d)$ , each with the operator norm at most 1. Since the natural “intermediate Banach spaces” of $L^1(\mathbb{R}^d)$ and $L^2(\mathbb{R}^d)$ are $L^p(\mathbb{R}^d)$ for $1 < p < 2$ , an interpolation theorem for our example should give us a norm estimate of the Fourier transform operator on $L^p(\mathbb{R}^d)$ .

Note, however, that the $L^1$ Fourier transform and the $L^2$ Fourier transform are, formally speaking, two different operators. We must find a way to describe an operator that takes, in a suitable sense, multiple domains. Taking a cue from how we defined the $L^2$ Fourier transform, we shall consider an operator defined on a collection of functions that is dense in each $L^p$ .

Definition. Let $(X,\mu)$ and $(Y,\nu)$ be $\sigma$ -finite measure spaces and fix a vector space $D$ of $\mu$ -measurable complex-valued functions on $X$ that

contains characteristic functions of sets of finite measure, and
is closed under truncation of $f \in D$ , viz., a $\mu$ -measurable complex-valued function $g$ defined by
$\displaystyle g(x) = \begin{cases} f(x) & \mbox{ if } r_1 < |f(x)| \leq r_2; \\ 0 & \mbox{ otherwise}; \end{cases}$

for some nonnegative real numbers $r_1$ and $r_2$ . Given $1 \leq p,q \leq \infty$ , a linear operator $T$ on $D$ into a vector space of $\nu$ -measurable complex-valued functions on $Y$ is of type $(p,q)$ if there exists a constant $k>0$ such that

$\displaystyle \|Tf\|_q \leq k \|f\|_p$

for all $f \in D \cap L^p(X)$ . The smallest such $k$ is the $(p,q)$ -norm of $T$ .

Since $D$ is dense in $L^p(X)$ , each linear operator $T$ of type $(p,q)$ can be extended uniquely to the operator $T':L^p(X) \to L^q(Y)$ with the same operator norm. With this definition, we can now state the Riesz-Thorin interpolation theorem as follows:

Theorem 3 (Riesz-Thorin). If a linear operator $T$ is of type $(p_0,q_0)$ with $(p_0,q_0)$ -norm $k_0$ and of type $(p_1,q_1)$ with $(p_1,q_1)$ -norm $k_1$ , then $T$ is of type $(p_t,q_t)$ with $(p_t,q_t)$ -norm $k_t \leq k_0^{1-t} k_1^t$ for each $0 \leq t \leq 1$ , where

$\displaystyle p_t^{-1} = (1-t)p_0^{-1} + p_1^{-1};$

$\displaystyle q_t^{-1} = (1-t)q_0^{-1} + tq_1^{-1}.$

We shall need the following lemma, which is the complex-analytic method we have advertised:

Lemma 4 (Hadamard’s three lines lemma). Let $F$ be a holomorphic function in the interior of the closed strip

$\displaystyle S = \{z \in \mathbb{C}: 0 \leq \mathrm{Re} z \leq 1\}$

and a bounded continuous function on $S$ . If $|F(z)| \leq m_0$ on the line $\mathrm{Im} z = 0$ and if $|F(z)| \leq m_1$ on the line $\mathrm{Im} z = 1$ , then, for each $0 \leq t \leq 1$ , we have the inequality

$\displaystyle |F(z)| \leq m_0^{1-t} m_1^t$

on the line $\mathrm{Im} z = t$ .

The proof of the lemma is an instance of a widely-used maximum modulus principle argument, known as the Phragmén–Lindelöf principle.

Proof of Lemma. We define holomorphic functions

$\displaystyle G(z) = \frac{F(z)}{m_0^{1-z}m_1^z} \mbox{ and } G_n(z) = G(z)e^{(z^2-1)/n},$

so that $G_n \to G$ as $n \to \infty$ . We wish to show that $|G(z)| \leq 1$ on $S$ .

We first note that $|G(z)| \leq 1$ on the lines $\mathrm{Im} z = 0$ and $\mathrm{Im} z = 1$ . Moreover, $F$ is bounded above on $S$ , and $m_0^{1-z} m_1^z$ is bounded below on $S$ , hence we have the bound $|G(z)| \leq M$ on $S$ . Noting the inequality

$\displaystyle |G_n(z)| \leq Me^{-\mathrm{Im}(z)^2 / n} e^{(\mathrm{Re}(z)^2-1)/n} \leq Me^{-\mathrm{Im}(z)^2/n}$ ,

we see that $G_n(z)$ converges uniformly to 0 as $|y| \to \infty$ . We can therefore find $y_n > 0$ such that $|G_n(z)| \leq 1$ for all $|\mathrm{Im}(z)| \geq y_n$ and $0 \leq \mathrm{Re}(z) \leq 1$ , whence by the maximum modulus principle we have $|G_n(z)| \leq 1$ on the rectangle

$\displaystyle \{z \in \mathbb{C} : 0 \leq \mathrm{Re}(z) \leq 1 \mbox{ and } -y_n \leq \mathrm{Im}(z) \leq y_n\}$ .

It follows that $|G_n(z)| \leq 1$ on $S$ for each $n \in \mathbb{N}$ , and letting $n \to \infty$ yields the desired result. $\square$

We are now ready to present a proof of the Riesz-Thorin interpolation theorem. We write $p'$ to denote the conjugate exponent of $p$ , so that

$\displaystyle \frac{1}{p} + \frac{1}{p'} = 1.$

The following proof is taken from pp. 181-183 of [SW71]:

Proof of Theorem 3. Fix $0 \leq t \leq 1$ . For notational convenience, we let

$\displaystyle \alpha_0 = \frac{1}{p_0}, \alpha_1 = \frac{1}{p_1}, \alpha = \frac{1}{p_t}, \beta_0 = \frac{1}{q_0}, \beta_1 = \frac{1}{q_1}, \beta = \frac{1}{q_t}.$

With this notation, we set

$\displaystyle \alpha(z) = (1-z)\alpha_0 + z\alpha_1 \mbox{ and } \beta(z) = (1-z)\beta_0 + z\beta_1,$

so that

$\displaystyle \alpha(0) = \alpha_0, \alpha(1) = \alpha_1, \alpha(t) = \alpha, \beta(0) = \beta_0, \beta(1) = \beta_1, \beta(t) = \beta.$

We shall first prove the theorem for simple functions, which evidently belong to $D \cap L^p(X)$ . By the Riesz representation theorem, we have

$\displaystyle \|Tf\|_q = \sup \left| \int_Y (Tf) g \, d\nu \right|$

for each simple function $f$ , where the supremum is taken over all simple functions $g$ of $L^{q'}(Y)$ -norm at most one. Therefore, it suffices to show that

$\displaystyle \left| \int_Y (Tf)g \, d\nu \right| \leq k_0^{1-t}k_1^{t}\|f\|_p$

for each such $g$ . If $\|f\|_p = 0$ , then there is nothing to prove, and so we can assume by renormalization of $f$ that $\|f\|_p = 1$ . We therefore set out to establish $\displaystyle \left| \int_Y (Tf)g \, d\nu \right| \leq k_0^{1-t}k_1^{t}$ for all simple functions $g$ with $\|g\|_{q'}=1$ .

We now suppose that

$\displaystyle f = \sum_{j=1}^m a_j \chi_{E_j} \mbox{ and } g = \sum_{j=1}^n b_k \chi_{F_k}$

are two simple functions satisfying the above conditions. We also assume without loss of generality that $p_t < \infty$ and $q_t > 1$ , so that $\alpha > 0$ and $\beta < 1$ . Write $a_j = |a_j| e^{i\theta_j}$ and $b_k = |b_k| e^{i\varphi_k}$ and set

$\displaystyle f_z = \sum_{j=1}^m |a_j|^{\alpha(z)/\alpha}e^{i\theta_j} \chi_{E_j} \mbox{ and } g_z = \sum_{k=1}^m |b_k|^{(1-\beta(z))/(1-\beta)}e^{i\varphi_k}\chi_{F_k}$

for each $z \in \mathbb{C}$ . Then

$\displaystyle F(z) = \int_Y (Tf_z) g_z \, d\nu$

is an entire function such that

$\displaystyle F(t) = \int_Y (Tf) g \, d\nu.$

We note, in particular, that $F$ is holomorphic in the interior of $S$ and is continuous on $S$ . Since $T$ is linear, we see that

$\displaystyle F(z) = \sum_{j=1}^m\sum_{k=1}^n |a_j|^{\alpha(z)/\alpha} |b_k|^{(1-\beta(z))/(1-\beta)}\left(e^{i(\theta_j + \varphi_k)} \int_Y (T\chi_{E_j}) \chi_{F_k} \right),$

whence $F$ is bounded on $S$ .

We now furnish a bound for $F$ on the lines $\mathrm{Re} z = 0$ and $\mathrm{Re} z = 1$ . Note first that $|F(z)| \leq \|Tf_z\|_{q_0} \|g_z\|_{q'}$ by Hölder’s inequality. Observing the identities $\alpha(z) = \alpha_0 + z(\alpha_1 -\alpha_0)$ and $1-\beta(z) = (1-\beta_0) - z(\beta_1 - \beta_0)$ on the line $\mathrm{Re} z =0$ , we see that

$\displaystyle |f_z|^{p_0} = |e^{i \arg f} |f|^{z (\alpha_1-\alpha_0)/\alpha} |f|^{p/p_0}|^{p_0} = |f|^p;$

$\displaystyle |g_z|^{q_0'} = |e^{i \arg g} |g|^{-z (\beta_1 - \beta_0)/(1-\beta)} |g|^{q'/q_0'}|^{q_0'} = |g|^{q'}.$

Since $T$ is of type $(p_0,q_0)$ , it thus follows that

$\displaystyle \begin{array}{rcl} \displaystyle |F(z)| &\leq& \displaystyle \|Tf_z\|_{q_0} \|g_z\|_{q'} \\ &\leq& \displaystyle k_0 \|f_z\|_{p_0}\|g_z\|_{q_0'} \\ &=& \displaystyle k_0 \left( \int_X |f|^p \, d\mu \right)^{1/p_0} \left( \int_Y |g|^{q_0'} \, d\nu \right)^{1/q_0} \\ &\leq& \displaystyle k_0 \|f\|_p^{p/p_0} \|g\|_q^{q'/q_0'} \\ &\leq& \displaystyle k_0 \end{array}$

on the line $\mathrm{Re} z = 0$ . A similar computation establishes the bound $|F(z)| \leq k_1$ on the line $\mathrm{Re} z = 1$ , whence by Hadamard’s three lines theorem we have the inequality

$\displaystyle |F(z)| \leq k_0^{1-t}k_1^t$

on the line $\mathrm{Re} z = t$ . Setting $z = t$ , we have

$\displaystyle \left|\int_Y (Tf) g \, d\nu\right| = |F(t)| \leq k_0^{1-t} k_1^t,$

which is the desired inequality.

Having established the theorem for simple functions, we now prove the theorem for the general $f \in D \cap L^p(X)$ . To this end, we shall furnish a sequence $(f_n)_{n=1}^\infty$ of simple functions such that

$\displaystyle \lim_{n \to \infty} \|f_n - f\|_p = 0 \mbox{ and } \lim_{n \to \infty} (T f_n)(x) = (Tf)(x),$

for then Fatou’s lemma yields the inequality

$\displaystyle \|Tf\|_q \leq \lim_{n \to \infty} \|Tf_n\|_q \leq \lim_{n \to \infty} k_0^{1-t} k_1^t \|f_n\|_p = k_0^{1-t} k_1^t \|f\|_p.$

Therefore, the task of proving the theorem reduces to finding such a sequence.

We assume without loss of generality that $f \geq 0$ and $p_0 \leq p_1$ . Let $f^0$ be the truncation of $f$ , defined as

$\displaystyle f^0(x) = \begin{cases} f(x) & \mbox{ if } f(x) > 1; \\ 0 & \mbox{ if } f(x) \leq 1; \end{cases}$

and $f^1=f-f^0$ another truncation. $D$ contains all truncations of $f$ , and $(f^0)^{p_0}$ and $(f^1)^{p_1}$ are bounded by $f^p$ , hence $f^0 \in D \cap L^{p_0}(X)$ and $f^1 \in D \cap L^{p_1}(X)$ . We can find a monotonically increasing sequence $(g_m)_{m=1}^\infty$ converging to $f$ , which satisfies

$\displaystyle \lim_{m \to \infty} \|g_m - f\|_p = 0$

by the monotone convergence theorem. If $g_m^0$ and $g_m^1$ are truncations of $g_m$ defined in the same way as $f^0$ and $f^1$ , then we have

$\displaystyle \lim_{m \to \infty} \|g_m^0 - f^0\|_p = \lim_{m \to \infty} \|g_m^1 f^1\|_p = 0.$

Since $T$ is of types $(p_0,q_0)$ and $(p_1,q_1)$ , we have

$\displaystyle \lim_{m \to \infty} \|Tg_m^0 - Tf^0\|_{q_0} = \lim_{m \to \infty} \|Tg_m^1 Tf^1\|_{q_1}= 0.$

We can then find a subsequence of $(T g_m^0)_{m=1}^\infty$ converging almost everywhere to $Tf^0$ , whence we may as well assume that the full sequence converges almost everywhere to $Tf^0$ . Similarly, we can find a subsequence $(g_{m_n})_{n=1}^\infty$ of $(g_m)_{m=1}^\infty$ such that $(T g_{m_n}^1)_{n=1}^\infty$ converges almost everywhere to $Tf^1$ , whence the sequence $(f_n)_{n=1}^\infty$ defined by setting

$\displaystyle f_n = g_{m_n}^0 + g_{m_n}^1$

is the desired sequence. This completes the proof of the Riesz-Thorin interpolation theorem. $\square$

- 3. Applications: two inequalities -

We now return to the task of establishing the $L^p$ -boundedness of the Fourier transform. The Fourier transform operator on $L^1 \cap L^2$ is of type $(1,\infty)$ and of type $(2,2)$ with the norm at most one in each case. It then follows from the Riesz-Thorin interpolation theorem that the Fourier transform operator is of type $(p,p')$ with $(p,p')$ -norm at most one for all $1 \leq p \leq 2$ . We therefore have the following inequality:

Corollary 5 (Hausdorff-Young). If $1 \leq p \leq 2$ , then

$\displaystyle \|\hat{f}\|_{p'} \leq \|f\|_p$

for all $f \in L^p(\mathbb{R}^d)$ .

We also extend the classical inequality of Young, which can be stated as follows:

Theorem 6 (Young). If $l \leq r \leq \infty$ , then

$\displaystyle \|f * g\|_p \leq \|f\|_p \|g\|_1$

for all $f \in L^p[(\mathbb{R}^d)$ and $g \in L^1(\mathbb{R}^d)$ .

This implies that the convolution operator $Tg= f * g$ is of type $(1,p)$ with $(1,p)$ -norm at most $\|f\|_p$ . We also note that Hölder’s inequality yields the estimate

$\displaystyle \|f * g\|_{\infty} \leq \|f\|_{p} \|g\|_{p'},$

whence $T$ is of type $(p',\infty)$ with $(p',\infty)$ -norm at most $\|f\|_p$ . It then follows that $T$ is of type $\left( [(1-t) + tp']^{-1}, [(1-t)p]^{-1} \right)$ with the operator norm at most $\|f\|_p$ for all $t \in [0,1]$ . We therefore have the following inequality:

Corollary 7 (Young). If $1 \leq p,q,r \leq \infty$ such that

$\displaystyle \frac{1}{p} + \frac{1}{q} = 1 + \frac{1}{r},$

then

$\displaystyle \|f*g\|_r \leq \|f\|_p \|g\|_q$

for all $f \in L^p(\mathbb{R}^d)$ and $g \in L^q(\mathbb{R}^d).$

- 4. Interpolation on analytic families of operators -

We note that the proof of Riesz-Thorin was a study of the function

$\displaystyle F(z) = \int_Y (T f_z) g_z \, d\nu,$

where $F(z)$ was holomorphic in virtue of $f_z$ and $g_z$ depending holomorphically on the complex variable $z$ . The two endpoint norm estimates on the operator $T$ gave bounds on the lines $\mathrm{Re}(z) = 0$ and $\mathrm{Re}(z) = 1$ , whence the three lines lemma yielded the interpolated bound on $F(z)$ . The corresponding bound for $T$ was then obtained by a “duality argument”, using the Riesz representation theorem.

What if we add the $z$ to the operator $T$ and study

$\displaystyle G(z) = \int_Y (T_z f_z) g_z \, d\nu$

instead? If the operator “varies holomorphically” over the strip

$\displaystyle S = \{z \in \mathbb{C}: 0 \leq \mathrm{Re} z \leq 1\}$

so that $G(z)$ is holomorphic in $S$ , then we could argue as above to obtain an interpolation theorem that allows for varying operators. A collection of such operators is called an analytic family of operators, following the traditional practice of referring to complex analysis as the theory of analytic functions.

Formally, we suppose that $T_z$ , for each $z \in S$ , is a linear operator on the space of simple functions in $L^1(X)$ into measurable functions on $Y$ such that $(T_zf)g \in L^1(Y)$ for all simple functions $f$ in $L^1(X)$ and all simple functions $g$ in $L^1(Y)$ . The family $\{T_z : z \in S\}$ of operators is said to be admissible if the function

$\displaystyle G(z) = \int_Y (T_zf) g \, d\nu$

is holomorphic in the interior of $S$ , continuous on $S$ , and admits a constant $a < \pi$ such that

$\displaystyle \sup_{z \in S} e^{-a |\mathrm{Im}(z)|} \log \left| \int_Y (T_z f) g \, d\nu \right| < \infty.$

The interpolation theorem of Stein can then be stated as follows:

Theorem 8 (Stein). If $\{T_z : z = x+iy \in S\}$ is an admissible family of linear operators with the norm estimates

$\displaystyle \|T_{iy}f\|_{q_0} M_0(y) \|f\|_{p_0} \mbox{ and } \|T_{1+iy} f\|_{q_1} \leq M_1(y) \|f\|_{p_1}$

for all simple functions $f$ in $L^1(X)$ such that $1 \leq p_j,q_j \leq \infty$ and $M_j(y)$ satisfies

$\displaystyle \sup{-\infty < y < \infty} e^{-b|y|} \log M_j(y) < \infty$

for all $f$ and some $b < \pi$ , then, for $t \in [0,1]$ , there exists a constant $M_t$ such that

$\displaystyle \|T_t f\|_{q_t} \leq M_t \|f\|_{p_t}$

for all such simple functions f, where

$\displaystyle p_t^{-1} = (1-t)p_0^{-1} + p_1^{-1};$

$\displaystyle q_t^{-1} = (1-t)q_0^{-1} + tq_1^{-1}.$

In fact, we can take

$\displaystyle M_t = \exp \left( \frac{1}{2} \sin(\pi t) \int_{-\infty}^\infty \frac{\log M_0(y)}{\cosh \pi y - \cos \pi t} + \frac{ \log M_1(y)}{\cosh \pi y + \cos \pi t} \, dy \right).$

We will not discuss the formal proof here: see, for example, pp.206-209 of [SW71]. See also Terry Tao’s blog post, which offers a detailed discussion of the theorem.

Lecture 2. Weak Endpoint Estimates: The theorems of Marcinkiewicz and Fefferman-Stein

We now shift gears to study the classical real interpolation theorem of Marcinkiewicz. Unlike Stein’s interpolation theorem, the Marcinkiewicz interpolation theorem does not allow the operator to vary. The improvement, however, is that we may now interpolate with weaker endpoint conditions—the extent of which we shall make precise in what follows. Due to the real-variable nature of the proof, we may take the scalars to be either real or complex.

- 1. The Hardy-Littlewood maximal function -

Recall the differentiation theorem of Lebesgue, which states that we have the almost-everywhere pointwise convergence

$\displaystyle \lim_{\delta \to 0} \frac{1}{m(B_\delta(x))} \int_{B_\delta(x)} f(y) \, dy = f(x)$

for all $f \in L^1_{\mbox{loc}}(\mathbb{R}^d)$ . Crucial in establishing this theorem is the Hardy-Littlewood maximal function

$\displaystyle (Mf)(x) = \sup_{\delta > 0} \frac{1}{m(B_\delta(x))} \int_{B_\delta(x)}$

and the corresponding weak-type inequality

$\displaystyle m(\{x \in \mathbb{R}^d : Mf(x) > \alpha\}) \leq \frac{A}{\alpha} \|f\|_1,$

where $A$ is a constant independent of $\alpha$ and $f$ . This inequality cannot be improved for general $L^1(\mathbb{R}^d)$ -functions. Consider, for example, the action of the maximal operator on $(A/2) \chi_{(0,1)}$ , a one-dimensional characteristic function. Since

$\displaystyle m(\{x : M\chi_{(0,1)}(x) > \alpha\}) = \begin{cases} A/2 & \mbox{ if } \alpha < 1; \\ 0 & \mbox{ if } \alpha \geq 1; \end{cases}$

setting $\alpha = 2$ yields the equality.

Surprisingly, the weak $L^1$ -bound, combined with the $L^\infty$ -bound

$\displaystyle \|Mf\|_\infty \leq \|f\|_\infty,$

is sufficient to guarantee that the maximal operator can be extended to a bounded operator $M:L^p \to L^p$ for all $1 < p < \infty$ . The proof of this fact can be generalized considerably, and we are led to yet another interpolation theorem—the Marcinkiewicz interpolation theorem.

- 2. The Marcinkiewicz interpolation theorem -

Before we state the interpolation theorem, let us introduce a way to measure the size of a measurable function in a global manner.

Definition. The distribution function of a measurable function $f$ on a $\sigma$ -finite measure space is

$\displaystyle d_f(\alpha) = \mu(\{x : |f(x)| > \alpha\})$ .

With this notation, we can write the weak-type inequality of the Hardy-Littlewood maximal function as

$\displaystyle d_{Mf}(\alpha) \leq \frac{A\|f\|_1}{\alpha}.$

In general, we define weak-type operators as follows:

Definition. Let $(X,\mu)$ and $(Y, \nu)$ be $\sigma$ -finite measure spaces and fix a vector space $D$ of $\mu$ -measurable functions on $X$ that

contains characteristic functions of sets of finite measure, and
is closed under truncation of $f \in D$ .

An operator $T$ on $D$ into a vector space of $\nu$ -measurable functions is subadditive if

$\displaystyle |T(f_1 + f_2)(x)| \leq |(T f_1)(x)| + |(T f_2)(x)|$

for almost every $x \in Y$ , whenever $f_1, f_2 \in D$ . Given $1 \leq p\leq \infty$ and $1 \leq q < \infty$ , a subadditive operator $T$ is of weak type $(p,q)$ if there exists a constant $k > 0$ such that

$\displaystyle d_{Tf}(\alpha) \leq \left( \frac{k \|f\|_p}{\alpha} \right)^q$

for all $f \in D \cap L^p(X)$ . If $q = \infty$ , the above inequality is replaced by the condition

$\displaystyle \|Tf\|_q \leq k\|f\|_p.$

The smallest such $k$ is the weak $(p,q)$ -norm of $T$ .

Of course, $D$ is dense in $L^p(X)$ , and so each subadditive operator $T$ of weak type $(p,q)$ can be extended uniquely to a “bounded” operator on $L^p(X)$ with the same norm. What, then, is the target space of $T$ ? Evidently, we need a larger space than the classical Lebesgue spaces. Natural substitutes in this case are the weak Lebesgue spaces, which we define below:

Definition. For $0 < p < \infty$ , the weak Lebesgue space $L^{p,\infty}(X)$ is the collection of all $\mu$ -measurable functions $f$ such that

$\displaystyle \|f\|_{L^{p,\infty}} = \inf \left\{ k > 0 : d_f(\alpha) \leq \left( \frac{k}{\alpha} \right)^p \mbox{ for all } \alpha > 0\right\}$

is finite.

We remark that $\|\cdot\|_{L^{p,\infty}}$ is not a norm. In fact, the weak Lebesgue spaces are merely quasi-normed vector spaces. A subadditive operator $T$ of weak type $(p,q)$ can nevertheless be extended uniquely to an operator $T:L^p(X) \to L^{q,\infty}(Y)$ , which is bounded in the sense that $\|T f\|_{L^{q,\infty}} \leq k \|f\|_p$ for all $f \in L^p(X)$ , where $k$ is the weak $(p,q)$ -norm of $T$ .

We also note that the definition of linear operators of type $(p,q)$ can be extended in an obvious manner to include subadditive operators. Every operator of type $(p,q)$ is then of weak type $(p,q)$ . The interpolation theorem of Marcinkiewicz can now be stated as follows:

Theorem 9 (Marcinkiewicz). Fix $p_0,p_1,q_0,q_1$ such that

$1 \leq p_0 \leq q_0 \leq \infty$ ,
$1 \leq p_1 \leq q_1 \leq \infty$ ,
$p_0 < p_1$ , and
$q_0 \neq q_1$ .

If a subadditive operator $T$ is of weak type $(p_0,q_0)$ with weak $(p_0,q_0)$-norm $k_0$ and of weak type $(p_1,q_1)$ with weak $(p_1,q_1)$ -norm $k_1$ , then $T$ is of type $(p_t,q_t)$ for each $t \in [0,1]$ , where

$\displaystyle p_t^{-1} = (1-t)p_0^{-1} + p_1^{-1};$

$\displaystyle q_t^{-1} = (1-t)q_0^{-1} + tq_1^{-1}.$

In the interest of time, we will not present a proof of the interpolation theorem. Instead, we merely point out a few key features of the theorem. We first note that, unlike Riesz-Thorin, there is no nice estimate for the interpolated bound. It is known, for example, that the $(p_t,q_t)$ -norm $k_t$ satisfies the inequality

$\displaystyle k_t \leq 2 \left( \frac{p_t}{p_t - p_0} + \frac{p_t}{p_1 - p_0} \right)^{1/p_t} k_0^{\frac{\frac{1}{p_t} - \frac{1}{p_1}}{\frac{1}{p_0}-\frac{1}{p_1}}} k_1^{ \frac{ \frac{1}{p_0} - \frac{1}{p_t} }{ \frac{1}{p_0} - \frac{1}{p_1} } },$

provided that $p_0 = q_0$ and $p_1 = q_1$ . see, for example, pp. 31-34 of [Gra10] for a proof of this bound.

The classical proof of the theorem makes use of Hardy’s inequality and relies heavily on the notion of non-increasing rearrangement of a measurable function $h$ on $\mathbb{R}^n$ that preserves the distribution function and the $L^p$ -norm of $h$. Such a rearrangement is given by

$\displaystyle h^*(t) = \inf\{\alpha : d_h(\alpha) \leq t\}$

on $(0,\infty)$ . Appendix B of [Ste70] contains a proof of the interpolation theorem for $\mathbb{R}^d$ .

The proper setting for the weak-type endpoint estimates is theLorentz space $L^{p,q}(X,\mu)$ , which is a collection of $\mu$ -measurable functions on $X$ such that the Lorentz norm

$\displaystyle \|f\|_{L^{p,q}} = \begin{cases} \displaystyle \left( \int_0^\infty \left( t^{1/p} f^*(t) \right)^q \frac{dt}{t} \right)^{1/q} & \mbox{ if } q < \infty; \\ \displaystyle \sup_{t > 0} t^{1/p} f^*(t) & \mbox{ if } q = \infty; \end{cases}$

is finite—here $0 < p,q \leq \infty$. The off-diagonal Marcinkiewicz interpolation theorem, due to Zygmund, provides an interpolation result over Lorentz spaces and can be found in Chapter 5, section 3 of [SW71]. An improvement of this result, due to Grakafos, can be found in $\S1.4.4$ of [Gra10].

- 3. Application: the Hilbert transform -

We now apply the interpolation theorem to the study of the Hilbert transform, which is, in a sense, the only singular integral operator in one dimension. First, a definition:

Definition. The Hilbert transform of $f \in \mathscr{S}(\mathbb{R})$ is

$\displaystyle (Hf)(x) = \frac{1}{\pi} \lim_{\varepsilon \to 0} \int_{|y| \geq \varepsilon} \frac{f(x-y)}{y} \, dy = \frac{1}{\pi} \lim_{\varepsilon \to 0} \int_{|y| \geq \varepsilon} \frac{f(y)}{x-y} \, dy.$

If we set $K(x) = 1/x$ , then the Hilbert transform can be written as a convolution operator

$\displaystyle (Hf)(x) = (f * K)(x)$ .

In the sense of tempered distributions, the Fourier transform of $y \mapsto 1/\pi y$ is $\xi \mapsto - i \mathrm{sgn}(\xi)$ , and so

$\displaystyle \widehat{Hf}(\xi) = - \mathrm{sgn}(\xi) \hat{f}(\xi).$

It thus follows from Plancherel’s theorem that

$\displaystyle \|Hf\|_2 = \|\widehat{Hf}\|_2 = \|-i \mathrm{sgn}(\xi) \hat{f}(\xi)\|_2 = \|\hat{f}\|_2 = \|f\|_2,$

and so the Hilbert transform is of type (2,2).

It can also be shown that the Hilbert transform is of weak type (1,1). The proof, which can be found in pp. 187-188 of [SW71], requires a basic knowledge of Fourier analysis on tempered distributions and will not be reproduced here. We assume the weak bound here and apply the Marcinkiewicz interpolation theorem to obtain the interpolated bound

$\displaystyle \|Hf\|_p \leq A_p \|f\|_p$

for $1 < p \leq 2$ .

Using the above result, we can obtain, “by duality”, the $L^p$ -bound for all $2 < p < \infty$ . Recall that if $\psi \in L^1_{\mbox{loc}}(\mathbb{R})$ such that

$\displaystyle \sup_{ \substack{ \varphi \in \mathscr{C}_c(\mathbb{R}) \\ \|\varphi\|_{p'} \leq 1 } } \left| \int_{-\infty}^\infty \psi(x) \varphi(x) \, dx \right| = A < \infty,$

then $\psi \in L^p(\mathbb{R})$ and $\|\psi\|_p = A$ . We fix $2 < p < \infty$ , $f \in L^1(\mathbb{R}) \cap L^p(\mathbb{R})$ , and $\varphi \in L^{p'}(\mathbb{R})$ such that $\|\varphi\|_{p'} \leq 1$ . Since $latex K \in latex L^2(\mathbb{R})$, the double integral

$\displaystyle \int_{-\infty}^\infty \int_{-\infty}^\infty K(x-y) f(y) \varphi(x) \, dx \, dy$

converges absolutely. Therefore, we may rewrite the above integral as the iterated integral

$\displaystyle I = \int_{-\infty}^\infty f(y) \left( \int_{-\infty}^\infty K(x-y) \varphi(x) \, dx \right) \, dy.$

Using the bound for $p' \in (1,2)$ —with the kernel $K(-x)$ instead of $K(x)$ but with the same constant $A_{p'}$ —we can check that $\int K(x-y)\varphi(x) \, dx$ belongs to $L^p(\mathbb{R})$ . Furthermore, its $L^{p'}$ norm is bounded by $A_{p'}\|\varphi\|_{p'}$ , which is at most $A_{p'}$ . By Hölder’s inequality, we have

$\displaystyle \left| \int_{-\infty}^\infty (Hf)(x) \varphi(x) \, dx \right| = |I| \leq A_{p'}\|f\|_p,$

whence taking the supremum over all $\varphi$ yields

$\displaystyle \|Hf\|_p \leq A_{p'} \|f\|_p.$

The duality argument presented above is quite general and can be applied to a large class of operators known as singular integrals. Indeed, we have the following theorem:

Theorem 10 (Calderón–Zygmund). Let $K \in L^2(\mathbb{R}^d)$ . If $|\hat{K}(\xi)| \leq B$ for almost every $\xi \in \mathbb{R}^d$ and $|\nabla K(x)| \leq B/|x|^{d+1}$ , then the operator

$\displaystyle (Tf)(x) = \int_{\mathbb{R}^d} K(x-y) f(y) \, dy$

is of type $(p,p)$ for all $1 < p < \infty$ . The $(p,p)$ -norm of $T$ only depends on p, B, and d; in particular, it does not depend on $\|K\|_2$ .

The idea of the proof is as follows. The weak-type (2,2) bound is established via a Plancherel argument, and the famous Calderón–Zygmund decomposition is used to establish the weak-type (1,1) bound. Marcinkiewicz establishes the interpolated bound between (1,1) and (2,2), and a duality argument gives a bound for the rest. For a detailed discussion, see Chapter 2 of [Ste70].

- 4. Interpolation on Hardy spaces: $H^1$ and $BMO$ -

The idea of allowing for weaker endpoint estimates can be applied in the context of complex interpolation as well. As a concluding remark, we mention here the Fefferman’s extension of Stein’s interpolation theorem, which allows for varying operators. The natural substitutes for the Lebesgue spaces in this setting are the (real-variable) Hardy spaces, which behave much nicer under the action of various operators used frequently in harmonic analysis. Formally, the Hardy space $H^p(\mathbb{R}^d)$ is a collection of tempered distributions $f$ such that, for some $\Phi \in \mathscr{S}(\mathbb{R}^d)$ with $\int \Phi = 1$ , the maximal function

$\displaystyle (M_\Phi f)(x) = \sup_{t > 0} |(f*\Phi)(x)|$

is in $L^p(\mathbb{R}^d)$ . It turns out that $H^p(\mathbb{R}^d) = L^p(\mathbb{R}^d)$ for $1 < p < \infty$ .

Note, however, that $H^1(\mathbb{R}^d) \supseteq L^1(\mathbb{R}^d)$ . In particular, the Hilbert transform maps $L^1(\mathbb{R})$ into $H^1(\mathbb{R})$ . The interpolation theorem, as was introduced in [FS72], replaces the $L^1$ bound in Stein’s interpolation theorem with an $H^1$ bound. By a similar duality argument as the one given above, this interpolation theorem can be extended to $L^p$ for all $p>2$ . The key fact is that the dual of $H^1$ is the John-Nirenberg space of bounded mean oscillation (BMO)—this was one of the main theorems in [FS72].

We will not discuss the interpolation theory on Hardy spaces any further. See Chapters 3 and 4 of [Ste93] for a detailed exposition of real-variable Hardy-space theory: a Fefferman-Stein type interpolation theorem is proved in Chapter 4, section 5.

References

[FS72] Charles Fefferman and Elias M. Stein, “ $H^p$ spaces of several variables”, Acta Math. 129 (1972), no. 3-4, 137-193.
[Fol99] Gerald B. Folland, Real Analysis: Modern Techniques and Their Applications, second ed., John Wiley & Sons, 1999.
[Gra10] Loukas Grafakos, Classical Fourier Analysis, Springer, 2010.
[SS05] Elias M. Stein and Rami Shakarchi, Real Analysis: Measure Theory, Integration, and Hilbert Spaces, Princeton University Press, 2005.
[SS11] Elias M. Stein and Rami Shakarchi, Functional Analysis: Introduction to Further Topics in Analysis, Princeton University Press, 2011.
[Ste70] Elias M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, 1970.
[Ste93] Elias M. Stein, Harmonic Analysis, Princeton University Press, 1993.
[SW71] Elias M. Stein and Guido Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton University Press, 1971.
[Tao11] Terence Tao, “Stein’s interpolation theorem“, What’s new (http://terrytao.wordpress.com/), May 3, 2011.
[Wol03] Thomas H. Wolff, Lectures on Harmonic Analysis, American Mathematical Society, 2003.

Mark H. Kim

Interpolation on Lebesgue Spaces

Lecture 1. Strong Endpoint Estimates: The theorems of Riesz-Throin and Stein

Lecture 2. Weak Endpoint Estimates: The theorems of Marcinkiewicz and Fefferman-Stein

References

Leave a Reply Cancel reply

Recent Blog Posts

Recent Comments

Categories

Blogroll

Internets, The

Mathematics

Mark H. Kim

Interpolation on Lebesgue Spaces

Lecture 1. Strong Endpoint Estimates: The theorems of Riesz-Throin and Stein

Lecture 2. Weak Endpoint Estimates: The theorems of Marcinkiewicz and Fefferman-Stein

References

Leave a Reply Cancel reply

Recent Blog Posts

Recent Comments

Categories

Tags

Blogroll

Internets, The

Mathematics