# On the equivalence between Weak BMO and the space of derivatives of the
Zygmund class

Eddy Kwessi111Department of Mathematics, Trinity University, San Antonio, TX,
USA<EMAIL_ADDRESS>
###### Abstract

In this paper, we will discuss the space of functions of weak bounded mean
oscillation. In particular, we will show that this space is the dual space of
the special atom space, whose dual space was already known to be the space of
derivative of functions (in the sense of distribution) belonging to the
Zygmund class of functions. We show in particular that this proves that the
Hardy space $H^{1}$ strictly contains the space special atom space.

Keywords: BMO, Derivative, Distributions, Zygmund Class

AMS Subject Classification: 42B05, 42B30, 30B50, 30E20

##  1 Introduction

The space of functions of bounded means oscillation has taken central stage in
the mathematical literature after the work of Charles Fefferman [4] where he
showed that it is the dual space of the real Hardy space $\mathbb{H}^{1}$, a
long sought after result. Right after, Ronald Coifman [2] showed this result
using a different method. Essentially, he showed that $\mathbb{H}^{1}$ has an
atomic decomposition. De Souza [3] showed there is a subset $B^{1}$ of
$\mathbb{H}^{1}$, formed by special atoms that is contained in
$\mathbb{H}^{1}$. This space $B^{1}$ has the particularity that it contains
some functions whose Fourier series diverge. The question of whether $B^{1}$
is equivalent to $\mathbb{H}^{1}$ was never truly answered explicitly but it
was always suspected that the inclusion $B^{1}\subset\mathbb{H}^{1}$ was
strict, that is, there must be at least one function in $\mathbb{H}^{1}$ that
is not in $B^{1}$. However, such a function had never being constructed nor
given. Since the dual space $(\mathbb{H}^{1})^{*}$ of $\mathbb{H}^{1}$ is
$BMO$ and $B^{1}\subset\mathbb{H}^{1}$, it follows that the dual space
$(B^{1})^{*}$ of $B^{1}$ must be a superset of $BMO$. A natural superset
candidate of $BMO$ is therefore the space $BMO^{w}$ since $BMO\subset
BMO^{w}$. So essence, that $BMO^{w}$ is the dual of $B^{1}$ would also prove
that $B^{1}\subset\mathbb{H}^{1}$ with a strict inclusion. Moreover, it was
already proved that $(B^{1})^{*}\cong\Lambda_{*}^{\prime}$, where
$\Lambda_{*}^{\prime}$ is the space of derivative (in the sense of
distributions) of functions in the Zygmund class $\Lambda_{*}$, see for
example [3] and [9]. Therefore, if $(B^{1})^{*}\cong BMO^{w}$, then by
transition we would have $\Lambda^{\prime}_{*}\cong BMO^{w}$.
Henceforth, we will adopt the following notations:
$\mathbb{D}=\left\\{z\in\mathbb{C}:\left|z\right|<1\right\\}$ is the open unit
disk and let $\mathbb{T}=\left\\{z\in\mathbb{C}:\left|z\right|=1\right\\}$ is
the unit sphere. For an integrable function $f$ on a measurable set $A$, and
the Lebesgue measure $\lambda$ on $A$, we will write
$\displaystyle\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{A}f(\xi)d\lambda(\xi):=\frac{1}{\lambda(A)}\int_{A}f(\xi)d\lambda(\xi)$.
We will start by recalling the necessary definitions and important results.
The interested reader can see for example [5] for more information.

######  Definition 1.

Let $0<p<\infty$ be a real number. The Hardy Space
$\mathbb{H}^{p}:=\mathbb{H}^{p}(\mathbb{D})$ is the space of holomorphic
functions $f$ defined on $\mathbb{D}$ and satisfying

$\left\|f\right\|_{\mathbb{H}^{p}}:=\sup_{0\leq
r<1}\left(\frac{1}{2\pi}\int_{0}^{2\pi}\left|f(re^{i\xi})\right|^{p}d\lambda(\xi)\right)^{\frac{1}{p}}<\infty\;.$

Let $\displaystyle f\in L^{1}_{\mbox{loc}}(\mathbb{R}^{n}),Q$ be a hypercube
in $\mathbb{R}^{n}$, and $\lambda$ be the Lebesgue measure on $\mathbb{R}^{n}$
for some $n\in\mathbb{N}$.
Put

$\displaystyle f^{\\#}_{Q}=\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{Q}f(\xi)d\lambda(\xi),\quad
f_{Q}=\left|\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{Q}f(\xi)d\lambda(\xi)\right|\;.$

For $\displaystyle f\in L^{1}_{\mbox{loc}}(\mathbb{R}^{n})$ and
$x\in\mathbb{R}^{n}$, we define

$\displaystyle\displaystyle M^{\\#}(f)(x)$ $\displaystyle=$
$\displaystyle\sup_{Q\ni x}\;\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{Q}\left|f(\xi)-f^{\\#}_{Q}\right|d\lambda(\xi)$
(1.1) $\displaystyle M(f)(x)$ $\displaystyle=$ $\displaystyle\sup_{Q\ni
x}\;\left|\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{Q}[f(\xi)-f_{Q}]d\lambda(\xi)\right|$ (1.2)
$\displaystyle mf(x)$ $\displaystyle=$ $\displaystyle\sup_{Q\ni x}\;f_{Q}\;.$
(1.3)

where the supremum is taken over all hypercubes $Q$ containing $x$. Now we can
define the space of functions of bounded mean oscillation and its weak
counterpart.

######  Definition 2.

The space of functions of bounded mean oscillation is defined as the space of
locally integrable functions $f$ for which the operator $M^{\\#}$ is bounded,
that is,

$BMO(\mathbb{R}^{n})=\left\\{f\in
L^{1}_{\mbox{loc}}(\mathbb{R}^{n}):M^{\\#}(f)\in
L^{\infty}(\mathbb{R}^{n})\right\\}\;.$

We can endowed $BMO(\mathbb{R}^{n})$ with the norm

$\left\|f\right\|_{BMO(\mathbb{R}^{n})}=\left\|M^{\\#}(f)\right\|_{\infty}=M^{\\#}(f)(x)\;.$

The space of functions of weak bounded mean oscillation is defined as the
space of locally integrable functions $f$ for which the operator $M$ is
bounded, that is,

$BMO^{w}(\mathbb{R}^{n})=\left\\{f\in L^{1}_{\mbox{loc}}(\mathbb{R}^{n}):Mf\in
L^{\infty}(\mathbb{R}^{n})\right\\}\;.$

######  Remark 1.1.

It follows from the above definitions that $BMO(\mathbb{R}^{n})\subseteq
BMO^{w}(\mathbb{R}^{n})$.

Let us recall the definition of the space of functions of vanishing mean
oscillation $VMO(\mathbb{R}^{n})$ and introduce the space of functions of weak
vanishing mean oscillations $VMO^{w}(\mathbb{R}^{n})$.

######  Definition 3.

Let $f\in L^{1}_{loc}(\mathbb{R}^{n})$.

$\mbox{ $f\in VMO(R^{n})$ if }\quad\lim_{\lambda(Q)\to
0}\;\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{Q}\left|f(\xi)-f_{Q}\right|d\lambda(\xi)=0\;.$
$\mbox{$f\in VMO^{w}(\mathbb{R}^{n})$ if }\quad\lim_{\lambda(Q)\to
0}\;\left|\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{Q}[f(\xi)-f_{Q}]d\lambda(\xi)\right|=0\;.$

######  Remark 1.2.

It follows from the above definition that $VMO(\mathbb{R}^{n})$ is a subspace
of $VMO^{w}(\mathbb{R}^{n})$ which is itself a subspace of
$BMO^{w}(\mathbb{R}^{n})$. We will show (see Theorem 2.8 below) that $VMO^{w}$
is in fact closed subspace of $BMO^{w}$.

Henceforth, $BMO(\mathbb{R}^{n}),BMO^{w}(\mathbb{R}^{n})$, and
$VMO^{w}(\mathbb{R}^{n})$ will simply be referred to as $BMO,BMO^{w}$, and
$VMO^{w}$.

Now we consider $A(\mathbb{D})$ as the space of analytic functions defined on
the unit disk $\mathbb{D}$. Following the work of Girela in [6] on the space
of analytic functions of bounded means oscillations, we introduce their weak
counterpart. Before, we recall that the Poisson Kernel is defined as

$\displaystyle
P_{r}(\theta)=\mbox{Re}\left(\frac{1+re^{i\theta}}{1-re^{i\theta}}\right)\;.$

######  Definition 4.

The space of analytic functions of bounded mean oscillation is defined as

$\displaystyle BMOA(\mathbb{D})=\left\\{F\in A(\mathbb{D});\exists f\in
BMO(\mathbb{T}):F(re^{i\theta})=\frac{1}{2\pi}\int_{\mathbb{T}}P_{r}(\theta-\xi)f(e^{i\xi})d\lambda(\xi)\right\\}\;.$

We can endowed $BMO(\mathbb{D})$ with the norm

$\left\|F\right\|_{BMOA(\mathbb{D})}:=\left|F(0)\right|+\sup_{\underset{\theta\in\mathbb{T}}{0\leq
r<1}}\;\left(\frac{1}{2\pi}\int_{\mathbb{T}}P_{r}(\theta-\xi)\left|f(e^{i\xi})-F(re^{i\theta})\right|d\lambda(\xi)\right)<\infty\;.$

In other words, $BMOA(\mathbb{D})$ is the space of Poisson integrals of
functions in $BMO(\mathbb{T})$.

We can now define the space $BMOA^{w}$ of analytic function of weak bounded
mean oscillation.

######  Definition 5.

An analytic function $F$ on $\mathbb{D}$ is said to be of weak bounded mean
oscillation if there exists $f\in BMO^{w}(\mathbb{T})$ such that

$F(re^{i\theta})=\frac{1}{2\pi}\int_{\mathbb{T}}P_{r}(\theta-\xi)f(e^{i\xi})d\lambda(\xi)\;.$

We endow $BMOA^{w}(\mathbb{D})$ with the norm

$\left\|F\right\|_{BMOA^{w}(\mathbb{D})}:=\left|F(0)\right|+\sup_{\underset{\theta\in\mathbb{T}}{0\leq
r<1}}\;\left(\frac{1}{2\pi}\left|\int_{\mathbb{T}}P_{r}(\theta-\xi)\left[f(e^{i\xi})-F(re^{i\theta})\right]d\lambda(\xi)\right|\right)<\infty\;.$

We recall the definition of special atom space $B^{1}$, see [8] .

######  Definition 6.

For $n\geq 1$, we consider the hypercube of $\mathbb{R}^{n}$ given as
$\displaystyle J=\prod_{j=1}^{n}[a_{j}-h_{j},a_{j}+h_{j}]$ where $a_{j},h_{j}$
are real numbers with $h_{j}>0$. Let $\phi\in L^{1}(J)$ with
$\displaystyle\phi(J)=\int_{J}\phi(\xi)d\lambda(\xi)$ .
The special atom (of type 1) is a function $b:I\subseteq J\to\mathbb{R}$ such
that

$\displaystyle b(\xi)$ $\displaystyle=$ $\displaystyle 1\leavevmode\nobreak\
\textrm{on $J\backslash I$ or }$ $\displaystyle b(\xi)$ $\displaystyle=$
$\displaystyle\dfrac{1}{\phi(I)}\left[\chi_{{}_{R}}(\xi)-\chi_{{}_{L}}(\xi)\right]\;,$

where

  * •

$\displaystyle R=\bigcup_{j=1}^{2^{n-1}}I_{i_{j}}$ for some
$i_{1},i_{2},\cdots,i_{2^{n-1}}\in\left\\{1,2,\cdots,2^{n}\right\\}$ with
$i_{1}<i_{2}<\cdots<i_{2^{n-1}}$ and $L=I\backslash R$ .

  * •

$\left\\{I_{1},I_{2},\cdots,I_{2^{n}}\right\\}$ is the collection of sub-cubes
of $I$, cut by the hyperplanes $x_{1}=a_{1},x_{2}=a_{2},\cdots,x_{n}=a_{n}$.

  * •

$\chi_{A}$ represents the characteristic function of set $A$ .

######  Definition 7.

The special atom space is defined as

$B^{1}=\left\\{f:J\to\mathbb{R};f(\xi)=\sum_{n=0}^{\infty}\alpha_{n}b_{n}(\xi);\sum_{n=0}^{\infty}\left|\alpha_{n}\right|<\infty\right\\}\;,$

where the $b_{n}$’s are special atoms of type 1.
$B^{1}$ is endowed with the norm
$\displaystyle\left\|f\right\|_{B^{1}}=\inf\sum_{n=0}^{\infty}\left|\alpha_{n}\right|,\quad\mbox{where
the infimum is taken over all representations of $f$}\;.$

(a) | (b)
---|---
|
(c) |
|

Figure 1: Illustration of the the special atom, for $n=1$ in (a) , $n=2$ in
(b), and $n=3$ in (c).

Now we define the Zygmund class of functions.

######  Definition 8.

Let $k\in\mathbb{N}$. A function $f$ is said to be in the Zygmund class
$\Lambda_{*}^{k}(\mathbb{R}^{n})$ of functions of order $k$ if $f\in
C^{k-1}(\mathbb{R}^{n})$ and

$\left\|f\right\|_{\Lambda_{*}^{k}(\mathbb{R}^{n})}=\sum_{\left|\alpha\right|=k}\sup_{x,h}\;\frac{\left|\partial^{\alpha}f(x+h)+\partial^{\alpha}f(x-h)-2\partial^{\alpha}f(x)\right|}{\left|h\right|}<\infty\;.$

In particular, for $k=1$, we have
$\Lambda_{*}:=\Lambda_{*}^{1}(\mathbb{R}^{n})$, and hence

$\Lambda_{*}=\left\\{f\in
C^{0}(\mathbb{R}^{n}):\left\|f\right\|_{\Lambda_{*}}:=\sup_{x,h>0}\;\frac{\left|f(x+h)+f(x-h)-2f(x)\right|}{2h}<\infty\right\\}\;.$

One important note about the space $\Lambda_{*}$ is that it contains the so-
called Weierstrass functions that are known to be continuous everywhere but
nowhere differentiable. Therefore, the space $\Lambda_{*}^{k}$ is the space of
derivatives of functions in $\Lambda_{*}^{k-1}$, where the derivative is taken
in the sense of distributions. Another equivalent way to see $\Lambda_{*}^{k}$
is to consider functions of $\Lambda_{*}^{k-1}$ that are either differentiable
or limits of convolutions with the Poisson kernel, that is, $\displaystyle
f(\xi)=\lim_{r\to 1}(f*P_{r})(\xi)$ where $P_{r}(\theta)$ is the Poisson
kernel.

##  2 Main Results

Our first result is about the constant $f_{Q}$ in the definition of $BMO^{w}$.
If fact, the constant $f_{Q}$ can be replaced with any non-negative constant.
The same can be said about $BMO$ as well.

######  Proposition 2.1.

Let $f\in L^{1}_{loc}$ .

  1. (1)

For any non-negative real number $\alpha$, we have

$\sup_{Q\ni x}\;\left|\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{Q}[f(\xi)-f_{Q}]d\lambda(\xi)\right|\leq
2\sup_{Q\ni x}\;\left|\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{Q}[f(\xi)-\alpha]d\lambda(\xi)\right|\;.$ (2.1)

  2. (2)

$f\in BMO^{w}$ if and only if for any $x\in\mathbb{R}^{n}$ and any cube $Q\ni
x$, there exists is a non-negative number $\alpha\in Q$ such that

$\sup_{Q\ni x}\;\left|\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{Q}[f(\xi)-\alpha]d\lambda(\xi)\right|<\infty.$
(2.2)

###### Proof.

(1) To prove assertion (1), fix $x\in\mathbb{R}^{n}$ and a cube
$Q\subseteq\mathbb{R}^{n}$ containing $x$. Observe that for every non-negative
$\alpha$ we have,

$\left|\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{Q}[f(\xi)-f_{Q}]d\lambda(\xi)\right|\leq
2\left|\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{Q}[f(\xi)-\alpha]d\lambda(\xi)\right|\;.$

Indeed, for any non-negative real number $\alpha$, we have

$\displaystyle\left|\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{Q}[f(\xi)-f_{Q}]d\lambda(\xi)\right|$
$\displaystyle=$ $\displaystyle\left|\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{Q}[f(\xi)-\alpha]d\lambda(\xi)+\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{Q}[\alpha-f_{Q}]d\lambda(\xi)\right|$
$\displaystyle\leq$ $\displaystyle\left|\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{Q}[f(\xi)-\alpha]d\lambda(\xi)\right|+\left|\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{Q}[\alpha-f_{Q}]d\lambda(\xi)\right|$
$\displaystyle\leq$ $\displaystyle\left|\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{Q}[f(\xi)-\alpha]d\lambda(\xi)\right|+\left|\left|\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{Q}\alpha
d\lambda(\xi)\right|-\left|\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{Q}f(\xi)d\lambda(\xi)\right|\right|$
$\displaystyle\leq$ $\displaystyle\left|\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{Q}[f(\xi)-\alpha]d\lambda(\xi)\right|+\left|\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{Q}[\alpha-f(\xi)]d\lambda(\xi)\right|$
$\displaystyle\leq$ $\displaystyle
2\left|\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{Q}[f(\xi)-\alpha]d\lambda(\xi)\right|\;.$

To conclude, we taking the supremum over of all cubes $Q\ni x$.

(2) Assertion (2) follows immediately from (1). ∎

We can define two equivalent norms on $BMO^{w}$ and prove that endowed with
these norms, $BMO^{w}$ is in fact a Banach space.

######  Proposition 2.2.

Consider the following: For a every $f\in BMO^{w}$, we put

$\displaystyle\left\|f\right\|_{BMO^{w}}=\left\|mf\right\|_{\infty}+\left\|Mf\right\|_{\infty}\quad\mbox{and
\quad$\displaystyle\left\|f\right\|^{*}_{BMO^{w}}=\left\|mf\right\|_{\infty}+2\sup_{Q\ni
x}\;\inf_{\alpha>0}\;\left|\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{Q}[f(\xi)-f_{Q}]d\lambda(\xi)\right|$\;.}$

Then

$\left\|f\right\|_{BMO^{w}}\leq\left\|f\right\|^{*}_{BMO^{w}}\leq
2\left\|f\right\|_{BMO^{w}}\;.$

###### Proof.

The proof is an immediate consequence of Proposition 2.1. We note that the
proof can also be obtained from the Closed-graph theorem, but that will
require to first prove that endowed with the two norms, $BMO^{w}$ is a Banach
space. ∎

######  Theorem 2.3.

The space $(BMO^{w},\left\|\leavevmode\nobreak\ \cdot\leavevmode\nobreak\
\right\|_{BMO^{w}})$ is complete.

###### Proof.

(1). In the proof that $\left\|\leavevmode\nobreak\ \cdot\leavevmode\nobreak\
\right\|_{BMO^{w}}$ is a norm, homogeneity and the triangle inequality are
easy to prove. As for positivity, we note that
$\displaystyle\left\|f\right\|_{BMO^{w}}=0\Longleftrightarrow\sup_{Q\ni
x}\;f_{Q}=0$ and $f(\xi)=f_{Q}$ on all cubes $Q\ni x$. It follows immediately
that $f=0$ .

(2). Let $\left\\{f_{n}\right\\}_{n\in\mathbb{N}}$ be a Cauchy sequence in
$BMO^{w}$. Let $\epsilon>0$ and $N\in\mathbb{N}$ such that $\forall n,m\in N$,
we have $\left\|f_{n}-f_{m}\right\|_{BMO^{w}}<\epsilon$. That is,

$\displaystyle\sup_{Q\ni
x}\left[(f_{n}-f_{m})_{Q}+\left|\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{Q}[(f_{n}-f_{m})(\xi)-(f_{n}-f_{m})_{Q}]d\lambda(\xi)\right|\right]<\epsilon\;.$
(2.3)

In particular, from (2.3), we have that $\displaystyle\sup_{Q\ni
x}\;(f_{n}-f_{m})_{Q}<\epsilon$, therefore

$\displaystyle\left|f_{n,Q}-f_{m,Q}\right|$ $\displaystyle=$
$\displaystyle\left|\left|\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{Q}f_{n}(\xi)d\lambda(\xi)\right|-\left|\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{Q}f_{m}(\xi)d\lambda(\xi)\right|\right|$
$\displaystyle\leq$ $\displaystyle\left|\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{Q}[f_{n}-f_{m}](\xi)d\lambda(\xi)\right|=(f_{n}-f_{m})_{Q}$
$\displaystyle\leq$ $\displaystyle\sup_{Q\ni
x}\;(f_{n}-f_{m})_{Q}<\epsilon\;.$

Hence for fixed $Q$, $\left\\{f_{n,Q}\right\\}$ is Cauchy sequence in
$\mathbb{R}$. Let $\displaystyle f_{Q}=\lim_{n\to\infty}f_{n,Q}$. We note from
the above inequalities that

$(f_{n}-f_{m})_{Q}=\left|\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{Q}[f_{n}-f_{m}](\xi)d\lambda(\xi)\right|\geq\left|f_{n,Q}-f_{m,Q}\right|\geq
f_{n,Q}-f_{m,Q}\;.$

Therefore, given a cube $Q\subset\mathbb{R}^{n}$ containing $x$, we have

$\displaystyle\left|Mf_{n}(x)-Mf_{m}(x)\right|$ $\displaystyle\leq$
$\displaystyle\sup_{Q\ni x}\;\left|\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{Q}[(f_{n}(\xi)-f_{m}(\xi))-(f_{n,Q}-f_{m,Q})]d\lambda(\xi)\right|$
$\displaystyle\leq$ $\displaystyle\sup_{Q\ni
x}\;\left|\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{Q}[(f_{n}(\xi)-f_{m}(\xi))-(f_{n}-f_{m})_{Q}]d\lambda(\xi)\right|+\
\sup_{Q\ni x}\;\left|(f_{n}-f_{m})_{Q}-(f_{n,Q}-f_{m,Q})\right|$
$\displaystyle\leq$
$\displaystyle\left\|f_{n}-f_{m}\right\|_{BMO^{w}}+\sup_{Q\ni
x}\;\left|(f_{n,Q}-f_{m,Q})\right|<2\epsilon\;.$

It follows that $\left\\{Mf_{n}\right\\}_{n\in\mathbb{N}}$ is a Cauchy-
sequence in $L^{\infty}_{loc}$. Let $\displaystyle h=\lim_{n\to\infty}Mf_{n}$.
Since $L^{\infty}_{loc}\subset L^{1}_{loc}$, we have $h\in L^{1}_{loc}$ .

$\displaystyle h(x)$ $\displaystyle=$
$\displaystyle\lim_{n\to\infty}Mf_{n}(x)$ $\displaystyle=$
$\displaystyle\lim_{n\to\infty}\;\sup_{Q\ni
x}\left|\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{Q}[f_{n}(\xi)-f_{n,Q}]d\lambda(\xi)\right|$
$\displaystyle=$ $\displaystyle\sup_{Q\ni
x}\left|\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{Q}[\;\lim_{n\to\infty}f_{n}(\xi)-f_{Q}]d\lambda(\xi)\right|\;.$

Since $h(x)$ is finite on any cube $Q\ni x$, that follows that $\displaystyle
f(\xi)=\lim_{n\to\infty}f_{n}(\xi)$ is finite a.e. on $Q$. Thus $h=Mf$, for
some $f\in BMO^{w}$ and
$\displaystyle\lim_{n\to\infty}\left\|f_{n}-f\right\|_{BMO^{w}}\to 0$. ∎

######  Theorem 2.4 (Hölder’s type inequality).

Let $g\in BMO^{w}$ and a hyper-cube $J\subset\mathbb{R}^{n}$. Consider the
following operator $T_{g}:B^{1}\to\mathbb{R}$ given by $\displaystyle
T_{g}(f)=\int_{J}f(\xi)g(\xi)d\lambda(\xi)$. Then, $T_{g}\in(B^{1})^{*}$ with
$\left\|T_{g}\right\|_{(B^{1})^{*}}\leq\left\|g\right\|_{BMO^{w}}$ .
Moreover, the operator $H:BMO^{w}\to(B^{1})^{*}$ defined as $H(g)=T_{g}$ is
onto.

###### Proof.

By linearity of the integral, $T_{g}$ is a linear. To start, we study the
action of this operator on special atoms. Indeed, let $\xi\in I\subseteq J$
and suppose
$f(\xi)=b(\xi)=\frac{1}{\phi(I)}[\chi_{{}_{R}}(\xi)-\chi_{{}_{L}}(\xi)]$,
where $R,L$ are sub-cubes of $I$ such that $I=R\cup L$ and $R\cap
L=\emptyset$. Therefore we have:

$\displaystyle T_{g}(b)$ $\displaystyle=$
$\displaystyle\int_{J}b(\xi)g(\xi)d\lambda(\xi)=\int_{I}b(\xi)g(\xi)d\lambda(\xi)$
$\displaystyle=$
$\displaystyle\int_{I}b(\xi)[g(\xi)-g_{I}]d\lambda(\xi),\quad\mbox{since
$\displaystyle\int_{I}b(\xi)g_{I}d\lambda(\xi)=0$}\;.$

Taking the supremum over all $I\ni x$ and using the fact that
$b(\xi)\leq\frac{1}{\phi(I)}$, we have

$\left|T_{g}(b)\right|\leq\sup_{\underset{I\subseteq J}{I\ni
x}}\inf_{\alpha>0}\left|\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{I}[g(\xi)-\alpha]d\lambda(\xi)\right|\leq\left\|g\right\|_{BMO^{w}}\;.$
(2.4)

Now suppose $\displaystyle f(\xi)=\sum_{n=1}^{\infty}c_{n}b_{n}(\xi)$ with
$\displaystyle\sum_{n=1}^{\infty}\left|c_{n}\right|<\infty$ is an element of
$B^{1}$, where the $b_{n}$’s are special atoms defined on sub-cubes $I_{n}$ of
$J$ with $I_{n}=R_{n}\cup L_{n}$ and $R_{n}\cap L_{n}=\emptyset$. Let
$\displaystyle I=\bigcup_{n=1}^{\infty}I_{n}$. We have for $\alpha>0$

$\displaystyle T_{g}(f)$ $\displaystyle=$
$\displaystyle\int_{J}\left(\sum_{n=1}^{\infty}c_{n}b_{n}(\xi)\right)g(\xi)d\lambda(\xi)$
$\displaystyle=$
$\displaystyle\int_{I}\left(\sum_{n=1}^{\infty}c_{n}b_{n}(\xi)\right)\left[g(\xi)-\alpha\right]d\lambda(\xi)$
$\displaystyle=$
$\displaystyle\sum_{n=1}^{\infty}c_{n}\int_{I}b_{n}(\xi)\left[g(\xi)-\alpha\right]d\lambda(\xi)$
$\displaystyle=$
$\displaystyle\sum_{n=1}^{\infty}c_{n}\int_{I_{n}}b_{n}(\xi)\left[g(\xi)-\alpha\right]d\lambda(\xi)\;.$

It follows from (2.4) that

$\left|T_{g}(f)\right|\leq\left(\sum_{n=1}^{\infty}\left|c_{n}\right|\right)\cdot\left\|g\right\|_{BMO^{w}}\;.$

Talking the infimum over all representations of $f$ yields:

$\left|T_{g}(f)\right|\leq\left\|f\right\|_{B^{1}}\cdot\left\|g\right\|_{BMO^{w}}\;.$

Therefore $T_{g}$ is a bounded linear operator on $B^{1}$ with

$\left\|T_{g}\right\|_{(B^{1})^{*}}=\sup_{\left\|f\right\|_{B^{1}}=1}\left|T_{g}(f)\right|\leq\left\|g\right\|_{BMO^{w}}\;.$
(2.5)

Now suppose that $T$ is a bounded linear functional on $B^{1}$, that is,
$T\in(B^{1})^{*}$. We want to show that there exists a function $g\in BMO^{w}$
such that $\displaystyle T(f)=T_{g}(f)=\int_{J}f(\xi)g(\xi)d\lambda(\xi)$.
That $T\in(B^{1})^{*}$ implies the existence of an absolute constant $C$ such
that

$\left|T(f)\right|\leq C\left\|f\right\|_{B^{1}},\quad\forall f\in B^{1}.$
(2.6)

Recall that a function $G:J\to\mathbb{R}$ is said to be absolutely continuous
on $J$ if for every positive number $\epsilon$, there exists a positive number
$\delta$, such that for a finite sequence of pairwise disjoint sub-cubes
$I_{n}=(x_{n},y_{n})$ of $J$,

$\displaystyle\sum_{n=1}^{\infty}(y_{n}-x_{n})<\delta\implies\sum_{n=1}^{\infty}\left|G(y_{n})-G(x_{n})\right|<\epsilon\;.$
(2.7)

Suppose such a sequence of sub-cubes exists. Now consider
$G(x)=T\left(\chi_{{}_{[x-h,x+h)}}\right)$ for some real number $h>0$. Then,
given an cube $I_{n}$ and a real number $h_{n}$, we have by linearity of $T$
that

$G(y_{n})-G(x_{n})=T\left(\left[\chi_{{}_{[y_{n}-h_{n},y_{n}+h_{n})}}-\chi_{{}_{[x_{n}-h_{n},x_{n}+h_{n})}}\right]\right)\;.$

Now if we define $L_{n}=[x_{n}-h_{n},x_{n}+h_{n})$ and
$R_{n}=[y_{n}-h_{n},y_{n}+h_{n})$, we see that $R_{n}\cap L_{n}=\emptyset$
their union forms a single cube $J_{n}$ if we choose $h_{n}=(y_{n}-x_{n})/2$.
Moreover, in that case, the length of the cube $J_{n}$ is
$y_{n}+h_{n}-x_{n}+h_{n}=2(y_{n}-x_{n})$. Therefore by linearity

$G(y_{n})-G(x_{n})=2(y_{n}-x_{n})T\left(\frac{1}{2(y_{n}-x_{n})}\left[\chi_{{}_{[y_{n}-h_{n},y_{n}+h_{n})}}-\chi_{{}_{[x_{n}-h_{n},x_{n}+h_{n})}}\right]\right)\;.$

We observe that $\displaystyle
b_{n}(\xi)=\frac{1}{2(y_{n}-x_{n})}\left[\chi_{{}_{[y_{n}-h_{n},y_{n}+h_{n})}}(\xi)-\chi_{{}_{[x_{n}-h_{n},x_{n}+h_{n})}}(\xi)\right]$
is a special atom with norm in $B^{1}$ equal to 1. It follows that using (2.6)

$\left|G(y_{n})-G(x_{n})\right|\leq 2(y_{n}-x_{n})\left|T(b_{n})\right|\leq
2C(y_{n}-x_{n})\left\|b_{n}\right\|_{B^{1}}=2C(y_{n}-x_{n})\;.$

Hence,

$\sum_{n=1}^{\infty}\left|G(y_{n})-G(x_{n})\right|\leq
2C\sum_{n=1}^{\infty}(y_{n}-x_{n})\;.$

We conclude by noting that given $\epsilon>0$, (2.7) is satisfied if we choose
$\displaystyle\delta=\frac{\epsilon}{2C}$. We conclude that the function $G$
is absolutely continuous on $J$. Therefore, $G$ is differentiable almost
everywhere, that is, there exists $g\in L^{1}$ such that $\displaystyle
G(x)=\int_{a}^{x}g(\xi)d\lambda(\xi)$ for all cubes $I=[a,b]\subseteq J$. Let
$I\ni x$ be a sub-cube of $J$. Therefore, $\displaystyle\sup_{I\ni
x}\frac{1}{\lambda(I)}\left|\int_{I}g(\xi)d\lambda(\xi)\right|<\infty$ because
an absolutely continuous function is a function with bounded variation. Since
$g\in L^{1}$, we have that $\displaystyle
g_{I}=\frac{1}{\lambda(I)}\left|\int_{I}g(\xi)d\lambda(\xi)\right|<\infty.$ It
follows that

$Mg(x)=\sup_{I\ni
x}\frac{1}{\lambda(I)}\left|\int_{I}(g(\xi)-g_{I})d\lambda(\xi)\right|\leq\sup_{I\ni
x}\frac{1}{\lambda(I)}\left|\int_{I}g(\xi)d\lambda(\xi)\right|+\sup_{I\ni
x}\;g_{I}<\infty\;.$

This proves that $g\in BMO^{w}$. That is, $H:BMO^{w}\to(B^{1})^{*}$ is onto
with $H(g)=T=T_{g}$. Identifying $T$ with the $g$, it follows from (2.5) that

$\left\|g\right\|_{(B^{1})^{*}}=\left\|T\right\|_{(B^{1})^{*}}=\left\|T_{g}\right\|_{(B^{1})^{*}}\leq\left\|g\right\|_{BMO^{w}}\;.$
(2.8)

∎

######  Remark 2.5.

We observe that the above result can be obtained differently. In fact, we
recall that it was proved in [3] that the dual space of $B^{1}$ is equivalent
to $\Lambda_{*}^{2}$. Let $x\in J,h>0,I=[x-h,x+h]\subseteq J,L=[x-h,x)$, and
$R=[x,x+h]$. Let $\displaystyle
b(\xi)=\frac{1}{2h}[\chi_{{}_{R}}(\xi)-\chi_{{}_{L}}(\xi)]$. We have

$\displaystyle\left\|f\right\|_{(B^{1})^{*}}\cong\left\|f\right\|_{\Lambda_{*}^{2}}$
$\displaystyle=$ $\displaystyle\sup_{\underset{h>0}{I\ni
x}}\;\left|\int_{I}b(\xi)\partial f(\xi)d\lambda(\xi)\right|$ $\displaystyle=$
$\displaystyle\sup_{\underset{h>0}{I\ni x}}\;\left|\int_{I}b(\xi)(\partial
f(\xi)-\partial f_{I})d\lambda(\xi)\right|$ $\displaystyle\leq$
$\displaystyle\left\|f\right\|_{BMO^{w}}\;.$

This shows that $BMO^{w}\subseteq\Lambda_{*}^{2}\cong(B^{1})^{*}$.

######  Theorem 2.6.

The dual space $(B^{1})^{*}$ of $B^{1}$ is $BMO^{w}$ with
$\left\|g\right\|_{BMO^{w}}\cong\left\|g\right\|_{(B^{1})^{*}}$ .

To prove Theorem 2.6, we recall that $B^{1}(\mathbb{T})$ has an analytic
extension $B^{1}_{A}(\mathbb{D})$ to the unit disk. In fact, in [1], it was
shown that $B^{1}_{A}(\mathbb{D})$ consists of functions $F$ that are Poisson
integrals of functions in $B^{1}(\mathbb{T})$, that is, $\displaystyle
F(z)=\frac{1}{2\pi}\int_{-\pi}^{\pi}\frac{1+e^{-i\xi}z}{1-e^{-i\xi}z}f(e^{i\xi})d\lambda(\xi)$
where $f\in B^{1}(\mathbb{T})$. Moreover, the norm
$\displaystyle\left\|F\right\|_{B^{1}_{A}(\mathbb{D})}=\int_{0}^{1}\int_{-\pi}^{\pi}\left|F^{\prime}(z)\right|dz$
is equivalent to the norm $\left\|f\right\|_{B^{1}(\mathbb{T})}$. This allows
to identify $B^{1}_{A}(\mathbb{D})$ with $B^{1}(\mathbb{T})$ and thus the
following:

######  Proposition 2.7.

$B^{1}_{A}(\mathbb{D})$ can be identified with a closed subspace of
$L^{1}(\mathbb{T})$.

###### Proof.

Let $\left\\{f_{n}\right\\}_{n\in\mathbb{N}}$ be a sequence in
$B^{1}(\mathbb{T})$ that converges to $f$ in $L^{1}(\mathbb{T})$. We need to
show that the Poisson integral of $f$ is in $B^{1}_{A}(\mathbb{D})$. Since
$f_{n}\in B^{1}(\mathbb{T}),\forall\;n\in\mathbb{N}$, then $\displaystyle
F_{n}(z)=\frac{1}{2\pi}\int_{-\pi}^{\pi}\frac{1+e^{-i\xi}z}{1-e^{-i\xi}z}f_{n}(e^{i\xi})d\lambda(\xi)$
belongs to $B^{1}_{A}(\mathbb{D})$. Let $F(z)$ be the Poisson integral of $f$.
We note that if $z=e^{i\theta}$, then
$(\xi-\theta)^{2}-\frac{(\xi-\theta)^{4}}{2}\leq\left|1-e^{-i\xi}z\right|^{2}\leq(\xi-\theta)^{2}$.
Therefore, we have that $\displaystyle
F^{\prime}(z)=\frac{1}{2\pi}\int_{-\pi}^{\pi}\frac{2e^{-i\xi}}{(1-e^{-i\xi}z)^{2}}f(e^{i\xi})d\lambda(\xi)$
and
$\displaystyle\left|F^{\prime}(z)\right|\leq\frac{1}{2\pi}\int_{-\pi}^{\pi}\frac{2}{\left|1-e^{-i\xi}z\right|^{2}}\left|f(e^{i\xi})\right|d\lambda(\xi)\leq
C\left\|f\right\|_{L^{1}(\mathbb{T})}$. It follows that ,

$\displaystyle\left\|F_{n}-F\right\|_{B^{1}_{A}}=\int_{0}^{1}\int_{-\pi}^{\pi}\left|F^{\prime}_{n}(z)-F^{\prime}(z)\right|dz\leq
C\left\|f_{n}-f\right\|_{L^{1}(\mathbb{T})}.$

Since $f_{n}\to f$ in $L^{1}(\mathbb{T})$, it follows that $F\in B^{1}_{A}$
and $F_{n}\to F$ in $B^{1}_{A}(\mathbb{D})$. The result follows by identifying
$B^{1}_{A}(\mathbb{D})$ and $B^{1}(\mathbb{T})$. ∎

We note that there is an extension of this result to the polydisk
$\mathbb{D}^{n}$ and polysphere $\mathbb{T}^{n}$, see [7] .

###### Proof of theorem 2.6 .

It sufficient to show that there exists a constant $M>0$ such that
$\left\|T\right\|_{(B^{1}(\mathbb{T}))^{*}}\geq
M\left\|g\right\|_{BMO^{w}(\mathbb{T})}.$ The extension to $\mathbb{R}^{n}$ is
natural, using the results in [7, 8] . With Proposition 2.7, the proof follows
along the lines of the proof of Proposition 7.3 in [6].
Let $T\in(B^{1}(\mathbb{T}))^{*}$. Since $B^{1}(\mathbb{T})$ is a closed
subspace of $L^{1}(\mathbb{T})$ by Proposition 2.7, then by Hahn-Banach
Theorem, $T$ can be extended to a bounded linear operator
$T^{\prime}\in(L^{1}(\mathbb{T}))^{*}$ with
$\left\|T\right\|_{(B^{1}(\mathbb{T}))^{*}}=\left\|T^{\prime}\right\|_{(L^{1}(\mathbb{T}))^{*}}$.
Since $(L^{1}(\mathbb{T}))^{*}=L^{\infty}(\mathbb{T})$, then there exists
$g_{0}\in L^{\infty}(\mathbb{T})$ such that
$\left\|g\right\|_{L^{\infty}(\mathbb{T})}=\left\|T^{\prime}\right\|_{(L^{1})^{*}}=\left\|T\right\|_{(B^{1})^{*}}$
and $\displaystyle
T(f)=\frac{1}{2\pi}\int_{\mathbb{T}}\overline{g_{0}(e^{i\xi})}f(e^{i\xi})d\lambda(\xi)$,
for all $f\in B^{1}_{A}(\mathbb{D})$. Note that here, we identify $f$ with its
correspondent in $B^{1}_{A}(\mathbb{D})$. Now let
$\displaystyle\sum_{n\in\mathbb{N}}A_{n}e^{in\xi}$ be the Fourier series of
$g_{0}$. Since $g_{0}\in L^{\infty}(\mathbb{T})\subset L^{2}(\mathbb{T})$, we
have that $\displaystyle\sum_{n\in\mathbb{Z}}\left|A_{n}\right|^{2}<\infty$ .
This means that $g_{0}$ is holomorphic. Let

$g(z)=\frac{1}{2\pi
i}\int_{-\pi}^{\pi}\frac{g_{0}(e^{i\xi})}{e^{i\xi}-z}d(e^{i\xi})=\frac{1}{2\pi}\int_{-\pi}^{\pi}\frac{g_{0}(e^{i\xi})}{1-e^{-i\xi}z}d\xi\;.$

Since
$\displaystyle\frac{1}{1-e^{i\xi}z}=\sum_{n\in\mathbb{N}}e^{-in\xi}z^{n}$ and
$\displaystyle
A_{n}=\frac{1}{2\pi}\int_{-\pi}^{\pi}e^{-in\xi}g_{0}(e^{i\xi})d\lambda(\xi)$,
we have

$g(z)=\frac{1}{2\pi}\int_{-\pi}^{\pi}\frac{g_{0}(e^{i\xi})}{1-e^{-i\xi}z}d\xi=\sum_{n\in\mathbb{N}}A_{n}z^{n}\;.$

This implies that $g\in H^{2}(\mathbb{D})$. For $\theta\in\mathbb{R}$,
$\displaystyle\overline{g(e^{i\theta})}=\frac{1}{2\pi}\int_{-\pi}^{\pi}\frac{\overline{g_{0}(e^{i\xi})}}{1-e^{-i\xi}e^{i\theta}}d\lambda(\xi)$.
Moreover, given $f\in B^{1}_{A}(\mathbb{D})\subseteq H^{1}(\mathbb{D})$, and
using the Cauchy integral formula, we have:

$\displaystyle\frac{1}{2\pi}\int_{-\pi}^{\pi}\overline{g(e^{i\theta})}f(e^{i\theta})d\theta$
$\displaystyle=$
$\displaystyle\frac{1}{2\pi}\int_{-\pi}^{\pi}\overline{g_{0}(e^{i\theta})}\left(\frac{1}{2\pi}\int_{-\pi}^{\pi}\frac{f(e^{i\xi})}{1-e^{-i\xi}e^{i\theta}}d\lambda(\xi)\right)d\theta$
$\displaystyle=$
$\displaystyle\frac{1}{2\pi}\int_{-\pi}^{\pi}\overline{g_{0}(e^{i\theta})}f(e^{i\theta})d\theta$
$\displaystyle=$ $\displaystyle T(f)\;.$

We also observe that

$\displaystyle
g(z)=\frac{1}{2\pi}\int_{-\pi}^{\pi}\frac{1}{2}\left[\frac{1+e^{-i\xi}z}{1-e^{-i\xi}z}+1\right]g_{0}(e^{i\xi})d\xi=\frac{1}{2\pi}\int_{-\pi}^{\pi}\frac{1+e^{-i\xi}z}{1-e^{-i\xi}z}G_{0}(e^{i\xi})d\lambda(\xi),$
(2.9)

where $\displaystyle
G_{0}(e^{i\xi})=\frac{1}{2}\left(g_{0}(e^{i\xi})+\frac{1}{2\pi}\int_{-\pi}^{\pi}g_{0}(e^{i\xi})d\lambda(\xi)\right)=\frac{1}{2}\left(g_{0}(e^{i\xi})+A_{0}\right)\;.$
Put $G_{1}=\mbox{Re}(G_{0}),G_{2}=\mbox{Im}(G_{0})$, and

$U(z)=\frac{1}{2\pi}\int_{-\pi}^{\pi}\frac{1+e^{-i\xi}z}{1-e^{-i\xi}z}G_{1}(e^{i\xi})d\lambda(\xi);V(z)=\frac{1}{2\pi}\int_{-\pi}^{\pi}\frac{1+e^{-i\xi}z}{1-e^{-i\xi}z}G_{2}(e^{i\xi})d\lambda(\xi)\;.$

Then $g=U+iV$. Moreover, $U$ and $V$ are analytic in $\mathbb{D}$ since they
represent the Poisson integral of $G_{1},G_{2}\in
L^{\infty}(\mathbb{T})\subseteq B^{1}(\mathbb{T})$. Observe that $BMO\subseteq
BMO^{w}$. Moreover, Theorem 3.2 in [6] shows that
$\left\|g\right\|_{BMO}\cong\left\|g\right\|_{BMOA}$. It therefore follows
that there exists a constant $C>0$ such that

$C\left\|g\right\|_{BMO^{w}}\leq
C\left\|g\right\|_{BMO}\leq\left\|g\right\|_{BMOA}\leq\left\|T^{\prime}\right\|_{(L^{1})^{*}}=\left\|T\right\|_{(B^{1})^{*}}\;.$

∎

###  2.1 Discussion

We note that $BMO^{w}\subseteq\Lambda^{\prime}_{*}$ with
$\left\|f\right\|_{\Lambda^{\prime}_{*}}\leq C\left\|f\right\|_{BMO^{w}}$
where $\left\|g\right\|_{\Lambda^{\prime}_{*}}=\left\|f\right\|_{\Lambda_{*}}$
with $g^{\prime}=f$ in the sense of distributions. Since
$(B^{1})^{*}\cong\Lambda^{\prime}_{*}$, and from Theorem 2.6 above
$(B^{1})^{*}\cong BMO^{w}$, it follows that
$BMO^{w}\cong\Lambda^{\prime}_{*}\;.$ The consequence is that there exists
$c>0$ such that
$c\left\|f\right\|_{BMO^{w}}\leq\left\|f\right\|_{\Lambda^{\prime}_{*}}$, that
is, those two norms are equivalent. We finish by noting $\Lambda^{\prime}_{*}$
has an analytic characterization, so we would expect the analytic
characterization of $BMO^{w}$ to also be equivalent to that of
$\Lambda^{\prime}_{*}$.
Another takeaway from Theorem 2.6 is that it provides another proof that $BMO$
is strictly contained in $BMO^{w}$ otherwise we would have had
$B^{1}\cong\mathbb{H}^{1}$, which is not true. In other words, there exists
$f\in\mathbb{H}^{1}$ such that $f\notin B^{1}$.

Our next result is about the closeness of $VMO^{w}$ in $BMO^{w}$.

######  Theorem 2.8.

$VMO^{w}$ is a closed subspace of $BMO^{w}$.

###### Proof.

Let $\left\\{f_{n}\right\\}_{n\in\mathbb{N}}$ be a sequence in $VMO^{w}$ that
converges to $f\in BMO^{w}$. Let us prove that $f\in VMO^{w}$. That $f_{n}\to
f$ as $n\to\infty$ in $BMO^{w}$ is equivalent to
$\displaystyle\lim_{n\to\infty}\left\|f_{n}-f\right\|_{BMO^{w}}=0$. The latter
is also equivalent, by definition of the norm in $BMO^{w}$ to

$\mbox{$\displaystyle\lim_{n\to\infty}\;\sup_{Q\ni x}\;(f_{n}-f)_{Q}=0$
and}\quad\displaystyle\lim_{n\to\infty}M(f_{n}-f)(x)=0\;.$

Since $f\in BMO^{w}$, then $\displaystyle\sup_{Q\ni
x}\;\left|\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{Q}[f(\xi)-f_{Q}]d\lambda(\xi)\right|<\infty$.
Therefore

$\displaystyle\left|\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{Q}[f(\xi)-f_{Q}]d\lambda(\xi)\right|$
$\displaystyle=$ $\displaystyle\left|\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{Q}[f(\xi)-f_{n}(\xi)+f_{n}(\xi)-f_{n,Q}+f_{n,Q}-f_{Q}]d\lambda(\xi)\right|$
$\displaystyle\leq$ $\displaystyle\left|\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{Q}[f(\xi)-f_{n}(\xi)]d\lambda(\xi)\right|+\left|\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{Q}[f_{n}(\xi)-f_{n,Q}]d\lambda(\xi)\right|+\left|\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{Q}[f_{n,Q}-f_{Q}]d\lambda(\xi)\right|$
$\displaystyle\leq$ $\displaystyle\left|\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{Q}[f(\xi)-f_{n}(\xi)]d\lambda(\xi)\right|+\left|\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{Q}[f_{n}(\xi)-f_{n,Q}]d\lambda(\xi)\right|+\left|f_{n,Q}-f_{Q}\right|$
$\displaystyle\leq$ $\displaystyle\left|\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{Q}[f(\xi)-f_{n}(\xi)]d\lambda(\xi)\right|+\left\|f_{n}-f\right\|_{BMO^{w}}$
$\displaystyle\leq$
$\displaystyle\left\|f_{n}-f\right\|_{L^{1}_{loc}}+\left\|f_{n}-f\right\|_{BMO^{w}}\;.$

Since $\left\\{f_{n}\right\\}_{n\in\mathbb{N}}$ is a sequence of function is
$BMO^{w}$, we have that
$\displaystyle\lim_{n\to\infty}\;\left\|f_{n}-f\right\|_{BMO^{w}}=0$. Since
$BMO^{w}\subset L^{1}_{loc}$, we have that
$\displaystyle\lim_{n\to\infty}\left\|f_{n}-f\right\|_{L^{1}_{loc}}=0$. Hence
$VMO^{w}$ is a closed subspace of $BMO^{w}$. ∎

######  Remark 2.9.

We note that it was proved in [6] that $VMO$ is a closed subspace of $BMO$. A
stronger result even showed that if we restrict ourselves to $\mathbb{T}$,
then the space of complex-valued and continuous functions
$C(\mathbb{T})\subseteq VMO(\mathbb{T})$ and the $BMO(\mathbb{T})$-closure of
$C(\mathbb{T})$ is precisely $VMO(\mathbb{T})$. It turns out this result is
also true for $VMO^{w}$.

######  Theorem 2.10.

The $BMO^{w}$-closure of $C(\mathbb{T})$ is precisely $VMO^{w}(\mathbb{T})$,
that is

$\overline{C(\mathbb{T})}^{BMO^{w}}=VMO^{w}(\mathbb{T})\;.$

###### Proof.

We observe that $C(\mathbb{T})\subseteq VMO(\mathbb{T})\subseteq
VMO^{w}(\mathbb{T})\subseteq BMO^{w}(\mathbb{T})$. Therefore since
$VMO^{w}(\mathbb{T})$ is closed in $BMO^{w}(\mathbb{T})$, we have that

$\overline{C(\mathbb{T})}^{BMO^{w}(\mathbb{T})}\subseteq\overline{VMO^{w}(\mathbb{T})}^{BMO^{w}(\mathbb{T})}=VMO^{w}(\mathbb{T})\;.$

For the other direction , let $f\in VMO^{w}(\mathbb{T})$, and consider the
sequence $\left\\{R_{\epsilon}(f)\right\\}_{\epsilon>0}$ of rotations of $f$
by angle $\epsilon$, defined on $\mathbb{T}$ as
$R_{\epsilon}(f)(e^{i\theta})=f(e^{i(\theta-\epsilon)});\theta\in\mathbb{R}$.
Then from Theorem 2.1 in [6], we have that for all
$\epsilon>0,R_{\epsilon}(f)\in C(\mathbb{T})$ and
$\displaystyle\lim_{\epsilon\to
0}\left\|R_{\epsilon}(f)-f\right\|_{BMO(\mathbb{T})}=0$. Since
$BMO(\mathbb{T})\subseteq BMO^{w}(\mathbb{T})$, we also have that
$\displaystyle\lim_{\epsilon\to
0}\left\|R_{\epsilon}(f)-f\right\|_{BMO^{w}(\mathbb{T})}=0$. That is,
$f\in\overline{C(\mathbb{T})}^{BMO^{w}(\mathbb{T})}.$

∎

## Acknowledgment

This material is based upon work supported by the National Science Foundation
under Grant No. DMS-1440140, National Security Agency under Grant No.
H98230-20-1-0015, and the Sloan Grant under Grant No. G-2020-12602 while the
author participated in a program hosted by the Mathematical Sciences Research
Institute in Berkeley, California, during the summer of 2020.

## References

  * [1] S. Bloom and G. De Souza.  Atomic decomposition of generalized Lipschitz spaces.  Illinois Journal of Mathematics, pages 682–686, 1989.
  * [2] R. Coifman.  A real variable characterization of $h^{p}$.  Studia Mathematica, 51:269–274, 1974.
  * [3] G. De Souza.  Spaces formed by special atoms.  PhD thesis, SUNY at Albany, 1980.
  * [4] C. Fefferman.  Characterization of bounded mean oscillation.  Bulletin of the American Mathematical Society, 77(4):587–588, 1971\.
  * [5] T. Gill.  Banach spaces for the schwartz distributions.  Real Analysis and Exchange, 43(1):1–36, 2017.
  * [6] D. Girela.  Analytic functions of bounded mean oscillation.  Complex Functions Spaces (Merkrijärvi 1999), Univ. Joensuu Dept. Math. Rep. Ser.(4):61–170, 2001.
  * [7] E. Kwessi and G. De Souza.  Analytic characterization of high dimension weighted special atom spaces,.  Complex Variables and Elliptic Equations, 2020.
  * [8] E. Kwessi, G. De Souza, D. Ngalla, and N. Mariama.  The special atom space in higher dimensions.  Demonstratio Mathematica, 33:131–151, 2020.
  * [9] A. Zygmund.  Trigonometric Series, Vol. I, II.  Cambridge Mathematical Library, 2002.


# How far is Language Model from 100% Few-shot Named Entity Recognition in
Medical Domain

Mingchen Li, Rui Zhang
University of Minnesota, Twin Cities
{li003378<EMAIL_ADDRESS>

###### Abstract

Recent advancements in language models (LMs) have led to the emergence of
powerful models such as Small LMs111We define SLMs as pre-trained models with
fewer parameters compared to models like GPT-3/3.5/4, such as T5, BERT, and
others. (e.g., T5) and Large LMs (e.g., GPT-4). These models have demonstrated
exceptional capabilities across a wide range of tasks, such as name entity
recognition (NER) in the general domain. Nevertheless, their efficacy in the
medical section remains uncertain and the performance of medical NER always
needs high accuracy because of the particularity of the field. This paper aims
to provide a thorough investigation to compare the performance of LMs in
medical few-shot NER and answer How far is LMs from 100% Few-shot NER in
Medical Domain, and moreover to explore an effective entity recognizer to help
improve the NER performance. Based on our extensive experiments conducted on
16 NER models spanning from 2018 to 2023, our findings clearly indicate that
LLMs outperform SLMs in few-shot medical NER tasks, given the presence of
suitable examples and appropriate logical frameworks. Despite the overall
superiority of LLMs in few-shot medical NER tasks, it is important to note
that they still encounter some challenges, such as misidentification, wrong
template prediction, etc. Building on previous findings, we introduce a simple
and effective method called RT (Retrieving and Thinking), which serves as
retrievers, finding relevant examples, and as thinkers, employing a step-by-
step reasoning process. Experimental results show that our proposed RT
framework significantly outperforms the strong open baselines on the two open
medical benchmark datasets222The source code, data are available at
https://github.com/ToneLi/RT-Retrieving-and-Thinking..

##  1 Introduction

Researchers are increasingly interested in applying information extraction to
mine a vast quantity of unstructured information from electronic medical
records. These techniques can offer valuable perception and generate
substantial benefits for clinical research, such as drug discovery Zhang et
al. (2021), knowledge graphs building Li et al. (2020); Wu et al. (2023),
question answering Li and Ji (2022); Pugachev et al. (2023) and link
prediction Li et al. (2022); Zheng et al. (2023). Within the scope of medical
text mining, one of the most essential tasks in medical text mining is medical
named entity recognition (NER). However, existing supervised medical NER
models necessitate a substantial amount of human-annotated (i.e. medical
student, doctor) data. To tackle this issue, few-shot techniques have been
introduced to perform NER in resource-constrained settings by leveraging
auxiliary information or improving the discrimination between different
labels.

To address the challenges posed by the scarcity of available medical data,
recent studies propose to harness the power of SLMs or LLMs. When designing
models based on SLMs, the current methods primarily utilize nearest neighbor
inference Yang and Katiyar (2020); Das et al. (2022); Ji et al. (2022), prompt
tuning Huang et al. (2022); Liu et al. (2022), contrastive learning Zhang et
al. (2022); Das et al. (2022); Li et al. (2023) or generation methods Li and
Huang (2023); Wang et al. (2023). On the other hand, when designing the models
based on LLMs, the main approach is to employ the in-context learning (ICL)
Min et al. (2022) method to stimulate the recognition ability of LLMs. Despite
the impressive performance showcased by current works, there remains a lack of
comprehensive investigation comparing the performance of LMs in medical few-
shot NER. Otherwise, we found researcher tends to design the models to solve
the NER problems in the general domain, become of the issues of secret data,
even though these models can get high performance. More important, The NER
task in the medical domain demand near-perfect (100%) accuracy due to their
critical implications for human life. Consequently, determining the
suitability of these models for the medical domain presents a challenging
problem.

In this paper, our objective is to conduct a comprehensive evaluation of the
advantage and disadvantages of SLMs and LLMs on the medical few-shot NER
tasks, and then answer the following questions: 1) Which of the two, SLMs or
LLMs performs better in the few-shot medical NER task? 2) Is it possible to
transfer NER models trained for the general domain to the medical domain? 3)
Do the quantity and quality of annotation have any impact on the performance
of SLMs and LLMs? 4) How far is LM from 100% few-shot medical NER?

To answer these questions, we conduct an extensive empirical study involving
16 different few-shot NER models spanning from 2018 to 2023. Our study
specifically focuses on evaluating these models using two well-established
open medical NER datasets, ensuring a standardized and comprehensive analysis.
The results show that 1) LLMs demonstrate superior performance over SLMs when
the LLMs are provided with high-quality instructions. 2) Our findings indicate
that in order to successfully transfer a NER model from the general domain to
the medical domain, it is essential to pre-train the NER models specifically
in the medical domain. The NER in the medical domain presents the same
challenges, including flat/net entity recognition, abbreviation handling, and
recognizing long-word entities, etc. The pre-training enables the model to
acquire the necessary prior knowledge and context relevant to medical
entities. 3) The quantity of samples has a greater impact on SLMs compared to
LLMs. Additionally, when we examine the scenario where the training data is
only partially annotated, we observe that LLMs exhibit more consistent and
stable performance compared to SLMs. Regarding the aspect of quality, our
findings contradict certain studies that suggest improved performance for LLMs
with retrieved relevant samples instead of using random samples. Our results
indicate that the effectiveness of this approach varies depending on the
specific datasets employed in the evaluation. Otherwise, we found the
effectiveness of LLMs is greatly influenced by the careful selection of
appropriate examples and the application of sound entity recognition logic. In
our latest findings, we found the choice between using a combination of
positive and negative examples or exclusively positive examples has a definite
influence on the overall performance of LLMs with ICL. 4) We have identified
several factors that contribute to the inability of the current state-of-the-
art model to achieve 100% entity recognition. One reason is that NER models
face challenges in extracting long, special entities and Out-of-vocabulary
entities. Furthermore, the superior performance of LLMs is highly reliant on
pre-existing knowledge and contextual understanding, the failure to timely
update the knowledge can have a significant impact on the overall performance
and effectiveness of the entity recognition system. Otherwise, there are many
entities annotated in the training or testing dataset has more than one entity
type. For the full analyses, please check Section.7.

Based on some significant findings, we have proposed a novel few-shot entity
extractor called RT (Retrieving and Thinking). The main focus of RT is to
consider both the selection of appropriate examples and effective entity
recognition logic. The basic idea of RT involves utilizing a basic LLM to
identify the entity classes present in a given sentence. Subsequently, we
employ K-nearest neighbors (KNN) to retrieve relevant examples for the
sentences based on the recognized entity classes. Finally, following the
approach suggested by Wei et al. (2022), we design a logical entity
recognition process that enables the model to identify entities in a step-by-
step manner. We perform extensive experiments on two standard medical datasets
and 16 NER models for few-shot medical NER demonstrating the superiority of
our method over prior state-of-the-art methods. Our contributions are the
following:

  * •

We conduct an extensive empirical study comparing SLMs and LLMs on medical
few-shot NER tasks across 16 NER models during 2018-2023.

  * •

We propose RT, a new framework that uses Retrieving and Thinking strategies to
improve the performance of LLMs with ICLs on medical few-shot NER tasks.

  * •

We conduct a thorough analysis of our method, including an ablation study,
demonstrating the ineffectiveness of RT.

##  2 Related Work

###  2.1 Small Language Models (SLMs) in Medical Few-Shot NER

Few-shot named entity recognition is a task that aims to predict the label
(type) of an entity from insufficient labeled data. Most previous work in the
medical domain prefer to directly use medical language models such as
ClinicalBERT Huang et al. (2019), BioBERT Lee et al. (2020), and GatorTron
Yang et al. (2022) to solve the few-shot NER problem. A few studies Fritzler
et al. (2019); Ji et al. (2022) propose to utilize the prototype network Snell
et al. (2017) to catch the few-shot medical NER tasks. Inspired by the nearest
neighbor inference Wiseman and Stratos (2019), Yang and Katiyar (2020)
proposes NNshot and Structshot, which use the nearest neighbor to search for
the nearest label of each testing entity. In CONTaiNER Das et al. (2021), the
authors use contrastive learning to increase the discrimination for each label
and adopt Gaussian embeddings for each token to solve the Anisotropic property
in the few-shot NER task. The prompt-based method is also explored in this
task, such as Huang et al. (2022) uses the prompt to guide contrastive
learning. Zhang et al. (2022) proposes a span-based contrastive learning
method to handle the nested NER. In W-Procer Li et al. (2023), the authors use
a weighted prototypical contrastive learning method to solve the class
collision issue in few-shot medical NER. MetaNER Chen et al. (2023) proposes a
novel approach to enhance the performance of NER by pre-training SLMs using
instructions and demonstrations. As the popularity of LLMs continues to rise,
it becomes crucial to assess the effectiveness of SLMs, so in our work, we
provide a thorough investigation to compare the performance of SLMs and LLMs
in the medical few-shot NER.

###  2.2 Large Language Models (LLMs) in Medical Few-Shot NER

There are few works to explore the effectiveness of LLMs in Few-Shot NER, so,
in this section, we present relevant works exploring the use of Large Language
Models (LLMs) in the general domain. Furthermore, we evaluate the performance
of NER models using LLMs specifically in the medical domain. Ma et al. (2023)
introduces a filter-then-rank method by combing the advantage of SLMs and
LLMs. The SLM is employed to filter the candidate labels for each token, while
the instruction method is utilized to predict the final answer, which is
performed by the LLM. GPT-NER Wang et al. (2023) presents a novel approach
that enhances in-context learning performance by employing entity-level
embedding and self-verification. PromptNER Ashok and Lipton (2023) introduces
the concept of a chain of thought into Named Entity Recognition. In our work,
to improve the medical few-shot NER performance, we propose a method by
retrieving the good samples and providing the right entity recognition logic.

##  3 Small LMs v.s. Large LMs

In this section, we aim to provide a thorough investigation to compare the
performance of SLMs and LLMs in medical few-shot NER and answer How far are
LMs from 100% Few-shot NER in Medical Domain. To this end, we evaluate 16 NER
models on two standard few-shot NER datasets.

###  3.1 Problem Statement and Setups

In this section, we formalize the task of few-shot named entity recognition
(NER) and present a standardized evaluation setup to ensure a fair comparison
against previous SOTA models.

####  3.1.1 Few-Shot NER

In the traditional named entity recognition, given an input sentence of $i$
tokens $\bm{x}=\\{x_{1},x_{2},...,x_{i}\\}$, the NER model intends to assign
each token $x_{i}$ to its corresponding label $y_{i}$. In the few-shot NER
task, we assume that there are two same label sets $\\{L^{n}\\}$ of $n$ length
in the support set (training set) and query set (testing set). The $K$-shot
NER task is formally defined as follows: given the input sentence $\bm{x}$ and
a set of $K$-shot entities for each label class in $\\{L^{n}\\}$, it aims to
find the best sequence $\bm{y}=\\{y_{1},y_{2},...,y_{i}\\}$ for $\bm{x}$.
$K$-shot entities set contains $K$ entity examples for each entity class in
$\\{L^{n}\\}$.

####  3.1.2 Setups

For the evaluation scheme on few-shot NER, the current work Fritzler et al.
(2019); Hou et al. (2020) adapted the episode evaluation method which just
utilizes episode data from the test set. As shown in Das et al. (2021),
sampling the test episodes from the actual test set perturbs the real
distribution of testing data that may not reflect the actual performance. So,
same as the existing studies Das et al. (2021); Yang and Katiyar (2020); Huang
et al. (2022) about Few-shot NER, we employ the original test set of NCBI for
our prediction on LLMs. Due to the expensive nature of the GPT API, we have
chosen to sample 100 instances from the BC5CDR dataset as the test set for the
LLMs. We randomly selected one LLM to evaluate the LLM has a similar
performance under the whole test set and sampled 100 instances on BC5CDR.
However, for the SLMs, we continue to utilize the original test set of BC5CDR
and NCBI. we utilize greedy sampling Yang and Katiyar (2020) to sample the
support set. To clarify the tagging scheme setup, we utilize the "IO" tagging
scheme. Under this scheme, the "I" indicates that all tokens are inside an
entity, while "O" refers to all other tokens.

###  3.2 Dataset

Dataset | # Domain | # Class | # Sentence/ # Entity
---|---|---|---
BC5CDR | Medical | 2 | 13,938/ 28,545
NCBI | Medical | 4 | 7,287/ 7,025

Table 1:  Datasets Statistics. # Class refers to the number of entity classes
(types) that have been labeled in a dataset.

SLMs and LLMs are evaluated with two medical datasets, including BC5CDR Li et
al. (2016) and NCBI Doğan et al. (2014). Due to concerns arising from the I2B2
Stubbs and Uzuner (2015) user agreement, we refrained from conducting a
comparison of LLMs within the I2B2 dataset. Among these datasets, NCBI and
BC5CDR consist of 798 and 1500 public medical abstracts separately, all of
which are annotated with MeSH identifiers. Table 1 shows the detailed
statistical information of these three datasets.

###  3.3 SLMs, LLMs and Evaluation Metrics

####  3.3.1 Small Language Models

We provide a thorough investigation with several strong baselines based on the
state-of-the-art pre-trained few-shot NER models, including both Domain
Transfer and Domain non-Transfer methods. Specifically, the Domain Transfer
models have been trained on the OntoNotes 5.0 dataset, which comprises medical
NER knowledge. The evaluation of these models is conducted using the support
sets and test sets from BC5CDR, and NCBI. On the contrary, Domain non-Transfer
models differ in that they do not undergo additional pre-training using other
medical NER datasets.

we consider the following Domain Transfer models: (1) NNshotYang and Katiyar
(2020) is a method that uses nearest neighbor classification. (2)
StructshotYang and Katiyar (2020) is an improved version of NNshot that
combines nearest neighbor classification, abstract transition matrix, and
Viterbi algorithm. (3) ContaiNER Das et al. (2022) adopts contrastive learning
to estimate the distributional distance between entities’ vectors, which are
represented using Gaussian embeddings. (4) COPNER Huang et al. (2022)
leverages contrastive learning with prompt tuning to identify entities. (5)
EP-NET Ji et al. (2022) is a NER method based on the dispersedly distributed
prototypes network. (6) MetaNER Chen et al. (2023) proposes a novel approach
to enhance the performance of NER by pre-training SLMs using instructions and
demonstration.

we also consider the following Domain non-Transfer models: (1) ProtoBERT Huang
et al. (2022) is a few-shot NER method, which utilizes the prototypical
network Snell et al. (2017) and BERT model to infer the entity label. (2)
BINDER Zhang et al. (2022) employs span-based contrastive learning to
effectively push the entities of different types by optimizing both the entity
type encoder and sentence encoder. (3) LM-tagger (BERT Devlin et al. (2018),
ClinicalBERT Huang et al. (2019), BioBERT Lee et al. (2020) and GatorTron Yang
et al. (2022)) are traditional SLM-based methods that fine-tune the SLM on the
support set with the label classifier. Same as Yang and Katiyar (2020); Das et
al. (2022); Huang et al. (2022); Ji et al. (2022); Zhang et al. (2022), we
evaluate all the models based on the generative evaluation metric, Micro F1.

|  | 1-shot | 5-shot
---|---|---|---
| Approach | BC5CDR | NCBI | BC5CDR | NCBI
SLMs | BERT Devlin et al. (2018) | 10.58 | 12.33 | 35.60 | 26.532
ClinicalBERT Huang et al. (2019) | 19.96 | 8.27 | 39.25 | 24.47
NNShot Yang and Katiyar (2020) | 32.96 | 11.82 | 39.30 | 16.22
BioBERT Lee et al. (2020) | 36.78 | 27.19 | 47.04 | 35.88
StructShot Yang and Katiyar (2020) | 16.09 | 4.63 | 30.97 | 13.89
ContaiNER Das et al. (2022) | 37.25 | 16.51 | 41.21 | 26.83
COPNER Huang et al. (2022) | 36.36 | 15.54 | 42.78 | 24.23
ProtoBERT Huang et al. (2022) | 23.61 | 17.24 | 40.58 | 34.18
GatorTron Yang et al. (2022) | 26.97 | 35.00 | 55.44 | 37.64
BINDER Zhang et al. (2023) | 1.86 | 2.95 | 51.79 | 31.95
W-Procer Li et al. (2023) | 40.26 | 38.86 | 56.02 | 40.90
MetaNER Chen et al. (2023) | – | 40.01 | – | 44.92
LLMs | Vanilla ICL Li et al. (2023) | 43.90 | 43.48 | 44.72 | 43.83
PromptNER Ashok and Lipton (2023) | 92.84 | 81.29 | 93.33 | 84.06
GPT-NER* Wang et al. (2023) | 91.28 | 90.44 | 91.56 | 90.44
GPT-NER*(self-ve) Wang et al. (2023) | 91.28 | 90.79 | 91.63 | 90.67

Table 2:  The performance comparison of SLMs and LLMs in BC5CDR, and NCBI.
self-ve refers to self-verification.

####  3.3.2 Large Language Models

We adapt the current strongest
GPT-4333https://platform.openai.com/docs/models/gpt-4 rather than
GPT-3.5444https://platform.openai.com/docs/models/gpt-3-5 in our experiments
out of two primary reasons: (1) According to recent findings by Li et al.
(2023), GPT-4 demonstrates superior performance in few-shot medical named
entity recognition (NER) tasks compared to GPT-3.5. (2) GPT-4 is equipped with
a more extensive set of pre-existing knowledge and a larger number of training
parameters compared to its predecessor, GPT-3.5. By employing the respective
source code, we have implemented a set of robust few-shot named entity
recognition (NER) methods on the BC5CDR and NCBI datasets. All of these
methods are specifically designed based on in-context learning principles, and
GPT-4. Note that: The cost of running these models typically ranges from 6$ to
10$ per run by using the GPT-4.

1) Vanilla ICL Li et al. (2023) utilizes a set of common prompts that include
instructions and demonstrations (examples). These examples are generated by
randomly retrieving demonstrations from a database. 2) PromptNER Ashok and
Lipton (2023) follows a similar sample retrieval strategy as Vanilla ICL Li et
al. (2023). However, PromptNER incorporates the concept of a chain of thought
to enhance entity recognition. 3) GPT-NER*. GPT-NER Wang et al. (2023) uses a
method by retrieving the samples using the KNN method to improve the
performance of in-context learning. In the original GPT-NER implementation,
the authors did not provide distinct symbols for different labels. To address
this, we introduced special symbols in the output of our modified version
called GPT-NER*. We give an example in Figure 2. 4) GPT-NER* (self-
verification) Wang et al. (2023) follows GPT-NER* and adopts a new strategy to
verify the accuracy of the recognition.

Same as Yang and Katiyar (2020); Das et al. (2022); Huang et al. (2022); Ji et
al. (2022); Zhang et al. (2022), we evaluate all the models based on the
generative evaluation metric, Micro F1.

##  4 Comparison Results

We evaluate 16 approaches, 12 SLM-base NER models, and 4 LLM-based NER models
on 2 open medical NER datasets. We first conduct pivot experiments and observe
LLMs without good instructions will reduce the performance of NER. As
indicated in Table 2, the Vanilla ICL exhibit lower performance compared to
MetaNER and W-PROCER on the NCBI 5-shot and BC5CDR 5-shot datasets,
respectively. Conversely, PromptNER and GPT-NER achieve exceptionally high
performance on these datasets when compared to the results of SLMs.

Can we transfer the NER models from the general domain to the medical domain.
The utilization of BERT models trained on general encyclopedic data poses
challenges in accurately identifying medical entities within the medical
domain. Despite various research efforts to redesign NER models like NNshot
and StructShot, traditional SLM in the general domain still struggle to
achieve ideal results in this context. However, pre-training models
specifically on medical datasets, such as BioBERT, have shown promising
performance gains compared to traditional BERT models. Consequently,
researchers have focused on enhancing medical NER performance by pre-training
models in the medical domain, as observed in projects like GatorTron and
MetaNER. This suggests that equipping language models with domain-specific
prior knowledge is crucial for improving NER performance.

SLMs are more susceptible to the amount of training data. From Table 2, we can
see that the quantity of samples has a greater impact on SLMs compared to
LLMs. As an illustration, consider GPT-NER, which demonstrates comparable
performance on both the NCBI 1-shot and 5-shot datasets. However, in the case
of SLMs like MetaMER and ProtoBERT, the performance on the 5-shot dataset
significantly surpasses that of the model performance on the 1-shot dataset.
It shows that the SLMs are more susceptible to the amount of training data.

Handling problems related to partially annotated datasets is comparatively
easier for LLM.

Figure 1: LLM performance on without (w/o) mask operation and with (w/) mask
operation

.

In the study conducted by Li et al. (2023), they experimented by masking
certain labeled entities to simulate a partially annotated dataset. The
authors observed a 3-4 point decrease in NER performance before and after the
mask operation. So in this paper, to test the ability of LLM, we mask the
labeled entities on the 5-shot dataset, to make sure each label just has one
entity, and then compare the performance of LLMs on the masked dataset and
source 1-shot dataset. We use PromptNER in this experiment, and on the dataset
BC5CDR and NCBI. The results are shown in Figure. 1. The experiment revealed
that the partially annotated dataset had a minimal impact on the LLM.

##  5 More Analysis on LLMs

Figure 2: The output format of different ICL methods

###  5.1 Positive and Negative Samples in the Chain-of-Thought

Figure 3: LLM performance on without (w/o) negative samples and with (w/)
negative samples

Certain LLMs, such as PromptNER, which emphasizes the utilization of the chain of thought in the NER task, incorporate a design that includes both positive and negative samples within the instructions. As shown in Figure 2, the positive example is "Apomorphine | True | as it is Chemical" and the negative example is "impact | False | as it is a verb". It is important to evaluate the necessary of using the negative samples. In this work, we set a simple experiment by comparing the model performance with the model that just uses the positive samples. As shown in Figure 3, the results show that the performance of the model using the negative sample and positive samples is better than the model just using the positive samples. However, this evaluation is based on the assumption that the model solely employs a chain-of-thought strategy to enhance its generation output. In more intricate scenarios, further examination and discussion will be discussed in the future.

###  5.2 The Influence of Different Output Examples

Figure 4: (a) refers to the comparison of the chain of thought and tree of
thought. (b) The LLM performance of the chain of thought and tree of thought.

In Figure. 2, we give a detailed example of output on different ICL
strategies, Vanilla ICL, the chain of thought, and the tree of thought Yao et
al. (2023). For the idea of the tree of thought, we design the examples
(samples) in the instruction as shown on the left of Figure 2. We have
identified that the reason for the poorer performance on Vanilla ICL is the
model’s difficulty in predicting the dictionary symbol, such as "$]$",
and"$\\}$". By the comparison result in Figure 4 and Table 2, we found the
chain of thought has a better performance than the tree of thought. The reason
is that the tree of thought is more suitable for the complex reasoning task,
such as mathematical calculation. The impressive results achieved by PromptNER
and GPT-NER also highlight the superiority of the chain of thought and KNN
retrieval method in the context of medical few-shot NER tasks.

###  5.3 LLMs are good for labeling entities, but not good for extracting
entities

| 1-shot | 5-shot
---|---|---
| BC5CDR | NCBI | BC5CDR | NCBI
P1 | 92.84 | 81.29 | 93.33 | 84.06
P2 | 99.06 | 95.92 | 99.06 | 95.92

Table 3:  Few-shot named entity recognition results in BC5CDR, and NCBI. In
P1, the model is the same as the PromptNER approach, while in P2, it assumes
that the PromptNER is aware of all entity mentions present in each sentence.

The NER process consists of two crucial steps: entity recognition and entity
labeling. In our study, as shown in Table 3, we have demonstrated that LLMs
exhibit a strong capability for accurately labeling entities by comparing the
results of P1 and P2. However, we have observed that recognizing the surface
name or identifying the explicit mentions of relevant entities within a
sentence can be challenging for LLMs. This particular finding is being left
for further research.

Figure 5: The overview about our framework RT.

##  6 Our Method

Based on the findings mentioned above, we introduce a novel approach called RT
(Retrieving and Thinking) and present it in Figure 5. The RT method comprises
two primary steps: (1) retrieving the most pertinent examples for the given
test sentence, which are incorporated as part of the instruction in the ICL.
This step is accomplished through the process of Retrieving. (2) guiding LLM
to recognize the entity gradually, demonstrating this progression as Thinking.
In the following sections, we provide a comprehensive explanation of each
component.

| 1-shot | 5-shot
---|---|---
Approach | BC5CDR | NCBI | BC5CDR | NCBI
PromptNER Ashok and Lipton (2023) | 92.84 | 81.29 | 93.33 | 84.06
GPT-NER* Wang et al. (2023) | 91.28 | 90.44 | 91.56 | 90.44
GPT-NER*(self-ve) Wang et al. (2023) | 91.28 | 90.79 | 91.63 | 90.67
RT (Our method) | 93.50 | 91.56 | 93.26 | 91.76

Table 4:  Few-shot medical NER results in BC5CDR, and NCBI. self-ve refers to
self-verification.

###  6.1 Retrieving

As depicted in Figure 5, the process of obtaining relevant examples involves
two distinct steps. The first step is represented by the red line in Figure 5,
while the second step is indicated by the blue line in Figure 5.

In the initial step, we employ a random selection of LLMs to determine the
labels that correspond to the test sentence. For this purpose, we utilize the
Vanilla ICL method proposed by Li et al. (2023). In Figure 5, the relevant
labels about the input sentence are Chemical and Disease. Subsequently, we
proceed to obtain candidate sentences (instances) for each label $l$ from the
development dataset. This is done by considering sentences $S$ that contain an
entity labeled as $l$. So the candidate sentences for label $l$ can be
represented as $l:\\{S_{1}^{l},S_{2}^{l},...,S_{n}^{l}\\}$, where $n$
represents the number of candidate sentences for a given label $l$.

In the subsequent step, we begin by embedding the input sentence $X$ into its
corresponding sentence embedding $\mathbf{X}$. Additionally, we generate
embeddings for the candidate sentences, denoted as
${\mathbf{S_{1}^{l}},\mathbf{S_{2}^{l}},...,\mathbf{S_{n}^{l}}}$. Next, we
calculate the similarity between the sentence embedding $\mathbf{X}$ and each
candidate embedding $\mathbf{S_{n}^{l}}$ for every label $l$. Based on these
similarity scores, we select the top $K$ most similar candidate sentences and
add them to the retrieved sentence set $C_{x}$ for the input sentence. To
ensure that each label has a limited number of entities (either one or five),
we utilize the Greedy Sampling approach to select the final examples from the
retrieved sentence set.

###  6.2 Thinking

In the implementation of this step, we incorporate thinking examples within
the instruction, as depicted in Figure 5. These thinking examples aid the
Language Labeling Model (LLM) in gradually recognizing medical entities.
Specifically, following a similar approach as PromptNER Ashok and Lipton
(2023), the first step involves making a judgment as to whether a particular
entity belongs to the candidate labels. Subsequently, in the second step, the
LLM provides a rationale explaining why the entity is associated with a
specific label.

###  6.3 Main Results

Table 4 presents the experimental results of various approaches based on the
Micro-F1. We have the following observations: (1) RT has an improved
performance on BC5CDR-1-shot, NCBI-1-shot, NCBI-5-shot over the best baselines
GPT-NER and PromptNER, and is almost neck to neck on BC5CDR-5-shot. Overall,
RT outperforms or achieves comparable exact-match F1 performances to the other
methods, demonstrating the effectiveness of our proposed method. (2) Our
method evaluates the effectiveness by retrieving a good example and guiding
the LLM to extract the entity by the right logic. (3) In our method, we
observed that RT achieves better performance on the NCBI dataset when
utilizing only positive examples (Choice 2 in Figure 5). Conversely, for the
BC5CDR dataset, Choice 1 proves to be more effective. These findings indicate
that different datasets have their own distinct examples that yield optimal
performance. Therefore, it is crucial to adapt the choice of examples based on
the specific dataset being addressed. By understanding and leveraging dataset-
specific examples, we can enhance the performance and applicability of the RT
method.

##  7 Error Analysis

We conducted an additional manual analysis to determine the reasons why RT
fails to achieve 100% accuracy in medical entity recognition on the test sets
of BC5CDR and NCBI. This analysis was performed separately for the 1-shot and
5-shot support sets. We have categorized the findings into different
categories, and examples for each category are presented in Table 5.

  * •

Unable to extract entities: Our analysis indicates that RT struggles to
comprehend the semantic information of input sentences and accurately identify
the corresponding entities.

  * •

Misidentification: RT often struggles to recognize and generate the entire
entity, instead only producing a partial representation of the gold entity.

  * •

Class collision: During our analysis, we observed that the test data is only
partially annotated, which prevents the RT gets 100% entity recognition.

  * •

Multi-label entities: During our analysis, we observed that some entities have
more than one label.

  * •

Unable to generate symbols: Despite RT’s ability to extract complex entities
like  severe combined immunodeficiency syndrome, it faces difficulties in
generating special symbols within certain entities. such as the gold entity is
B-cell non-Hodgkin’s lymphoma, the predicted entity is B-cell non-Hodgkin\"s
lymphoma.

  * •

Wrong Template Prediction: Despite being provided with output templates and
examples, RT still generates a considerable number of output errors that are
not based on the provided template.

Dataset | Unable to extract entities:
---|---
NCBI | Test sentence: |  Myotonic dystrophy protein kinase is involved in the modulation…
gold entities |  Myotonic dystrophy, muscular disorder, DM
predicted entities |  DM
| Misidentification:
NCBI | gold entities |  breast malignancies
predicted entity: |  breast
NCBI | gold entities |  non-familial cancers"
predicted entity: |  cancers
BC5CDR | gold entities |  ate-onset scleroderma renal crisis
predicted entity: |  scleroderma renal crisis
| Class collision:
BC5CDR | error description: |  acute cardiotoxicity is not annotated
| Multi-label entities:
NCBI | situation description: |  the entity DM has two labels SpecificDisease and Modifier
| Unable to generate symbols:
BC5CDR | gold entity: |  B-cell non-Hodgkin’s lymphoma
predicted entity: |  B-cell non-Hodgkin\"s lymphoma
| Wrong template prediction:
NCBI | gold template: |  </M>WD|True| as it is Modifier<M>
predicted template: |  </M>WD|True|as it is Modifier<M> cannot be homologous to </M>"

Table 5: The examples of the reason about RT cannot get 100% medical entity
recognition.

##  8 Conclusion

We have conducted an extensive empirical study comparing 16 SLMs and LLMs on
two open-domain medical NER datasets. We show that LLMs are good few-shot
medical NER extractors with good examples and reasonable extraction logical.
Building on some important findings, we propose a RT (Retrieving and Thinking)
method to improve the performance of medical NER. We also found it is true
that LLMs may encounter difficulties in extracting entities with 100%
accuracy. There are several reasons for this, including misidentification,
wrong template prediction, etc.

## Acknowledgements

This work was supported by the National Institutes of Health’s National Center
for Complementary and Integrative Health grant number R01AT009457 and National
Institute on Aging grant number R01AG078154. The content is solely the
responsibility of the authors and does not represent the official views of the
National Institutes of Health.

## References

  * Ashok and Lipton (2023) Dhananjay Ashok and Zachary C Lipton. 2023.  Promptner: Prompting for named entity recognition.  _arXiv preprint arXiv:2305.15444_.
  * Chen et al. (2023) Jiawei Chen, Yaojie Lu, Hongyu Lin, Jie Lou, Wei Jia, Dai Dai, Hua Wu, Boxi Cao, Xianpei Han, and Le Sun. 2023.  Learning in-context learning for named entity recognition.  _arXiv preprint arXiv:2305.11038_.
  * Das et al. (2021) Sarkar Snigdha Sarathi Das, Arzoo Katiyar, Rebecca J Passonneau, and Rui Zhang. 2021\.  Container: Few-shot named entity recognition via contrastive learning.  _arXiv preprint arXiv:2109.07589_.
  * Das et al. (2022) Sarkar Snigdha Sarathi Das, Arzoo Katiyar, Rebecca J Passonneau, and Rui Zhang. 2022\.  Container: Few-shot named entity recognition via contrastive learning.  _arXiv preprint arXiv:2109.07589_.
  * Devlin et al. (2018) Jacob Devlin, Ming-Wei Chang, Kenton Lee, and Kristina Toutanova. 2018.  Bert: Pre-training of deep bidirectional transformers for language understanding.  _arXiv preprint arXiv:1810.04805_.
  * Doğan et al. (2014) Rezarta Islamaj Doğan, Robert Leaman, and Zhiyong Lu. 2014.  Ncbi disease corpus: a resource for disease name recognition and concept normalization.  _Journal of biomedical informatics_ , 47:1–10.
  * Fritzler et al. (2019) Alexander Fritzler, Varvara Logacheva, and Maksim Kretov. 2019.  Few-shot classification in named entity recognition task.  In _Proceedings of the 34th ACM/SIGAPP Symposium on Applied Computing_ , pages 993–1000.
  * Hou et al. (2020) Yutai Hou, Wanxiang Che, Yongkui Lai, Zhihan Zhou, Yijia Liu, Han Liu, and Ting Liu. 2020.  Few-shot slot tagging with collapsed dependency transfer and label-enhanced task-adaptive projection network.  _arXiv preprint arXiv:2006.05702_.
  * Huang et al. (2019) Kexin Huang, Jaan Altosaar, and Rajesh Ranganath. 2019.  Clinicalbert: Modeling clinical notes and predicting hospital readmission.  _arXiv preprint arXiv:1904.05342_.
  * Huang et al. (2022) Yucheng Huang, Kai He, Yige Wang, Xianli Zhang, Tieliang Gong, Rui Mao, and Chen Li. 2022.  Copner: Contrastive learning with prompt guiding for few-shot named entity recognition.  In _Proceedings of the 29th International conference on computational linguistics_ , pages 2515–2527.
  * Ji et al. (2022) Bin Ji, Shasha Li, Shaoduo Gan, Jie Yu, Jun Ma, and Huijun Liu. 2022.  Few-shot named entity recognition with entity-level prototypical network enhanced by dispersedly distributed prototypes.  _arXiv preprint arXiv:2208.08023_.
  * Lee et al. (2020) Jinhyuk Lee, Wonjin Yoon, Sungdong Kim, Donghyeon Kim, Sunkyu Kim, Chan Ho So, and Jaewoo Kang. 2020.  Biobert: a pre-trained biomedical language representation model for biomedical text mining.  _Bioinformatics_ , 36(4):1234–1240.
  * Li et al. (2016) Jiao Li, Yueping Sun, Robin J Johnson, Daniela Sciaky, Chih-Hsuan Wei, Robert Leaman, Allan Peter Davis, Carolyn J Mattingly, Thomas C Wiegers, and Zhiyong Lu. 2016.  Biocreative v cdr task corpus: a resource for chemical disease relation extraction.  _Database_ , 2016.
  * Li et al. (2022) Mingchen Li, Junfan Chen, Samuel Mensah, Nikolaos Aletras, Xiulong Yang, and Yang Ye. 2022.  A hierarchical n-gram framework for zero-shot link prediction.  _arXiv preprint arXiv:2204.10293_.
  * Li and Huang (2023) Mingchen Li and Lifu Huang. 2023.  Understand the dynamic world: An end-to-end knowledge informed framework for open domain entity state tracking.  _arXiv preprint arXiv:2304.13854_.
  * Li and Ji (2022) Mingchen Li and Jonathan Shihao Ji. 2022.  Semantic structure based query graph prediction for question answering over knowledge graph.  _arXiv preprint arXiv:2204.10194_.
  * Li et al. (2023) Mingchen Li, Yang Ye, Jeremy Yeung, Huixue Zhou, Huaiyuan Chu, and Rui Zhang. 2023\.  W-procer: Weighted prototypical contrastive learning for medical few-shot named entity recognition.  _arXiv preprint arXiv:2305.18624_.
  * Li et al. (2020) Mingchen Li, Zili Zhou, and Yanna Wang. 2020.  Multi-fusion chinese wordnet (mcw): Compound of machine learning and manual correction.  _arXiv preprint arXiv:2002.01761_.
  * Liu et al. (2022) Andy T Liu, Wei Xiao, Henghui Zhu, Dejiao Zhang, Shang-Wen Li, and Andrew Arnold. 2022.  Qaner: Prompting question answering models for few-shot named entity recognition.  _arXiv preprint arXiv:2203.01543_.
  * Ma et al. (2023) Yubo Ma, Yixin Cao, YongChing Hong, and Aixin Sun. 2023.  Large language model is not a good few-shot information extractor, but a good reranker for hard samples!  _arXiv preprint arXiv:2303.08559_.
  * Min et al. (2022) Sewon Min, Xinxi Lyu, Ari Holtzman, Mikel Artetxe, Mike Lewis, Hannaneh Hajishirzi, and Luke Zettlemoyer. 2022.  Rethinking the role of demonstrations: What makes in-context learning work?  _arXiv preprint arXiv:2202.12837_.
  * Pugachev et al. (2023) Alexander Pugachev, Ekaterina Artemova, Alexander Bondarenko, and Pavel Braslavski. 2023.  Consumer health question answering using off-the-shelf components.  In _European Conference on Information Retrieval_ , pages 571–579. Springer.
  * Snell et al. (2017) Jake Snell, Kevin Swersky, and Richard Zemel. 2017.  Prototypical networks for few-shot learning.  _Advances in neural information processing systems_ , 30.
  * Stubbs and Uzuner (2015) Amber Stubbs and Özlem Uzuner. 2015.  Annotating longitudinal clinical narratives for de-identification: The 2014 i2b2/uthealth corpus.  _Journal of biomedical informatics_ , 58:S20–S29.
  * Wang et al. (2023) Shuhe Wang, Xiaofei Sun, Xiaoya Li, Rongbin Ouyang, Fei Wu, Tianwei Zhang, Jiwei Li, and Guoyin Wang. 2023.  Gpt-ner: Named entity recognition via large language models.  _arXiv preprint arXiv:2304.10428_.
  * Wei et al. (2022) Jason Wei, Xuezhi Wang, Dale Schuurmans, Maarten Bosma, Ed Chi, Quoc Le, and Denny Zhou. 2022.  Chain of thought prompting elicits reasoning in large language models.  _arXiv preprint arXiv:2201.11903_.
  * Wiseman and Stratos (2019) Sam Wiseman and Karl Stratos. 2019.  Label-agnostic sequence labeling by copying nearest neighbors.  _arXiv preprint arXiv:1906.04225_.
  * Wu et al. (2023) Xuehong Wu, Junwen Duan, Yi Pan, and Min Li. 2023.  Medical knowledge graph: Data sources, construction, reasoning, and applications.  _Big Data Mining and Analytics_ , 6(2):201–217.
  * Yang et al. (2022) Xi Yang, Aokun Chen, Nima PourNejatian, Hoo Chang Shin, Kaleb E Smith, Christopher Parisien, Colin Compas, Cheryl Martin, Anthony B Costa, Mona G Flores, et al. 2022.  A large language model for electronic health records.  _npj Digital Medicine_ , 5(1):194.
  * Yang and Katiyar (2020) Yi Yang and Arzoo Katiyar. 2020.  Simple and effective few-shot named entity recognition with structured nearest neighbor learning.  _arXiv preprint arXiv:2010.02405_.
  * Yao et al. (2023) Shunyu Yao, Dian Yu, Jeffrey Zhao, Izhak Shafran, Thomas L Griffiths, Yuan Cao, and Karthik Narasimhan. 2023.  Tree of thoughts: Deliberate problem solving with large language models.  _arXiv preprint arXiv:2305.10601_.
  * Zhang et al. (2021) Rui Zhang, Dimitar Hristovski, Dalton Schutte, Andrej Kastrin, Marcelo Fiszman, and Halil Kilicoglu. 2021.  Drug repurposing for covid-19 via knowledge graph completion.  _Journal of biomedical informatics_ , 115:103696.
  * Zhang et al. (2022) Sheng Zhang, Hao Cheng, Jianfeng Gao, and Hoifung Poon. 2022.  Optimizing bi-encoder for named entity recognition via contrastive learning.  _arXiv preprint arXiv:2208.14565_.
  * Zhang et al. (2023) Sheng Zhang, Hao Cheng, Jianfeng Gao, and Hoifung Poon. 2023.  Optimizing bi-encoder for named entity recognition via contrastive learning.  _arXiv preprint arXiv:2208.14565_.
  * Zheng et al. (2023) Kai Zheng, Xin-Lu Zhang, Lei Wang, Zhu-Hong You, Bo-Ya Ji, Xiao Liang, and Zheng-Wei Li. 2023.  Sprda: a link prediction approach based on the structural perturbation to infer disease-associated piwi-interacting rnas.  _Briefings in Bioinformatics_ , 24(1):bbac498.