Boundary behavior of Hardy spaces on angular domains

Abstract

The aim of this paper is to characterize the behavior of boundary limits of functions in Hardy space on angular domains. We investigate that the Cauchy integral of a function $f\in L^{p}(\partial D_a)$ is in $H^{p}(D_a).$ We also prove that the functions in $H^{p}(D_a)$ are the Cauchy integral of their non-tangential boundary limits. In addition, we establish the orthogonality of non-tangential boundary limits of functions in $H^{p}(D_a)$ and $H^{q}(D_a)$.

Appendix: The proof of Lemma 2.6

For the convenience of readers, we restate the proof of Lemma 2.6 in this appendix. In [2], the proof is divided into two cases, namely, $1<p\le 2$ and $2\le p<\infty$.

We begin with the case $1<p\le 2$. Assume that f is a nonnegative function with compact support and $\{t:f(t)\ne 0\}\subseteq [-a,a]$. For simplicity, we denote

$$\begin{aligned} F(z):=Cf(z)=u+iv. \end{aligned}$$

We have $u>0$ and $u^{p}, |F|^{p}\in C^{2}({\mathbb {C}}_{+})$. Notice that

$$\begin{aligned} 2\frac{\partial u}{\partial z}=F^{'}, \end{aligned}$$

we thus obtain

$$\begin{aligned} \Delta u^{p}=p(p-1)|F'|^{2}u^{p-2}\ \ \text{ and } \ \ \Delta |F|^{p}=p^{2}|F'|^{2}|F|^{p-2}. \end{aligned}$$

Let

$$\begin{aligned} h(z)=\frac{p}{p-1}(u(z))^{p}-|F(z)|^{p}, \end{aligned}$$

then $h\in C^{2}({\mathbb {C}}_{+})$ and

$$\begin{aligned} \Delta h(z)=p^{2}|F'|^{2}(u^{p-2}-|F|^{p-2})\ge 0, \end{aligned}$$

which mean h(z) is subharmonic on ${\mathbb {C}}_{+}.$ According to the Minkowski inequality, we have

$$\begin{aligned} \int _{-\infty }^{+\infty }|h(x+iy)|^{p}dx&\le \frac{2p}{p-1}\Vert F_{y}\Vert _{L^p}^p\\&\le \frac{2p}{p-1}\left( \int _{-\infty }^{+\infty }|f(t)|\left( \int _{-\infty }^{+\infty } \frac{1}{|t-x-iy|^p}dx\right) ^{\frac{1}{p}}dt\right) ^p\\&\le (2a)^{\frac{p}{q}}\frac{2p}{p-1}\frac{1}{y^{\frac{p}{q}}}\Vert f\Vert _{L^{p}}^{p} \int _{-\infty }^{+\infty }\frac{1}{(1+s^2)^{\frac{p}{2}}}ds\\&< \infty , \end{aligned}$$

where $\frac{1}{p}+\frac{1}{p}=1$. Therefore, the function

$$\begin{aligned} \int _{-\infty }^{+\infty }|h(x+iy)|^{p}dx, y\in (0,\infty ) \end{aligned}$$

decreases monotonously to 0 as y tends to $\infty$. This implies

$$\begin{aligned} \frac{p}{p-1}\int _{-\infty }^{+\infty }(u(x+iy))^{p}dx\ge \int _{-\infty }^{+\infty }|F(x+iy)|^{p}dx=\Vert F_y\Vert _{L^{p}}^{p}. \end{aligned}$$

Moreover,

$$\begin{aligned} \frac{p}{p-1}\Vert f\Vert _{L^{p}}^{p}\ge \Vert Cf\Vert _{H^{p}_1}^{p}. \end{aligned}$$

Let $1<p\le 2$ and f be the function with compact support and $f\not \equiv 0$. Suppose that $\{t:f(t)\ne 0\}\subseteq [-a,a]$. Notice that

$$\begin{aligned} f(z)=P_y*f(x)=P_y*f^{+}(x)-P_y*f^{-}(x), \end{aligned}$$

where $P_y(x)=\frac{1}{\pi }\frac{y}{x^2+y^2}$ is the Poisson kernel on ${\mathbb {C}}_+,$$f^+=\max \{f,0\},$$f^-=\max \{0, -f\}$ and $f*g(x)=\int _{\infty }^{+\infty }f(x-t)g(t)dt.$ Thus both $P_y*f^{+}$ and $P_y*f^{-}$ are nonnegative harmonic functions on ${\mathbb {C}}_+$ and

$$\begin{aligned} Cf(z)=Cf^+(z)-Cf^-(z). \end{aligned}$$

Therefore,

$$\begin{aligned} \Vert Cf\Vert _{H^{p}}&\le \Vert Cf^+\Vert _{H^{p}}+\Vert Cf^-\Vert _{H^{p}}\\&\le \left( \frac{p}{p-1}\right) ^{\frac{1}{p}}(\Vert f^+\Vert _{L^{p}}+\Vert f^-\Vert _{L^{p}})\\&\le 2\left( \frac{p}{p-1}\right) ^{\frac{1}{p}}\Vert f\Vert _{L^{p}}. \end{aligned}$$

Now let $1<p\le 2,$$f\in L^{p}$ and $f\not \equiv 0$. We consider the following cut off functions

$$\begin{aligned} f_{n}(t)=\left\{ \begin{aligned}&f(t), \quad |t|\le n,\\&0, \qquad \ |t|> n. \end{aligned} \right. \end{aligned}$$

We thus obtain

$$\begin{aligned} \Vert f_n-f\Vert _{L^{p}}\rightarrow 0,\quad (n\rightarrow \infty ) \end{aligned}$$

and

$$\begin{aligned} \Vert Cf_n(\cdot +iy)-Cf(\cdot +iy)\Vert _{L^{p}}\rightarrow 0,\quad (n\rightarrow \infty ). \end{aligned}$$

According to the Fatou lemma, we have

$$\begin{aligned} \int _{-\infty }^{+\infty }|Cf(x+iy)|^{p}dx&\le \liminf _{n\rightarrow \infty } \int _{-\infty }^{+\infty }|Cf_{n}(x+iy)|^{p}dx\\&\le \liminf _{n\rightarrow \infty } \frac{p2^p}{p-1}\Vert f_{n}\Vert _{L^{p}}^{p}\\&=\frac{p2^p}{p-1}\Vert f\Vert _{L^{p}}^{p}. \end{aligned}$$

Now we complete the proof of Lemma 2.6 for the case $1<p\le 2.$

Let $2<p<\infty ,$ it is not hard to verify that Cf(z) is holomorphic on ${\mathbb {C}}_{+}$ for $f\in L^{p}({\mathbb {R}}).$ For any $y>0$, we denote

$$\begin{aligned} L_{y}(g)=\int _{-\infty }^{+\infty }g(x)\overline{Cf(x+iy)}dx,\ \ g\in L^{q}({\mathbb {R}}). \end{aligned}$$

We thus have

$$\begin{aligned} L_{y}(g)=\int _{-\infty }^{+\infty }\overline{f(t)}Cg(t+iy)dt,\ \ g\in L^{q}({\mathbb {R}}). \end{aligned}$$

Therefore,

$$\begin{aligned} |L_{g}|\le \Vert Cg\Vert _{L^{q}}\Vert f\Vert _{L^{p}}\le 2\left( \frac{p}{p-1}\right) ^{\frac{1}{q}}\Vert f\Vert _{L^{p}}\Vert g\Vert _{L^{q}}. \end{aligned}$$

As a consequence,

$$\begin{aligned} \Vert L_{y}\Vert ^p\le \int _{-\infty }^{+\infty }|Cf(x+iy)|^pdx \le 2^p\left( \frac{p}{p-1}\right) ^{\frac{p}{q}}\Vert f\Vert _{L^{p}}^{p} =2^pp^{p-1}\Vert f\Vert _{L^{p}}^{p}. \end{aligned}$$

We complete the proof of Lemma 2.6.