Denote by $ {\mathbb T} $ the boundary of the open unit disk $ {\mathbb{D}} $ in the complex plane $ {\mathbb{C}} $. Let $ H({\mathbb{D}}) $ be the space of analytic functions in $ {\mathbb{D}} $. For $ 0<p<\infty $, the Hardy space $ H^p({\mathbb{D}}) $ consists of functions $ f\in H({\mathbb{D}}) $ such that

$ \|f\|_{H^p} = \sup\limits_{0<r<1} \left( {\frac{1}{{2\pi }}\int_0^{2\pi } | f(r{e^{i\theta }}){|^p}d\theta } \right)^{1/p} < \infty $

Let $ H^\infty $ be the space of bounded analytic function on $ {\mathbb{D}} $ consisting of functions $ f\in H({\mathbb{D}}) $ with

$ \|f\|_\infty = \sup\limits_{z\in{\mathbb{D}}}|f(z)|<\infty. $

We refer to [1, 2] for $ H^p $ and $ H^\infty $ spaces.

For $ \lambda\in (0, 1] $, denote by $ {\mathcal L}^{2, \lambda}({\mathbb T}) $ the Morrey space of all Lebesgue measurable functions $ f $ on $ {\mathbb T} $ that satisfy

$ \parallel f{\parallel _{{\mathcal{L}^{2,\lambda }}(\mathbb{T})}} = \mathop {\sup }\limits_{I \subseteq \mathbb{T}} {\left( {|I{|^{ - \lambda }}\int_I | f(\zeta ) - {f_I}{|^2}|d\zeta |} \right)^{\frac{1}{2}}} < \infty , $

where $ |I| $ denotes the length of the arc $ I $ and

${f_I} = \frac{1}{{|I|}}\int_I f (\zeta )|d\zeta |.$

Clearly, $ {\mathcal L}^{2, 1}({\mathbb T}) $ coincides with $ {\rm BMO}({\mathbb T}) $, the space of functions with bounded mean oscillation on $ {\mathbb T} $ (cf. [3, 4]). Similar to a norm on $ {\rm BMO}({\mathbb T}) $ given in [4, p. 68], a norm on $ {\mathcal L}^{2, \lambda}({\mathbb T}) $ can be defined by

$ |||f|||_{{\mathcal L}^{2, \lambda}({\mathbb T})} = \left|\int_{\mathbb T} f |d\zeta|\right|+\|f\|_{{\mathcal L}^{2, \lambda}({\mathbb T})}. $

From Xiao's monograph [5, p. 52],

$ {\rm BMO}({\mathbb T})\subseteq {\mathcal L}^{2, \lambda_1}({\mathbb T})\subseteq {\mathcal L}^{2, \lambda_2}({\mathbb T}) \subseteq L^2({\mathbb T}), \ \ 0<\lambda_2<\lambda_1<1. $

It is well known that if $ f\in H^2 $, then its non-tangential limit $ f(\zeta) $ exists almost everywhere for $ \zeta\in {\mathbb T} $. For $ \lambda\in (0, 1] $, the analytic Morrey space $ {\mathcal L}^{2, \lambda}({\mathbb{D}}) $ is the set of $ f\in H^2 $ with $ f(\zeta)\in {\mathcal L}^{2, \lambda}({\mathbb T}) $. It is clear that $ {\mathcal L}^{2, 1}({\mathbb{D}}) $ is $ {\rm BMOA} $, the analytic space of functions with bounded mean oscillation (cf. [3, 4]). For $ \lambda\in (0, 1] $, $ {\mathcal L}^{2, \lambda}({\mathbb{D}}) $ is located between $ {\rm BMOA} $ and $ H^2 $. It is worth mentioning that there exists a isomorphism relation between analytic Morrey spaces and Möbius invariant $ {\mathcal{Q}}_p $ spaces via fractional order derivatives of functions (see [6]). Recall that for $ 0< p<\infty $, a function $ f\in H({\mathbb{D}}) $ belongs to the space $ {{\mathcal{Q}}_p} $ if

$ \sup\limits_{a\in{\mathbb{D}}}\, \int_{{\mathbb{D}}} |f'(z)|^{2}\left(1-|\sigma_a(z)|^2\right)^pdA(z)<\infty, $

where $ dA $ is the area Lebesgue measure on $ {\mathbb{D}} $ and ${\sigma _a}(z) = \frac{{a - z}}{{1 - \bar az}}$ is the Möbius transformation of the unit disk $ {\mathbb{D}} $ interchanging $ a $ and $ 0 $. See [5,7 ] for a general exposition on $ {\mathcal{Q}}_p $ spaces. Recently, the interest in $ {\mathcal L}^{2, \lambda}({\mathbb{D}}) $ spaces grew rapidly (cf. [8-13]).

An important problem of studying function spaces is to characterize the multipliers of such spaces. For a Banach function space $ X $, denote by $ M(X) $ the class of all multipliers on $ X $. Namely,

$ M(X) = \{f\in X: fg\in X \ \text{for all }\ g\in X\}. $

Bao and Pau [14] characterized boundary multipliers of $ {\mathcal{Q}}_p $ spaces. Stegenga [15] described multipliers of $ {\rm BMO}({\mathbb T}) $ which is equal to $ {\mathcal L}^{2, 1}({\mathbb T}) $. It is natural to consider multipliers of $ {\mathcal L}^{2, \lambda}({\mathbb T}) $ with $ \lambda \in (0, 1) $ in this paper.

Given a function $ \varphi\in L^{2}({\mathbb T}) $. Let $ T_{\varphi} $ be the Toeplitz operator on $ H^2 $ with symbol $ \varphi $ defined by

$ T_{\varphi}f(z) = \frac{1}{2\pi }\int_{{\mathbb T}}\frac{\varphi(\zeta)f(\zeta)}{1-\overline{\zeta}z}|d\zeta|, \ \ f\in H^2, \ \ z\in {\mathbb{D}}. $

For the study of Toeplitz operators on Hardy spaces and Bergman spaces, see, for example, [16, 17]. We refer to [9] for the results of Toeplitz operators on $ {\mathcal L}^{2, \lambda}({\mathbb{D}}) $ spaces.

The aim of this paper is to consider boundary multiplies and Toeplitz operators associated with analytic Morrey spaces. In Section 2, using a characterization of $ {\mathcal L}^{2, \lambda}({\mathbb T}) $ in terms of functions with absolute values, we characterize the multipliers of $ {\mathcal L}^{2, \lambda}({\mathbb T}) $. In Section 3, we characterize the boundedness and compactness of Toeplitz operators from Hardy spaces to analytic Morrey spaces.

Throughout this paper, we write $ a\lesssim b $ if there exists a constant $ C $ such that $ a\leq Cb $. Also, the symbol $ a\thickapprox b $ means that $ a\lesssim b\lesssim a $.

By the study of certain integral operators on analytic Morrey spaces, Li, Liu and Lou [8] proved that $ M(\mathcal{L}^{2, \lambda}({\mathbb{D}})) = H^\infty $. In this section, applying a characterization of $ \mathcal{L}^{2, \lambda}({\mathbb T}) $ in terms of absolute values of functions, we characterize $ M(\mathcal{L}^{2, \lambda}({\mathbb T})) $, boundary multiplies of analytic Morrey spaces.

Given $ f\in L^2({\mathbb T}) $, let $ \widehat{f} $ be the Poisson extension of $ f $. Namely,

$ \widehat{f}(z) = \int_{{\mathbb T}} f(\zeta)d\mu_z(\zeta), \ z\in {\mathbb{D}}, $

where

$ d\mu_z(\zeta) = \frac{1-|z|^2}{2\pi|\zeta-z|^2}|d\zeta|. $

Let $ 0<\lambda<1 $. From [5, p.52], $ f\in \mathcal{L}^{2, \lambda}({\mathbb T}) $ if and only if

$ \begin{equation} \sup\limits_{z\in {\mathbb{D}}}(1-|z|^{2})^{1-\lambda}\int_{\mathbb{D}} |\nabla \widehat{f}(w)|^2 (1-|\sigma_z(w)|^2) dA(w)<\infty, \end{equation} $

(2.1)

where $ \nabla $ is the Laplace operator. Also, $ f\in \mathcal{L}^{2, \lambda}({\mathbb{D}}) $ if and only if

$ \begin{equation} \|f\|_{\mathcal{L}^{2, \lambda}({\mathbb{D}})} = \sup\limits_{z\in {\mathbb{D}}}(1-|z|^{2})^{1-\lambda}\int_{\mathbb{D}} |f'(w)|^2 (1-|\sigma_z(w)|^2) dA(w)<\infty. \end{equation} $

(2.2)

We need the following useful inequality (see [18, Lemma 2.5]).

Lemma A Suppose that $ s>-1 $, $ r $, $ t\geq 0 $, and $ r+t-s>2 $. If $ t<s+2<r $, then

$ \int_{\mathbb{D}}\frac{(1-|w|^2)^s}{|1-\overline{w}z|^r |1-\overline{w}\zeta|^t}dA(w)\lesssim \frac{(1-|z|^{2})^{2+s-r}}{|1-\overline{z}\zeta|^{t}} $

for all $ z $, $ \zeta\in {\mathbb{D}} $.

Now we characterize $ \mathcal{L}^{2, \lambda}({\mathbb T}) $ via absolute values of functions as follows. See [9] for the analytic version of the following result.

Theorem 2.1 Let $ 0<\lambda<1 $ and $ f\in L^2({\mathbb T}) $. Then $ f\in \mathcal{L}^{2, \lambda}({\mathbb T}) $ if and only if

$ \sup\limits_{a\in \mathbb{D}}(1-|a|^{2})^{1-\lambda}\int_{\mathbb{D}}\int_{{\mathbb T}}|f(\zeta)|^{2}d\mu_{z}(\zeta)|\sigma_{a}'(z)|^{2}dA(z)<\infty. $

Proof Let $ f\in L^2({\mathbb T}) $. It is well known (cf. [19, p. 564]) that

$ \int_{{\mathbb T}}|f(\zeta)|^2d\mu_z(\zeta)-|\widehat{f}(z)|^2\approx \int_{\mathbb{D}} |\nabla \widehat{f}(w)|^2 (1-|\sigma_z(w)|^2) dA(w) $

for all $ z\in {\mathbb{D}} $. Combining this with the Fubini theorem, we obtain that for any $ a\in {\mathbb{D}} $,

$ \begin{eqnarray*} &\; &\int_{\mathbb{D}} \int_{{\mathbb T}}|f(\zeta)|^2d\mu_z(\zeta) |\sigma_{a}'(z)|^{2}dA(z)-\int_{\mathbb{D}} |\widehat{f}(z)|^2 |\sigma_{a}'(z)|^{2}dA(z)\\ &\approx& \int_{\mathbb{D}} |\nabla \widehat{f}(w)|^2 \int_{\mathbb{D}} (1-|\sigma_z(w)|^2) |\sigma_{a}'(z)|^{2}dA(z)dA(w)\\ &\approx& (1-|a|^2)^2 \int_{\mathbb{D}} |\nabla \widehat{f}(w)|^2 (1-|w|^2) \int_{\mathbb{D}} \frac{1-|z|^2}{|1-\overline{z}w|^2|1-\overline{a}z|^4}dA(z)dA(w). \end{eqnarray*} $

By Lemma A and the same argument in [19, p. 563], we get that

$ \int_{\mathbb{D}} \frac{1-|z|^2}{|1-\overline{z}w|^2|1-\overline{a}z|^4}dA(z)\approx \frac{1}{(1-|a|^2)|1-\overline{a}w|^2}. $

Thus,

$ \begin{eqnarray} &\; &\int_{\mathbb{D}} \int_{{\mathbb T}}|f(\zeta)|^2d\mu_z(\zeta) |\sigma_{a}'(z)|^{2}dA(z)-\int_{\mathbb{D}} |\widehat{f}(z)|^2 |\sigma_{a}'(z)|^{2}dA(z)\\ &\approx& \int_{\mathbb{D}} |\nabla \widehat{f}(w)|^2 \frac{(1-|a|^2)(1-|w|^2)}{|1-\overline{a}w|^2}dA(w)\\ &\approx& \int_{\mathbb{D}} |\nabla \widehat{f}(w)|^2 (1-|\sigma_a(w)|^2) dA(w) \end{eqnarray} $

(2.3)

for all $ a\in {\mathbb{D}} $.

Let

$ \sup\limits_{a\in \mathbb{D}}(1-|a|^{2})^{1-\lambda}\int_{\mathbb{D}}\int_{{\mathbb T}}|f(\zeta)|^{2}d\mu_{z}(\zeta)|\sigma_{a}'(z)|^{2}dA(z)<\infty. $

From (1) and (3), we get that $ f\in\mathcal{L}^{2, \lambda}({\mathbb T}) $.

On the other hand, let $ f\in\mathcal{L}^{2, \lambda}({\mathbb T}) $. Without loss of generality, we may assume that $ f $ is real valued. Denote by $ \widetilde{f} $ the harmonic conjugate function of $ \widehat{f} $. Set $ g = \widehat{f}+i\widetilde{f} $. The Cauchy-Riemann equations give $ |\nabla \widehat{f}(z)|\thickapprox |g'(z)| $. Thus $ g\in \mathcal{L}^{2, \lambda}({\mathbb{D}}) $. By the growth estimates of functions in $ \mathcal{L}^{2, \lambda}({\mathbb{D}}) $ (cf. [8, Lemma 2]), one gets that

$ |\widehat{f}(z)|\leq |g(z)|\lesssim (1-|z|)^{\frac{\lambda-1}{2}}\|g\|_{\mathcal{L}^{2, \lambda}({\mathbb{D}})} $

for all $ z\in {\mathbb{D}} $. Consequently,

$ \begin{eqnarray*} &\; &\sup\limits_{a\in \mathbb{D}}(1-|a|^{2})^{1-\lambda}\int_{\mathbb{D}} |\widehat{f}(z)|^2 |\sigma_{a}'(z)|^{2}dA(z)\\ &\lesssim& \|g\|_{\mathcal{L}^{2, \lambda}({\mathbb{D}})}^2 \sup\limits_{a\in \mathbb{D}}(1-|a|^{2})^{3-\lambda} \int_{\mathbb{D}} \frac{(1-|z|)^{\lambda-1}}{|1-\overline{a}z|^4}dA(z)\\ &\lesssim& \|g\|_{\mathcal{L}^{2, \lambda}({\mathbb{D}})}^2. \end{eqnarray*} $

Combining this with (2.3), $ f\in\mathcal{L}^{2, \lambda}({\mathbb T}) $, we get that

$ \sup\limits_{a\in \mathbb{D}}(1-|a|^{2})^{1-\lambda}\int_{\mathbb{D}}\int_{{\mathbb T}}|f(\zeta)|^{2}d\mu_{z}(\zeta)|\sigma_{a}'(z)|^{2}dA(z)<\infty. $

The proof is completed.

Denote by $ L^\infty({\mathbb T}) $ the space of essentially bounded measurable functions on $ {\mathbb T} $. Using Theorem 2.1, we characterize multipliers of $ \mathcal{L}^{2, \lambda}({\mathbb T}) $ as follows.

Theorem 2.2 Let $ 0<\lambda<1 $. Then $ M(\mathcal{L}^{2, \lambda}({\mathbb T})) = L^\infty({\mathbb T}) $.

Proof Let $ f\in L^\infty({\mathbb T}) $ and $ g\in \mathcal{L}^{2, \lambda}({\mathbb T}) $. From Theorem 2.1, one gets that

$ \begin{eqnarray*} &\; &\sup\limits_{a\in \mathbb{D}}(1-|a|^{2})^{1-\lambda}\int_{\mathbb{D}}\int_{{\mathbb T}}|f(\zeta)g(\zeta)|^{2}d\mu_{z}(\zeta)|\sigma_{a}'(z)|^{2}dA(z)\\ &\leq& \|f\|_{L^\infty({\mathbb T})}\sup\limits_{a\in \mathbb{D}}(1-|a|^{2})^{1-\lambda}\int_{\mathbb{D}}\int_{{\mathbb T}}|g(\zeta)|^{2}d\mu_{z}(\zeta)|\sigma_{a}'(z)|^{2}dA(z)\\ &<&\infty. \end{eqnarray*} $

Applying Theorem again, we know that $ fg\in \mathcal{L}^{2, \lambda}({\mathbb T}) $. Thus $ L^\infty({\mathbb T})\subseteq M(\mathcal{L}^{2, \lambda}({\mathbb T})) $.

On the other hand, let $ f\in M(\mathcal{L}^{2, \lambda}({\mathbb T})) $. By the closed graph theorem, there exists a positive constant $ C $ such that $ |||fg|||_{\mathcal{L}^{2, \lambda}({\mathbb T})}\leq C |||g|||_{\mathcal{L}^{2, \lambda}({\mathbb T})} $ for any $ g\in \mathcal{L}^{2, \lambda}({\mathbb T}) $. Set $ h = f/C $. Clearly, $ h\in \mathcal{L}^{2, \lambda}({\mathbb T}) $. We deduce that $ |||h^n|||_{\mathcal{L}^{2, \lambda}({\mathbb T})}\leq |||h|||_{\mathcal{L}^{2, \lambda}({\mathbb T})} $ for all positive integer $ n $. As mentioned in Section 1, $ \mathcal{L}^{2, \lambda}({\mathbb T}) \subseteq L^2({\mathbb T}) $. Form the closed graph theorem again, there exists a positive constant $ C_1 $ satisfying

$ \left(\int_{\mathbb T} |f(\zeta)|^2 |d\zeta|\right)^{1/2}\leq C_1 |||f|||_{\mathcal{L}^{2, \lambda}({\mathbb T})} $

for all $ f\in \mathcal{L}^{2, \lambda}({\mathbb T}) $. Consequently,

$ \left(\int_{\mathbb T} |h^n(\zeta)|^2 |d\zeta|\right)^{1/2}\leq C_1 |||h^n|||_{\mathcal{L}^{2, \lambda}({\mathbb T})}\leq C_1|||h|||_{\mathcal{L}^{2, \lambda}({\mathbb T})}<\infty. $

Since $ n $ is arbitrary, we know that $ h\in L^\infty({\mathbb T}) $. Hence $ f\in L^\infty({\mathbb T}) $. The proof is completed.

In this section, we characterize the boundedness and compactness of Toeplitz operators from the Hardy space $ H^p $ to the analytic Morrey space $ \mathcal{L}^{2, 1-\frac{2}{p}}(\mathbb{D}) $ for $ 2<p<\infty $. Toeplitz operators on analytic Morrey spaces were investigated in [9]. We refer to [8] for the study of some integral operators from $ H^p $ to $ \mathcal{L}^{2, 1-\frac{2}{p}}(\mathbb{D}) $ for $ 2<p<\infty $.

Following [9], we use a norm of $ {\mathcal L}^{2, \lambda}({\mathbb{D}}) $, $ \lambda\in (0, 1) $, defined by

$ |||f|||_{\mathcal{L}^{2, \lambda}({\mathbb{D}})} = |f(0)|+\sup\limits_{w\in \mathbb{D}}\Big((1-|w|^{2})^{1-\lambda}\int_{\mathbb{D}}\int_{{\mathbb T}}|f(\zeta)|^{2}d\mu_{z}(\zeta)|\sigma_{w}'(z)|^{2}dA(z)\Big)^{1/2}. $

The following well-known lemma can be found in [17].

Lemma B Suppose $ s>0 $ and $ t>-1 $. Then there exists a positive constant $ C $ such that

$ \int_\mathbb{D} {\frac{{{{(1 - |w{|^2})}^t}}}{{|1 - \bar zw{|^{2 + t + s}}}}} dA(w) \leqslant \frac{C}{{{{(1 - |z{|^2})}^s}}} $

for all $ z\in {\mathbb{D}} $.

Applying some well-known results of Toeplitz operators and composition operators on Hardy spaces, we characterize the boundedness of $ T_{\varphi} $ from Hardy spaces to analytic Morrey spaces as follows.

Theorem 3.3 Let $ 2<p<\infty $ and $ \varphi \in L^2({\mathbb T}) $. Then the Toeplitz operator $ T_{\varphi} $ is bounded from $ H^p $ to $ \mathcal{L}^{2, 1-\frac{2}{p}}({\mathbb T}) $ if and only if $ \varphi\in L^{\infty}({\mathbb T}) $.

Proof Suppose that $ T_{\varphi} $ is bounded from $ H^p $ to $ \mathcal{L}^{2, 1-\frac{2}{p}}({\mathbb{D}}) $. For $ b\in {\mathbb{D}} $, let

$ f_{b}(z) = \frac{(1-|b|^{2})^{1-\frac{1}{p}}}{1-\overline{b}z}, \ \ z \in {\mathbb{D}}. $

Note that $ p>2 $. By the well known estimates in [20, p. 9], one gets that

$ \mathop {\sup }\limits_{b \in \mathbb{D}} \int_0^{2\pi } | {f_b}({e^{i\theta }}){|^p}d\theta = \mathop {\sup }\limits_{b \in \mathbb{D}} {(1 - |b{|^2})^{p - 1}}\int_0^{2\pi } {\frac{1}{{|1 - \bar b{e^{i\theta }}{|^p}}}} d\theta<\infty. $

Thus functions $ f_{b} $ belong to $ H^{p} $ uniformly for all $ b\in {\mathbb{D}} $. Consequently,

$ \begin{equation} \nonumber \begin{split} \infty>|||T_{\varphi}f_{b}|||_{\mathcal{L}^{2, 1-\frac{2}{p}}({\mathbb{D}})}^{2}&\gtrsim (1-|b|^{2})^{\frac{2}{p}}\int_{\mathbb{D}}\int_{{\mathbb T}}|T_{\varphi}f_{b}(\zeta)|^{2}d\mu_{z}(\zeta)|\sigma_{b}'(z)|^{2}dA(z) \\&\approx(1-|b|^{2})^{\frac{2}{p}}\int_{\mathbb{D}}\int_{{\mathbb T}}|T_{\varphi}f_{b}(\zeta)|^{2}d\mu_{\sigma_{b}(z)}(\zeta)dA(z) \\&\gtrsim(1-|b|^{2})^{\frac{2}{p}}\int_{\mathbb{D}}|T_{\varphi}f_{b}(\sigma_{b}(z))|^{2}dA(z) \\&\gtrsim(1-|b|^{2})^{\frac{2}{p}}|T_{\varphi}f_{b}(b)|^{2}. \end{split} \end{equation} $

Note that

$ (1-|b|^{2})^{\frac{2}{p}}|T_{\varphi}f_{b}(b)|^{2} = \frac{1}{4\pi^2}\left|\int_{\mathbb T} \frac{\varphi(\zeta)(1-|b|^2)}{(1-\overline{\zeta}b)(1-\overline{b}\zeta)}|d\zeta|\right|^2 = |\hat{\varphi}(b)|^{2}. $

Thus $ \sup\limits_{b\in {\mathbb{D}}}|\hat{\varphi}(b)|<\infty $. By [1, p. 5], $ \varphi\in L^\infty({\mathbb T}) $.

On the other hand, let $ \varphi\in L^{\infty}({\mathbb T}) $. It is well known that $ T_{\varphi} $ is bounded on $ H^{p} $ (cf. [21-23]). Namely, $ \|T_{\varphi}g\|_{H^p}\lesssim \|\varphi\|_{L^\infty({\mathbb T})}\|g\|_{H^p} $ for all $ g\in H^p $. Let $ f\in H^p $, we deduce that

$ \begin{equation} \nonumber \begin{split} &|||T_{\varphi}f|||_{\mathcal{L}^{2, 1-\frac{2}{p}}({\mathbb{D}})}\\ = & |T_{\varphi}f(0)|+\sup\limits_{w\in \mathbb{D}}\Big((1-|w|^{2})^{\frac{2}{p}}\int_{\mathbb{D}}\int_{{\mathbb T}}\Big|\frac{1}{2\pi} \int_{{\mathbb T}}\frac{\varphi(\zeta)f(\zeta)}{1-\overline{\zeta}\xi}|d\zeta|\Big|^{2}d\mu_{z}(\xi) |\sigma'_w(z)|^{2} dA(z)\Big)^{1/2} \\\lesssim& \|\varphi\|_{L^{\infty}({\mathbb T})}\|f\|_{H^{1}}+\sup\limits_{w\in \mathbb{D}}\Big((1-|w|^{2})^{\frac{2}{p}}\int_{\mathbb{D}}\int_{{\mathbb T}}\Big|\frac{1}{2\pi} \int_{{\mathbb T}}\frac{\varphi(\zeta)f(\zeta)}{1-\overline{\zeta}\sigma_{z}(\xi)}|d\zeta|\Big|^{2}|d\xi| |\sigma'_w(z)|^{2}dA(z) \Big)^{1/2} \\\lesssim& \|\varphi\|_{L^{\infty}({\mathbb T})}\|f\|_{H^{p}}+ \sup\limits_{w\in \mathbb{D}}\Big((1-|w|^{2})^{\frac{2}{p}}\int_{\mathbb{D}}\|(T_{\varphi}f)\circ\sigma_{z}\|_{H^{2}}^{2} |\sigma'_w(z)|^{2}dA(z) \Big)^{1/2} \\\lesssim& \|\varphi\|_{L^{\infty}({\mathbb T})}\|f\|_{H^{p}}+ \sup\limits_{w\in \mathbb{D}}\Big((1-|w|^{2})^{\frac{2}{p}}\int_{\mathbb{D}}\|(T_{\varphi}f)\circ\sigma_{z}\|_{H^{p}}^{2} |\sigma'_w(z)|^{2}dA(z) \Big)^{1/2}. \end{split} \end{equation} $

By the well-known characterization of composition operators on $ H^p $ (cf. [17, Theorem 11.12]), we get that

$ \|(T_{\varphi}f)\circ\sigma_{z}\|_{H^{p}}\leq \left(\frac{1+|z|}{1-|z|}\right)^{1/p}\|T_{\varphi}f\|_{H^{p}}. $

Thus,

$ \begin{eqnarray*} &&|||T_{\varphi}f|||_{\mathcal{L}^{2, 1-\frac{2}{p}}({\mathbb{D}})}\\&\lesssim& \|\varphi\|_{L^{\infty}({\mathbb T})}\|f\|_{H^{p}}+ \sup\limits_{w\in \mathbb{D}}\Big((1-|w|^{2})^{\frac{2}{p}}\|T_{\varphi}f\|_{H^{p}}^{2}\int_{\mathbb{D}} (1-|z|)^{-2/p} |\sigma'_w(z)|^{2}dA(z) \Big)^{1/2}\\ &\lesssim& \|\varphi\|_{L^{\infty}({\mathbb T})}\|f\|_{H^{p}}+\|\varphi\|_{L^{\infty}({\mathbb T})}\|f\|_{H^p} \sup\limits_{w\in \mathbb{D}}\Big((1-|w|^{2})^{\frac{2}{p}+2}\int_{\mathbb{D}} \frac{(1-|z|)^{-2/p}} {|1-\overline{w}z|^{4}}dA(z) \Big)^{1/2}. \end{eqnarray*} $

Note that $ p>2 $. By Lemma B, we get that $ |||T_{\varphi}f|||_{\mathcal{L}^{2, 1-\frac{2}{p}}({\mathbb{D}})}\lesssim\|\varphi\|_{L^{\infty}({\mathbb T})}\|f\|_{H^{p}}. $ The proof is completed.

We characterize the compactness of Toeplitz operators from $ H^p $ to $ \mathcal{L}^{2, 1-\frac{2}{p}}({\mathbb T}) $ as follows.

Theorem 3.4 Let $ 2<p<\infty $ and $ \varphi \in L^\infty({\mathbb T}) $. Then the Toeplitz operator $ T_{\varphi} $ is compact from $ H^p $ to $ \mathcal{L}^{2, 1-\frac{2}{p}}({\mathbb{D}}) $ if and only if $ \varphi = 0 $.

Proof It suffices to prove the necessity. Let $ \{a_{n}\}_{n = 1}^\infty\subseteq {\mathbb{D}} $ be a sequence such that $ |a_{n}|\rightarrow 1 $ as $ n\rightarrow \infty $. Set

$ f_{n}(z) = \frac{(1-|a_{n}|^{2})^{1-\frac{1}{p}}}{1-\overline{a_{n}}z}, \ \ z\in {\mathbb T}. $

As explained in the proof of Theorem 3.3, $ \sup\limits_n \|f_{n}\|_{H^{p}}<\infty $. Clearly, $ f_{n}\rightarrow 0 $ uniformly on compact subsets of $ {\mathbb{D}} $ as $ n\rightarrow \infty $. Since $ T_{\varphi} $ is compact, we get that

$ \lim\limits_{n\rightarrow\infty}\|T_{\varphi}f_{n}\|_{\mathcal{L}^{2, 1-\frac{2}{p}}({\mathbb{D}})} = 0. $

By the proof of Theorem 3.3, one gets that $ |\hat{\varphi}(a_{n})|\lesssim \|T_{\varphi}f_{n}\|_{\mathcal{L}^{2, 1-\frac{2}{p}}({\mathbb{D}})} $ for all $ n $. Consequently, $ |\hat{\varphi}(a_{n})|\rightarrow 0, \ \ n\rightarrow \infty. $ Since $ a_{n} $ is arbitrary and $ \hat{\varphi} $ is harmonic, by the maximum principle, $ \hat{\varphi}\equiv0 $ on $ {\mathbb{D}} $. Hence $ \varphi = 0 $ on $ {\mathbb T} $. We finish the proof.