Harmonic extension of Q-type space related to logarithmic functions

Abstract

This paper studies a class of Q-type spaces Q_{\mathrm{l}\mathrm{o}\mathrm{g},\lambda }^m\left({\mathbb{R}}^n\right)related to logarithmic functions. We first investigate some basic properties of Q_{\mathrm{l}\mathrm{o}\mathrm{g},\lambda }^m\left({\mathbb{R}}^n\right). Further, by the aid of Poisson integral and harmonic function spaces H_{\mathrm{l}\mathrm{o}\mathrm{g},\lambda }^m\left({\mathbb{R}}_{+}^{n+1}\right), the harmonic extension of Q_{\mathrm{l}\mathrm{o}\mathrm{g},\lambda }^m\left({\mathbb{R}}^n\right)and the boundary value problem of H_{\mathrm{l}\mathrm{o}\mathrm{g},\lambda }^m\left({\mathbb{R}}_{+}^{n+1}\right)are obtained.

Keywords

Q type space / Poisson integral / harmonic extension

1 Introduction

In recent years, function spaces related to logarithmic functions have garnered a great deal of attention. In 2016, Cobos et al. [3] defined the Besov space B_{p,q}^{0,b} with zero-order classical smoothness and logarithmic smoothness of exponent b using the difference method. This space is defined as follows.

Definition 1.1　Let n\in \mathbb{N},h\in {\mathbb{R}}^n,k\in \mathbb{N}. Let

\{\begin{array}{l}\left({\mathrm{\Delta }}_h^{1}f\right)\left(x\right):=f(x+h)-f\left(x\right),\\ \left({\mathrm{\Delta }}_h^{k+1}f\right)\left(x\right):={\mathrm{\Delta }}_h^1\left({\mathrm{\Delta }}_h^k\right)f\left(x\right),x\in {\mathbb{R}}^n,\\ {\omega }_k(f,t{)}_p:=\underset{\left|h\right|\le t}{\mathrm{s}\mathrm{u}\mathrm{p}} \Vert {\mathrm{\Delta }}_h^{k}f\Vert L_p\Vert ,t>0.\end{array}

Assume , , then a Lebesgue measurable function f on {\mathbb{R}}^n belongs to B_{p,q}^{0,b} if and only if

Based on the work in [3], Liu et al. [11] considered various functional and geometric properties of the logarithmic Sobolev capacity {\mathrm{C}\mathrm{a}\mathrm{p}}_{\mathrm{l}\mathrm{o}\mathrm{g},\gamma ,p}, generated by the logarithmic Sobolev space W_p^{\mathrm{l}\mathrm{o}\mathrm{g},\gamma }, where (\gamma ,p)\in (0,\mathrm{\infty })\times [1,\mathrm{\infty }). This new non-linear, non-homogeneous, and non-trivial capacity is closely related to the logarithmic Hausdorff capacity.

As a generalization of the bounded mean oscillation space and Campanato-Morrey space, Q-type spaces and their generalized forms have been extensively studied in recent decades. Initially, Q-type spaces were proposed as a generalization of the analytic function space BMOA(\mathbb{D}), see [1]. In 2000, Essén et al. [5] extended Q-type spaces to the Euclidean space {\mathbb{R}}^n, establishing the function space Q_{\alpha }\left({\mathbb{R}}^n\right), which is defined as follows.

Definition 1.2　Let , n\ge 3. A measurable function f belongs to Q_{\alpha }\left({\mathbb{R}}^n\right) if and only if

where the supremum sup_I is taken over all cubes I in {\mathbb{R}}^n with edge length \ell \left(I\right) and parallel to the coordinate axes.

Q-type spaces have many important applications in harmonic analysis and the study of partial differential equations, see [4, 6, 9, 10, 14-16]. Inspired by the above works, in this paper, we study a class of Q-type spaces related to logarithmic functions, defined as follows.

Definition 1.3　Suppose \lambda >0, n>m>0, n\ge 2. f\in L_{\mathrm{l}\mathrm{o}\mathrm{c}}^2\left({\mathbb{R}}^n\right) belongs to Q_{\mathrm{l}\mathrm{o}\mathrm{g},\lambda }^m\left({\mathbb{R}}^n\right) if and only if

where the supremum sup_I is taken over all cubes I in {\mathbb{R}}^n with edge length \ell \left(I\right) and parallel to the coordinate axes.

The classical Q-type space Q_{\alpha }\left({\mathbb{R}}^n\right) has the following harmonic extension properties. Essén et al. [5] proved that f\in Q_{\alpha }\left({\mathbb{R}}^n\right) if and only if the Poisson integral of f, u(x,t)=f\ast p_t\left(x\right)u\left(x\right) is a harmonic function satisfying the following (1.1):

(1.1){\int }_{S\left(I\right)} |\mathrm{\nabla }u(x,t){|}^2{t}^{1-2\alpha }\mathrm{d}x\mathrm{d}t\le M(\ell \left(I\right){)}^{n-2\alpha },

where and I is any cube in {\mathbb{R}}^n. Here p_t denotes the Poisson kernel, i.e.,

p_t\left(x\right):=\frac{c_{n}t}{\left(\right|x{|}^2+t^2{)}^{\frac{n+1}{2}}},c_n=\frac{\mathrm{\Gamma }\left(\frac{n+1}{2}\right)}{{\pi }^{\frac{n}{2}}\mathrm{\Gamma }\left(\frac{1}{2}\right)}.

Accordingly, the Poisson integral of f is defined as

P_{t}f(x,t):=p_t\ast f\left(x\right)=c_n{\int }_{{\mathbb{R}}^n} \frac{f\left(y\right)t}{\left(\right|x-y{|}^2+t^2{)}^{\frac{n+1}{2}}}\mathrm{d}y,

where f\ast g denotes the convolution of f and g. Denote the gradient of f(x,t) as

{\mathrm{\nabla }}_{x,t}f(x,t)=\left(\frac{\mathrm{\partial }f(x,t)}{\mathrm{\partial }{x}_1},\frac{\mathrm{\partial }f(x,t)}{\mathrm{\partial }{x}_2},\dots ,\frac{\mathrm{\partial }f(x,t)}{\mathrm{\partial }{x}_n},\frac{\mathrm{\partial }f(x,t)}{\mathrm{\partial }t}\right).

A very natural question is whether Q_{\mathrm{l}\mathrm{o}\mathrm{g},\lambda }^m\left({\mathbb{R}}^n\right) introduced above can establish harmonic extension results similar to those in [5].

The main content of this paper is as follows. In Section 2, we present some basic properties of Q_{\mathrm{l}\mathrm{o}\mathrm{g},\lambda }^m\left({\mathbb{R}}^n\right), and discuss its relationship with Campanato-type spaces. Section 3 establishes Hardy-type inequalities and Stegenga-type estimates related to logarithmic functions. In Section 4, based on the above results, we extend Q_{\mathrm{l}\mathrm{o}\mathrm{g},\lambda }^m\left({\mathbb{R}}^n\right) to {\mathbb{R}}_{+}^{n+1} using the Poisson kernel pt(\cdot ). Let f be a measurable function on {\mathbb{R}}^nsatisfying

(1.2)

Using the harmonic extension P_t:f\to p_t\ast f, this paper proves that if f\in Q_{\mathrm{l}\mathrm{o}\mathrm{g},\lambda }^m\left({\mathbb{R}}^n\right), then p_t\ast f belongs to the harmonic function space H_{\mathrm{l}\mathrm{o}\mathrm{g},\lambda }^m in the upper half-space. Section 5 primarily studies the boundary value problem for H_{\mathrm{l}\mathrm{o}\mathrm{g},\lambda }^m\left({\mathbb{R}}_{+}^{n+1}\right).

The above results indicate that if , n>m>0, n\ge 2, and u(x,t)=p_t\ast f\left(x\right), then the following three conditions are equivalent:

In this paper, U\approx V means that there exists a constant c>0 such that c^{-1}V\le U\le cV, where the right-hand inequality can be written as \text{U}\lesssim V. Similarly, V\ge cU is written as V\gtrsim U. C^k\left({\mathbb{R}}^n\right) denotes the set of all functions f:{\mathbb{R}}^n\to \mathbb{R} with continuous partial derivatives up to order k. C_c^{\mathrm{\infty }}\left({\mathbb{R}}^n\right) denotes the set of all functions f:{\mathbb{R}}^n\to \mathbb{R} that have infinite-order continuous derivatives and compact support. L_{\mathrm{l}\mathrm{o}\mathrm{c}}^2\left({\mathbb{R}}^n\right) denotes the set of locally integrable functions f:{\mathbb{R}}^n\to \mathbb{R} satisfying .

2 Basic properties of {\mathit{Q}}_{\mathbf{l}\mathbf{o}\mathbf{g}\mathbf{,}\mathit{\lambda }}^{\mathbf{2},\mathit{m}}\left({\mathbb{R}}^{\mathit{n}}\right)

As is well known, the classical Q-type space Q_{\alpha }\left({\mathbb{R}}^n\right) has affine invariance. We consider whether Q_{\mathrm{l}\mathrm{o}\mathrm{g},\lambda }^m\left({\mathbb{R}}^n\right) also possesses similar properties. Let I be a cube centered at x_Iwith edge length \ell \left(I\right). Let tI(t>0) be a cube centered at x_I with edge length t\ell \left(I\right). We have the following proposition.

Proposition 2.1 [2]　 Suppose \lambda >0, n>m>0, n\ge 2.

(i) Q_{\mathrm{l}\mathrm{o}\mathrm{g},\lambda }^m\left({\mathbb{R}}^n\right) has translation invariance;

(ii) If f_r\left(x\right)=r^{\frac{m}{2}}f\left(rx\right), r>0, then \Vert f_r{\Vert }_{Q_{\mathrm{l}\mathrm{o}\mathrm{g},\lambda }^m\left({\mathbb{R}}^n\right)}=\Vert f{\Vert }_{Q_{\mathrm{l}\mathrm{o}\mathrm{g},\lambda }^m\left({\mathbb{R}}^n\right)};

(iii) Q_{\mathrm{l}\mathrm{o}\mathrm{g},\lambda }^m\left({\mathbb{R}}^n\right) has rotational invariance.

Proof　(i) For any f\in Q_{\mathrm{l}\mathrm{o}\mathrm{g},\lambda }^m\left({\mathbb{R}}^n\right) and x_0\in {\mathbb{R}}^n, we have

\begin{array}{rl}& \underset{I}{\mathrm{s}\mathrm{u}\mathrm{p}}(\ell (I){)}^{-n\lambda -m}{\int }_I {\int }_I \frac{\left|f\right(x+x_0)-f(y+x_0){|}^2}{|x-y{|}^{2n-m}}{\left(\log \frac{\mathrm{e}\sqrt{n}\ell \left(I\right)}{|x-y|}\right)}^2\mathrm{d}x\mathrm{d}y\\ & =\underset{I}{\mathrm{s}\mathrm{u}\mathrm{p}}(\ell (I){)}^{-n\lambda -m}{\int }_{I+x_0} {\int }_{I+x_0} \frac{\left|f\right(\xi )-f(\eta ){|}^2}{|\xi -\eta {|}^{2n-m}}{\left(\log \frac{\mathrm{e}\sqrt{n}\ell (I+x_0)}{|\xi -\eta |}\right)}^2\mathrm{d}\xi \mathrm{d}\eta ,\end{array}

which implies \Vert f(\cdot +x_0){\Vert }_{Q_{\mathrm{l}\mathrm{o}\mathrm{g},\lambda }^m\left({\mathbb{R}}^n\right)}=\Vert f{\Vert }_{Q_{\mathrm{l}\mathrm{o}\mathrm{g},\lambda }^m\left({\mathbb{R}}^n\right)}. Therefore, Q_{\mathrm{l}\mathrm{o}\mathrm{g},\lambda }^m\left({\mathbb{R}}^n\right) has translation invariance.

(ii) If f\in Q_{\mathrm{l}\mathrm{o}\mathrm{g},\lambda }^m\left({\mathbb{R}}^n\right)and x_0\in {\mathbb{R}}^n, then by a change of variables, we obtain

\begin{array}{rl}& \underset{I}{\mathrm{s}\mathrm{u}\mathrm{p}} (\ell (I){)}^{-n\lambda -m}{\int }_I {\int }_I \frac{\left|f_r\right(x)-f_r(y){|}^2}{|x-y{|}^{2n-m}}{\left(\log \frac{\mathrm{e}\sqrt{n}\ell \left(I\right)}{|x-y|}\right)}^2\mathrm{d}x\mathrm{d}y\\ & =\underset{I}{\mathrm{s}\mathrm{u}\mathrm{p}} (\ell (I){)}^{-n\lambda -m}{\int }_{rI} {\int }_{rI} \frac{\left|r^{\frac{m}{2}}f\right(\xi )-r^{\frac{m}{2}}f(\eta ){|}^2}{r^m|\xi -\eta {|}^{2n-m}}{\left(\log \frac{\mathrm{e}\sqrt{n}r\ell \left(I\right)}{|\xi -\eta |}\right)}^2\mathrm{d}\xi \mathrm{d}\eta \\ & =\underset{I}{\mathrm{s}\mathrm{u}\mathrm{p}}(\ell (I){)}^{-n\lambda -m}{\int }_{rI} {\int }_{rI} \frac{\left|f\right(\xi )-f(\eta ){|}^2}{|\xi -\eta {|}^{2n-m}}{\left(\log \frac{\mathrm{e}\sqrt{n}\ell \left(rI\right)}{|\xi -\eta |}\right)}^2\mathrm{d}\xi \mathrm{d}\eta ,\end{array}

which implies \Vert f_r{\Vert }_{Q_{\mathrm{l}\mathrm{o}\mathrm{g},\lambda }^m\left({\mathbb{R}}^n\right)}=\Vert f{\Vert }_{Q_{\mathrm{l}\mathrm{o}\mathrm{g},\lambda }^m\left({\mathbb{R}}^n\right)}.

(iii) Let I be a cube centered at x_I with edge length \ell \left(I\right). It is known that I\subset B(x_I,\sqrt{n}\ell (I\left)\right). For any n-th order orthogonal matrix M and x\in {\mathbb{R}}^n, we have \left|x\right|=\left|x\mathcal{M}\right|. Let \psi \left(x\right)=x\mathcal{M}. For any ball B\subset {\mathbb{R}}^n with radius r(B), we obtain

\begin{array}{rl}& \underset{B}{\mathrm{s}\mathrm{u}\mathrm{p}} (\ell (I){)}^{-n\lambda -m}{\int }_B {\int }_B \frac{\left|f\right(x\mathcal{M})-f(y\mathcal{M}){|}^2}{|x-y{|}^{2n-m}}{\left(\log \frac{\mathrm{e}\sqrt{n}r\left(B\right)}{|x-y|}\right)}^2\mathrm{d}x\mathrm{d}y\\ & =\underset{B}{\mathrm{s}\mathrm{u}\mathrm{p}} (\ell (I){)}^{-n\lambda -m}{\int }_{\{\xi :\xi =x\mathcal{M},x\in B\}} {\int }_{\{\eta :\eta =y\mathcal{M},y\in B\}} \frac{\left|f\right(\xi )-f(\eta ){|}^2}{|\xi -\eta {|}^{2n-m}}{\left(\mathrm{l}\mathrm{o}\mathrm{g}\frac{\mathrm{e}\sqrt{n}r\left(B\right)}{|\xi -\eta |}\right)}^2\mathrm{d}\xi \mathrm{d}\eta \\ & =\underset{B}{\mathrm{s}\mathrm{u}\mathrm{p}} (\ell (I){)}^{-n\lambda -m}{\int }_B {\int }_B \frac{\left|f\right(x)-f(y){|}^2}{|x-y{|}^{2n-m}}{\left(\log \frac{\mathrm{e}\sqrt{n}r\left(B\right)}{|x-y|}\right)}^2\mathrm{d}x\mathrm{d}y,\end{array}

which implies \Vert f\circ \psi {\Vert }_{Q_{\mathrm{l}\mathrm{o}\mathrm{g},\lambda }^m\left({\mathbb{R}}^n\right)}=\Vert f{\Vert }_{Q_{\mathrm{l}\mathrm{o}\mathrm{g},\lambda }^m\left({\mathbb{R}}^n\right)}. Therefore, Q_{\mathrm{l}\mathrm{o}\mathrm{g},\lambda }^m\left({\mathbb{R}}^n\right) has rotational invariance.

Through a change of variables, we can prove that Q_{\mathrm{l}\mathrm{o}\mathrm{g},\lambda }^m\left({\mathbb{R}}^n\right) has the following equivalent definition.

Proposition 2.2　 Let \lambda >0, n>m>0,n\ge 2. Then f\in Q_{\mathrm{l}\mathrm{o}\mathrm{g},\lambda }^m\left({\mathbb{R}}^n\right) if and only if

(2.1)

where the supremum sup_I is taken over all cubes I in {\mathbb{R}}^n with edge length \ell \left(I\right) and parallel to the coordinate axes.

Next, we discuss the non-triviality of Q_{\mathrm{l}\mathrm{o}\mathrm{g},\lambda }^m\left({\mathbb{R}}^n\right) and introduce the space {\mathrm{C}\mathrm{I}\mathrm{S}}_{\gamma }\left({\mathbb{R}}^n\right) as follows.

Definition 2.1　Let \gamma >0, f\in C^1\left({\mathbb{R}}^n\right), and {\mathrm{\nabla }}_{x}f\left(x\right)=(\frac{\mathrm{\partial }f\left(x\right)}{\mathrm{\partial }{x}_1},\frac{\mathrm{\partial }f\left(x\right)}{\mathrm{\partial }{x}_2},\dots ,\frac{\mathrm{\partial }f\left(x\right)}{\mathrm{\partial }{x}_n}). If the function f satisfies

then f\in {\mathrm{C}\mathrm{I}\mathrm{S}}_{\gamma }\left({\mathbb{R}}^n\right). Through Li et al.’s wavelet characterization of the Triebel-Lizorkin type modulation space {\dot{F}}_{p,q}^{{\gamma }_1,{\gamma }_2}\left({\mathbb{R}}^n\right) obtained in [8, Theorem 3.1], it can be seen that {\mathrm{C}\mathrm{I}\mathrm{S}}_{\gamma }\left({\mathbb{R}}^n\right) is a special case of the Triebel-Lizorkin type modulation space {\dot{F}}_{p,q}^{{\gamma }_1,{\gamma }_2}\left({\mathbb{R}}^n\right).

Proposition 2.3　 Let , n>m>n-1, and n\ge 2. Then {\mathrm{C}\mathrm{I}\mathrm{S}}_{\gamma }\left({\mathbb{R}}^n\right)={\dot{F}}_{2,2}^{0,1+n-n-\frac{\gamma n}{2}}\left({\mathbb{R}}^n\right).

Clearly, constants belong to Q_{log,\lambda }^m\left({\mathbb{R}}^n\right).

Theorem 2.1　 Let , n>m>n-1, and n\ge 2. Then Q_{\mathrm{l}\mathrm{o}\mathrm{g},\lambda }^m\left({\mathbb{R}}^n\right) is non-trivial if and only if

(2.2)

Furthermore, if (2.2) holds, then {\mathrm{C}\mathrm{I}\mathrm{S}}_{\lambda }\left({\mathbb{R}}^n\right)\subseteq Q_{\mathrm{l}\mathrm{o}\mathrm{g},\lambda }^m\left({\mathbb{R}}^n\right)

Proof　Let f\in {\mathrm{C}\mathrm{I}\mathrm{S}}_{\lambda }\left({\mathbb{R}}^n\right). Since

\left|f\right(x+z)-f(x\left)\right|\le {\int }_0^1 \left|{\mathrm{\nabla }}_{x}f\right(x+tz\left)\right|\left|z\right|\mathrm{d}t,

by using variable substitution and the Minkowski inequality, we obtain

This shows that {\mathrm{C}\mathrm{I}\mathrm{S}}_{\lambda }\left({\mathbb{R}}^n\right)\subseteq Q_{\mathrm{l}\mathrm{o}\mathrm{g},\lambda }^m\left({\mathbb{R}}^n\right). Hence, Q_{\mathrm{l}\mathrm{o}\mathrm{g},\lambda }^m\left({\mathbb{R}}^n\right) is non-trivial.

To prove the converse, it suffices to show that if {\int }_0^{\sqrt{n}}(\mathrm{l}\mathrm{o}\mathrm{g}\frac{\mathrm{e}\sqrt{n}}{t}{)}^2{t}^{m-n+1}\mathrm{d}t=\mathrm{\infty }, then Q_{\mathrm{l}\mathrm{o}\mathrm{g},\lambda }^m\left({\mathbb{R}}^n\right) is trivial. Assume f\in Q_{\mathrm{l}\mathrm{o}\mathrm{g},\lambda }^m\left({\mathbb{R}}^n\right)\cap C^1\left({\mathbb{R}}^n\right) is not identically constant. Without loss of generality, assume fis a real-valued function, then there exists a point x=(x_1,x_2,\dots ,x_n) such that {\mathrm{\nabla }}_{x}f\left(x\right)\ne 0. According to Householder reflection [13], there exists an orthogonal matrix \mathcal{M}=\left(a_{ij}\right), i,j=1,2,\dots ,n, such that

\begin{array}{rl}{\mathrm{\nabla }}_{x}f\left(x\right)\mathcal{M}& =\left(\sum _{i=1}^n \frac{\mathrm{\partial }f\left(x\right)}{\mathrm{\partial }{x}_i}{a}_{i1},\sum _{i=1}^n \frac{\mathrm{\partial }f\left(x\right)}{\mathrm{\partial }{x}_i}{a}_{i2},\dots ,\sum _{i=1}^n \frac{\mathrm{\partial }f\left(x\right)}{\mathrm{\partial }{x}_i}{a}_{in}\right)\\ & =\left(\right|{\mathrm{\nabla }}_{x}f\left(x\right)|,0,\dots ,0).\end{array}

Let \omega \left(x\right)=f\left(x{\mathcal{M}}^{\mathrm{\top }}\right), where {\mathcal{M}}^{\mathrm{\top }} is the transpose of \mathcal{M} and det\left({\mathcal{M}}^{\mathrm{\top }}\right)\ne 0. According to the rotational invariance of Q_{\mathrm{l}\mathrm{o}\mathrm{g},\lambda }^m\left({\mathbb{R}}^n\right), we have \omega \in Q_{\mathrm{l}\mathrm{o}\mathrm{g},\lambda }^m\left({\mathbb{R}}^n\right). There exists a point y=(y_1,y_2,\dots ,y_n) such that y{\overline{\mathcal{M}}}^{\mathrm{\top }}=x, implying x_j=\sum _{i=1}^n y_i{a}_{ji}. Additionally,

\{\begin{array}{rl}& \frac{\mathrm{\partial }\omega \left(x\right)}{\mathrm{\partial }{y}_1}=\frac{\mathrm{\partial }f\left(x{\mathcal{M}}^{\mathrm{\top }}\right)}{\mathrm{\partial }{y}_1}=\sum _{j=1}^n \frac{\mathrm{\partial }f\left(x{\mathcal{M}}^{\mathrm{\top }}\right)}{\mathrm{\partial }{x}_j}{a}_{j1}=\sum _{j=1}^n \frac{\mathrm{\partial }f\left(x\right)}{\mathrm{\partial }{x}_j}{a}_{j1}=\left|{\mathrm{\nabla }}_{x}f\right(x\left)\right|;\\ & \frac{\mathrm{\partial }\omega \left(x\right)}{\mathrm{\partial }{y}_i}=\frac{\mathrm{\partial }f\left(x{\mathcal{M}}^{\mathrm{\top }}\right)}{\mathrm{\partial }{y}_i}=\sum _{j=1}^n \frac{\mathrm{\partial }f\left(x{\mathcal{M}}^{\mathrm{\top }}\right)}{\mathrm{\partial }{x}_j}{a}_{ji}=0,i\ge 2.\end{array}

Therefore, {\mathrm{\nabla }}_x\omega \left(x\right)=\left(\right|{\mathrm{\nabla }}_{x}f\left(x\right)|,0,\dots ,0). It is noted that \omega \in C^1\left({\mathbb{R}}^n\right). For any \epsilon >0, there exists a positive constant δ and a smaller cube I centered at x_0, such that if , then . Let \epsilon =\frac{\left|{\mathrm{\nabla }}_{x}f\right(x\left)\right|}{3}, we have \frac{\mathrm{\partial }\omega \left(y\right)}{\mathrm{\partial }{y}_1}>2\delta and for j\ge 2. Define

If z\in A, then \left|z\right|\approx z_1. If x,y\in I and x-y\in A, by the mean value theorem, \text{|}\omega \left(x\right)-\omega \left(y\right)|\gtrsim \delta |x_1-y_1|. Hence,

which shows that \omega \notin Q_{\mathrm{l}\mathrm{o}\mathrm{g},\lambda }^m\left({\mathbb{R}}^n\right), contradicting our assumption. Therefore, if

{\int }_0^{\sqrt{n}} {\left(\mathrm{l}\mathrm{o}\mathrm{g}\frac{\mathrm{e}\sqrt{n}}{t}\right)}^2{t}^{m-n+1}\mathrm{d}t=\mathrm{\infty },

then Q_{\mathrm{l}\mathrm{o}\mathrm{g},\lambda }^m\left({\mathbb{R}}^n\right)\cap C^1\left({\mathbb{R}}^n\right) is trivial. For f\in Q_{\mathrm{l}\mathrm{o}\mathrm{g},\lambda }^m\left({\mathbb{R}}^n\right) and \omega \in L^1\left({\mathbb{R}}^n\right), using Minkowski’s inequality and translation invariance, we obtain f\ast \omega \in Q_{\mathrm{l}\mathrm{o}\mathrm{g},\lambda }^m\left({\mathbb{R}}^n\right) satisfying

\Vert f\ast \omega {\Vert }_{Q_{\mathrm{l}\mathrm{o}\mathrm{g},\lambda }^m\left({\mathbb{R}}^n\right)}\le \Vert f{\Vert }_{Q_{\mathrm{l}\mathrm{o}\mathrm{g},\lambda }^m\left({\mathbb{R}}^n\right)}{\int }_{{\mathbb{R}}^n} \left|\omega \right(y\left)\right|\mathrm{d}y.

In particular, if ω is a smooth function with compact support, then f\ast \omega \in Q_{\mathrm{l}\mathrm{o}\mathrm{g},\lambda }^m\left({\mathbb{R}}^n\right)\cap C^1\left({\mathbb{R}}^n\right) is almost everywhere constant. According to [5], there exists a non-negative sequence ω_n such that {\int }_{{\mathbb{R}}^n} {\omega }_n\left(x\right)\mathrm{d}x=1 and \mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}{\omega }_n converges to 0 and f\ast {\omega }_n\to f almost everywhere. Hence, f is almost everywhere constant. This completes the proof of Theorem 2.1. □

Next, we prove that Q_{\mathrm{l}\mathrm{o}\mathrm{g},\lambda }^m\left({\mathbb{R}}^n\right) is a subspace of the Campanato space. Let f_I=\left|I{|}^{-1}{\int }_I f\right(y)\mathrm{d}y.

Definition 2.2　Let\gamma >0. If f\in L_{\mathrm{l}\mathrm{o}\mathrm{c}}^2\left({\mathbb{R}}^n\right) satisfies

then f is said to belong to the Campanato space {\mathcal{L}}_{\gamma }\left({\mathbb{R}}^n\right).

Note 2.1　Through simple calculations, an equivalent characterization of the Campanato space {\mathcal{L}}_{\gamma }\left({\mathbb{R}}^n\right) can be derived. For \gamma >0, f\in {\mathcal{L}}_{\gamma }\left({\mathbb{R}}^n\right) if and only if

where the supremum sup_I is taken over all cubes I in {\mathbb{R}}^n with edge length \ell \left(I\right) and parallel to the coordinate axes.

We can establish an inclusion relationship between Q_{\mathrm{l}\mathrm{o}\mathrm{g},\lambda }^m\left({\mathbb{R}}^n\right) and {\mathcal{L}}_{\lambda +1}\left({\mathbb{R}}^n\right).

Theorem 2.2　 Let \lambda >0, n>m>0, and n\ge 2. Q_{\mathrm{l}\mathrm{o}\mathrm{g},\lambda }^m\left({\mathbb{R}}^n\right) is a subspace of {\mathcal{L}}_{\lambda +1}\left({\mathbb{R}}^n\right).

Proof　Let f\in Q_{\mathrm{l}\mathrm{o}\mathrm{g},\lambda }^m\left({\mathbb{R}}^n\right). For any cube I and x,y\in I, if \xi >0 is sufficiently small, then the set \{z\in I,\mathrm{m}\mathrm{i}\mathrm{n}(|x-z|,|y-z|)>\xi \ell (I\left)\right\} has a Lebesgue measure greater than C_{\xi ,n}\left|I\right|, where C_{\xi ,n}=1-\frac{\mathrm{e}{\xi }^n{\pi }^{n/2}}{n\mathrm{\Gamma }(n/2)}. Thus,

\begin{array}{rl}& {\int }_I \mathrm{m}\mathrm{i}\mathrm{n}\left\{{\left(\frac{|x-z|}{\ell \left(I\right)}\right)}^m{\left(\mathrm{l}\mathrm{o}\mathrm{g}\frac{\mathrm{e}\sqrt{n}\ell \left(I\right)}{|x-z|}\right)}^2,{\left(\frac{|y-z|}{\ell \left(I\right)}\right)}^m{\left(\mathrm{l}\mathrm{o}\mathrm{g}\frac{\mathrm{e}\sqrt{n}\ell \left(I\right)}{|y-z|}\right)}^2\right\}\mathrm{d}z\\ & \ge {\int }_{\{z\in I,\mathrm{m}\mathrm{i}\mathrm{n}\{|x-z|,|y-z|\}>\xi \ell (I)\}} \mathrm{m}\mathrm{i}\mathrm{n}\{\ell (I{)}^{-m}|x-z{|}^m{\left(\mathrm{l}\mathrm{o}\mathrm{g}\frac{\mathrm{e}\sqrt{n}\ell \left(I\right)}{|x-z|}\right)}^2,\\ & \ell (I{)}^{-m}|y-z{|}^m{\left(\log \frac{\mathrm{e}\sqrt{n}\ell \left(I\right)}{|y-z|}\right)}^2\}\mathrm{d}z\\ & \ge C_{\xi ,n}\left|I\right|{\xi }^m{\left(\log \frac{\mathrm{e}\sqrt{n}}{\xi }\right)}^2.\end{array}

Noting |x-y{|}^{2n}\le n^n|I{|}^2, we obtain

For sufficiently small ξ > 0, we conclude

Therefore, Q_{\mathrm{l}\mathrm{o}\mathrm{g},\lambda }^m\left({\mathbb{R}}^n\right)\subseteq {\mathcal{L}}_{\lambda +1}\left({\mathbb{R}}^n\right). This completes the proof of Theorem 2.2. □

3 Hardy-type inequalities and Stegenga-type estimates

To study the harmonic extension of Q_{\mathrm{l}\mathrm{o}\mathrm{g},\lambda }^m\left({\mathbb{R}}^n\right) and boundary value problems for H_{\mathrm{l}\mathrm{o}\mathrm{g},\lambda }^m\left({\mathbb{R}}_{+}^{n+1}\right), the following Hardy-type inequalities are necessary.

Lemma 3.1 [7]　 Let , \frac{1}{p}+\frac{1}{q}=1, and . Suppose the non-negative functions \mu and h are measurable on (0,b). For all measurable functionsf\ge 0, the following Hardy-type inequalities hold.

(i) {\int }_0^a \left({\int }_0^s f\right(t\left)\mathrm{d}t{)}^{p}u\right(s)\mathrm{d}s\le C{\int }_0^a f^p(s\left)v\right(s)\mathrm{d}s holds if and only if

(ii) {\int }_0^a \left({\int }_s^a f\right(t\left)\mathrm{d}t{)}^{p}u\right(s)\mathrm{d}s\le C{\int }_0^a f^p(s\left)v\right(s)\mathrm{d}s holds if and only if

where the constants C in (i) and (ii) depend only on p, A, and B.

Next, we provide a Stegenga-type estimate.

Lemma 3.2　 Let , , and n\ge 2. Suppose f\in L_{\mathrm{l}\mathrm{o}\mathrm{c}}^2\left({\mathbb{R}}^n\right) satisfies condition (1.2), I and J are cubes centered at x₀ in {\mathbb{R}}^n with \ell \left(J\right)=3\ell \left(I\right). Then the following inequality holds:

(\ell (J){)}^{-n\lambda -m}{\int }_{S\left(I\right)}|{\mathrm{\nabla }}_{x,t}u(x,t){|}^2{\left(\mathrm{l}\mathrm{o}\mathrm{g}\frac{\mathrm{e}\sqrt{n}\ell \left(I\right)}{t}\right)}^2{t}^{m-n+1}\mathrm{d}x\mathrm{d}t\lesssim D_1+D_2+D_3,

where

\{\begin{array}{rl}& D_1:=(\ell (J){)}^{-n\lambda -m}{\int }_{\left|y\right|\le \sqrt{n}\ell \left(J\right)} {\int }_{x\in J} |f\left(x\right)\\ & -f(x+y){|}^2|y{|}^{m-2n}{\left(\log \frac{\mathrm{e}\sqrt{n}\ell \left(J\right)}{\left|y\right|}\right)}^2\mathrm{d}x\mathrm{d}y;\\ & D_2:=\left({\int }_0^{\sqrt{n}} t^{m-n+1}{\left(\mathrm{l}\mathrm{o}\mathrm{g}\frac{\mathrm{e}\sqrt{n}}{t}\right)}^2\mathrm{d}t+{\int }_{\frac{1}{10}}^{\mathrm{\infty }} t^{m-n-1}{\left(\mathrm{l}\mathrm{o}\mathrm{g}\frac{\mathrm{e}\sqrt{n}}{t}\right)}^2\mathrm{d}t\right)\\ & \times (\ell (J){)}^{-n(\lambda +1)}{\int }_{y\in J} |f\left(y\right)-f_J{|}^2\mathrm{d}y;\\ & D_3:=(\ell (J){)}^{-n\lambda -m}{\int }_0^1 t^{m-n+1}{\left(\log \frac{\mathrm{e}\sqrt{n}}{t}\right)}^2\mathrm{d}t{\left(\ell (J){\int }_{{\mathbb{R}}^n\setminus \frac{2}{3}J} \frac{\left|f\right(y)-f_J|}{|y{|}^{n+1}}\mathrm{d}y\right)}^2.\end{array}

Proof　Without loss of generality, assume x₀ = 0. Let \eta be a function satisfying 0\le \eta \le 1, with \eta =1 on \frac{2}{3}J, \mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}\eta \subseteq \frac{3}{4}J and \left|\eta \right(x)-\eta (y\left)\right|\lesssim \ell (J{)}^{-1}|x-y|, for x,y\in {\mathbb{R}}^n. Similar to [12], we decompose f into three parts: f=f_1+f_2+f_{31}, where f_1=f_J, f_2=(f-f_J)\eta, and f_3=(f-f_J)(1-\eta ). Let u=u_1+u_2+u_3, where u_1=p_t\ast f_1, u_2=p_t\ast f_2, and u_3=p_t\ast f_3. Note that

\frac{\mathrm{\partial }{p}_t\left(y\right)}{\mathrm{\partial }{y}_j}=-\frac{c_{n}t{y}_j}{\left(\right|y{|}^2+t^2{)}^{\frac{n+3}{2}}}.

Since \left(\right|y{|}^2+t^2{)}^{\frac{1}{2}}\ge t and \left(\right|y{|}^2+t^2{)}^{\frac{1}{2}}\ge \left|y_j\right|, forj=1,2,\dots ,n, we have

\left|\frac{\mathrm{\partial }{p}_t\left(y\right)}{\mathrm{\partial }{y}_j}\right|\le \frac{t\left|y_j\right|}{\left(\right|y{|}^2+t^2{)}^{\frac{n+1}{2}}\left(\right|y{|}^2+t^2{)}^{\frac{1}{2}}\left(\right|y{|}^2+t^2{)}^{\frac{1}{2}}}\lesssim \frac{1}{\left(\right|y{|}^2+t^2{)}^{\frac{n+1}{2}}},

which implies

\left|\frac{\mathrm{\partial }{p}_t\left(y\right)}{\mathrm{\partial }{y}_j}\right|\lesssim t^{-n-1},\left|\frac{\mathrm{\partial }{p}_t\left(y\right)}{\mathrm{\partial }{y}_j}\right|\lesssim |y{|}^{-n-1}.

Given {\int }_{{\mathbb{R}}^n} \frac{\mathrm{\partial }{p}_t\left(y\right)}{\mathrm{\partial }{y}_j}\mathrm{d}y=0, by variable substitution, we obtain

\frac{\mathrm{\partial }u(x,t)}{\mathrm{\partial }{x}_j}={\int }_{{\mathbb{R}}^n} \frac{\mathrm{\partial }{p}_t(x-y)}{\mathrm{\partial }{x}_j}f\left(y\right)\mathrm{d}y={\int }_{{\mathbb{R}}^n} \frac{\mathrm{\partial }{p}_t\left(y\right)}{\mathrm{\partial }{y}_j}\left(f\right(x)-f(x+y\left)\right)\mathrm{d}y.

Therefore,

\begin{array}{rl}{\Vert \frac{\mathrm{\partial }u(x,t)}{\mathrm{\partial }{x}_j}\Vert }_{L^2\left({\mathbb{R}}^n\right)}& \lesssim t^{-n-1}{\int }_{\left|y\right|\le t} \Vert f(x+y)-f\left(x\right){\Vert }_{L^2\left({\mathbb{R}}^n\right)}\mathrm{d}y\\ & +{\int }_{\left|y\right|>t} \Vert f(x+y)-f\left(x\right){\Vert }_{L^2\left({\mathbb{R}}^n\right)}|y{|}^{-n-1}\mathrm{d}y.\end{array}

Let y=r\xi \in {\mathbb{R}}^n such that r=\left|y\right| and \left|\xi \right|=1. Let

\mathrm{\Omega }\left(r\right):={\int }_{\left|\xi \right|=1} {\left({\int }_{{\mathbb{R}}^n} \left|f\right(x+r\xi )-f(x){|}^2\mathrm{d}x\right)}^{\frac{1}{2}}\mathrm{d}\xi .

We can derive that

{\Vert \frac{\mathrm{\partial }u(x,t)}{\mathrm{\partial }{x}_j}\Vert }_{L^2\left({\mathbb{R}}^n\right)}\lesssim t^{-n-1}{\int }_0^t \mathrm{\Omega }\left(r\right)r^{n-1}\mathrm{d}r+{\int }_t^{\mathrm{\infty }} \mathrm{\Omega }\left(r\right)r^{-2}\mathrm{d}r.

Thus,

(\ell (I){)}^{-n\lambda -m}{\int }_{S\left(I\right)} {\left|\frac{\mathrm{\partial }u(x,t)}{\mathrm{\partial }{x}_j}\right|}^2{\left(\mathrm{l}\mathrm{o}\mathrm{g}\frac{\mathrm{e}\sqrt{n}\ell \left(I\right)}{t}\right)}^2{t}^{m-n+1}\mathrm{d}x\mathrm{d}t\lesssim (\ell (I){)}^{-n(\lambda +1)}(B_1+B_2),

where

\{\begin{array}{rl}& B_1={\int }_0^1 t^{m-3n-1}{\left(\log \frac{\mathrm{e}\sqrt{n}}{t}\right)}^2{\left({\int }_0^t \mathrm{\Omega }(\ell (I\left)r\right)r^{n-1}\mathrm{d}r\right)}^2\mathrm{d}t;\\ & B_2={\int }_0^1 t^{m-n+1}{\left(\log \frac{\mathrm{e}\sqrt{n}}{t}\right)}^2{\left({\int }_t^{\mathrm{\infty }} \mathrm{\Omega }(\ell (I)r)r^{-2}\mathrm{d}r\right)}^2\mathrm{d}t.\end{array}

For B₁, note that

According to Lemma 3.1, it is evident that

B_1\lesssim {\int }_0^1 t^{m-n-1}{\left(\mathrm{l}\mathrm{o}\mathrm{g}\frac{\text{e}\sqrt{n}}{t}\right)}^2(\mathrm{\Omega }\left(\ell \right(I\left)t\right){)}^2\frac{1}{2}t.

For B₂, we estimate as follows: Since

By Lemma 3.1, we have

\begin{array}{rl}{B}_2& \lesssim {\int }_0^{\mathrm{\infty }} t^{m-n-1}{\left(\log \frac{\mathrm{e}\sqrt{n}}{t}\right)}^2{\left({\int }_t^{\mathrm{\infty }} \mathrm{\Omega }(\ell (I\left)r\right)r^{-2}\mathrm{d}r\right)}^2\mathrm{d}t\\ & \lesssim {\int }_0^{\mathrm{\infty }} t^{m-n+1}{\left(\log \frac{\mathrm{e}\sqrt{n}}{t}\right)}^2(\mathrm{\Omega }\left(\ell \right(I\left)t\right){)}^2\mathrm{d}t.\end{array}

Therefore, by Holder’s inequality and spherical coordinate transformation, we obtain

\begin{array}{rl}& (\ell (I){)}^{-n\lambda -m}{\int }_{S\left(I\right)} {\left|\frac{\mathrm{\partial }u(x,t)}{\mathrm{\partial }{x}_j}\right|}^2{\left(\log \frac{\mathrm{e}\sqrt{n}\ell \left(I\right)}{t}\right)}^2{t}^{m-n+1}\mathrm{d}x\mathrm{d}t\\ & \lesssim (\ell (I){)}^{-n\lambda -m}{\int }_0^{\mathrm{\infty }} t^{m-n-1}{\left(\log \frac{\mathrm{e}\sqrt{n}\ell \left(I\right)}{t}\right)}^2{\int }_{\left|\xi \right|=1} \Vert f(x+t\xi )-f\left(x\right){\Vert }_{L^2\left({\mathbb{R}}^n\right)}^2\mathrm{d}\xi \mathrm{d}t\\ & \approx (\ell (I){)}^{-n\lambda -m}{\int }_{{\mathbb{R}}^n} {\int }_{{\mathbb{R}}^n}|f(x+y)-f\left(x\right){|}^2|y{|}^{m-2n}{\left(\log \frac{\mathrm{e}\sqrt{n}\ell \left(I\right)}{\left|y\right|}\right)}^2\mathrm{d}x\mathrm{d}y.\end{array}

Note

\frac{\mathrm{\partial }{p}_t\left(y\right)}{\mathrm{\partial }t}\lesssim \frac{\left(\right|y{|}^2+t^2)}{\left(\right|y{|}^2+t^2{)}^{\frac{n+3}{2}}}\lesssim \frac{1}{\left(\right|y{|}^2+t^2{)}^{\frac{n+1}{2}}}.

This implies \left|\frac{\mathrm{\partial }{p}_t\left(y\right)}{\mathrm{\partial }t}\right|\lesssim t^{-n-1} and \left|\frac{\mathrm{\partial }{p}_t\left(y\right)}{\mathrm{\partial }t}\right|\lesssim |y{|}^{-n-1}. Furthermore, since {\int }_{{\mathbb{R}}^n} \frac{\mathrm{\partial }{p}_t\left(y\right)}{\mathrm{\partial }t}\mathrm{d}y=0, it follows that

\left|\frac{\mathrm{\partial }u(x,t)}{\mathrm{\partial }t}\right|\le {\int }_{{\mathbb{R}}^n} \left|\frac{\mathrm{\partial }{p}_t\left(y\right)}{\mathrm{\partial }t}\right|\left|f\right(x+y)-f(x\left)\right|\mathrm{d}y.

Similarly, we can prove that

\begin{array}{rl}& (\ell (I){)}^{-n\lambda -m}{\int }_{S\left(I\right)} {\left|\frac{\mathrm{\partial }u(x,t)}{\mathrm{\partial }t}\right|}^2{\left(\mathrm{l}\mathrm{o}\mathrm{g}\frac{\mathrm{e}\sqrt{n}\ell \left(I\right)}{t}\right)}^2{t}^{m-n+1}\mathrm{d}x\mathrm{d}t\\ & \lesssim (\ell (I){)}^{-n\lambda -m}{\int }_{{\mathbb{R}}^n} {\int }_{{\mathbb{R}}^n} |f(x+y)-f\left(x\right){|}^2|y{|}^{m-2n}{\left(\log \frac{\mathrm{e}\sqrt{n}\ell \left(I\right)}{\left|y\right|}\right)}^2\mathrm{d}x\mathrm{d}y.\end{array}

Thus,

(\ell (J){)}^{-n\lambda -m}{\int }_{S\left(I\right)}|{\mathrm{\nabla }}_{x,t}{u}_2(x,t){|}^2{\left(\mathrm{l}\mathrm{o}\mathrm{g}\frac{\mathrm{e}\sqrt{n}\ell \left(I\right)}{t}\right)}^2{t}^{m-n+1}\mathrm{d}x\mathrm{d}t\lesssim C_1+C_2+C_3,

where

\{\begin{array}{rl}& C_1:=(\ell (J){)}^{-n\lambda -m}{\int }_{y\in J} {\int }_{x\in J} \frac{\left|f_2\right(x)-f_2(y){|}^2}{|x-y{|}^{2n-m}}{\left(\mathrm{l}\mathrm{o}\mathrm{g}\frac{\mathrm{e}\sqrt{n}\ell \left(I\right)}{|x-y|}\right)}^2\mathrm{d}x\mathrm{d}y;\\ & C_2:=(\ell (J){)}^{-n\lambda -m}{\int }_{y\in \frac{4}{5}J} {\int }_{x\notin J} \frac{\left|f_2\right(x)-f_2(y){|}^2}{|x-y{|}^{2n-m}}{\left(\mathrm{l}\mathrm{o}\mathrm{g}\frac{\mathrm{e}\sqrt{n}\ell \left(I\right)}{|x-y|}\right)}^2\mathrm{d}x\mathrm{d}y;\\ & C_3:=(\ell (J){)}^{-n\lambda -m}{\int }_{y\notin J} {\int }_{x\in \frac{4}{5}J} \frac{\left|f_2\right(x)-f_2(y){|}^2}{|x-y{|}^{2n-m}}{\left(\mathrm{l}\mathrm{o}\mathrm{g}\frac{\mathrm{e}\sqrt{n}\ell \left(I\right)}{|x-y|}\right)}^2\mathrm{d}x\mathrm{d}y.\end{array}

For C₁, it is clear that |f_2(x)-f_2(y)|\lesssim |f(x)-f(y)|+\ell (J{)}^{-1}|x-y||f(y)-f_J|. Therefore,

\begin{array}{rl}& (\ell (J){)}^{-m-2}{\int }_{\left|y\right|\le \sqrt{n}\ell \left(J\right)} {\int }_{x\in J}|f(x+y)-f_J{|}^2|y{|}^{m-2(n-1)}{\left(\log \frac{\mathrm{e}\sqrt{n}\ell \left(I\right)}{\left|y\right|}\right)}^2\mathrm{d}x\mathrm{d}y\\ & \lesssim (\ell (J){)}^{-m}{\int }_{\left|y\right|\le \sqrt{n}\ell \left(J\right)} {\int }_{x\in J}|f(x+y)-f\left(x\right){|}^2|y{|}^{m-2n}{\left(\log \frac{\mathrm{e}\sqrt{n}\ell \left(I\right)}{\left|y\right|}\right)}^2\mathrm{d}x\mathrm{d}y\\ & +(\ell (J){)}^{-n}{\int }_0^{\sqrt{n}} {\int }_{x\in J}|f\left(x\right)-f_J{|}^2{t}^{m-n+1}{\left(\log \frac{\mathrm{e}\sqrt{n}}{t}\right)}^2\mathrm{d}x\mathrm{d}t.\end{array}

Therefore,

\begin{array}{rl}& C_1\lesssim (\ell (J){)}^{-n\lambda -m}{\int }_{y\in J} {\int }_{x\in J}{(\left|f\right(x)-f(y\left)\right|+\ell (J{)}^{-1}|x-y||f\left(y\right)-f_J|)}^2\\ & \times |x-y{|}^{m-2n}{\left(\log \frac{\mathrm{e}\sqrt{n}\ell \left(J\right)}{|x-y|}\right)}^2\mathrm{d}x\mathrm{d}y\\ & \lesssim (\ell (J){)}^{-n\lambda -m}{\int }_{\left|y\right|\le \sqrt{n}\ell \left(J\right)} {\int }_{x\in J}|f\left(x\right)\\ & -f(x+y){|}^2|y{|}^{m-2n}{\left(\log \frac{\mathrm{e}\sqrt{n}\ell \left(J\right)}{\left|y\right|}\right)}^2\mathrm{d}x\mathrm{d}y\\ & +(\ell (J){)}^{-n(\lambda +1)}{\int }_0^{\sqrt{n}} {\int }_{x\in J}|f\left(x\right)-f_J{|}^2{t}^{m-n+1}{\left(\log \frac{\mathrm{e}\sqrt{n}}{t}\right)}^2\mathrm{d}x\mathrm{d}t.\end{array}

For C₂, if x\notin J and y\in \frac{4}{5}J, then , and

\begin{array}{rl}{C}_2& \lesssim (\ell (J){)}^{-n\lambda -m}{\int }_{y\in \frac{4}{5}J} {\int }_{\left|z\right|>\frac{1}{10}\ell \left(J\right)}|f\left(y\right)-f_J{|}^2|z{|}^{m-2n}{\left(\log \frac{\mathrm{e}\sqrt{n}\ell \left(J\right)}{\left|z\right|}\right)}^2\mathrm{d}z\mathrm{d}y\\ & \lesssim (\ell (J){)}^{-n(\lambda +1)}{\int }_{y\in \frac{4}{5}J} {\int }_{\frac{1}{10}}^{\mathrm{\infty }}|f\left(y\right)-f_J{|}^2{t}^{m-n-1}{\left(\log \frac{\mathrm{e}\sqrt{n}}{t}\right)}^2\mathrm{d}t\mathrm{d}y.\end{array}

Similarly, analogous to C₂, we can show that

C_3\lesssim (\ell (J){)}^{-n(\lambda +1)}{\int }_{y\in \frac{4}{5}J} {\int }_{\frac{1}{10}}^{\mathrm{\infty }}|f\left(y\right)-f_J{|}^2{t}^{m-n-1}{\left(\mathrm{l}\mathrm{o}\mathrm{g}\frac{\mathrm{e}\sqrt{n}}{t}\right)}^2\mathrm{d}t\mathrm{d}y.

Thus,

\begin{array}{rl}& (\ell (J){)}^{-n\lambda -m}{\int }_{S\left(I\right)}|{\mathrm{\nabla }}_{x,t}{u}_2(x,t){|}^2{\left(\log \frac{\mathrm{e}\sqrt{n}\ell \left(I\right)}{t}\right)}^2{t}^{m-n+1}\mathrm{d}x\mathrm{d}t\\ & \lesssim (\ell (J){)}^{-n\lambda -m}{\int }_{\left|y\right|\le \sqrt{n}\ell \left(J\right)} {\int }_{x\in J}|f\left(x\right)-f(x+y){|}^2|y{|}^{m-2n}{\left(\log \frac{\mathrm{e}\sqrt{n}\ell \left(J\right)}{\left|y\right|}\right)}^2\mathrm{d}x\mathrm{d}y\\ & +\left({\int }_0^{\sqrt{n}} t^{m-n+1}{\left(\log \frac{\mathrm{e}\sqrt{n}}{t}\right)}^2\mathrm{d}t+{\int }_{\frac{1}{10}}^{\mathrm{\infty }} t^{m-n-1}{\left(\log \frac{\mathrm{e}\sqrt{n}}{t}\right)}^2\mathrm{d}t\right)\\ & \times (\ell (J){)}^{-n(\lambda +1)}{\int }_{y\in J}|f\left(y\right)-f_J{|}^2\mathrm{d}y.\end{array}

If (x, t)∈S(I) and y\in {\mathbb{R}}^n\setminus \frac{2}{3}J, considering \left|{\mathrm{\nabla }}_{x,t}{p}_t\right(x-y\left)\right|\lesssim |y{|}^{-n-1},|f_3\left(y\right)|\lesssim |f\left(y\right)-f_J|, we have

\begin{array}{rl}& (\ell (J){)}^{-n\lambda -m}{\int }_{S\left(I\right)}|{\mathrm{\nabla }}_{x,t}{u}_3(x,t){|}^2{\left(\log \frac{\mathrm{e}\sqrt{n}\ell \left(I\right)}{t}\right)}^2{t}^{m-n+1}\mathrm{d}x\mathrm{d}t\\ & \le (\ell (J){)}^{-n\lambda -m}{\int }_{S\left(I\right)} t^{m-n+1}{\left(\log \frac{\mathrm{e}\sqrt{n}\ell \left(I\right)}{t}\right)}^2{\left({\int }_{{\mathbb{R}}^n}|{\mathrm{\nabla }}_{x,t}{p}_t(x-y)\left|\right|f_3\left(y\right)|\mathrm{d}y\right)}^2\mathrm{d}x\mathrm{d}t\\ & \lesssim (\ell (J){)}^{-n\lambda }{\int }_0^1 t^{m-n+1}{\left(\log \frac{\mathrm{e}\sqrt{n}}{t}\right)}^2\mathrm{d}t{\left(\ell (J){\int }_{{\mathbb{R}}^n\setminus \frac{2}{2}J} \frac{\left|f\right(y)-f_J|}{|y{|}^{n+1}}\mathrm{d}y\right)}^2.\end{array}

Therefore,

(\ell (J){)}^{-n\lambda -m}{\int }_{S\left(I\right)}|{\mathrm{\nabla }}_{x,t}u(x,t){|}^2{\left(\mathrm{l}\mathrm{o}\mathrm{g}\frac{\mathrm{e}\sqrt{n}\ell \left(I\right)}{t}\right)}^2{t}^{m-n+1}\mathrm{d}x\mathrm{d}t\lesssim D_1+D_2+D_3,

where

\{\begin{array}{rl}& D_1:=(\ell (J){)}^{-n\lambda -m}{\int }_{\left|y\right|\le \sqrt{n}\ell \left(J\right)} {\int }_{x\in J}|f\left(x\right)\\ & -f(x+y){|}^2|y{|}^{m-2n}{\left(\log \frac{\mathrm{e}\sqrt{n}\ell \left(J\right)}{\left|y\right|}\right)}^2\mathrm{d}x\mathrm{d}y;\\ & D_2:=\left({\int }_0^{\sqrt{n}} t^{m-n+1}{\left(\mathrm{l}\mathrm{o}\mathrm{g}\frac{\mathrm{e}\sqrt{n}}{t}\right)}^2\mathrm{d}t+{\int }_{10}^{-\mathrm{\infty }} t^{m-n-1}{\left(\mathrm{l}\mathrm{o}\mathrm{g}\frac{\mathrm{e}\sqrt{n}}{t}\right)}^2\mathrm{d}t\right)\\ & \times (\ell (J){)}^{-n(\lambda +1)}{\int }_{y\in J}|f\left(y\right)-f_J{|}^2\mathrm{d}y;\\ & D_3:=(\ell (J){)}^{-n\lambda }{\int }_0^1 t^{m-n+1}{\left(\log \frac{\mathrm{e}\sqrt{n}}{t}\right)}^2\mathrm{d}t{\left(\ell (J){\int }_{{\mathbb{R}}^n\setminus \frac{2}{3}J} \frac{\left|f\right(y)-f_J|}{|y{|}^{n+1}}\mathrm{d}y\right)}^2.\end{array}

4 Harmonic extension of {\mathit{Q}}_{\mathbf{l}\mathbf{o}\mathbf{g},\lambda }^{\mathit{m}}\left({\mathbb{R}}^{\mathit{n}}\right)

Let I be a cube in {\mathbb{R}}^n, and {\mathbb{R}}_{+}^{n+1}:={\mathbb{R}}^n\times (0,\mathrm{\infty }). For any cube I\subseteq {\mathbb{R}}^n, a Carleson cube on I is defined as

In this section, we introduce the following harmonic function space.

Definition 4.1　Let \lambda >0, n>m>0, and n\ge 2. A harmonic function u(\cdot ,\cdot ) belongs to H_{\mathrm{l}\mathrm{o}\mathrm{g},\lambda }^m\left({\mathbb{R}}_{+}^{n+1}\right) if and only if

It can be readily shown that the harmonic function space defined above has the following equivalent characterization.

Proposition 4.1　 Let ，n>m>0，n\ge 2. Then u\in H_{\mathrm{l}\mathrm{o}\mathrm{g},\lambda }^m\left({\mathbb{R}}_{+}^{n+1}\right) if and only if

(4.1)

Proof　First, assume u(\cdot ,\cdot )\in H_{\mathrm{l}\mathrm{o}\mathrm{g},\lambda }^m\left({\mathbb{R}}_{+}^{n+1}\right). Let I be a cube with edge length \ell \left(I\right), centered at y, and parallel to the coordinate axes. Set s=\frac{\ell \left(I\right)}{2}. For (x,t)\in S\left(I\right), we have |x-y{|}^2+(s+t{)}^2\le \frac{n+9}{4}\left(\ell \right(I){)}^2. Thus,

\frac{t}{\ell \left(I\right)}\lesssim \frac{C_{1}t{s}^n}{\left(\right|x-y{|}^2+(s+t{)}^2{)}^{\frac{n+1}{2}}},

where C_1=2^{-1}(n+9{)}^{\frac{n+1}{2}}>0. Hence, there exists an integer k₀ such that 2^{k_0}\le C_1\le 2^{k_0+1}. Therefore,

\begin{array}{rl}& (\ell (I){)}^{-n\lambda -m}{\int }_{S\left(I\right)} {\left(\mathrm{l}\mathrm{o}\mathrm{g}\frac{\mathrm{e}\sqrt{n}\ell \left(I\right)}{t}\right)}^2{t}^{m-n+1}\mathrm{d}x\mathrm{d}t\\ & \lesssim (\ell (I){)}^{-n\lambda }{\int }_0^{\ell \left(I\right)} {\int }_I s^{mn}(|x-y{|}^2+(s+t{)}^2{)}^{-\frac{m(n+1)}{2}}\\ & \times {\left(\mathrm{l}\mathrm{o}\mathrm{g}\frac{\mathrm{e}\sqrt{n}\left(\right|x-y{|}^2+(s+t{)}^2{)}^{\frac{n+1}{2}}}{s^{n}t}\right)}^2{t}^{m-n+1}\left|{\mathrm{\nabla }}_{x,t}u\right(x,t){|}^2\mathrm{d}x\mathrm{d}t\\ & \lesssim \Vert u{\Vert }_{H_{\mathrm{l}\mathrm{o}\mathrm{g},\lambda }^m\left({\mathbb{R}}_{+}^{n+1}\right)}.\end{array}

Therefore, u(\cdot ,\cdot ) satisfies (4.1).

On the contrary, suppose u(\cdot ,\cdot ) (4.1). For any fixed point (y,s)\in {\mathbb{R}}_{+}^{n+1}, let I be a cube centered at y, with edge length 2s, and parallel to the coordinate axes. For any positive integer m, denote I_m as a cube centered at the same point with I, with edge length 2^m\ell \left(I\right). S\left(I_m\right) represents the corresponding Carleson cube. Use {\chi }_{S\left(I_m\right)} to denote the characteristic function of S\left(I_m\right). Let

\begin{array}{rl}& \underset{(y,s)\in {\mathbb{R}}_{+}^{n+1}}{\mathrm{s}\mathrm{u}\mathrm{p}}(\ell (I){)}^{-n\lambda }{\int }_{{\mathbb{R}}_{+}^{n+1}} s^{mn}(|x-y{|}^2+(s+t{)}^2{)}^{-\frac{m(n+1)}{2}}\\ & \times {\left(\mathrm{l}\mathrm{o}\mathrm{g}\frac{\mathrm{e}\sqrt{n}\left(\right|x-y{|}^2+(s+t{)}^2{)}^{\frac{n+1}{2}}}{s^{n}t}\right)}^2{t}^{m-n+1}\left|{\mathrm{\nabla }}_{x,t}u\right(x,t){|}^2\mathrm{d}x\mathrm{d}t\\ & \lesssim A_0+\sum _{m=1}^{\mathrm{\infty }} A_m,\end{array}

where

\{\begin{array}{rl}& A_0:=\underset{(y,s)\in {\mathbb{R}}_{+}^{n+1}}{\mathrm{s}\mathrm{u}\mathrm{p}} (\ell (I){)}^{-n\lambda }{\int }_{{\mathbb{R}}_{+}^{n+1}} s^{mn}(|x-y{|}^2+(s+t{)}^2{)}^{-\frac{m(n+1)}{2}}\\ & \times {\left(\mathrm{l}\mathrm{o}\mathrm{g}\frac{\mathrm{e}\sqrt{n}\left(\right|x-y{|}^2+(s+t{)}^2{)}^{\frac{n+1}{2}}}{s^{n}t}\right)}^2\\ & {\chi }_{S\left(I\right)}(x,t)t^{m-n+1}\left|{\mathrm{\nabla }}_{x,t}u\right(x,t){|}^2\mathrm{d}x\mathrm{d}t;\\ & A_m:=\underset{(y,s)\in {\mathbb{R}}_{+}^{n+1}}{\mathrm{s}\mathrm{u}\mathrm{p}}(\ell (I){)}^{-n\lambda }{\int }_{{\mathbb{R}}_{+}^{n+1}} s^{mn}(|x-y{|}^2+(s+t{)}^2{)}^{-\frac{m(n+1)}{2}}\\ & \times {\left(\mathrm{l}\mathrm{o}\mathrm{g}\frac{\mathrm{e}\sqrt{n}\left(\right|x-y{|}^2+(s+t{)}^2{)}^{\frac{n+1}{2}}}{s^{n}t}\right)}^2\\ & {\chi }_{S\left(I_m\right)\setminus S\left(I_{m-1}\right)}(x,t)t^{m-n+1}\left|{\mathrm{\nabla }}_{x,t}u\right(x,t){|}^2\mathrm{d}x\mathrm{d}t.\end{array}

For A_0, we have

\begin{array}{rl}& A_0\lesssim \underset{(y,s)\in {\mathbb{R}}_{+}^{n+1}}{\mathrm{s}\mathrm{u}\mathrm{p}}(\ell (I){)}^{-n\lambda }{\int }_{{\mathbb{R}}_{+}^{n+1}} s^{mn}(|x-y{|}^2+(s+t{)}^2{)}^{-\frac{m(n+1)}{2}}\\ & \times {\left(\mathrm{l}\mathrm{o}\mathrm{g}\frac{\mathrm{e}\sqrt{n}\left(\right|x-y{|}^2+(s+t{)}^2{)}^{\frac{n+1}{2}}}{s^{n}t}\right)}^2{\chi }_I\left(x\right){\chi }_{[0,\ell (I\left)\right]}\left(t\right)t^{m-n+1}\left|{\mathrm{\nabla }}_{x,t}u\right(x,t){|}^2\mathrm{d}x\mathrm{d}t\\ & \lesssim \underset{I\subset {\mathbb{R}}^n}{\mathrm{s}\mathrm{u}\mathrm{p}}(\ell (I){)}^{-n\lambda -m}{\int }_{S\left(I\right)} {\left(\mathrm{l}\mathrm{o}\mathrm{g}\frac{\mathrm{e}\sqrt{n}\ell \left(I\right)}{t}\right)}^2{t}^{m-n+1}\mathrm{d}x\mathrm{d}t.\end{array}

For A_m, if (x,t)\in S\left(I_m\right)\setminus S\left(I_{m-1}\right), then (x,t)\in S_1\cup S_2, where

Then, A_m\lesssim B_{m,1}+B_{m,2}, where

\{\begin{array}{rl}& B_{m,1}:=(\ell (I){)}^{-n\lambda }{\int }_{{\mathbb{R}}_{+}^{n+1}} s^{mn}(|x-y{|}^2+(s+t{)}^2{)}^{-\frac{m(n+1)}{2}}{\chi }_{I_m}\left(x\right){\chi }_{\left[2^{m-1}\ell \right(I),2^m\ell (I\left)\right]}\left(t\right)\\ & \times {\left(\mathrm{l}\mathrm{o}\mathrm{g}\frac{\mathrm{e}\sqrt{n}\left(\right|x-y{|}^2+(s+t{)}^2{)}^{\frac{n+1}{2}}}{s^{n}t}\right)}^2{t}^{m-n+1}\left|{\mathrm{\nabla }}_{x,t}u\right(x,t){|}^2\mathrm{d}x\mathrm{d}t;\\ & B_{m,2}:=\underset{(y,s)\in {\mathbb{R}}_{+}^{n+1}}{\mathrm{s}\mathrm{u}\mathrm{p}}(\ell (I){)}^{-n\lambda }{\int }_{{\mathbb{R}}_{+}^{n+1}} s^{mn}(|x-y{|}^2\\ & +(s+t{)}^2{)}^{-\frac{m(n+1)}{2}}{\chi }_{I_m\setminus I_{m-1}}\left(x\right){\chi }_{[0,2^m\ell (I\left)\right]}\left(t\right)\\ & \times {\left(\mathrm{l}\mathrm{o}\mathrm{g}\frac{\mathrm{e}\sqrt{n}\left(\right|x-y{|}^2+(s+t{)}^2{)}^{\frac{n+1}{2}}}{s^{n}t}\right)}^2{t}^{m-n+1}\left|{\mathrm{\nabla }}_{x,t}u\right(x,t){|}^2\mathrm{d}x\mathrm{d}t.\end{array}

First, we estimate B_{m,1}. Let

Thus,

\begin{array}{rl}{B}_{m,1}& \lesssim (\ell (I){)}^{-n\lambda -m}{\int }_{2^{m-1}\ell \left(I\right)}^{2^m\ell \left(I\right)} \sum _{m=1}^{\mathrm{\infty }} {\int }_{I_m} {\left(\mathrm{l}\mathrm{o}\mathrm{g}\frac{\mathrm{e}\sqrt{n}\ell \left(I\right)}{t}\right)}^2{t}^{m-n+1}|{\mathrm{\nabla }}_{x,t}u(x,t){|}^2\mathrm{d}x\mathrm{d}t\\ & \lesssim \sum _{m=1}^{\mathrm{\infty }} 2^{mn\lambda }\mathrm{\Psi }\left(2^{-mn}\right)(2^m\ell (I){)}^{-n\lambda -m}{\int }_{2^{m-1}\ell \left(I\right)}^{2^m\ell \left(I\right)} {\int }_{I_m} {\left(\mathrm{l}\mathrm{o}\mathrm{g}\frac{\mathrm{e}\sqrt{n}\ell \left(I\right)}{t}\right)}^2{t}^{m-n+1}\\ & \times \left|{\mathrm{\nabla }}_{x,t}u\right(x,t){|}^2\mathrm{d}x\mathrm{d}t.\end{array}

Because

where the last inequality uses and

Therefore,

Similarly,

Thus, this completes the proof of Proposition 4.1. □

Next, we provide the harmonic extension of Q_{\mathrm{l}\mathrm{o}\mathrm{g},\lambda }^m\left({\mathbb{R}}^n\right).

Theorem 4.1　 Suppose f\in L_{\mathrm{l}\mathrm{o}\mathrm{c}}^2\left({\mathbb{R}}^n\right) and satisfies Inequality (1.2). Then for , , n\ge 2, if f\in Q_{\mathrm{l}\mathrm{o}\mathrm{g},\lambda }^m\left({\mathbb{R}}^n\right), then u(x,t)=f\ast p_t\in H_{\mathrm{l}\mathrm{o}\mathrm{g},\lambda }^m\left({\mathbb{R}}_{+}^{n+1}\right).

Proof　Assume f\in Q_{\mathrm{l}\mathrm{o}\mathrm{g},\lambda }^m\left({\mathbb{R}}^n\right). According to Lemma 3.2, it is evident that D_1\lesssim \Vert f{\Vert }_{Q_{\mathrm{l}\mathrm{o}\mathrm{g},\lambda }^m\left({\mathbb{R}}^n\right)}^2. Using Holder’s inequality and straightforward calculations, we obtain

D_2\lesssim (\ell (J){)}^{-n(\lambda +1)}{\int }_{y\in J}|f\left(y\right)-f_J{|}^2\mathrm{d}y\lesssim (\ell (J){)}^{-n(\lambda +2)}{\int }_J {\int }_J|f\left(x\right)-f\left(y\right){|}^2\mathrm{d}x\mathrm{d}y.

According to Theorem 2.2, Q_{\mathrm{l}\mathrm{o}\mathrm{g},\lambda }^m\left({\mathbb{R}}^n\right)\subseteq {\mathcal{L}}_{\lambda +1}\left({\mathbb{R}}^n\right). Therefore, D_2\le \Vert f{\Vert }_{Q_{\mathrm{l}\mathrm{o}\mathrm{g},\lambda }^m\left({\mathbb{R}}^n\right)}^2. Finally, estimating D₃,

\begin{array}{rl}{D}_3& \lesssim (\ell (J){)}^{-n\lambda +2}{\left({\int }_{{\mathbb{R}}^n\setminus (2/3)J} \frac{\left|f\right(x)-f_J|}{|x-x_0{|}^{n+1}}\mathrm{d}x\right)}^2\\ & \lesssim (\ell (J){)}^{-n\lambda +2}{\left(\sum _{k=1}^{\mathrm{\infty }} {\int }_{2^k\ell \left(J\right)\le |x-x_0|\le 2^{k+1}\ell \left(J\right)} \frac{\left|f\right(x)-f_J|}{|x-x_0{|}^{n+1}}\mathrm{d}x\right)}^2\\ & \le (\ell (J){)}^{-n\lambda +2}(\sum _{k=1}^{\mathrm{\infty }} \frac{1}{\left(2^k\ell \right(J){)}^{n+1}}{\int }_{2^{k+1}J} \left|f\right(x)-f_{2^{k+1}J}|\mathrm{d}x\\ & +\sum _{k=1}^{\mathrm{\infty }} \sum _{i=0}^k \frac{1}{\left(2^k\ell \right(J){)}^{n+1}}{\int }_{2^{k+1}J} |f_{2^{i+1}J}-f_{2^{i}J}|\mathrm{d}x{)}^2.\end{array}

Because

{\int }_{2^{k+1}J} \left|f\right(x)-f_{2^{k+1}J}{|}^2\mathrm{d}x\lesssim (2^{k+1}\ell \left(J\right){)}^{-n}{\int }_{2^{k+1}J} {\int }_{2^{k+1}J} \left|f\right(x)-f(y){|}^2\mathrm{d}x\mathrm{d}y,

we have

\begin{array}{rl}& (2^k\ell (J){)}^{-n(1+\frac{\lambda }{2})}{\int }_{2^{k+1}J}|f\left(x\right)-f_{2^{k+1}J}|\mathrm{d}x\\ & \lesssim {\left((2^{k+1}\ell (J){)}^{-n(2+\lambda )}{\int }_{2^{k+1}J} {\int }_{2^{k+1}J}|f\left(x\right)-f\left(y\right){|}^2\mathrm{d}x\mathrm{d}y\right)}^{\frac{1}{2}}\lesssim \Vert f{\Vert }_{Q_{\mathrm{l}\mathrm{o}\mathrm{g},\lambda }^m\left({\mathbb{R}}^n\right)},\end{array}

indicating that

\begin{array}{rl}& (\ell (J){)}^{-\frac{n\lambda }{2}+1}\sum _{k=1}^{\mathrm{\infty }}(2^{k}l\left(J\right){)}^{-n-1}{\int }_{2^{k+1}J} \left|f\right(x)-f_{2^{k+1}J}|\mathrm{d}x\\ & \lesssim \sum _{k=1}^{\mathrm{\infty }} 2^{k(\frac{\lambda n}{2}-1)}\Vert f{\Vert }_{Q_{\mathrm{l}\mathrm{o}\mathrm{g},\lambda }^m\left({\mathbb{R}}^n\right)}\lesssim \Vert f{\Vert }_{Q_{\mathrm{l}\mathrm{o}\mathrm{g},\lambda }^m\left({\mathbb{R}}^n\right)}.\end{array}

For i=0,1,2,\dots ,k, we have

|f_{2^{t+1}J}-f_{2^{t}J}|\lesssim (2^{i+1}\ell (J){)}^{-n}{\int }_{2^{t+1}J}|f\left(x\right)-f_{2^{t+1}J}|\mathrm{d}x\lesssim (2^{i+1}\ell \left(J\right){)}^{\frac{n\lambda }{2}}\Vert f{\Vert }_{Q_{\mathrm{l}\mathrm{o}\mathrm{g},\lambda }^m\left({\mathbb{R}}^n\right)}.

Therefore,

\begin{array}{rl}& (\ell (J){)}^{-\frac{n\lambda }{2}+1}\sum _{k=1}^{\mathrm{\infty }} \sum _{i=0}^k(2^{k}l\left(J\right){)}^{-n-1}{\int }_{2^{k+1}J} |f_{2^{i+1}J}-f_{2^{i}J}|\mathrm{d}x\\ & \lesssim \sum _{k=1}^{\mathrm{\infty }} \sum _{i=0}^k 2^{-k+\frac{tn\lambda }{2}}\Vert f{\Vert }_{Q_{\mathrm{l}\mathrm{o}\mathrm{g},\lambda }^m\left({\mathbb{R}}^n\right)}\lesssim \Vert f{\Vert }_{Q_{\mathrm{l}\mathrm{o}\mathrm{g},\lambda }^m\left({\mathbb{R}}^n\right)}.\end{array}

Thus, D_3\lesssim \Vert f{\Vert }_{Q_{\mathrm{l}\mathrm{o}\mathrm{g},\lambda }^m\left({\mathbb{R}}^n\right)}^2, which implies

(\ell (J){)}^{-n\lambda -m}{\int }_{S\left(I\right)}|{\mathrm{\nabla }}_{x,t}u(x,t){|}^2{\left(\mathrm{l}\mathrm{o}\mathrm{g}\frac{\text{e}\sqrt{n}\ell \left(I\right)}{t}\right)}^2{t}^{m-n+1}\text{d}x\text{d}t\lesssim \Vert f{\Vert }_{Q_{\mathrm{l}\mathrm{o}\mathrm{g},\lambda }^m\left({\mathbb{R}}^n\right)}^2.

Therefore, u(x,t)\in H_{\mathrm{l}\mathrm{o}\mathrm{g},\lambda }^m\left({\mathbb{R}}_{+}^{n+1}\right). □

5 Boundary value problems

Next, we investigate the boundary value problems for H_{\mathrm{l}\mathrm{o}\mathrm{g},\lambda }^m\left({\mathbb{R}}_{+}^{n+1}\right).

Theorem 5.1　 Suppose f\in L_{\mathrm{l}\mathrm{o}\mathrm{c}}^2\left({\mathbb{R}}^n\right) and satisfies Inequality (1.2). Then for , ,n\ge 2, if u(x,t)=f\ast p_t\in H_{\mathrm{l}\mathrm{o}\mathrm{g},\lambda }^m\left({\mathbb{R}}_{+}^{n+1}\right), then f\in Q_{\mathrm{l}\mathrm{o}\mathrm{g},\lambda }^m\left({\mathbb{R}}^n\right).

Proof　According to the equivalent form (2.1) of Q_{\mathrm{l}\mathrm{o}\mathrm{g},\lambda }^m\left({\mathbb{R}}^n\right), let u(x,t) be the Poisson integral of f\left(x\right). Then

where

For A_1, according to u(x,|y\left|\right)-u\left(x\right)={\int }_0^{\left|y\right|} \frac{\mathrm{\partial }u(x,t)}{\mathrm{\partial }t}\mathrm{d}t, using the Minkowski inequality, we obtain

\begin{array}{rl}& {\left({\int }_I \left|u\right(x,\left|y\right|)-u(x){|}^2\mathrm{d}x\right)}^{\frac{1}{2}}\le {\int }_0^{\left|y\right|} {\left({\int }_I {\left|\frac{\mathrm{\partial }u(x,t)}{\mathrm{\partial }t}\right|}^2\mathrm{d}x\right)}^{\frac{1}{2}}\mathrm{d}t\\ & \le {\int }_0^{\left|y\right|} {\left({\int }_I \left|{\mathrm{\nabla }}_{x,t}u\right(x,t){|}^2\mathrm{d}x\right)}^{\frac{1}{2}}\mathrm{d}t.\end{array}

Since

(5.1)

based on Lemma 3.1 and (5.1), we have

For A_2, it is easy to see that

{\int }_I \left|u\right(x+y,\left|y\right|)-u(x+y){|}^2\mathrm{d}x\le {\int }_{3I} |u(x,|y\left|\right)-u\left(x\right){|}^2\mathrm{d}x.

Similarly, we have

For A_3, noting that

\left|u\right(x+y,\left|y\right|)-u(x,\left|y\right|\left)\right|\le {\int }_0^{\left|y\right|}\left|\frac{\mathrm{\partial }u}{\mathrm{\partial }t}\left(x+\frac{ty}{\left|y\right|},|y|\right)\right|\mathrm{d}t,

again, using the Minkowski inequality, we get

\begin{array}{rl}& {\left({\int }_I \left|u\right(x+y,\left|y\right|)-u(x,\left|y\right|){|}^2\mathrm{d}x\right)}^{\frac{1}{2}}\le {\int }_0^{\left|y\right|} {\left({\int }_I {\left|\frac{\mathrm{\partial }u}{\mathrm{\partial }t}\left(x+\frac{ty}{\left|y\right|},|y|\right)\right|}^2\mathrm{d}x\right)}^{\frac{1}{2}}\mathrm{d}t\\ & \le \left|y\right|{\left({\int }_{3I}|{\mathrm{\nabla }}_{x,t}u(x,{\left|y\left|\right)\right|}^2\mathrm{d}x\right)}^{\frac{1}{2}}.\end{array}

Thus,

Therefore, f\in Q_{\mathrm{l}\mathrm{o}\mathrm{g},\lambda }^m\left({\mathbb{R}}^n\right). This completes the proof of Theorem 5.1. □

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