UDC 517.581

MATHEMATICS

A. A. BONAMI

ON CONJUGATE HARMONIC FUNCTIONS OF SEVERAL VARIABLES

(Presented by Academician V. I. Smirnov on 18 XI 1969)

Denote by (\mathbf{x}=(x_1,\ldots,x_n)) the points of (n)-dimensional Euclidean space (\mathbb{R}^n); (y) is a real variable; ((\mathbf{x},y)=(x_1,\ldots,x_n,y)) are the points of the Cartesian product (\mathbb{R}^n\times\mathbb{R}=\mathbb{R}^{n+1}); (\mathbf{t}\cdot\mathbf{x}=t_1x_1+\cdots+t_nx_n); (|\mathbf{x}|^2=\mathbf{x}\cdot\mathbf{x}); (dt) is the element of Lebesgue measure in (\mathbb{R}^n). In the half-space (\mathbb{R}^{n+1}_+=\mathbb{R}^n\times(0,+\infty)) we shall consider harmonic vectors (\mathbf{F}(\mathbf{x},y)=(U,V_1,\ldots,V_n)), i.e. ((^1)) vector-functions ((\mathbf{x},y)), whose components are harmonic functions satisfying the generalized Cauchy—Riemann conditions

[
\frac{\partial U}{\partial y}+\sum_{k=1}^{n}\frac{\partial V_k}{\partial x_k}=0,\quad
\frac{\partial U}{\partial x_k}=\frac{\partial V_k}{\partial y},\quad
\frac{\partial V_k}{\partial x_j}=\frac{\partial V_j}{\partial x_k},\quad
k\ne j;\ k,j=1,2,\ldots,n.
]

In the present paper the theorems proved by Hardy, Littlewood ((^2)) and Kawata ((^3)) in the plane are generalized to the multidimensional case.

Denote

[
M(y)=M(\mathbf{F},y)=\max_{\mathbf{x}\in\mathbb{R}^n}|\mathbf{F}(\mathbf{x},y)|;
\tag{1}
]

[
M_p(y)=M_p(\mathbf{F},y)=\left{\int_{\mathbb{R}^n}|\mathbf{F}(\mathbf{x},y)|^p\,dx\right}^{1/p}.
\tag{2}
]

Definition 1. We shall say that a harmonic vector (\mathbf{F}(\mathbf{x},y)) in (\mathbb{R}^{n+1}_+) belongs to the class (H^p), (p>0), if (M_p(\mathbf{F},y)<C), where (C) does not depend on (y).

Definition 2. We shall say that a harmonic vector (\mathbf{F}(\mathbf{x},y)) in (\mathbb{R}^{n+1}_+) belongs to the class (S^p), (p>0), if for every (y_0>0) there exists a constant (C(y_0,\mathbf{F})) such that, for all (y\ge y_0), (M_p(\mathbf{F},y)\le C(y_0,\mathbf{F})).

As is known ((^1)), for (p\ge(n-1)/n) harmonic vectors (\mathbf{F}(\mathbf{x},y)\in H^p) have finite boundary values (f(\mathbf{x})) almost everywhere in (\mathbb{R}^n) as (y\downarrow0), and for (p>(n-1)/n) convergence in the mean with exponent (p) also holds.

Theorem 1. Let (f(\mathbf{x})=(u,v_1,\ldots,v_n)\in L^2(\mathbb{R}^n)). Then the necessary and sufficient condition for (f(\mathbf{x})) to be the boundary value as (y\downarrow0) of a harmonic vector (\mathbf{F}(\mathbf{x},y)) of class (H^2) in (\mathbb{R}^{n+1}_+) is that the Fourier transform of the function (f(\mathbf{x})) can be represented in the form

[
g(\mathbf{x})\left(e_0+i\sum_{k=1}^{n}e_k\frac{t_k}{|\mathbf{t}|}\right),
\tag{3}
]

where (e_0,e_1,\ldots,e_n) are the units of the Clifford algebra, and

[
g(\mathbf{x})=\operatorname*{l.i.m.}{T\to\infty}\frac{1}{(2\pi)^{n/2}}\int\,dt.}|<T}u(\mathbf{t})e^{i\mathbf{x}\cdot\mathbf{t}
]

The proof is obtained by applying the Plancherel theorem to the representation of (\mathbf F(\mathbf x,y)) by means of the Poisson integral. As a result one also obtains the following integral representation for harmonic vectors of class (\mathbf H^2) in (\mathbf R_+^{n+1})

[
\mathbf F(\mathbf x,y)=\frac{1}{(2\pi)^{n/2}}\int_{\mathbf R^n} g(\mathbf t)e^{-y|\mathbf t|}
\left(e_0+i\sum_{k=1}^n e_k\frac{t_k}{|\mathbf t|}\right)e^{-i\mathbf x\cdot \mathbf t}\,d\mathbf t .
\tag{4}
]

Lemma. If (p\ge (n-1)/n,\ a\ge 0,\ \mathbf F(\mathbf x,y)=(U,V_1,\ldots,V_n)) is a harmonic vector of class (S^p) in (\mathbf R_+^{n+1}), then from the inequality

[
M_p(U,y)\le Cy^{-a}
\tag{5}
]

it follows that

[
|\mathbf F(\mathbf x,y)|\le BCy^{-B},
\tag{6}
]

where (B) and (C) are positive constants.

Proof. We first consider the case ((n-1)/n\le p\le 1). Choose (C=1,\ y_0>0), and consider the harmonic vector (\mathbf\Phi(\mathbf x,y)=\mathbf F(\mathbf x,y+y_0)). According to (4), (\mathbf\Phi(\mathbf x,y)=(W_0,W_1,\ldots,W_n)) belongs to (\mathbf H^1\cap\mathbf H^2) in (\mathbf R_+^{n+1}). Let (G(\mathbf t)) be the Fourier transform of the boundary values (W_0(\mathbf x)) of the function (W_0(\mathbf x,y)). Then

[
G(\mathbf t)e^{-y(\mathbf t)}=\frac{1}{(2\pi)^{n/2}}\int_{\mathbf R^n} W_0(\mathbf x,y)e^{i\mathbf x\cdot \mathbf t}\,d\mathbf t .
\tag{7}
]

If (p=1), then from (4) and (5) it follows that

[
|\mathbf F(\mathbf x,y+y_0)|\le
\frac{2}{(2\pi)^{n/2}}\int_{\mathbf R^n}|G(\mathbf t)|e^{-(y+y_0)|\mathbf t|}\,d\mathbf t
\le
\frac{2}{(2\pi)^n}(y+y_0)^{-a}\int_{\mathbf R^n}e^{-y_0|\mathbf t|}\,d\mathbf t,
]

or (|\mathbf F(\mathbf x,y+y_0)|=O((y+y_0)^{-a-n})).

If ((n-1)/n\le p<1), then from (7) and (5) we have

[
|G(\mathbf t)|e^{-y|\mathbf t|}\le \frac{1}{(2\pi)^{n/2}}M^{1-p}(U,\eta)\eta^{-ap},\qquad \eta=y+y_0 .
\tag{8}
]

Let (\eta_1=y_1+y_0>y+y_0=\eta). Then

[
|\mathbf F(\mathbf x,\eta)|\le
\frac{2}{(2\pi)^{n/2}}M^{1-p}(U,\eta)\int_{\mathbf R^n}e^{-(y_1-y)|\mathbf t|}\,d\mathbf t,
\tag{9}
]

[
|\mathbf F(\mathbf x,\eta)|\le KM^{1-p}(\mathbf F,\eta)\eta^{-ap}(\eta_1-\eta)^{-n},\qquad K=\mathrm{const}.
]

Using the results of the paper ((^2)), from (9) we obtain

[
M(\mathbf F,y)\le By^{-B}.
]

Theorem 2. If a harmonic vector (\mathbf F(\mathbf x,y)=(U,V_1,\ldots,V_n)) of class (S^p) in (\mathbf R_+^{n+1}), (p\ge (n-1)/n,\ a\ge 0,\ q>p), then from

[
M_p(U,y)\le Cy^{-a}
\tag{10}
]

it follows that

[
M_q(\mathbf F,y)\le BCy^{-a-n/p+n/q}.
\tag{11}
]

In particular, for (q=\infty),

[
M(\mathbf F,y)\le BCy^{-a-n/p}.
\tag{12}
]

Here (B,C) are constants independent of (y).

Proof. We shall first carry it out for ((n-1)/n \le p \le 1). It suffices to prove (12). Put
(\tau=1/\eta_1,\quad t=2/\eta_1=1/\eta,\quad h(\tau)=\ln(\eta_1^{a+n/p}M(\eta_1)),\quad b=1-p.)

Then, using (9),

[
h(\tau)-bh(2\tau)\le (a+n/p)\ln(\eta_1/\eta)+n\ln(\eta/(\eta_1-\eta))<B .
]

On the basis of (2), either (h(\tau)) is bounded by a constant, and the theorem is true, or

[
\varlimsup_{\tau\to\infty} h(\tau)/\ln\tau=\infty,
]

but the latter contradicts the lemma.

The case (p>1) is obtained with the aid of the following two theorems.

Theorem 3. Let (\mathbf F(\mathbf x,y)=(U,V_1,\ldots,V_n)) be a harmonic vector of class (S^p(\mathbf R^{n+1}_+)), (p>1). There exists a constant (C_p), depending only on (p), such that for every (y>0) the inequality

[
|\mathbf F(\mathbf x,y)|_p\le C_p|U(\mathbf x,y)|_p
]

holds.

Theorem 4. If (p\ge (n-1)/n), (a\ge0), (q>p), and a harmonic vector (\mathbf F(\mathbf x,y)) belongs to the class (S^p) in (\mathbf R^{n+1}_+), then from the condition

[
M_p(\mathbf F,y)=O(y^{-a})
\tag{13}
]

it follows that

[
M_q(\mathbf F,y)=O(y^{-a-n/p+n/q}).
\tag{14}
]

In particular, for (q=\infty),

[
M(\mathbf F,y)=O(y^{-a-n/p}).
\tag{15}
]

Proof for ((n-1)/n\le p\le1) follows from the part of the proof of Theorem 2 already carried out. Let (p>1) and take (\eta>0). In the half-space (\mathbf R^n\times(\eta,+\infty)) the function (\mathbf F(\mathbf x,y)\in H^p), and therefore, by (1), is representable by the Poisson–Lebesgue integral

[
\mathbf F(\mathbf x,y)=\frac1{c_n}\int_{\mathbf R^n}
\frac{\mathbf F(\mathbf t,\eta)(y-\eta)}
{\left(|\mathbf x-\mathbf t|^2+(y-\eta)^2\right)^{(n+1)/2}}\,d\mathbf t,
\qquad
c_n=\frac{\pi^{(n+1)/2}}{\Gamma((n+1)/2)} .
]

With the aid of Hölder’s inequality we have

[
|\mathbf F(\mathbf x,y)|\le M_p(\mathbf F,\eta)\frac1{c_n}
\left{\int_{\mathbf R^n}
\frac{(y-\eta)^{p'}}
{\left(|\mathbf x-\mathbf t|^2+(y-\eta)^2\right)^{(n+1)p'/2}}\,d\mathbf t
\right}^{1/p'},
]

where (p'=p/(p-1)), and hence

[
|\mathbf F(\mathbf x,y)|=O(y^{-a-n/p}).
]

To prove (11), we note on the basis of (10) that

[
M_q(\mathbf F,y)\le M^{(q-p)/q}(\mathbf F,y)\,M_p^{p/q}\le By^{-a-n/p+n/q}.
]

Vladimir State Pedagogical Institute
named after P. I. Lebedev-Polyanskii

Received
24 X 1969

REFERENCES

¹ E. M. Stein, G. Weiss, Acta Math., 103, No. 1–2, 25 (1960).
² G. H. Hardy, J. E. Littlewood, J. reine u. angew. Math., 167, 405 (1932).
³ T. Kawata, Japan. J. Math., 13, No. 3, 421 (1936).
⁴ U. Kuran, Proc. London Math. Soc., (3), 16, 478 (1966).