HARDY SPACES AND PARTIAL DERIVATIVES OF CONJUGATE HARMONIC FUNCTIONS

ANATOLY RYABOGIN AND DMITRY RYABOGIN

Abstract. In this paper we give necessary and sufficient conditions for a harmonic vector and all its partial derivatives to belong to H^{p}(R_{+}^{n+1}) for all p > 0.

1. Introduction and statements of main results

In this article we study the following problem: what can we say about conjugate harmonic functions in R_{+}^{n+1} = R^{n}×(0, ∞), provided we are given certain restrictions, imposed on partial derivatives of a harmonic vector F,

F = (U(x, y), V_{1}(x, y), V_{2}(x, y), ..., V_{n}(x, y)), (x, y) ∈ R_{+}^{n+1}.

We refer the reader to the classical works [11], [3], [13], [15], [5], [2], [4], [16] for the history and different results related to this problem and classes S^{p}(R_{+}^{n+1}), h^{p}(R_{+}^{n+1}), H^{p}(R_{+}^{n+1}), (all definitions are given in Section 2).

We give necessary and sufficient conditions for a harmonic vector and all its partial derivatives up to the order k to belong to H^{p}(R_{+}^{n+1}), p > 0. Our main result is

Theorem 1. Let 0 < p < q. The harmonic vector F = (U, V_{1}, ..., V_{n}) and all its partial derivatives of the order ≤ k belong to H^{r}, p ≤ r ≤ q, if and only if

(1)  1) M_{p}(y + 1, F) ≤ C,  2) ∫_{R^{n}} ( sup_{η≥y} |D_{n+1}^{k}U(x, η)|)^{q}dx ≤ C.

The case p < (n − 1)/n leads to additional technical difficulties, since |F |^{p} is subharmonic, provided p ≥ (n − 1)/n, [13]. One of the methods of the proof is the application of classes S^{p}(R_{+}^{n+1}) together with the Lagrange mean-value Theorem. We also use the boundary behaviour of the conjugate harmonic functions. We note that the first condition in (1) is natural not only due to the decomposition of the function into two parts, “S^{p}(R_{+}^{n+1})” and “H^{p}(R_{+}^{n+1})” (see Section 3, Proof of Theorem 5). In fact, it, together with the second condition, implies M_{p}(y + y_{0}, F) ≤ C(y_{0}) ∀y_{0} > 0 (see Section 4, Lemma 7).

The paper is organized as follows. In section 2 we give all necessary definitions and auxiliary results used in the sequel. Section 3 is devoted to the results needed for the proof of Theorem 1, and in section 4 we prove Theorem 1. For convenience of the reader we split our proofs into elementary Lemmata.

1991 Mathematics Subject Classification. Primary 30E25, secondary 42B25.

Key words and phrases. Hardy spaces, subharmonic functions.

2. Auxiliary results

We begin with the definition of classes S^{p}(R_{+}^{n+1}) and h^{p}(R_{+}^{n+1}).

Let U(x, y) be a harmonic function in R_{+}^{n+1} ≡ R^{n} × (0, ∞). We say that the vector-function V (x, y) = (V_{1}(x, y), ..., V_{n}(x, y)) is the conjugate of U(x, y) in the sense of M. Riesz [12], [14], if V_{k}(x, y), k = 1, ..., n are harmonic functions, satisfying the generalized Cauchy-Riemann conditions:

\frac{∂U}{∂y} + ∑_{k=1}^{n} \frac{∂V_{k}}{∂x_{k}} = 0,  \frac{∂V_{i}}{∂x_{k}} = \frac{∂V_{k}}{∂x_{i}} ,  \frac{∂U}{∂x_{i}} = \frac{∂V_{i}}{∂y} ,  i ≠ k, k = 1, ..., n.

If U(x, y) and V (x, y) are conjugate in R_{+}^{n+1} in the above sense, then the vectorfunction

F (x, y) = (U(x, y), V (x, y)) = (U(x, y), V_{1}(x, y), ..., V_{n}(x, y))

is called a harmonic vector.

Define

M_{p}(y) = M_{p}(y, F) = { ∫_{R^{n}} |F (x, y)|^{p}dx}^{1/p}, p > 0.

Definition 1 ([1], [7]). We say that F (x, y) ∈ S^{p}(R_{+}^{n+1}), p > 0 if for any y_{0} > 0 there exists a constant C(y_{0}, F), such that ∀y ≥ y_{0}, M_{p}(y, F) ≤ C(y_{0}). In particular, if C is independent of y_{0}, then F (x, y) ∈ h^{p}(R_{+}^{n+1}).

Now we define the space H^{p}(R_{+}^{n+1}). We follow the work of Fefferman and Stein[4]. Let U(x, y) be a harmonic function in R_{+}^{n+1}, and let U_{j_{1}j_{2}j_{3}...j_{k}} denote a component of a symmetric tensor of rank k, 0 ≤ j_{i} ≤ n, i = 1, ..., n. Suppose also that the trace of our tensor is zero, meaning

∑_{j=0}^{n} U_{jjj_{3}...j_{k}}(x, y) = 0, ∀j_{3}, ..., j_{k}.

The tensor of rank k + 1 can be obtained from the above tensor of rank k by passing to its gradient:

U_{j_{1}j_{2}...j_{k}j_{k+1}}(x, y) = \frac{∂}{∂x_{j_{k+1}}} (U_{j_{1}j_{2}j_{3}...j_{k}}(x, y)), x_{0} = y, 0 ≤ j_{k+1} ≤ n.

Definition 2 ([4]).We say that U(x, y) ∈ H^{p}(R_{+}^{n+1}), p > 0, if there exists a tensor of rank k of the above type with the properties:

U_{0...0}(x, y) = U(x, y),  sup_{y>0} ∫_{R^{n}} ( ∑_{(j)} U_{(j)}^{2}(x, y))^{p/2}dx < ∞,  (j) = (j_{1}, ...j_{k}).

It is well-known that the function ( ∑_{(j)} U_{(j)}^{2}(x, y))^{p/2} is subharmonic for p ≥ p_{k} =

(n − k)/(n + k − 1), see [3],[4],[14].

We will use the “radial” and nontangential maximal functions:

F^{+}(x) = sup_{y>0} |F (x, y)|, N_{α}(F)(x^{0}) = sup_{(x,y)∈Γ_{α}(x^{0})} |F (x, y)|.

H^{p} SPACES OF CONJUGATE HARMONIC FUNCTIONS 3

Here

Γ_{α}(x^{0}) = {(x, y) ∈ R_{+}^{n+1} : |x − x^{0}| < αy}, α > 0,

is an infinite cone with the vertex at x^{0}. It is well-known [4] that

F (x, y) ∈ H^{p}(R_{+}^{n+1}) ⟺ N_{α}(F)(x) ∈ L^{p} ⟺ F^{+}(x) ∈ L^{p}, p > 0.

We also define the weak maximal function

W F (x, y) = sup_{ζ≥y} |F (x, ζ)|, y > 0.

The above expression is understood as folows: we fix x, and for fixed y we find the supremum over all ζ ≥ y.

We will repeatedly use the following results.

Lemma 1. ([4], p.173). Suppose u(x, y) is harmonic in R_{+}^{n+1}, and for some p, 0 < p < ∞,

sup_{y>0} ∫_{R^{n}} |u(x, y)|^{p}dx < ∞,

then

(2)  sup_{x∈R^{n}} |u(x, y)| ≤ Ay^{−n/p},  0 < y < ∞.

Theorem 2. ([5], p. 268). Let 0 < p ≤ 1, k ∈ N, and let u : R_{+}^{n+1} → R be a harmonic function such that

u(x, t) ⇒_{t→∞}^{x} 0, K_{k,p} ≡ ∫_{R_{+}^{n+1}} t^{kp−1}|D_{n+1}^{k}u(x, t)|^{p}dxdt < C.

Then u(x, 0) = lim_{t→0+} u(x, t) exists for almost all x ∈ R^{n}, and for all t ≥ 0,

∫_{R^{n}} |u(x, t)|^{p}dx ≤ AC(k, n, p)K_{k,p}.

Theorem 3. ([5], p. 269). Let m ∈ N, p ≥ (n − 1)/(m + n − 1) (if n = 1 we suppose p > 0), and let u : R_{+}^{n+1} → R be harmonic. Then, for all t > 0,

∫_{R^{n}} |∇^{m}u(x, t)|^{p}dx ≤ A(m, n, p)t^{−mp−1} ∫_{t/2}^{3t/2}ds ∫_{R^{n}} |u(x, t)|^{p}dx.

Corollary 1. ([5], p. 270). Let p, m be as in Theorem 3, let b > 0, and let u : R_{+}^{n+1} → R be a harmonic function such that for all t > 0

∫_{R^{n}} |u(x, t)|^{p}dx ≤ Ct^{−b}.

Then

∫_{R^{n}} |∇^{m}u(x, t)|^{p}dx ≤ A(b, m, n, p)Ct^{−b−mp}, (t > 0).

In fact, the choice of p in Theorem 2 and Corollary 1 may be independent of m (see Lemma 2).

Theorem 4. [10]. Let p > 0 and let F (x, y) = (U, V_{1}, ..., V_{n}) be a harmonic vector satisfying

(3) 1) V_{i}(x, y) ⇒_{y→∞}^{x} 0, i = 1, ..., n,  2) M_{p}(y, U) ≤ C,  3) |U| ≤ C.

Then F ∈ H^{r}, r > p.

Notation. We denote by D_{i}^{k}f(x, y) the partial derivative of the function f of the order k with respect to x_{i}, i = 1, 2, ..., n + 1. M(f)(x) denotes the usual Hardy-Littlewood maximal function of f(x). The notation f(x, y) ⇒_{y→∞}^{x} 0 means that f(x, y) converges to 0 uniformly with respect to x, provided y → ∞, ∇^{k}f(x) = (\frac{∂^{k}f(x)}{∂x_{1}^{k}}, ..., \frac{∂^{k}f(x)}{∂x_{n}^{k}}). Everywhere below the constants A(k, n), C, K depend only on the parameters pointed in parentheses, and may be different from time to time.

3. Auxiliary lemmata for the proof of Theorem 1.

The main results of this section are Theorem 5 and Theorem 7. Our first auxiliary result shows that in Theorem 3 and Corollary 1 the choice of p > 0 may be independent on m ∈ N. We include it here for convenience of the reader.

Lemma 2. Let p > 0 and let F = (U, V_{1}, ..., V_{n}) be such that V_{i} ⇒_{y→∞}^{x} 0, i = 1, ..., n, M_{p}(y, U) ≤ C. Then

M_{p}(y, ∇^{k}F) ≤ ACy^{−k}, k ∈ N.

Proof. By induction on k. Let k = 1. Fix p > 0 and let l = inf{j ∈ N : p ≥ (n − 1)/(j + n − 1)}. Let ϕ_{ij}(x, y) be a coordinate of ∇V_{i}(x, y), j = 1, ..., n + 1, x_{n+1} = y, i = 0, ..., n, V_{0} = U. Since ∇V_{i}(x, y) ⇒_{y→∞}^{x} 0, we may use the following relation (see [5] or [4])

ϕ_{ij}(x, y) = \frac{1}{(2l − 2)!} ∫_{y}^{∞}(s − y)^{2l−2}D_{n+1}^{2l−1}ϕ_{ij}(x, s)ds =

\frac{1}{(2l − 2)!} ∫_{0}^{∞}s^{2l−2}D_{n+1}^{2l−1}ϕ_{ij}(x, s + y)ds.

We have |ϕ_{ij}(x, y)| ≤ h_{ij}(x, y), where

h_{ij}(x, y) ≡ \frac{1}{(2l − 2)!} ∫_{0}^{∞}s^{2l−2}|∇^{l}D_{n+1}^{l−1}ϕ_{ij}(x, s + y)|^{p}ds.

Theorem 3 of[5] implies (take w = ∇^{l}D_{n+1}^{l−1}ϕ_{ij}, a = 2l − 1, A = A(l, n, p)),

∫_{R^{n}} |ϕ_{ij}(x, y)|^{p}dx ≤ ∫_{R^{n}} |h_{ij}(x, y)|^{p}dx ≤ A ∫_{0}^{∞}s^{(2l−1)p−1}ds ∫_{R^{n}} |∇^{l}D_{n+1}^{l−1}ϕ_{ij}(x, s + y)|^{p}dx.

Since D_{n+1}^{l−1}ϕ_{ij}(x, y) is the l−th derivative of V_{i}, we use Theorem 3 to get

(4)  ∫_{R^{n}} |∇^{l}D_{n+1}^{l−1}ϕ_{ij}(x, y)|^{p}dx ≤ ∫_{R^{n}} |∇^{2l}F (x, y)|^{p}dx ≤ C y^{−2lp}.

This gives

∫_{R^{n}} |ϕ_{ij}(x, y)|^{p}dx ≤ A(l, n, p)C ∫_{0}^{∞}s^{(2l−1)p−1}(s + y)^{−2lp}ds = A(l, n, p)Cy^{−p},

and the first induction step is proved.

Assume that the statement is true for k − 1. Then M_{p}(y, ∇^{k−1}F) ≤ ACy^{−(k−1)}. To prove it for k we define l as above and apply Corollary 1 with b = k − 1, m = 1, u = ∇^{k−1}F. □

Lemma 3. Let D_{i}U ⇒_{y→∞}^{x} 0, i = 1, ..., n, and let

(5)  M_{p}(y, D_{n+1}U) ≤ Cy^{−1}

for some p > 0. Then

(6) ∫_{R^{n}} ( sup_{y>0} |D_{n+1}U(x, y + y_{0})|)^{p}dx ≤ AC,  ∀y_{0} > 0, A = A(n, p, y_{0}).

Proof. Let p > 1. Then (see [12])

∥ sup_{y>0} |D_{n+1}U(·, y + y_{0})|∥_{p} ≤ CM_{p}(y + y_{0}, D_{n+1}U) ≤ AC, A = A(n, p, y_{0}).

Now let 0 < p ≤ 1. Assume that

(7)  ∫_{0}^{∞}∫_{R^{n}} s^{p−1}|∇^{2}U(x, s + y_{0})|^{p}dxds ≡ ∫_{0}^{1} ∫_{R^{n}} + ∫_{1}^{∞}∫_{R^{n}} < C(y_{0}) < ∞.

Then Lemma 2, Theorem 2 (with D_{n+1}U instead of u and k = 1), and the tensor representation of D_{n+1}U from [4], imply D_{n+1}U(x, y + y_{0}) ∈ H^{p}. This gives (6).

Thus, we have to show (7). By Theorem 3, (5) yields

(8)  M_{p}(y, ∇^{2}U) ≤ ACy^{−2}.

Then, the first integral in the right-hand side of (7) is finite, since

M_{p}(y + y_{0}, ∇^{2}U) ≤ AC(y + y_{0})^{−2} ≤ AC(y_{0}), ∀y_{0} > 0.

On the other hand,

∫_{1}^{∞}s^{p−1}ds ∫_{R^{n}} |∇^{2}U(x, s + y_{0})|^{p}dx ≤ AC ∫_{1}^{∞}s^{p−1}(s + y_{0})^{−2p}ds ≤ AC(y_{0}) < ∞.

□

Let V_{i} be components of the harmonic vector F = (U, V_{1}, ..., V_{n}). By the meanvalue theorem,

(9)  V_{i}(x, y) = V_{i}(x, y + 1) − D_{n+1}V_{i}(x, y + θ_{i}),  0 < θ_{i}(x, y) < 1.

Lemma 4. Let F = (U, V_{1}, ..., V_{n}) ∈ H^{q} and let θ_{i} be as in (9), i = 0, ..., n. Then

(10)  ∫_{R^{n}} ( sup_{y>0} |D_{i}U(x, y + θ_{i})|)^{q}dx ≤ AC.

Proof. We have

(11)  sup_{y>0} |V_{i}(x, y)| ≤ sup_{y>0} |V_{i}(x, y + 1)| + sup_{y>0} |D_{n+1}V_{i}(x, y + θ_{i})|,
(12)  sup_{y>0} |D_{n+1}V_{i}(x, y + θ_{i})| ≤ 2F^{+}(x),

where i = 0, ..., n, V_{0} = U. Then, (12) implies

(13)  ∫_{R^{n}} ( sup_{y>0} |D_{n+1}V_{i}(x, y + θ_{i})|)^{q}dx ≤ AC.

To get the desired result we apply the Cauchy-Riemann equations D_{i}U(x, y) = D_{n+1}V_{i}(x, y). □

In the following result we observe that θ_{i} > 0 on a set, controlled by estimate (2). Then we use the fact that under conditions (14) the supremum sup_{y>0} |F (x, y)| is

attained at the boundary.

Theorem 5. Let 0 < p < q and let F = (U, V_{1}, ..., V_{n}) be a harmonic vector in R_{+}^{n+1}. Then F ∈ h^{p}, provided

(14)  1) M_{p}(y + 1, F) ≤ C,  2) M_{p}(y, D_{n+1}U) ≤ Cy^{−1},  3) F ∈ H^{q}.

Moreover, F ∈ H^{r} ∀r : p ≤ r ≤ q if and only if conditions (14) are valid.

Proof. We prove at first that (14) implies F ∈ h^{p}. Let θ_{i} be as in (9). Due to 2), (6), and 1), it is enough to show that sup_{y>0} |D_{n+1}V_{i}(·, y + θ_{i}(·, y))| ∈ L^{p}(R^{n}). This will

follow from the Cauchy-Riemann equations and

(15)  ∫_{R^{n}} ( sup_{y>0} |D_{i}U(x, y + θ_{i}(x, y))|)^{p}dx < ∞.

Thus, we prove (15). Condition 3) and Lemma 1 imply

(16)  sup_{x∈R^{n}} |D_{i}U(x, y + θ_{i})| ≤ sup_{x∈R^{n}} \frac{AC}{(y + θ_{i})^{1+n/q}} ≤ AC(y + α_{i}(y))^{−1−n/q},

where α_{i}(y) = inf_{x∈R^{n}} θ_{i}(x, y). Define

(17)  L_{i} = \{x ∈ R^{n} : sup_{y>0} |D_{i}U(x, y + θ_{i})| ≤ sup_{y>0} \frac{AC}{(y + α_{i}(y))^{1+n/q}} ≤ 1\}.

Then ∀x ∈ CL_{i} (the complement of L_{i}) we have sup_{y>0} |D_{i}U(x, y + θ_{i})| > 1. Then

(18) ∫_{CL_{i}} ( sup_{y>0} |D_{i}U(x, y + θ_{i}(x, y))|)^{p}dx ≤ ∫_{R^{n}} ( sup_{y>0} |D_{i}U(x, y + θ_{i}(x, y))|)^{q}dx ≤ AC

due to condition 3) and Lemma 4.

We estimate the integral in (15) over L_{i}. Observe that for fixed x ∈ L_{i},

sup_{y>0} |D_{i}U(x, y + θ_{i}(x, y))| ≤ sup_{y>0} |D_{i}U(x, y + α_{i}(y))|,

and α_{i} ≡ inf_{y>0} α_{i}(y) > 0. We put γ = min_{i=0,...,n} α_{i} > 0, and take any 0 < y_{0} ≤ γ. Then

(19) ∫_{L_{i}} ( sup_{y>0} |D_{i}U(x, y + θ_{i}(x, y))|)^{p}dx ≤ ∫_{R^{n}} ( sup_{y>0} |∇U(x, y + y_{0})|)^{p}dx ≤ AC.

Indeed, condition 3) and Theorem 3 imply V_{i}(x, y) ⇒_{y→∞}^{x} 0. The same is true for all partial derivatives of V_{i}, i = 0, ..., n, V_{0} = U. Now the second inequality in (19) follows from 2) and Lemma 3. Taking into account (19), (18), we get (15). Thus, F ∈ h^{p}.

Now, F ∈ H^{r} ∀r : p ≤ r ≤ q, implies (14) by Theorem 3. We prove the converse statement. It is enough to show that F ∈ H^{p}, or (sup_{y>0} |F (·, y)|)^{p} ∈ L^{1}(R^{n}). Since

F ∈ h^{p}, we apply the Fatou Lemma to have

(20)  ∫_{R^{n}} |F (x, 0)|^{p}dx = ∫_{R^{n}} ( lim_{y→0} |F (x, y)|)^{p}dx ≤ lim_{y→0} M_{p}(y, F) ≤ C.

We claim that sup_{y>0} |F (x, y)| = |F (x, 0)| and our result follows from (20). Since for

fixed x_{0} ∈ R^{n} the function W F (x_{0}, y) ≡ sup_{η≥y} |F (x_{0}, η)| is nonincreasing in y, we have

sup_{y>0} |F (x_{0}, y)| = sup_{y>0} sup_{η≥y} |F (x_{0}, η)| = lim_{y→0} sup_{η≥y} |F (x_{0}, η)| = |F (x_{0}, 0)|.

Even if sup_{y>0} |F (x_{0}, y)| = |F (x_{0}, y_{0})| for some y_{0} > 0, then W F (x_{0}, y) = W F (x_{0}, y_{0})

for all 0 ≤ y ≤ y_{0}, and we may put |F (x_{0}, y_{0})| = |F (x_{0}, 0)|. □

Theorem 6. Let 0 < p < q, k ∈ N, and let F = (U, V_{1}, ..., V_{n}) be a harmonic vector in R_{+}^{n+1}. Then F ∈ H^{r} ∀r : p ≤ r ≤ q if and only if

(21)  1) M_{p}(y + 1, F) ≤ C,  2) M_{p}(y, D_{n+1}^{k}U) ≤ Cy^{−k},  3) F ∈ H^{q}.

Proof. By Corollary 1, and the inverse statement, proved in [9], conditions (14), (21) are equivalent. □

Theorem 7. Let p > 0 and let F = (U, V_{1}, ..., V_{n}) be a harmonic vector in R_{+}^{n+1} such that

(22)  1) M_{p}(y + 1, F) ≤ C,  2) M_{p}(y, U) ≤ C,  3) |U(x, y)| ≤ C.

Then F ∈ H^{r}, r ≥ p.

Proof. By Theorem 4 we have F ∈ H^{r}, r > p. Let r = p. By Lemma 1 and Theorem 3 we have M_{p}(y, ▽^{k}U) ≤ Cy^{−k}, and we may use Theorem 6. □

Lemma 5. Let 0 < p < q and let F = (U, V_{1}, ..., V_{n}) be a harmonic vector in R_{+}^{n+1} such that

(23)  1) M_{p}(y + 1, F) ≤ C,  2) ∫_{R^{n}} ( sup_{η≥y} |D_{n+1}U(x, η)|)^{q}dx ≤ C.

Then F ∈ H^{q}.

Proof. By the mean-value theorem,

sup_{y>0} |U(x, y)| ≤ sup_{y>0} |U(x, y + 1)| + sup_{y>0} |∇U(x, y)|,

and it is enough to show that (sup_{y>0} |U(·, y + 1)|)^{p} ∈ L^{1}. To prove this, we apply

Theorem 3, the mean-value theorem again,

|U(x, y + 1)| ≤ |U(x, y + 2)| + sup_{y>0} |D_{n+1}U(x, y + 1 + θ)|,

and observe that the conditions of Theorem 7 are satisfied with y+1 instead of y. □

4. Proof of Theorem 1.

The proof is given in two lemmata presented below.

Lemma 6. Let k ∈ N, 0 < p < q and let F = (U, V_{1}, ..., V_{n}) be a harmonic vector in R_{+}^{n+1}. Then F and all its partial derivatives up to order k belong to H^{r}, p ≤ r ≤ q, if and only if

(24) 1) M_{p}(y + y_{0}, F) ≤ C(y_{0}) ∀y_{0} > 0,  2) ∫_{R^{n}} ( sup_{η≥y} |D_{n+1}^{k}U(x, η)|)^{q}dx ≤ C.

Proof. The only if part is trivial. We prove if by induction. Let k = 1. We show at first that F ∈ H^{r}, p ≤ r ≤ q. By Lemma 5 we have F ∈ H^{q}, and it is enough to show that F ∈ H^{p}. To this end, we apply the mean-value theorem and repeat the proof of Theorem 5 begining with (15). As in Theorem 5 we define L_{i}, (see (17)), and (18) follows from 2). The last estimate in (19) follows from Theorem 3 and Lemma 7. Conditions (22) are satisfied with ∇U instead of F, D_{n+1}U instead of U, and y + y_{0} instead of y.

To show that all partial derivatives of the first order belong to H^{r}, p ≤ r ≤ q one has to proceed as above by changing ∇U by ∇^{2}U, and D_{n+1}^{2}U by D_{n+1}U.

Assume that the statement is true for k − 1, and we have to prove it for k. By Theorem 3 we have 1) with D_{n+1}^{k−1}U instead of F, and the result follows. □

Lemma 7. Conditions of the theorem are equivalent to conditions of the previous lemma.

Proof. It is enough to prove that M_{p}(y + 1, F) ≤ C and 2) of (24) imply 1) of (24). This will follow from F ∈ H^{q}. Since V_{i}(x, y) ⇒_{y→∞}^{x} 0, i = 1, ..., n, it is enough to show that U ∈ H^{q}. We will subsequently show that all D_{n+1}^{k−1}U, D_{n+1}^{k−2}U, ..., U ∈ H^{q}. In fact, we prove that D_{n+1}^{k−1}U ∈ H^{q}. The proof of D_{n+1}^{k−2}U, ..., U ∈ H^{q} is similar.

Observe that D_{n+1}^{k−1}U ∈ H^{q} follows from D_{n+1}^{k−1}U ∈ h^{q}. Indeed, let D_{n+1}^{k−1}U ∈ h^{q}. By the mean-value theorem we have

sup_{y>0} |D_{n+1}^{k−1}U(x, y)| ≤ sup_{y>0} |D_{n+1}^{k−1}U(x, y + 1)| + sup_{y>0} |D_{n+1}^{k}U(x, y)|,

and we may apply Lemma 7, (with p = q, D_{n+1}^{k−1}U(x, y + 1) instead of U(x, y), ∇^{k−1}F instead of F), to obtain

∫_{R^{n}} ( sup_{y>0} |D_{n+1}^{k−1}U(x, y + 1)|)^{p}dx < ∞.

The assumption D_{n+1}^{k−1}U ∈ h^{q} and Lemma 1 are used to satisfy the third condition of Lemma 7. The above inequality gives D_{n+1}^{k−1}U ∈ H^{q}.

Thus, it remains to prove that D_{n+1}^{k−1}U(x, y) ∈ h^{q}. By the mean-value theorem,

|D_{n+1}^{k−1}U(x, y)| ≤ |D_{n+1}^{k−1}U(x, y + 1)| + sup_{y>0} |D_{n+1}^{k}U(x, y)|,

and it is enough to prove that D_{n+1}^{k−1}U(x, y + 1) ∈ h^{q}. Again, by the mean-value theorem,

|D_{n+1}^{k−1}U(x, y + 1)| ≤ |D_{n+1}^{k−1}U(x, y + 1 + 1)| + sup_{y>0} |D_{n+1}^{k}U(x, y)|,

but now we may use the assumption M_{p}(y + 1, F) ≤ C to show that for r ≥ p, M_{r}(y + 2, D_{n+1}^{k−1}U) ≤ C. To this end, we apply Theorem 3, Lemma 1, and take y + 1 instead of y. □

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[3] A. P. Calderón and A. Zygmund, On higher gradients of harmonic functions, Studia Math. 24 (1964), No. 2, 211-266.

[4] C. Fefferman, E. M. Stein, H^{p} spaces of several variables, Acta Math., 129, No 3, (1972), 137-193.

[5] T. M. Flett, Inequalities for the p−th mean values of harmonic and subharmonic functions with p ≤ 1, Proc. Lon. Math. Soc., Ser. 3, 20, No. 3, (1970), 249-275.

[6] V. I. Krylov, On functions regular in the half-plane, Math., Sb., (1939), 6(48), pp.95-138.

[7] U. Kuran, Classes of subharmonic functions in R^{n} × (0, ∞), Proc. Lond. Math. Soc., Ser 3., 16(1966), No. 3, 473-492.

[8] I. Privalov, Subharmonic functions, Moscow, 1937.

[9] A. Ryabogin, Conjugate harmonic functions of the Hardy class. (Russian), Izv. Vyssh. Uchebn. Zaved. Mat. 1991, , no. 9, 47-53; translation in Soviet Math. (Iz. VUZ) 35 (1991), no. 9, 46-51.

[10] A. Ryabogin, Boundary values of conjugate harmonic functions of several variables, (Russian), Izv. Vyssh. Uchebn. Zaved. Mat. (1980), no. 12, 50-54.

[11] E. D. Solomentsev, On classes of subharmonic functions in the half-space, Notes of Moscow State Univ., 10, (1958).

[12] E. M. Stein, Singular integrals and differentiability properties of functions, Princeton University Press, Princeton NJ, 1970.

[13] E. M. Stein and G. Weiss, On the theory of harmonic functions of several variables, Acta Math., 103 (1960), pp. 26-62.

[14] E. M. Stein and G. Weiss, Generalization of the Caushy-Riemann equations and representation of the rotation group, Amer. J. Math.,90(1968), 163-196.

[15] E. M. Stein and G. Weiss, An introduction to harmonic analysis on Euclidean spaces, Princeton University Press, Princeton NJ, 1969.

[16] T. Wolff, Counterexamples with harmonic gradient in R^{3}, Essays in honor of E. M. Stein, Princeton Mathematical Series, 42 (1995), 321-384.

[17] A. Zygmund, Trigonometric series, 2nd ed., Cambridge University Press, Cambridge, 1968.

Anatoly Ryabogin, Department of Mathematics, Ben Gurion University of the Negev, P.O.B. 653, Be’er Sheva 84105, Israel

Dmitry Ryabogin, Department of Mathematics, Kansas State University, Manhattan, KS 66506-2602, USA

  E-mail address: ryabs@math.ksu.edu