Embeddings of harmonic mixed norm spaces on smoothly bounded domains in ℝn

Abstract The main result of this paper is the embedding Bβs,r(Ω)↪Bβ+(n−1)(1s−1s1)s1,r1(Ω), $$\begin{array}{} \displaystyle \mathcal{B}^{s,r}_\beta({\it\Omega})\hookrightarrow \mathcal{B}^{s_1,r_1}_{\beta+(n-1)\big(\frac 1s-\frac 1{s_1}\big)}({\it\Omega}), \end{array}$$ 0 < r ≤ r1 ≤ ∞, 0 < s ≤ s1 ≤ ∞, β > –1, of harmonic functions mixed norm spaces on a smoothly bounded domain Ω ⊂ ℝn. We also extend a result on boundedness, in mixed norm, of a maximal function-type operator from the case of the unit disc and the unit ball to general domains in ℝn.


Introduction and preliminaries
The embedding theorems for harmonic or analytic function spaces with mixed norm have been studied extensively, especially in the case of the unit disc, where rst results are due to Hardy and Littlewood [1,2]. In the case of analytic functions such theorems were proved for general bounded strictly pseudoconvex domains in C n , see [3]. Mixed norm spaces of harmonic and analytic functions on the upper half plane were investigated in [4,5], some of the methods we use here can be traced to these papers. For harmonic functions many authors considered embeddings of mixed norm spaces on B n or upper half-space H n , see for example [6] for B n , [7][8][9] for R n , or [10] for H n . However, it seems that the case of more general domains was not treated.
In this paper we prove an embedding theorem for mixed norm spaces of harmonic functions, Theorem 1 below, in the setting of bounded C domains. This result generalizes Theorem 1.1 (iv) from [6]. In addition, we consider a maximal function-type operator u → u × and prove its boundedness with respect to mixed norm in the class of quasi-nearly subharmonic functions u, see Theorem 2 below.
We note that the operator u × was discussed, in the case of the unit disc, in [11], and the corresponding result in Ω ⊂ R n is Theorem 2; see also a related result in [12] for weighted harmonic Bergman spaces on B n .
We denote the Lebesgue measure on R n by dV and the Lebesgue measure of a measurable set E ⊂ R n by |E|. The surface measure on ∂Ω is denoted by dσ. B(a, r) denotes the usual Euclidean ball in R n , with center at a ∈ R n and radius r > . We also use a standard convention: C denotes a constant which can actually change its value from one occurrence to the next one. Also, for positive quantities A and B, A B means that cA ≤ B ≤ CA for some constants < c ≤ C < ∞.
In this paper we work with a bounded domain Ω ⊂ R n with C boundary. We x a de ning function ρ for Ω, which means ρ ∈ C (R n ), Ω = {x ∈ R n : ρ(x) > }, ∂Ω = {x ∈ R n : ρ(x) = } and ∇ρ(ξ ) ≠ for all ξ ∈ ∂Ω. We note that By well known Tubular Neighborhood Theorem, there is a neighborhood U of ∂Ω and there is a Cdi eomorphism χ : . For a given measurable complex valued function f de ned on U ∩ Ω (or Ω), we de ne f : If u , u ∈ h(Ω) and u = u on U ∩ Ω, then u = u on Ω. We set, by a slight abuse of notation, u = (u| U∩Ω )˜. By the above remark, if u = u , then u = u for u , u ∈ h(Ω).
Next we de ne certain spaces of functions on Ω and ∂Ω × ( , r ) which are a natural generalization of classical mixed norm spaces on the unit ball. For a Borel measurable function f on Ω or Ω ∩ U we set with the usual modi cation for s = ∞. Also for a Borel measurable function g on ∂Ω × ( , r ) we set again with the usual modi cation for s = ∞. Now we have a mixed norm space as the space of Borel measurable function g on ∂Ω × ( , r ) such that the following (quasi) norm of g is nite ||g|| L s,r β = ||t β Ms(g, t)|| L r (( ,r ), dt t ) . The main object of study in this paper is the following space of harmonic functions with the following (quasi) norm Here < s, r ≤ ∞ and β > − . Note that these spaces are trivial for β ≤ − . Di erent choice of a de ning function ρ and a di erent choice of tubular neighborhood map χ lead to di erent, but equivalent norms and the same mixed norm spaces. For every point ξ on the boundary of Ω and t > we de ne a "ball" B ∂Ω t (ξ ) with center at point ξ ∈ ∂Ω and radius t > by Note that the following area estimate is valid: We also consider a "cylinder" in Ω centered at φ(ξ , t): We have the following two-sided volume estimate: We de ne a metric on ∂Ω × R by In fact, these C di eomorphisms have continuous and bounded partial derivatives. Hence, without loss of generality, we can assume that χ and φ are Lipshitz continuous, i.e. there are constants < l ≤ L < ∞ such that Therefore, for any non-negative and measurable f on Ω ∩ U we have: This is, in view of (1.1), a generalization of (1.2). Let r = min( r , r L ). Let us prove the following inclusions: The rst inclusion is equivalent to the following one: which proves a stronger inclusion: Working within V has certain advanteges: one can always consider Q(ξ , t) when φ(ξ , t) ∈ V and, within V, one can use inclusions (1.4).
The following lemma, due to Fe erman and Stein (see [13]), states that |u| p has subharmonic behavior for any p > .

Lemma 1. Let u ∈ h(Ω) and let B = B(z, r) ⊂ Ω. Then
where C is a constant which depends only on p and n.
The above lemma combined with (1.2) and (1.4) gives the next result: t) is a cylinder in Ω, where ξ ∈ ∂Ω, < t ≤ r , and assume h is harmonic in Ω. Then for every p > there is a constant C > that depends only on p and n such that Remark 1. In the above constructions one can use segment [( − δ), ( + δ)], where < δ < instead of [ t , t ] (the case δ = ). In particulatr, Lemma 2 is valid in this case, of course, the constant C depends on δ as well.

Main results
The main result of the paper is: The following lemma is a special case of Theorem 1, where s = s , r = ∞: Proof. Let us x u ∈ B s,r β (Ω). We treat separately the cases < s ≤ r < ∞ and < r < s < ∞. Assume < s ≤ r < ∞. For < t < r we obtain, by Lemma 2 and (1.3), the following estimate: Integrating over ξ ∈ ∂Ω and applying Fubini's theorem we obtain For a xed τ we have, again applying Fubini's theorem and (1.1):

We use the above inequality and (1.2) to estimate inner integrals in (2.2):
note that we also used τ t for t ≤ τ ≤ t . Next we use Hölder's inequality with exponent r s ≥ and get Therefore we obtained Our next goal is to obtain the crucial estimate (2.3) also in the second case, i.e. for < r ≤ s < ∞. Let us set p = s/r ≥ . We x < t < r and, as in the rst case, see (2.1), we obtain from Lemma 2 the following estimate: This gives, using (1.2): Now we integrate with respect to dσ(ξ ) and obtain:
We will use a duality argument: let us x ψ ∈ L q (∂Ω, dσ(ξ )), ||ψ||q ≤ , where /p + /q = . Then we have Combining the above estimates we obtain and, by duality, this gives ||φτ|| L p (∂Ω,dσ(ξ )) ≤ t n− M r s (u, τ). Using (2.6) and remembering that t τ for t/ ≤ τ ≤ t/ we nally obtain which means we proved (2.3) also in the case < r ≤ s. Thus, again using τ t, in both cases we have: and consequently ||u|| B s,∞ β (Ω) ≤ C||u|| B s,r β (Ω) . In order to proceed from this special case of Theorem 1 to the full scope of Theorem 1 we need to investigate a class of quasi-nearly subharmonic functions. A key result in this direction is Theorem 2 below.
Let, for K ≥ , QNS K (W) denote the class of nonnegative, locally bounded Borel measurable functions u on a domain W ⊂ R n satisfying Functions in the class QNS(W) = K≥ QNS K (W) are called quasi-nearly subharmonic functions. We need the next result, which generalizes Lemma 1. Theorem A [14,15] Let < p < ∞. If u ∈ QNS(W), then u p ∈ QNS(W). More precisely, if u ∈ QNS K (W), then u p ∈ QNS K (W), where K depends only on K, n and p. Let u × is a function de ned on Ω ∩ U.
Using Remark 1 and estimates (1.1) and (1.2) one easily proves that we have: where K depends only on K, n and Lipschitz constants L, l of χ, φ. This means that for u ∈ QNS K (Ω) we have: As already noted, this version of maximal operator was studied in [11,12]. The space L p,q β (Ω ∩ U) consists of all measurable functions f : In other words, ||f || L p,q The following theorem is a result on boundedness of u → u × in the class of quasi-nearly subharmonic functions. It will be used in the proof of our main result, Theorem 1 Proof. Since u is locally bounded, we only have to prove the implication u ∈ L s,r β ⇒ u × ∈ L s,r β . Assume that < s < r < ∞. Since u s is, by Theorem A, a QNS function we have, using (2.7) (u × (φ(ξ , t))) s ≤ C |Q(ξ , t)| t t B ∂Ω t (ξ ) u s (φ(η, τ))dσ(η)dτ. (2.8) Integration over ξ ∈ ∂Ω gives: ∂Ω (u × (φ(ξ , t))) s dσ(ξ ) ≤ C |Q(ξ , t)| If r < s < ∞, we have as in (2.8) |u × (φ(ξ , t))| r ≤ C |Q(ξ , t)| We nish this paper with a proof of Theorem 1.