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

Miloš Arsenović; Tanja Jovanović

doi:10.1515/math-2019-0108

Article Open Access

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

Miloš Arsenović and Tanja Jovanović

Published/Copyright: November 8, 2019

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$Open Mathematics$

From the journal Open Mathematics Volume 17 Issue 1

Abstract

The main result of this paper is the embedding

Bβs,r(Ω)↪Bβ+(n−1)(1s−1s1)s1,r1(Ω),

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.

Keywords: mixed norm spaces; harmonic functions spaces; maximal functions

MSC 2010: 31B05; 42B25; 42B35

1 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 first 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 ℂⁿ, 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 𝔹ⁿ or upper half-space ℍⁿ, see for example [6] for 𝔹ⁿ, [7, 8, 9] for ℝⁿ, or [10] for ℍⁿ. 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 C¹ 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 ⊂ ℝⁿ is Theorem 2; see also a related result in [12] for weighted harmonic Bergman spaces on 𝔹ⁿ.

We denote the Lebesgue measure on ℝⁿ by dV and the Lebesgue measure of a measurable set E ⊂ ℝⁿ by |E|. The surface measure on ∂Ω is denoted by dσ. B(a, r) denotes the usual Euclidean ball in ℝⁿ, with center at a ∈ ℝⁿ and radius r > 0. 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 0 < c ≤ C < ∞.

In this paper we work with a bounded domain Ω ⊂ ℝⁿ with C¹ boundary. We fix a defining function ρ for Ω, which means ρ ∈ C¹(ℝⁿ), Ω = {x ∈ ℝⁿ : ρ(x) > 0}, ∂Ω = {x ∈ ℝⁿ : ρ(x) = 0} and ∇ρ(ξ) ≠ 0 for all ξ ∈ ∂Ω. We note that

ρ(x)≍dist(x,∂Ω)forx∈Ω.

By well known Tubular Neighborhood Theorem, there is a neighborhood U of ∂Ω and there is a C¹-diffeomorphism χ : U ⟶ ∂Ω × (–r₀, r₀) such that χ(∂Ω) = ∂Ω × {0}, χ(U ∩ Ω) = ∂Ω × (0, r₀). We set φ = χ^–1 and, for –r₀ < t < r₀, Γ_t = φ(∂Ω × {t}). For a given measurable complex valued function f defined on U ∩ Ω (or Ω), we define f͠ : ∂Ω × (0, r₀) ⟶ ℂ by f͠(ξ, t) = f(φ(ξ, t)).

Let h(Ω) = {u : Ω → ℂ | u is harmonic in Ω}. If u₁, u₂ ∈ h(Ω) and u₁ = u₂ on U ∩ Ω, then u₁ = u₂ on Ω. We set, by a slight abuse of notation, u͠ = (u|_U∩Ω)^~. By the above remark, if u͠₁ = u͠₂, then u₁ = u₂ for u₁, u₂ ∈ h(Ω).

Next we define certain spaces of functions on Ω and ∂Ω × (0, r₀) 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

Ms(f,t)=∫∂Ω|f~(ξ,t)|sdσ(ξ)1s,0<s<∞,0<t<r0,

with the usual modification for s = ∞. Also for a Borel measurable function g on ∂Ω × (0, r₀) we set

M~s(g,t)=∫∂Ω|g(ξ,t)|sdσ(ξ)1s,0<s<∞,0<t<r0,

again with the usual modification for s = ∞. Now we have a mixed norm space

Lβs,r=Lβs,r(∂Ω×(0,r0)),0<s,r≤∞,β∈R,

as the space of Borel measurable function g on ∂Ω × (0, r₀) such that the following (quasi) norm of g is finite

||g||Lβs,r=||tβM~s(g,t)||Lr((0,r0),dtt).

The main object of study in this paper is the following space of harmonic functions

Bβs,r(Ω)={u∈h(Ω):u~∈Lβs,r(∂Ω×(0,r0))},

with the following (quasi) norm

||u||Bβs,r(Ω)=||u~||Lβs,r.

Here 0 < s, r ≤ ∞ and β > –1. Note that these spaces are trivial for β ≤ –1. Different choice of a defining function ρ and a different choice of tubular neighborhood map χ lead to different, but equivalent norms and the same mixed norm spaces.

For every point ξ on the boundary of Ω and t > 0 we define a “ball” Bt∂Ω (ξ) with center at point ξ ∈ ∂Ω and radius t > 0 by

Bt∂Ω(ξ)={η∈∂Ω:|ξ−η|≤t}.

Note that the following area estimate is valid:

σ(Bt∂Ω(ξ))≍tn−1,0<t≤diam(∂Ω). (1.1)

We also consider a “cylinder” in Ω centered at φ(ξ, t):

Q(ξ,t)=z∈Ω∩U|χ(z)∈Bt∂Ω(ξ)×t2,3t2,ξ∈∂Ω,0<t<2r03.

We have the following two-sided volume estimate:

|Q(ξ,t)|≍tn,0<t≤diam(∂Ω). (1.2)

We define a metric on ∂Ω × ℝ by

d∂Ω×R((ξ1,t1),(ξ2,t2))=|ξ1−ξ2|2+|t1−t2|2,

for (ξ₁, t₁), (ξ₂, t₂) in ∂Ω × ℝ. It is easy to see that χ : U₁ → ∂Ω × [–r₁, r₁] and φ : ∂Ω × [–r₁, r₁] → U₁ are Lipschitz continuous for any r₁ ∈ (0, r₀), where U₁ = φ(∂Ω × [–r₁, r₁]). In fact, these C¹ diffeomorphisms 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 0 < l ≤ L < ∞ such that

l|z−w|≤d∂Ω×R(χ(z),χ(w))≤L|z−w|,

for all z, w ∈ U. Also, there are constants 0 < c ≤ C < ∞ such that for any measurable E ⊂ U we have

c(dσ×dt)(χ(E))≤|E|≤C(dσ×dt)(χ(E)).

Therefore, for any non-negative and measurable f on Ω ∩ U we have:

∫U∩ΩfdV≍∫∂Ω∫0r0f~dσdt. (1.3)

This is, in view of (1.1), a generalization of (1.2).

Let r2=min(2r03,r02L). Let us prove the following inclusions:

B¯φ(ξ,t),t2L⊂Q(ξ,t)⊂Bφ(ξ,t),2tl,ξ∈∂Ω,0<t≤r2. (1.4)

The first inclusion is equivalent to the following one:

χB¯φ(ξ,t),t2L⊂χ(Q(ξ,t))=Bt∂Ω(ξ)×t2,3t2.

Now, for z∈B¯(φ(ξ,t),t2L) we have

d∂Ω×R(χ(z),(ξ,t))=d∂Ω×R(χ(z),χ(φ(ξ,t)))≤L|z−φ(ξ,t)|≤Lt2L=t2,

which proves a stronger inclusion:

χB¯φ(ξ,t),t2L⊂Bt/2∂Ω(ξ)×t2,3t2.

Similarly one proves Q(ξ, t) ⊂ B(φ(ξ, t), 2t/l).

Let us set

V=φ(∂Ω×(0,r2))⊂Ω∩U. (1.5)

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 Fefferman and Stein (see [13]), states that |u|^p has subharmonic behavior for any p > 0.

Lemma 1

Let u ∈ h(Ω) and let B = B(z, r) ⊂ Ω. Then

|u(z)|p≤C|B|∫B|u|pdV,

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:

Lemma 2

Suppose Q(ξ, t) is a cylinder in Ω, where ξ ∈ ∂Ω, 0 < t ≤ r₂, and assume h is harmonic in Ω. Then for every p > 0 there is a constant C > 0 that depends only on p and n such that

|u(φ(ξ,t))|p≤C|Q(ξ,t)|∫Q(ξ,t)|u|pdV.

Remark 1

In the above constructions one can use segment [(1 – δ), (1 + δ)], where 0 < δ < 1 instead of [t2,3t2] (the case δ = 12 ). In particulatr, Lemma 2 is valid in this case, of course, the constant C depends on δ as well.

2 Main results

The main result of the paper is:

Theorem 1

For 0 < s ≤ s₁ ≤ ∞ and 0 < r ≤ r₁ ≤ ∞ we have a continuous embedding

Bβs,r(Ω)↪Bβ1s1,r1(Ω),

where β1=β+(n−1)(1s−1s1).

The following lemma is a special case of Theorem 1, where s = s₁, r₁ = ∞:

Lemma 3

Suppose 0 < r ≤ ∞ and β > –1, then we have Bβs,r (Ω) ↪ Bβs,∞ (Ω).

Proof

Let us fix u ∈ Bβs,r (Ω). We treat separately the cases 0 < s ≤ r < ∞ and 0 < r < s < ∞.

Assume 0 < s ≤ r < ∞. For 0 < t < r₂ we obtain, by Lemma 2 and (1.3), the following estimate:

|u~(ξ,t)|s≤C|Q(ξ,t)|∫t23t2∫Bt∂Ω(ξ)|u~(η,τ)|sdσ(η)dτ. (2.1)

Integrating over ξ ∈ ∂Ω and applying Fubini’s theorem we obtain

∫∂Ω|u~(ξ,t)|sdσ(ξ)≤C|Q(ξ,t)|∫t23t2∫∂Ω∫Bt∂Ω(ξ)|u~(η,τ)|sdσ(η)dσ(ξ)dτ. (2.2)

For a fixed τ we have, again applying Fubini’s theorem and (1.1):

∫∂Ω∫Bt∂Ω(ξ)|u~(η,τ)|sdσ(η)dσ(ξ)=∫∂Ω|u~(η,τ)|s∫∂ΩχBt∂Ω(ξ)dσ(ξ)dσ(η)≤Cτn−1∫∂Ω|u~(η,τ)|sdσ(η).

We use the above inequality and (1.2) to estimate inner integrals in (2.2):

Mss(u,t)≤Ctn∫t23t2τn−1∫∂Ω|u~(η,τ)|sdσ(η)dτ≤C∫t23t2Mss(u,τ)dττ,

note that we also used τ≍t for t2≤τ≤3t2. Next we use Hölder’s inequality with exponent rs≥1 and get

Mss(u,τ)≤C∫t23t2Msr(u,τ)dττsr∫t23t2dττ1−sr=C∫t23t2Msr(u,τ)dττsr.

Therefore we obtained

Ms(u,t)≤C∫t23t2Msr(u,τ)dττ1r,0<t<r2. (2.3)

Our next goal is to obtain the crucial estimate (2.3) also in the second case, i.e. for 0 < r ≤ s < ∞. Let us set p = s/r ≥ 1. We fix 0 < t < r₂ and, as in the first case, see (2.1), we obtain from Lemma 2 the following estimate:

|u~(ξ,t)|r≤C|Q(ξ,t)|∫t23t2∫Bt∂Ω(ξ)|u~(η,τ)|rdσ(η)dτ. (2.4)

This gives, using (1.2):

|u~(ξ,t)|s≤(Ctn)p(∫t23t2∫Bt∂Ω(ξ)|u~(η,τ)|rdσ(η)dτ)p.

Now we integrate with respect to dσ(ξ) and obtain:

Mss(u,t)≤(Ctn)p∫∂Ω(∫t23t2∫Bt∂Ω(ξ)|u~(η,τ)|rdσ(η)dτ)pdσ(ξ),

which gives

Msr(u,t)≤Ctn(∫∂Ω(∫t23t2∫Bt∂Ω(ξ)|u~(η,τ)|rdσ(η)dτ)pdσ(ξ))1p.

Now we use Minkowski’s integral inequality with exponent p = s/r and obtain

Msr(u,t)≤Ctn∫t23t2(∫∂Ω(∫Bt∂Ω(ξ)|u~(η,τ)|rdσ(η))pdσ(ξ))1pdτ.

We set

φτ(ξ)=∫Bt∂Ω(ξ)|u~(η,τ)|rdσ(η), (2.5)

and write the above estimate as

Msr(u,t)≤Ctn∫t23t2∫∂Ωφτp(ξ)dσ(ξ)1pdτ=Ctn∫t23t2||φτ||Lp(∂Ω,dσ(ξ))dτ. (2.6)

Next we want to estimate the L^P(∂Ω, dσ) norm of φ_τ, where t/2 ≤ τ ≤ 3t/2, to that end we define a function θ : ∂Ω × ∂Ω → ℝ by

θ(ξ,η)=1,|ξ−η|≤t0,|ξ−η|>t,

clearly θ(ξ, η) = θ(η, ξ) and

φτ(ξ)=∫∂Ωθ(ξ,η)|u~(η,τ)|rdσ(η).

We will use a duality argument: let us fix ψ ∈ L^q(∂Ω, dσ(ξ)), ||ψ||_q ≤ 1, where 1/p + 1/q = 1. Then we have

∫∂Ωφτ(ξ)ψ(ξ)dσ(ξ)=|∫∂Ω∫∂Ω|u~(η,τ)|rθ(ξ,η)|ψ(ξ)|dσ(η)dσ(ξ)|≤∫∂Ω∫∂Ω|u~(η,τ)|rθ1p(ξ,η)θ1q(ξ,η)|ψ(ξ)|dσ(ξ)dσ(η)≤AB,

where

A=∫∂Ω∫∂Ω(u~(η,τ))prθ(ξ,η)dσ(ξ)dσ(η)1p≤tn−1p∫∂Ω|u~(η,τ)|sdσ(η)r/s,

B=∫∂Ω∫∂Ω|ψ(ξ)|qθ(ξ,η)dσ(ξ)dσ(η)1q≤tn−1q∥ψ∥q≤tn−1q.

Combining the above estimates we obtain

∫∂Ωφτ(ξ)ψ(ξ)dσ(ξ)≤tn−1Msr(u,τ),∥ψ∥q≤1,

and, by duality, this gives ||φτ||Lp(∂Ω,dσ(ξ))≤tn−1Msr(u,τ). Using (2.6) and remembering that t ≍ τ for t/2 ≤ τ ≤ 3t/2 we finally obtain

Msr(u,t)≤C∫t23t2Msr(u,τ)dττ,

which means we proved (2.3) also in the case 0 < r ≤ s. Thus, again using τ ≍ t, in both cases we have:

tβMsr(u,t)≤C∫t23t2τβMsr(u,τ)dττ≤C||u||Bβs,r(Ω)r,0<t<r2,

and consequently ||u||Bβs,∞(Ω)≤C||u||Bβs,r(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 ≥ 1, QNS_K(W) denote the class of nonnegative, locally bounded Borel measurable functions u on a domain W ⊂ ℝⁿ satisfying

u(x)≤K|B(x,r)|∫B(x,r)udV,B(x,r)⊂W.

Functions in the class QNS(W) = ⋃_K≥1 QNS_K(W) are called quasi-nearly subharmonic functions. We need the next result, which generalizes Lemma 1.

Theorem A

[14, 15] Let 0 < p < ∞. If u ∈ QNS(W), then u^p ∈ QNS(W). More precisely, if u ∈ QNS_K(W), then u^p ∈ QNS_K₁(W), where K₁ depends only on K, n and p.

Let

u×(φ(ξ,t))=sup3t4≤τ≤tu(φ(ξ,τ)),(ξ,t)∈∂Ω×(0,r0),

u^× is a function defined on Ω ∩ U.

Using Remark 1 and estimates (1.1) and (1.2) one easily proves that we have:

sup34t≤τ≤t|u~(ξ,τ)|≤K1|Q(ξ,t)|∫Q(ξ,t)udV,u∈QNSK(Ω),ξ∈∂Ω,0<t≤r2,

where K₁ depends only on K, n and Lipschitz constants L, l of χ, φ. This means that for u ∈ QNS_K(Ω) we have:

u×(φ(ξ,t))≤K1|Q(ξ,t)|∫Q(ξ,t)udV,ξ∈∂Ω,0<t≤r2. (2.7)

As already noted, this version of maximal operator was studied in [11, 12].

The space Lβp,q (Ω ∩ U) consists of all measurable functions f : Ω ∩ U ⟶ ℂ such that

||f||Lβp,q=(∫0r0(∫∂Ω|f~(ξ,t)pdσ(ξ))qptβq−1dtt)1q<∞.

In other words, ||f||Lβp,q=||f~||Lβp,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

Theorem 2

Let 0 < s, r ≤ ∞ and β > –1. A function u ∈ QNS_K(Ω ∩ U) belongs to Lβs,r (Ω ∩ U) if and only if u^× belongs to Lβs,r (V). Moreover we have

||u×||Lβs,r(V)≤C||u||Lβs,r(Ω∩U),

where C depends on K and Ω but is independent of u.

Proof

Since u is locally bounded, we only have to prove the implication u ∈ Lβs,r ⇒ u^× ∈ Lβs,r . Assume that 0 < 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)|∫t23t2∫Bt∂Ω(ξ)us(φ(η,τ))dσ(η)dτ. (2.8)

Integration over ξ ∈ ∂Ω gives:

∫∂Ω(u×(φ(ξ,t)))sdσ(ξ)≤C|Q(ξ,t)|∫∂Ω∫t23t2∫Bt∂Ω(ξ)us(φ(η,τ))dσ(η)dτdσ(ξ).

Arguing as in the proof of Lemma 3 we obtain

Mss(u×,t)≤Ctn∫t23t2τn−1∫∂Ω|u(φ(η,τ))|sdσ(η)dτ≤C∫t23t2Mss(u,τ)dττ.

Then we use Hölder’s inequality with exponent rs and obtain

Ms(u×,t)≤C∫t23t2Msr(u,τ)dττ1r.

If r < s < ∞, we have as in (2.8)

which gives

Msr(u×,t)≤Ctn(∫∂Ω(∫t23t2∫Bt∂Ω(ξ)ur(φ(η,τ))dσ(η)dτ)srdσ(ξ))rs.

Arguing as in Lemma 3 we get

Msr(u×,t)≤C∫t23t2Msr(u,τ)dττ,0<t<r2.

Multiplying by t^β and integrating over 0 < t < r₂ gives

||u×||Lβs,rr=∫0r2tβrMsr(u×,t)dtt≤C∫0r2tβr∫t23t2Msr(u,τ)dττdtt≤C∫0r2tβr∫0r0χ[t2,3t2](τ)Msr(u,t)dττdtt≤C∫0r0τβrMsr(u,τ)dττ=C||u||Lβs,rr.

□

Theorem 3

Let 0 < s < s₁ ≤ ∞, 0 < r ≤ ∞ and β > –1. If a function u belongs to QNS_K(Ω ∩ U) ∩ Lβs,r , then it belongs to Lβ1s1,r (Ω ∩ U), where β1=β+(n−1)(1s−1s1), and we have ||u||Lβ1s1,r≤C||u||Lβs,r, where C is a constant independent of u.

Proof

Let u ∈ QNS_K ∩ Lβs,r . Then, by Theorem A, u^s ∈ QNS_K₁, and it easily follows that:

M∞(u,t)≤Ctn−1ssupt2≤τ≤3t2Ms(u,τ),3t2<r0.

Therefore, we obtain an estimate:

Ms1s1(u,t)=∫∂Ωus1−s(φ(ξ,t))us(φ(ξ,t))dσ(ξ)≤M∞s1−s(u,t)Mss(u,t).

Then

Ms1(u,t)≤Ctn−1ss1−ss1supt2≤τ≤3t2Ms(u,τ)=Ct(n−1)(1s−1s1)supt2≤τ≤3t2Ms(u,τ). (2.9)

Since [t2,3t2]⊂⋃j=14Δj, where Δj=(34)jt2,(34)j−13t2 we have

supt2≤τ≤3t2Ms(u,τ)≤∑j=14supτ∈ΔjMs(u,τ)≤∑j=14Msu×,3j−14j−13t2. (2.10)

Therefore, using (2.9) and (2.10), we obtain

Ms1(u,t)tβ+(n−1)(1s−1s1)≤C∑j=14tβMsu×,3j−14j−13t2. (2.11)

Now the result follows from the previous theorem.□

We finish this paper with a proof of Theorem 1.

Proof of Theorem 1

Let u ∈ Bβs,r (Ω). Then |u| is subharmonic and therefore in QNS₁(Ω). Now |u| ∈ Lβs,r , because of u ∈ Bβs,r (Ω). Hence, by Theorem 3, |u| ∈ Lβ1s1,r . Since u is harmonic, this means u ∈ Bβ1s1,r (Ω). Lemma 3 gives us u ∈ Bβ1s1,∞ (Ω) and hence u ∈ Bβ1s1,r1 (Ω). Therefore Bβs,r (Ω) ⊂ Bβ1s1,r1 (Ω). The continuity of the embedding Bβs,r (Ω) ↪ Bβ1s1,r1 (Ω) follows from the estimates given in Theorem 3 and Lemma 3, or from the Closed Graph Theorem.□

Acknowledgements

The authors are grateful to the referee who pointed out many inaccuracies and whose comments improved the presentation of results.

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Received: 2018-06-10

Accepted: 2019-09-27

Published Online: 2019-11-08

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https://doi.org/10.1515/math-2019-0108

Keywords for this article

mixed norm spaces; harmonic functions spaces; maximal functions

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