Mixed norm spaces and RM(p, q) spaces

Revista Matemática Complutense https://doi.org/10.1007/s13163-023-00466-1 Mixed norm spaces and RM(p,q)spaces Tanausú Aguilar-Hernández1 Received: 7 June 2022 / Accepted: 24 April 2023 © The Author(s) 2023 Abstract In this paper we present the containment relationship between the spaces of analytic functions with average radial integrability RM(p,q)and a family of mixed norm spaces. Keywords Mixed norm spaces ·Radial integrability ·Bergman projection Mathematics Subject Classification Primary 30H20 ·Secondary 46E15 ·46B10 1 Introduction Let Dbe the unit disc in the complex plane and H(D)be the collection of the analytic functions in D. A good number of meaningful spaces of analytic functions in D, such as the classical Hardy spaces, may be defined in terms of integral means or in terms of the integrability with respect to the Lebesgue area measure, maybe with a certain weight (see, i.e., [6,9–11]). The most classical examples in this situation are the Hardy spaces but also the mixed norm spaces Hq,pdefined explicitly by Flett in [7,8] and, nowadays, widely studied (see [11]). Let us recall that a holomorphic function fin the unit disc Dbelongs to Hq,pif 1 02π 0|f(reiθ)|qdθ 2πp/q dr1/p <+∞.(1.1) In other less studied cases, the belonging to a Banach space of analytic functions is determined by the average radial integrability. Maybe the most well-known space This research was supported in part by Ministerio de Economía y Competitividad, Spain, MTM2015-63699-P, and Junta de Andalucía, FQM133. BTanausú Aguilar-Hernández [email protected] 1Departamento de Matemática Aplicada II and IMUS, Escuela Técnica Superior de Ingeniería, Universidad de Sevilla, Camino de los Descubrimientos, s/n, 41092 Sevilla, Spain 123 T. Aguilar-Hernández in this situation is the spaces of bounded radial variation BRV, a topic that goes back to Zygmund and where many different authors have work (see, i.e., [4,12,14]). This space of analytic function with bounded radial variation consists of those holomorphic functions g∈H(D)such that sup θ1 0|g(teiθ)|dt <+∞. It is clear that any function g∈BRV induces a bounded integral operator Tg(f)(z)=z 0f(ξ)g(ξ)dξon H∞. It was conjectured in [3] that such a boundedness only holds if g∈BRV (see also [5]). In [13], it was shown that there are functions g/∈BRV such that Tgis bounded on H∞, but in case g is univalent Tgis bounded on H∞if and only if g∈BRV. Other different situation where the radial integrability plays an important role is in the Féjer-Riesz theorem which says that if fbelongs to the Hardy space Hpthen sup θ1 0|f(reiθ)|pdr≤1 2fp Hp. Recently, a family of spaces of holomorphic functions in the unit disc with average radial integrability, denoted by RM(p,q), has been studied in [1,2]. These spaces are formed by the analytic functions such that 2π 01 0|f(reiθ)|pdrq/pdθ 2π1/q <+∞ (1.2) for 0 <p,q<+∞. If either por qis infinity, we change the integral by the essential supremum, respectively. This family contains the classical Hardy spaces Hq(when p=+∞) and Bergman spaces Ap(when p=q). Looking at (1.1) and (1.2), a natural question that has been raised is the containment relationship with the mixed norm spaces Hq,pand RM(p,q). Notice that by Fubini’s theorem, it is clear that RM(p,p)=Hp,pfor all 1 <p<∞, and by Minkowski’s inequality, RM(p,q)⊂Hq,p,if p>q, Hq,p⊂RM(p,q), if q>p.(1.3) Therefore, a natural question is whether or not both embeddings in (1.3) are strict. The main result in this paper give a positive answer to this question: Theorem 1.1 Let 1<p,q<+∞. (a) If p >q, then RM(p,q)Hq,p. (b) If q >p, then Hq,pRM(p,q). 123 Mixed norm spaces... Throughout the paper the letter C=C(·)will denote an absolute constant whose value depends on the parameters indicated in the parenthesis, and may change from one occurrence to another. We will use the notation abif there exists a constant C=C(·)>0 such that a≤Cb, and abis understood in an analogous manner. In particular, if aband ab, then we will write ab. 2 First definitions We start by recalling the definition of the spaces with average radial integrability RM(p,q). These spaces are formed by those holomorphic functions in Dsuch that taking the p-norm in every radius and then the q-norm, the result is a finite number. More precisely, we give the following definition. Definition 2.1 Let 0 <p,q<+∞. We define the spaces of analytic functions RM(p,q)=f∈H(D)ρ p,q(f)<+∞ where ρp,q(f)=1 2π2π 01 0|f(reit)|pdrq/p dt1/q ,if p,q<+∞. Remark 2.2 Notice that the analyticity is not required to define the norm. So that, we will use this definition for measurable functions with the aim of simplifying the notation and to facilitate the reading. It is easy to see that RM(p,q)endowed with the norm ρp,qis a Banach space whenever p,q≥1. One example of functions in RM(p,q), that we will play an important role in the main result of this work, is the following family of functions. Proposition 2.3 Let 0<p,q≤+∞. Let α∈Dand β>1 p+1 q, then there are constants C j=Cj(p,q,β),j=1,2, such that C1(1−|α|) 1 p+1 q−β≤ρp,q((1−αz)−β)≤C2(1−|α|) 1 p+1 q−β, where we are using the main branch of the logarithm to define w−β. Proof Let 0 <p,q<+∞. We may assume without loss of generality that 1/2≤ α<1. Let us estimate the quantity |1−αreiθ|2for points around 1. If 0 <θ<1−α and 1/2<r<1, then |1−αreiθ|2(1−rα)2.If1/2>θ>1−α, then |1−αreiθ|2θ2,1>r≥1−θ α, (1−rα)2,1/2<r≤1−θ α. 123 T. Aguilar-Hernández First of all, let us see that ρp,q((1−αz)−β)(1−|α|) 1 p+1 q−βif β>1 p+1 q. Using the symmetry in θand the monotonicity in θand r,wehave 2π 01 0 dr |1−αreiθ|βpq/p dθ≤22+q/pπ1/2 01 1/2 dr |1−αreiθ|βpq/p dθ Therefore, 2π 01 0 dr |1−αreiθ|βpq/p dθ ≤22+q pπ1−α 01 1/2 dr |1−αreiθ|βpq/p dθ+1/2 1−α1 1/2 dr |1−αreiθ|βpq/p dθ ≤22+q pπ1/2 1−α1 1−θ α dr θβp+1−θ α 1/2 dr (1−rα)βpq/p dθ+1−α 01 1/2 dr (1−rα)βpq/p dθ ≤22+q pπ1/2 1−αθ−(1−α) αθβp+1 α(β p−1)1 θβp−1−1 (1−α/2)βp−1q/p dθ +22+q pπ1−α 01 α(β p−1)1 (1−α)βp−1−1 (1−α/2)βp−1q/p dθ ≤4π2βp α(β p−1)q/p1/2 1−α 1 θβq−q/pdθ+1 (1−α)βq−q/p−1 ≤4π2βp α(β p−1)q/pβq−q/p βq−q/p−1(1−α)1+q/p−βq. Now, we will show that ρp,q((1−αz)−β)(1−|α|) 1 p+1 q−βif β>1 p+1 q. Since for 0 <θ<1−αand 0 ≤r<1 one have that |1−αreiθ|≤√2(1−αr),wehave 2π 01 0 dr |1−αreiθ|βpq/p dθ≥1 (√2)βq1−α 01 0 dr (1−αr)βpq/p dθ ≥1 (√2)βq(β p−1)q/pαq/p1−α 01 (1−α)βp−1−1q/p dθ ≥1−(1/2)βp−1 (√2)βp(β p−1)α q/p (1−α)1+q/p−βq. If por qis ∞, the proof follows similarly.  Now, we continue with the definition of a particular case of the mixed normed spaces. Bearing in mind the RM(p,q)spaces, they are defined interchanging the order of integration, that is, they are formed by those holomorphic functions in Dsuch that taking the q-norm in every circle and then the p-norm, the result is a finite number. More precisely, 123 Mixed norm spaces... Definition 2.4 Let 0 <p,q<+∞. We define the mixed norm spaces Hq,p={f∈H(D):fHq,p<+∞} where fHq,p=1 02π 0|f(reiθ)|qdθ 2πp/q dr1/p . Remark 2.5 As in the definition of the RM(p,q)spaces, the analyticity is not required to define the norm. Thus, we will use this definition for measurable functions with the aim of simplifying the notation and to facilitate the reading. By Fubini’s theorem, it is clear that fHp,p=ρp,p(f). Thus, it is a natural question to analyse what happens when p= q. This is the aim of this work. Along the same lines as for the RM(p,q)spaces, we have one example of functions that belong to these mixed norm spaces is the following family of analytic functions. Proposition 2.6 Let 0<p,q<+∞. Let α∈Dand β>1 p+1 q, then there are constants C j=Cj(p,q,β),j=1,2, such that C1(1−|α|) 1 p+1 q−β≤(1−αz)−βHq,p≤C2(1−|α|) 1 p+1 q−β, where we are using the main branch of the logarithm to define w−β. 3 Containment relationships We proceed with the presentation of the containment relationship between the RM(p,q)spaces and a particular case of the mixed norm spaces, which we have defined above. Bearing in mind the standard argument of [1, Corollary 4.8, p. 29], one can prove an analogous duality result for the mixed norm spaces by means of the boundedness of the weighted Bergman projection: Proposition 3.1 [11, Corollary 7.3.4, p. 153] Let 1≤p,q<+∞ and 1/p<γ+1. Then the operator Pγf(z)=(γ +1)D (1−|w|2)γf(w) (1−zw)2+γdA(w) 123 T. Aguilar-Hernández is a bounded projection mapping Lq,ponto Hq,p, where Lq,pis the corresponding space of equivalence classes of measurable functions. In particular (γ=0), we have the Bergman projection P maps Lq,ponto Hq,p, when 1<p<+∞. The proof of the next result follows the same scheme as for the spaces RM(p,q) [1, Corollary 4.8, p. 29]. Proposition 3.2 Let 1<p,q<+∞. Then (Hq,p)∗∼ =Hq,p, where 1 p+1 p=1 and 1 q+1 q=1. The antiisomorphism between Hq,pand (Hq,p)∗is given by the operator g→ λg where λgis defined by λg(f)=D f(z)g(z)dA(z), f∈Hq,p. Now we are ready to prove Theorem 1.1. Proof of Theorem 1.1 (a) If p≥qthen RM(p,q)⊂Hq,p, because using Minkowski’s integral inequality we have 1 02π 0|f(reiθ)|qdθ 2πp/q dr1/p ≤2π 01 0|f(reiθ)|pdrq/pdθ 2π1/q . Let us see that RM(p,q)Hq,pif p>q. Consider the functions uδ(z)=δ (1+δ−z)1+1 p+1 q ,z∈D, where 0 <δ<1/2. One can see that uδHq,pρp,q(uδ)1, because, for α∈D, (1−αz)−1−1 p−1 qHq,p(1−|α|)−1 (see Proposition 2.6) and ρp,q(1−αz)−1−1 p−1 q(1−|α|)−1 (see Proposition 2.3). Let {δn}be a sequence of positive numbers such that n2δn<1 4for all n≥1, δn/n2p>(n+1)δ1/2 n+1for all n≥1, and ∞ j=1j2δj<2. Define the sets An:= reiθ:|θ−θn|≤n2δnr∈[1−nδ1/2 n,1−δn/n2p], 123 Mixed norm spaces... where θ1=δ1and θn−θn−1=1 n2+(n−1)2δn−1+n2δn,forn≥2. Observe that An={reiθr∈In,θ∈Jn}where the sets {In}are pairwise disjoint and so are the sets {Jn}. Let us check that ρp,q(uδn(ze−iθn)χD\An(z)) 1 n2. Firstly, notice that ρp,quδn(ze−iθn)χ{0<|w|<1−nδ1/2 n}(z) =⎛ ⎜ ⎝2π 0⎛ ⎝1−nδ1/2 n 0 δp n |1+δn−reiθ|p1+1 p+1 qdr⎞ ⎠ q/p dθ 2π⎞ ⎟ ⎠ 1/q ≤⎛ ⎝1−nδ1/2 n 0 δp n (1+δn−r)p1+1 p+1 qdr⎞ ⎠ 1/p ≤δn (nδ1/2 n+δn)1+1 q ≤δn (nδ1/2 n)1+1 q≤1 n2.(3.1) In the next inequalities we use that, for |θ|≤1 and r≥1−nδ1/2 n |1+δn−reiθ|2=(1+δn−r)2+2r(1+δn)(1−cos(θ)) ≥(1+δn−r)2+rθ2 2 ≥(1+δn−r)2+(1−nδ1/2 n)θ2 2≥(1+δn−r)2+θ2 4. Thus, it follows ρq p,quδn(ze−iθn)χ{reiθ:1−nδ1/2 n<r<1,|θ−θn|>n2δn}(z) =2π n2δn⎛ ⎝1 1−nδ1/2 n δp n |1+δn−reiθ|p1+1 p+1 qdr⎞ ⎠ q/p dθ 2π ≤2π−n2δn 1−n2δn1 n2δn⎛ ⎝1 1−nδ1/2 n δp n |1+δn−reiθ|p1+1 p+1 qdr⎞ ⎠ q/p dθ 2π, since the inner integral is a decreasing function in θ. Therefore, ρq p,quδn(ze−iθn)χ{reiθ:1−nδ1/2 n<r<1,|θ−θn|>n2δn}(z) ≤23q+1n2δn+nδ1/2 n n2δn⎛ ⎝1+δn−θ 1−nδ1/2 n δp n (1+δn−r)p1+1 p+1 qdr +1 1+δn−θ δp n θp1+1 p+1 qdrq/p dθ 123 T. Aguilar-Hernández +23q+11 n2δn+nδ1/2 n1 1−nδ1/2 n δp n θp1+1 p+1 qdrq/p dθ ≤23q+1n2δn+nδ1/2 n n2δn⎛ ⎝ 1 p1+1 q δp n θp1+1 q+δp n θp1+1 q⎞ ⎠ q/p dθ +23q+11 n2δn+nδ1/2 n δq+q 2p nnq/p θq1+1 p+1 qdθ ≤23q+12q/pδq nn2δn+nδ1/2 n n2δn 1 θq1+1 q+23q+1δq n1 n2δn+nδ1/2 n 1 θq1+1 qdθ ≤23q+1+q pδq n1 n2δn 1 θq1+1 qdθ≤23q+1+q pδq n (n2δn)q≤23q+21 n2q.(3.2) Similarly, we obtain ρq p,quδn(ze−iθn)χ{reiθ:1−δn n2p<r<1,|θ−θn|<n2δn}(z) =2n2δn 0⎛ ⎝1 1−δn n2p δp n |1+δn−reiθ|p1+1 p+1 qdr⎞ ⎠ q/p dθ 2π ≤2δn 0⎛ ⎝1 1−δn n2p δp n (1+δn−r)p1+1 p+1 qdr⎞ ⎠ q/p dθ 2π +23q+1n2δn δn⎛ ⎝1 1−δn n2p δp n θp1+1 p+1 qdr⎞ ⎠ q/p dθ 2π ≤2δq nδn n2pq/pδn 0 1 δ q1+1 p+1 q n dθ 2π+23q+1δq nδn n2pq/pn2δn δn 1 θq1+1 p+1 q dθ 2π. ≤8q+1 2πδq nδn n2pq/p1 δ q1+1 p n =8q+1 2π 1 n2q.(3.3) Using inequalities (3.1), (3.2), and (3.3) we deduce uδn(ze−iθn)χD\An(z)Hq,p≤ρp,q(uδn(ze−iθn)χD\An(z)) 1 n2. Now, by the very definition of An,thesets{An}are pairwise disjoint. Define the functions gn(z)=uδn(ze−iθn)χD\An(z)and fn(z)=uδn(ze−iθn)χAn(z)such that uδn(ze−iθn)=fn(z)+gn(z). As we have seen, we have that ρp,q(gn)1 n2and 123 Mixed norm spaces... gnHq,p1 n2for n∈N. In addition, one can see that ρp,q(fn)fnHq,p1 because ρp,q(uδn(ze−iθn)) uδn(ze−iθn)Hq,p1. Given a measure space (, A,μ). We have that for any sequence of measurable functions hn:→Cwhose supports are pairwise disjoint, it holds that  n hn(w) s dμ(w) = n|hn(w)|sdμ(w) for all s>0. Then, using this fact twice (one in each variable, first with s=qand then with s=p/q), we obtain   αnfn  Hq,p=1 0|αn|p2π 0fn(reiθ) qdθ 2πp/q dr1/p =|αn|pfnp Hq,p1/p |αn|p1/p . By the same reason, ρp,qαnfn=2π 0|αn|q1 0fn(reiθ) p drq/pdθ 2π1/q =|αn|qρq p,q(fn)1/q |αn|q1/q . Hence, if we consider the function Fm(z):= m n=1uδn(ze−iθn)we obtain that ρp,q(Fm)≤ρp,qm  n=1 fn+ρp,qm  n=1 gnm1/q+ m  n=1 1 n2 ≤m1/q+π2 6≤1+π2 6m1/q and ρp,q(Fm)≥ρp,qm  n=1 fn−ρp,qm  n=1 gnm1/q for mbig enough. So that ρp,q(Fm)m1/q. In the same way for the norm in Hq,p, it follows that FmHq,pm1/pusing the same argument. Therefore, if it were true that RM(p,q)=Hq,p, then we would have ρp,q(Fm)FmHq,p.Butifp>qthis is impossible, because this implies that m1/pm1/qfor all m∈N. Thus, we conclude (a). 123