# Parabolic Fractional Maximal Operator in Modified Parabolic Morrey Spaces

Authors: Vagif S. Guliyev; Kamala R. Rahimova
Journal: Journal of Function Spaces and Applications
 (2012)

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## Abstract

We prove that the parabolic fractional maximal operatorMαP, 0≤α<γ,  is bounded from the modified parabolic Morrey space M̃1,λ,P(ℝn) to the weak modified parabolic Morrey space WM̃q,λ,P(ℝn) if and only if α/γ≤1-1/q≤α/(γ-λ) and from M̃p,λ,P(ℝn) to M̃q,λ,P(ℝn) if and only if α/γ≤1/p-1/q≤α/(γ-λ). Here γ=trP is the homogeneous dimension on ℝn. In the limiting case (γ-λ)/α≤p≤γ/α we prove that the operator MαP is bounded from M̃p,λ,P(ℝn) to L∞(ℝn). As an application, we prove the boundedness of MαP from the parabolic Besov-modified Morrey spaces BM̃pθ,λs(ℝn) to BM̃qθ,λs(ℝn). As other applications, we establish the boundedness of some Schrödinger-ype operators on modified parabolic Morrey spaces related to certain nonnegative potentials belonging to the reverse Hölder class.


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## Body

## 1. Introduction

The theory of boundedness of classical operators of the real analysis, such as the maximal operator, the fractional maximal operators, the fractional integral operators, and the singular integral operators, from one weighted Lebesgue space to another one is well studied by now. These results have good applications in the theory of partial differential equations. However, in the theory of partial differential equations, along with Morrey spaces, modified Morrey spaces also play an important role (see [1, 2]).Forx∈ℝn and  r>0, we denote by B(x,r)  the open ball centered at x of radius r and by ∁B(x,r) denote its complement. Let |B(x,r)| be the Lebesgue measure of the ball B(x,r).LetP  be a real n×n  matrix, all of whose eigenvalues have positive real part. Let At=tP(t>0) and set γ=trP. Then, there exists a quasi-distance ρ associated with P such that(a)
ρ(Atx)=tρ(x),t>0, for every x∈ℝn;(b)
ρ(0)=0,ρ(x-y)=ρ(y-x)≥0, andρ(x-y)≤k(ρ(x-z)+ρ(y-z));(c)
dx=ργ-1dσ(w)dρ, where ρ=ρ(x), w=Aρ-1x, and dσ(w) is a measure on the ellipsoid {w:ρ(w)=1}.Then,{ℝn,ρ,dx}  becomes a space of homogeneous type in the sense of Coifman-Weiss. Moreover, we always assume the following properties on ρ.(d)
For everyx,
(1.1)c1|x|α1≤ρ(x)≤c2|x|α2ifρ(x)≥1,c3|x|α3≤ρ(x)≤c4|x|α4ifρ(x)≤1,ρ(θx)≤ρ(x)for0<θ<1.Hereαi and ci(i=1,…,4) are some positive constants. Similar properties hold for ρ* which is associated with the matrix P*. Here P* is the adjoint matrix of P.There are some important examples for the above spaces.(1)
Let(Px,x)≥(x,x)(x∈ℝn). In this case, ρ(x) is defined by the unique solution of |At-1x|=1, and k=1. This space is just the one studied by Calderón and Torchinsky in [3].(2)
LetP be a diagonal matrix with positive diagonal entries and let t=ρ(x), x∈ℝn be the unique solution of |At-1x|=1.
(a)
If all diagonal entries are greater than or equal to1, this space was studied by Fabes and Rivière [4]. More precisely they studied the weak (1,1) and Lp estimates of the singular integral operators on this space in 1966.(b)
If there are diagonal entries smaller than1, then ρ satisfies the above (a)–(d) with k>1.Thusℝn, endowed with the metric ρ, defines a homogeneous metric space [4, 5]. The balls with respect to ρ, centered at x of radius r, are just the ellipsoids ℰ(x,r)={y∈ℝn:ρ(x-y)<r}, with the Lebesgue measure |ℰ(x,r)|=vnrγ, where vn is the volume of the unit ellipsoid in ℝn. Let also ∁ℰ(x,r)=ℝn∖ℰ(x,r) be the complement of ℰ(x,r). If P=I, then clearly ρ(x)=|x| and ℰI(x,r)=B(x,r). Note that in the standard parabolic case P0=diag(1,…,1,2) we have
(1.2)ρ(x)=|x′|2+|x′|4+xn22,x=(x′,xn).
Let  f∈L1loc(ℝn). The parabolic fractional maximal function  MαPf and the parabolic fractional integral  IαPf are defined by
(1.3)MαPf(x)=supt>0|ℰ(x,t)|-1+α/γ∫ℰ(x,t)|f(y)|dy,0≤α<γ.IαPf(x)=∫ℝnf(y)ρ(x-y)γ-αdy,0<α<γ.Ifα=0, then MP≡M0P is the parabolic maximal operator. If P=I, then Mα≡MαI is the fractional maximal operator, M≡M0I is the Hardy-Littlewood maximal operator and Iα≡IαI is the Riesz potential.It is well known that the fractional maximal operator, the fractional integral operator, and Calderón-Zygmund operators play an important role in harmonic analysis (see [6–9]).Definition 1.1.
Let1≤p<∞, 0≤λ≤γ, [t]1=min{1,t}. We denote by Mp,λ,P(ℝn) the parabolic Morrey space, and by M~p,λ,P(ℝn) the modified parabolic Morrey space the set of locally integrable functions f(x), x∈ℝn, with the finite norms
(1.4)∥f∥Mp,λ,P=supx∈ℝn,t>0(t-λ∫ℰ(x,t)|f(y)|pdy)1/p,∥f∥M~p,λ,P=supx∈ℝn,t>0([t]1-λ∫ℰ(x,t)|f(y)|pdy)1/p,
respectively.
Note that(1.5)M~p,0,P(ℝn)=Mp,0,P(ℝn)=Lp(ℝn),M~p,λ,P(ℝn)⊂≻Mp,λ,P(ℝn)∩Lp(ℝn),max{∥f∥Mp,λ,P,∥f∥Lp}≤∥f∥M~p,λ,P,
and if λ<0 or λ>γ, then Mp,λ,P(ℝn)=M~p,λ,P(ℝn)=Θ, where the symbol ⊂≻ means continuous embedding (let X,Y be the normed spaces, then by definition X⊂≻Y means that there exists C>0 such that ∥x∥Y≤C∥x∥X for allx∈X) and Θ is the set of all functions equivalent to 0 on ℝn.Definition 1.2 (see [10–14]).
Let1≤p<∞, 0≤λ≤γ. We denote by WMp,λ,P(ℝn) the weak parabolic Morrey space and by WM~p,λ,P(ℝn) the modified weak parabolic Morrey space the set of locally integrable functions f(x),x∈ℝn with finite norms:
(1.6)∥f∥WMp,λ,P=supr>0rsupx∈ℝn,t>0(t-λ|{y∈ℰ(x,t):|f(y)|>r}|)1/p,∥f∥WM~p,λ,P=supr>0rsupx∈ℝn,t>0([t]1-λ|{y∈ℰ(x,t):|f(y)|>r}|)1/p,
respectively.Note that(1.7)WLp(ℝn)=WMp,0,P(ℝn)=WM~p,0,P(ℝn),Mp,λ,P(ℝn)⊂WMp,λ,P(ℝn),∥f∥WMp,λ,P≤∥f∥Mp,λ,P,M~p,λ,P(ℝn)⊂WM~p,λ,P(ℝn),∥f∥WM~p,λ,P≤∥f∥M~p,λ,P.
If P=I, then Mp,λ(ℝn)≡Mp,λ,I(ℝn) is the classical Morrey spaces [15] and M~p,λ(ℝn)≡M~p,λ,I(ℝn) is the modified Morrey spaces [2].Note that the parabolic generalized Morrey spaces are defined as follows (see, e.g., [16–18], etc.)Definition 1.3.
Letφ(x,r) be a positive measurable function on  ℝn×(0,∞) and 1≤p<∞. We denote by Mp,φ,P≡Mp,φ,P(ℝn) the parabolic generalized Morrey space, the space of all functions f∈Lploc(ℝn) with finite quasinorm:
(1.8)∥f∥Mp,φ,P=supx∈ℝn,t>0φ(x,t)-1|ℰ(x,t)|-1/p∥f∥Lp(ℰ(x,t)).
Notice that if we let φ(x,t)=[t]1λ/p|ℰ(x,t)|1/p, then we obtain the modified Morrey norm.
The anisotropic result by Hardy-Littlewood-Sobolev states that if1<p<q<∞, then IαP is bounded from Lp(ℝn) to Lq(ℝn) if and only if α=γ(1/p-1/q) and for p=1<q<∞,IαP is bounded from  L1(ℝn) to  WLq(ℝn) if and only if  α=γ(1-1/q). Spanne (see [19]) and Adams [1] studied boundedness of the Riesz potential Iα for 0<α<n in Morrey spaces Mp,λ. Later on Chiarenza and Frasca [20] reproved boundedness of the Riesz potential Iα in these spaces. By more general results of Guliyev [21] (see also [17, 18, 22, 23]) one can obtain the following generalization of the results in [1, 19, 20] to the anisotropic case.Theorem A.
Let0<α<γ and 0≤λ<γ-α, 1≤p<(γ-λ)/α.(1)
If1<p<(γ-λ)/α, then the condition 1/p-1/q=α/(γ-λ) is necessary and sufficient for the boundedness of the operator IαP from Mp,λ,P(ℝn)  to Mq,λ,P(ℝn).(2)
Ifp=1, then the condition 1-1/q=α/(γ-λ) is necessary and sufficient for the boundedness of the operator IαP from M1,λ,P(ℝn) to WMq,λ,P(ℝn).Ifα=γ/p-γ/q,  then λ=0  and the statement of Theorem A reduces to the aforementioned result by anisotropic version of Hardy-Littlewood-Sobolev.Recall that, for0<α<γ,
(1.9)MαPf(x)≤vnα/γ-1IαP(|f|)(x),
hence Theorem A also implies the boundedness of the fractional maximal operator MαP. It is known that the parabolic maximal operator MP is also bounded on Mp,λ,P for all 1<p<∞ and 0<λ<γ (see, e.g. [24]), whose isotropic counterpart was proved by Chiarenza and Frasca [20].In this paper we study the parabolic fractional maximal integralMαPf in the modified parabolic Morrey space M~p,λ,P(ℝn). In the case p=1 we prove that the operator MαP is bounded from M~1,λ,P(ℝn) to WM~q,λ,P(ℝn) if and only if, α/γ≤1-1/q≤α/(γ-λ). In the case 1<p<(γ-λ)/α we prove that the operator MαP is bounded from M~p,λ,P(ℝn) to M~q,λ,P(ℝn) if and only if, α/γ≤1/p-1/q≤α/(γ-λ). In the limiting case (γ-λ)/α≤p≤γ/α we prove that the operator MαP is bounded from M~p,λ,P(ℝn) to L∞(ℝn).The structure of the paper is as follows. In Section1 the boundedness of the maximal operator in modified Morrey space M~p,λ,P is proved. The main result of the paper is the Hardy-Littlewood-Sobolev inequality in modified parabolic Morrey space for the parabolic fractional maximal operator, established in Section 2. In Section 3 by using the (M~p,λ,P,M~q,λ,P) boundedness of the parabolic fractional maximal operators we establish the boundedness of some Schrödinger type operators on modified Morrey spaces related to certain nonnegative potentials belonging to the reverse Hölder class.ByA≲B we mean that A≤CB with some positive constant C independent of appropriate quantities. If A≲B and B≲A, we write A≈B and say that A and B are equivalent.


## 2. Main Results

Theorem 2.1.
Let0<α<γ, 0≤λ<γ-α,  and 1≤p≤(γ-λ)/α.(1)
If1<p<(γ-λ)/α, then the condition α/γ≤1/p-1/q≤α/(γ-λ) is necessary and sufficient for the boundedness of the operator MαP from M~p,λ,P(ℝn) to M~q,λ,P(ℝn).(2)
Ifp=1<(γ-λ)/α, then the condition α/γ≤1-1/q≤α/(γ-λ) is necessary and sufficient for the boundedness of the operator MαP from M~1,λ,P(ℝn) to WM~q,λ,P(ℝn).(3)
If(γ-λ)/α≤p≤γ/α, then the operator MαP is bounded from M~p,λ,P(ℝn) to L∞(ℝn).Besov-Morrey (and Triebel-Lizorkin-Morrey) spaces attracted some attention in these two decades. Kozono and Yamazaki [25] and Mazzucato [26] used these spaces in the theory of Navier-Stokes equations. Some properties of the spaces including the wavelet characterizations were described in the papers by Sawano [27, 28], Sawano and Tanaka [29, 30], Tang and Xu [31]. The most systematic and general approach can certainly be found in the very recent book [32] of Yuan et al., we also recommend this monograph for further up-to-date references on this subject.In the following theorem we prove the boundedness ofMαP in the parabolic Besov-modified Morrey spaces on ℝn (see [33]), whose norm is given by
(2.1)BM~pθ,λ,Ps(ℝn)={∫ℝn∥f(x+·)-f(·)∥M~p,λ,Pθρ(x)γ+sθ(∫ℝn∥f(x+·)-f(·)∥M~p,λ,Pθρ(x)γ+sθdx)1/θf:∥f∥BM~pθ,λ,Ps=∥f∥M~p,λ,P+(∫ℝn∥f(x+·)-f(·)∥M~p,λ,Pθρ(x)γ+sθdx)1/θ<∞},
where 1≤p,θ≤∞, 0<s<1 and 0≤λ<γ.These spaces generalize certain Besov-Morrey and Triebel-Lizorkin-Morrey spaces. As a general theory of Besov-Triebel-Lizorkin spaces, the Besov-Morrey and Triebel-Lizorkin-Morrey spaces are introduced due to the study of Navier-Stokes equations and attract some attention in recent years. Another scales of generalized Besov and Triebel-Lizorkin spaces, the Besov-type space and Triebel-Lizorkin-type space, were introduced by Yang and Yuan in [34, 35] and proved therein to be closely related to the theory of Q spaces. For further developments and applications of these spaces, we also refer to [32, 34–39].Theorem 2.2.
Let0≤α<γ, 0≤λ<γ-α and 1≤p<(γ-λ)/α. If α/Q≤1/p-1/q≤α/(Q-λ), 1≤θ≤∞, and 0<s<1, then the operator MαP is bounded from the space BM~pθ,λ,Ps(ℝn) to BM~qθ,λ,Ps(ℝn). More precisely, there is a constant C>0 such that
(2.2)∥MαPf∥BM~qθ,λ,Ps≤C∥f∥BM~pθ,λ,Ps
holds for all f∈BM~pθ,λ,Ps(ℝn).


## 3. Some Several Embeddings

Lemma 3.1.
Let1≤p<∞, 0≤λ≤γ. Then
(3.1)M~p,λ,P(ℝn)=Mp,λ,P(ℝn)∩Lp(ℝn),∥f∥M~p,λ,P=max{∥f∥Mp,λ,P,∥f∥Lp}.Proof.
Letf∈M~p,λ,P(ℝn). Then from (1.5) we have that f∈Mp,λ,P(ℝn)∩Lp(ℝn), and max{∥f∥Mp,λ,P, ∥f∥Lp}≤∥f∥M~p,λ,P.
Let nowf∈Mp,λ,P(ℝn)∩Lp(ℝn). Then
(3.2)∥f∥M~p,λ,P=supx∈ℝn,t>0([t]1-λ∫ℰ(x,t)|f(y)|pdy)1/p=max{supx∈ℝn,0<t≤1(t-λ∫ℰ(x,t)|f(y)|pdy)1/p,supx∈ℝn,t>1(∫ℰ(x,t)|f(y)|pdy)1/p}≤max{∥f∥Mp,λ,P,∥f∥Lp}.
Therefore, f∈M~p,λ,P(ℝn) and the embedding Mp,λ,P(ℝn)∩Lp(ℝn)⊂≻M~p,λ,P(ℝn) is valid.
ThusM~p,λ,P(ℝn)=Mp,λ,P(ℝn)∩Lp(ℝn) and max{∥f∥Mp,λ,P,∥f∥Lp}=∥f∥M~p,λ,P.The following statement can be proved analogously.Lemma 3.2.
Let1≤p<∞, 0≤λ≤γ, then
(3.3)WM~p,λ,P(ℝn)=WMp,λ,P(ℝn)∩WLp(ℝn),∥f∥WM~p,λ,P=max{∥f∥WMp,λ,P,∥f∥WLp}.Lemma 3.3.
Let0<α<γ, and 0≤λ≤γ-α. Then
(3.4)M(γ-λ)/α,λ,P(ℝn)⊂≻M1,γ-α,P(ℝn),∥f∥M1,γ-α,P≤vn1-α/(γ-λ)∥f∥M(γ-λ)/α,λ,P.Proof.
Let0<α<γ, 0≤λ≤γ-α, f∈M(γ-λ)/α,λ,P(ℝ^n). By the Hölder's inequality we have
(3.5)∫ℰ(x,t)|f(y)|dy≤(∫ℰ(x,t)|f(y)|(γ-λ)/αdy)α/(γ-λ)(∫ℰ(x,t)dy)1-α/(γ-λ)≤(vntγ)1-α/(γ-λ)(∫ℰ(x,t)|f(y)|(γ-λ)/αdy)α/(γ-λ).
Moreover,
(3.6)tα-γ∫ℰ(x,t)|f(y)|dy≤vn1-α/(γ-λ)t-αλ/(γ-λ)(∫ℰ(x,t)|f(y)|(γ-λ)/αdy)α/(γ-λ)≤vn1-α/(γ-λ)(t-λ∫ℰ(x,t)|f(y)|(γ-λ)/αdy)α/(γ-λ)≤vn1-α/(γ-λ)∥f∥M(γ-λ)/α,λ,P,
thereforef∈M1,γ-α,P(ℝn) and
(3.7)∥f∥M1,γ-α,P≤vn1-α/(γ-λ)∥f∥M(γ-λ)/α,λ,P.Lemma 3.4.
Let0<α<γ, 0≤λ≤γ-α. Then for (γ-λ)/α≤p≤γ/α.
(3.8)M~p,λ,P(ℝn)⊂≻M1,γ-α,P(ℝn),∥f∥M1,γ-α,P≤vn1/p'∥f∥M~p,λ,P.Proof.
Let0<α<γ, 0≤λ≤γ-α, f∈M~p,λ,P(ℝn), and (γ-λ)/α≤p≤γ/α. By the Hölder's inequality we have
(3.9)∥f∥M1,γ-α,P=supx∈ℝn,t>0tα-γ∫ℰ(x,t)|f(y)|dy≤vn1/p'supx∈ℝn,t>0tα-γ/p[t]1λ/p([t]1-λ∫ℰ(x,t)|f(y)|pdy)1/p≤vn1/p′∥f∥M~p,λ,Psupt>0tα-γ/p[t]1λ/p=vn1/p′∥f∥M~p,λ,Pmax{sup0<t≤1tα-(γ-λ)/p,supt>1tα-γ/p}=vn1/p′∥f∥M~p,λ,P,
therefore f∈M1,γ-α,P(ℝn) and
(3.10)∥f∥M1,γ-α,P≤vn1/p′∥f∥M~p,λ,P.For the0≤α<γ we define the following fractional maximal functions:
(3.11)Mp,αPf(x)≡(MαP|f|p)1/p(x)=supt>0(|ℰ(x,t)|-1+α/γ∫ℰ(x,t)|f(y)|pdy)1/p.
In the case α=0 we denote  Mp,0Pf  is simply denoted by MpPf.Lemma 3.5.
Let1≤p<∞,0≤α<γ, and f∈Mp,γ-α,P(ℝn). Then Mp,αPf∈L∞(ℝn) and
(3.12)∥Mp,αPf∥L∞=vn(α/γ-1)(1/p)∥f∥Mp,γ-α,P.Proof.
We have the following.(3.13)∥Mp,αPf∥L∞=vn(α/γ-1)(1/p)supx∈ℝn,t>0(tα-γ∫ℰ(x,t)|f(y)|pdy)1/p=vn(α/γ-1)(1/p)∥f∥Mp,γ-α,P.From Lemmas3.1 and 3.5 we get the following.Lemma 3.6.
Let1≤p<∞, 0≤α<γ and  f∈M~p,γ-α,P(ℝn). Then Mp,αPf∈L∞(ℝn)  and
(3.14)∥Mp,αPf∥L∞≤vn(α/γ-1)(1/p)∥f∥M~p,γ-α,P.In the caseα=0 from Lemmas 3.5 and 3.6 one gets that for the MpPf  the following property is valid.Corollary 3.7.
Let1≤p<∞ and f∈L∞(ℝn). Then MpPf∈L∞(ℝn) and
(3.15)∥MpPf∥L∞=vn-1/p∥f∥L∞.In the casep=1 from Lemmas 3.3 and 3.5 we get for the MαPf the following property is valid.Corollary 3.8.
Let0≤α<γ,  0≤λ≤γ-α and f∈M(γ-λ)/α,λ,P(ℝ^n). Then MαPf∈L∞(ℝn) and
(3.16)∥MαPf∥L∞=vnα/γ-1∥f∥M1,γ-α,P≤vnα/γ-α/(n-λ)∥f∥M(γ-λ)/α,λ,P.From Lemmas3.4 and 3.6 one  gets that for MαPf the following property is valid.Corollary 3.9.
Let0≤α<γ, 0≤λ≤γ-α and f∈M~p,λ,P(ℝn). Then MαPf∈L∞(ℝn) for (γ-λ)/α≤p≤γ/α and
(3.17)∥MαPf∥L∞=vnα/γ-1∥f∥M1,γ-α,P≤vnα/γ-1/p∥f∥M~p,λ,P.


## 4.M~p,λ,P Boundedness of the Parabolic Maximal Operator

In this section we study theM~p,λ,P boundedness of the maximal operator MP.Theorem 4.1 (see [24]).
(
1
) If f∈M1,λ,P(ℝn), 0≤λ<γ, then MPf∈WM1,λ,P(ℝn) and
(4.1)∥MPf∥WM1,λ,P≤Cλ∥f∥M1,λ,P,
where Cλ depends only on n and λ.
(
2
) If f∈Mp,λ,P(ℝn), 1<p<∞, 0≤λ<γ, then MPf∈Mp,λ,P(ℝn) and
(4.2)∥MPf∥Mp,λ,P≤Cp,λ,P∥f∥Mp,λ,P,
where Cp,λ,P depends only on n, p, λ, and P.Applying Theorem4.1, one  obtains the following result.Theorem 4.2.
(
1
) If f∈M~1,λ,P(ℝn), 0≤λ<γ, then MPf∈WM~1,λ,P(ℝn) and
(4.3)∥MPf∥WM~1,λ,P≤C¯1,λ,P∥f∥M~1,λ,P,
where C¯1,λ,P depends only on λ and n.
(
2
) If f∈M~p,λ,P(ℝn), 1<p<∞, 0≤λ<γ, then MPf∈M~p,λ,P(ℝn) and
(4.4)∥MPf∥M~p,λ,P≤C¯p,λ,P∥f∥M~p,λ,P,
where C¯p,λ,P depends only on p, λ, P, and n.Proof.
It is obvious that (see Lemmas3.1 and 3.2)
(4.5)∥MPf∥M~p,λ,P=max{∥MPf∥Mp,λ,P,∥MPf∥Lp}
for 1<p<∞ and
(4.6)∥MPf∥WL~1,λ,P=max{∥MPf∥WM1,λ,P,∥MPf∥WL1}
for p=1.
Let1<p<∞. By the boundedness of MP on Lp(ℝn) (see, e.g. [3]) and from Theorem 4.1 we get
(4.7)∥MPf∥M~p,λ,P≤max{Cp,Cp,λ,P}∥f∥M~p,λ,P.Letp=1. By the boundedness of MP from L1(ℝn) to WL1(ℝn) (see, e.g., [3]) and from Theorem 4.1 we have
(4.8)∥MPf∥WM~1,λ,P≤max{C1,C1,λ,P}∥f∥M~1,λ,P.


## 5.   Proof of Main Results

Proof of Theorem2.1.
(1)Sufficiency. Let r>0, 0<α<γ, 0<λ<γ-α, f∈M~p,λ,P(ℝn), and 1<p< (γ-λ)/α. Then
(5.1)MαPf(x)≤supt≤r|ℰ(x,t)|-1+α/γ∫ℰ(x,t)|f(y)|dy+supt>r|ℰ(x,t)|-1+α/γ∫ℰ(x,t)|f(y)|dy≡A(x,r)+C(x,r).ForA(x,r) we get
(5.2)|A(x,r)|≲supt≤rtα|ℰ(x,t)|-1∫ℰ(x,t)|f(y)|dy≤rαMPf(x).By the Hölder inequality(5.3)C(x,r)≈supt>rtα-γ∫ℰ(x,t)|f(y)|dy≤min{rα-γ/p∥f∥Lp,rα-(γ-λ)/p∥f∥Mp,λ,P}.Thus for allr>0(5.4)|MαPf(x)|≲min{rαMPf(x)+rα-γ/p∥f∥M~p,λ,P,rαMPf(x)+tα-(γ-λ)/p∥f∥M~p,λ,P}.Minimizing with respect tor, at
(5.5)r=[(MPf(x))-1∥f∥M~p,λ,P]p/(γ-λ),r=[(MPf(x))-1∥f∥M~p,λ,P]p/n,we have(5.6)|MαPf(x)|≲min{(MPf(x)∥f∥M~p,λ,P)1-pα/(γ-λ),(MPf(x)∥f∥M~p,λ,P)1-pα/γ}∥f∥M~p,λ,P.Then(5.7)|MαPf(x)|≲(MPf(x))p/q∥f∥M~p,λ,P1-p/q.Hence, by Theorem4.2, we have
(5.8)∫ℰ(x,t)|MαPf(y)|qdy≲∥f∥M~p,λ,Pq-p∫ℰ(x,t)(MPf(y))pdy≲[t]1λ∥f∥M~p,λ,Pq,which implies thatMαP is bounded from M~p,λ,P(ℝn) to M~q,λ,P(ℝn).Necessity. Let 1<p<γ-λ/α,f∈M~p,λ,P(ℝn) and assume that MαP is bounded from M~p,λ,P(ℝn) to M~q,λ,P(ℝn).Defineft(x)=:f(tx), [t]1,+=max{1,t}. Then(5.9)∥ft∥M~p,λ,P=supr>0,x∈ℝn([r]1-λ∫ℰ(x,r)‍|ft(y)|pdy)1/p=t-γ/psupx∈ℝn,r>0([r]1-λ∫ℰ(x,tr)‍|f(y)|pdy)1/p=t-γ/psupr>0([tr]1[r]1)λ/psupr>0,x∈ℝn([tr]1-λ∫ℰ(x,tr)‍|f(y)|pdy)1/p=t-γ/p[t]1,+λ/p∥f∥M~p,λ,P,MαPft(x)=t-αMαPf(tx),∥MαPft∥M~q,λ,P=t-αsupx∈ℝn,r>0([r]1-λ∫ℰ(x,r)‍|MαPf(ty)|qdy)1/q=t-α-γ/qsupr>0([tr]1[r]1)λ/qsupr>0,x∈ℝn([tr]1-λ∫ℰ(tx,tr)‍|MαPf(y)|qdy)1/q=t-α-γ/q[t]1,+λ/q∥MαPf∥M~q,λ,P.By the boundedness ofMαP from M~p,λ,P(ℝn) to M~q,λ,P(ℝn)(5.10)∥MαPf∥L~q,λ,P=tα+γ/q[t]1,+-λ/q∥MαPft∥L~q,λ,P≤tα+γ/q[t]1,+-λ/q∥ft∥L~p,λ,P≲tα+γ/q-γ/p[t]1,+λ/p-λ/q∥f∥M~p,λ,P.If1/p<1/q+α/γ, then by letting t→0 we have ∥MαPf∥M~q,λ,P=0 for all f∈M~p,λ,P(ℝn).As well as if1/p>1/q+α/(γ-λ), then at t→∞ we obtain ∥MαPf∥M~q,λ,P=0 for all f∈M~p,λ,P(ℝn).Thereforeα/γ≤1/p-1/q≤α/(γ-λ).(2)Sufficiency. Let f∈M~1,λ,P(ℝn). From (5.1) we have
(5.11)|{y∈ℰ(x,t):|MαPf(y)|>2β}|≤|{y∈ℰ(x,t):|A(y,t)|>β}|+|{y∈ℰ(x,t):|C(y,t)|>β}|.Then(5.12)C(y,t)≈supτ>ttα-γ∫ℰ(y,τ)|f(y)|dy≤min{tα-γ∥f∥L1,tα-(γ-λ)∥f∥M1,λ,P}.≲[t]1λtα-γ∥f∥M~1,λ,P.Taking into account inequality (5.2) and Theorem 4.2, we have
(5.13)|{y∈ℰ(x,t):|A(y,t)|>β}|≤|{y∈ℰ(x,t):MPf(y)>βC1tα}|≤C2tαβ·[t]1λ∥f∥M~1,λ,P,
where C2=C1·C1,λ,P and thus if C2[2t]1λtα-γ∥f∥M~1,λ,P=β, then |C(y,t)|≤β and consequently, |{y∈ℰ(x,t):|C(y,t)|>β}|=0.In the case2t<1(5.14)C2[2t]1λtα-γ∥f∥M~1,λ,P=β⟺∥f∥M~1,λ,Pβ≈tγ-α-λ⟺tα≈(∥f∥M~1,λ,Pβ)α/(γ-λ-α)⟺tα∥f∥M~1,λ,Pβ≈(∥f∥M~1,λ,Pβ)(γ-λ)/(γ-λ-α),then(5.15)|{y∈ℰ(x,t):|MαPf(y)|>2β}|≲1β[t]1λtα∥f∥M~1,λ,P≈[t]1λ(∥f∥M~1,λ,Pβ)(γ-λ)/(γ-λ-α).In the case2t≥1(5.16)C2[2t]1λtα-γ∥f∥M~1,λ,P=β⟺∥f∥M~1,λ,Pβ≈tγ-α⟺tα≈(∥f∥M~1,λ,Pβ)α/(γ-α)⟺tα∥f∥M~1,λ,Pβ≈(∥f∥M~1,λ,Pβ)α/(γ-α),then(5.17)|{y∈ℰ(x,t):|MαPf(y)|>2β}|≲1β[t]1λtα∥f∥M~1,λ,P≈[t]1λ(∥f∥M~1,λ,Pβ)γ/(γ-α).Finally we have(5.18)|{y∈ℰ(x,t):|MαPf(y)|>2β}|≲[t]1λmin{(∥f∥M~1,λ,Pβ)(γ-λ)/(γ-λ-α),(∥f∥M~1,λ,Pβ)γ/(γ-α)}≲[t]1λ(1β∥f∥M~1,λ,P)q.Necessity. Let MαP be bounded from M~1,λ,P(ℝn) to WM~q,λ,P(ℝn). We have
(5.19)∥MαPft∥WL~q,λ,P=supr>0rsupx∈ℝn,τ>0([τ]1-λ∫{y∈ℰ(x,τ):|MαPft(y)|>r}dy)1/q=supr>0rsupx∈ℝn,τ>0([τ]1-λ∫{y∈ℰ(tx,τ):|MαPf(ty)|>rtα}dy)1/q=t-α-γ/qsupτ>0([tτ]1[τ]1)λ/qsupr>0rtα×supx∈ℝn,τ>0([tτ]1-λ∫{y∈ℰ(x,tτ):|MαPf(y)|>rtα}dy)1/q=t-α-γ/q[t]1,+λ/q∥MαPf∥WM~q,λ,P.By the boundedness ofMαP from M~1,λ,P(ℝn) to WM~q,λ,P(ℝn)(5.20)∥MαPf∥WL~q,λ,P≤C1,q,λtα+γ/q-n[t]1,+λ-λ/q∥f∥M~1,λ,P,
where C1,q,λ depends only on q, λ, and n.If1<1/q+α/γ, then by letting t→0 we have ∥MαPf∥WM~q,λ,P=0 for all f∈M~1,λ,P(ℝn), which is impossible.Similarly, if1>1/q+α/(γ-λ), then for t→∞ we obtain ∥MαPf∥WM~q,λ,P=0 for all f∈M~1,λ,P(ℝn), which is impossible.Thereforeα/γ≤1-1/q≤α/(γ-λ).Thus Theorem2.1 is proved.Proof of Theorem2.2.
By the definition of the parabolic Besov-modified Morrey spaces onℝn it suffices to show that
(5.21)∥τyMαPf-MαPf∥M~q,λ,P≲∥τyf-f∥M~p,λ,P,
where  τyf(x)=f(x+y).
It is easy to see thatτyf commutes with MαP, that is, τyMαPf=MαP(τyf). Hence we obtain
(5.22)|τyMαPf-MαPf|=|MαP(τyf)-MαPf|≤MαP(|τyf-f|).
Taking M~p,λ,P-norm on both sides of the last inequality, we obtain the desired result by using the boundedness of MαP from  M~p,λ,P(ℝn) to  M~q,λ,P(ℝn).
Thus the proof of the Theorem2.2 is completed.


## 6. Parabolic Schrödinger-Type OperatorsVμ(∂/∂t-Δ+V)-βand Vμ∇2(∂/∂t-Δ+V)-β

In this section we consider the parabolic Schrödinger operator(6.1)∂∂t-Δ+Vonℝn+1,
where V=V(x,t) is a nonnegative potential which belongs to the parabolic reverse Hölder class Bq(ℝn+1). Examples of such potentials are all positive polynomials but also singular functions like max{|x|,t1/2}α for α>-(n+2)/q. We prove the modified parabolic Morrey space M~p,λ,P0(ℝn+1)→ M~q,λ,P0(ℝn+1) estimates for the operators Vμ(∂/∂t-Δ+V)-β and Vμ∇2(∂/∂t-Δ+V)-β, where P0=diag(1,…,1,2).The investigation of Schrödinger operators on the Euclidean spaceℝn with nonnegative potentials which belong to the reverse Hölder class has attracted attention of a number of authors (cf. [40–42]). Shen [41] studied the Schrödinger operator -Δ+V, assuming that the nonnegative potential V belongs to the reverse Hölder class Bq(ℝn) for q≥n/2 and he proved the Lp boundedness of the operators (-Δ+V)iμ, ∇2(-Δ+V)-1, ∇(-Δ+V)-1/2, and ∇(-Δ+V)-1. Kurata and Sugano generalized Shen’s results to uniformly elliptic operators in [43]. Sugano [44] also extended some results of Shen to the operator Vμ(-Δ+V)-β, 0≤μ≤β≤1 and Vμ∇(-Δ+V)-β, 0≤μ≤1/2≤β≤1 and β-μ≥1/2. Following Shen's approach, Gao and Jiang extend the results to the parabolic case. In [45], they consider the parabolic operator ∂/∂t-Δ+V where V∈Bq(ℝn+1) is a nonnegative potential depending only on the space variables and, under the assumptions n≥3 and p>(n+2)/2, they proved the boundedness of V(∂/∂t-Δ+V)-1 in Lp(ℝn+1).The main purpose of this section is to investigate the modified parabolic Morrey spaceM~p,λ,P0(ℝn+1)→M~q,λ,P0(ℝn+1) boundedness of the operators
(6.2)𝒯1=Vμ(∂∂t-Δ+V)-β,0≤μ≤β≤1,𝒯2=Vμ∇2(∂∂t-Δ+V)-β,0≤μ≤12≤β≤1,β-μ≥12.
Note that the operator ∇2(∂/∂t-Δ+V)-1 in [45] is the special case of 𝒯2.It is worth pointing out that we need to establish pointwise estimates for𝒯1, 𝒯2 and their adjoint operators by using the estimates of fundamental solution for the Schrödinger operator on ℝn+1 in [45]. And we prove the modified parabolic Morrey space M~p,λ,P0(ℝn+1)→M~q,λ,P0(ℝn+1) boundedness of the parabolic fractional maximal operators.Definition 6.1.
(1) A nonnegative locallyLq integrable function V on ℝn+1 is said to belong to the parabolic reverse Hölder class Bq(ℝn+1)(1<q<∞) if there exists C>0 such that the reverse Hölder inequality
(6.3)(1|K|∫KV(y,τ)qdydτ)1/q≤C|K|∫KV(y,τ)dydτ
holds for every parabolic cylinder
(6.4)K=K((x,t),r)={(y,τ)∈ℝn+1:|xi-yi|<r,|t-τ|<r2,i=1,…,n}
of center (x,t) and radius r in ℝn+1.
(2) LetV=V(x,t)≥0. We say V∈B∞(ℝn+1), if there exists a constant C>0 such that
(6.5)∥V∥L∞(K)≤C|K|∫KV(y,τ)dydτ
holds for every parabolic cylinder K=K((x,t),r) in ℝn+1.Clearly,B∞(ℝn+1)⊂Bq(ℝn+1) for 1<q<∞. But it is important that the Bq(ℝn+1) class has a property of “self-improvement”, that is, if V∈Bq(ℝn+1), then V∈Bq+ε(ℝn+1) for some ε>0 (see [46]).By the functional calculus, we may write, for all0<β<1,
(6.6)(∂∂t-Δ+V)-β=1π∫0∞‍λ-β(∂∂t-Δ+V+λ)-1dλ.Letf∈C0∞(ℝn+1). From
(6.7)(∂∂t-Δ+V+λ)-1f(x,t)=∫ℝn+1Γ(x,t;y,τ;λ)f(y,τ)dydτ,
it follows that
(6.8)𝒯1f(x,t)=∫ℝn+1K1(x,t;y,τ)V(y,τ)μf(y,τ)dydτ,
where
(6.9)K1(x,t;y,τ)={1π∫0∞λ-βΓ(x,t;y,τ;λ)dλfor0<β<1Γ(x,t;y,τ;0)forβ=1.The following two pointwise estimates for𝒯1 and 𝒯2 which were proved in [42], Lemma 3.2 with the potential V∈B∞(ℝn+1).Theorem A.
SupposeV∈B∞(ℝn+1) and 0≤μ≤β≤1. Then, for any f∈C0∞(ℝn+1)(6.10)|𝒯1f(x,t)|≲MαP0f(x,t),
where α=2(β-μ).Theorem B.
SupposeV∈B∞(ℝn+1), 0≤μ≤1/2≤β≤1 and β-μ≥1/2. Then, for any f∈C0∞(ℝn+1)(6.11)|𝒯2f(x,t)|≲MαP0f(x,t),
where α=2(β-μ)-1.Note that the similar estimates for the adjoint operatorsT1* and T2* with the potential V∈Bq1 for some q1>(n+2)/2 are also valid (see [47]).Theorem C.
SupposeV∈Bq1(ℝn+1) for some q1>(n+2)/2, 0≤μ≤β≤1 and let 1/q2=1-μ/q1. Then there exists a constant C>0 such that
(6.12)|T1*f(x,t)|≤C(Mαq2P0(|f|q2)(x,t))1/q2,f∈C0∞(ℝn+1),
where α=2(β-μ).Theorem D.
SupposeV∈Bq1(ℝn+1) for some q1>(n+2)/2, 0≤μ≤1/2<β≤1, and β-μ≥1/2. And let
(6.13)1q2={1-μq1,ifq1>n+2,1-μ+1q1+1n+2,ifn+22<q1<n+2.Then there exists a constantC>0 such that
(6.14)|T2*f(x,t)|≤C(Mαq2P0(|f|q2)(x,t))1/q2,f∈C0∞(ℝn+1),
where α=2(β-μ)-1.The above theorems will yield the modified parabolic Morrey estimates for𝒯1 and 𝒯2.Corollary 6.2.
Assume thatV∈B∞(ℝn+1), and 0≤μ≤β≤1. Let 1≤p≤q<∞, 0≤λ<n+2 -2(β-μ), and
(6.15)2(β-μ)n+2≤1p-1q≤2(β-μ)n+2-λ.Then, for anyf∈C0∞(ℝn+1)(6.16)∥𝒯1f∥M~q,λ,P0≲∥f∥M~p,λ,P0forp>1,∥𝒯1f∥WM~q,λ,P0≲∥f∥M~1,λ,P0forp=1.Corollary 6.3.
Assume thatV∈B∞(ℝn+1), 0≤μ≤1/2≤β≤1 and β-μ≥1/2. Let 1≤p≤q<∞, 0≤λ<n+1-2(β-μ), and
(6.17)2(β-μ)-1n+2≤1p-1q≤2(β-μ)-1n+2-λ.Then, for anyf∈C0∞(ℝn+1)(6.18)∥𝒯2f∥M~q,λ,P0≲∥f∥M~p,λ,P0forp>1,∥𝒯2f∥WM~q,λ,P0≲∥f∥M~1,λ,P0forp=1.Corollary 6.4.
Assume thatV∈Bq1(ℝn+1) for q1>(n+2)/2 and 0≤μ≤β≤1. Let α=2(β-μ), 1≤ p<1/(μ/q1+α/(n+2)), α/(n+2)≤1/p-1/q≤α/(n+2-λ), 1/q2=1-μ/q1.
Then, for anyf∈C0∞(ℝn+1)(6.19)∥𝒯1f∥M~q,λ,P0≲∥f∥M~p,λ,P0forp>1,∥𝒯1f∥WM~q,λ,P0≲∥f∥M~1,λ,P0forp=1.Corollary 6.5.
Assume thatV∈Bq1(ℝn+1) for q1>(n+2)/2, and
(6.20)0≤μ≤12≤β≤1ifq1>n+2,0≤μ≤12<β≤1ifn+22<q1<n+2.Letα=2(β-μ)-1,β-μ≥1/2, 1≤p<1/(μ/q1+α/(n+2)), α/(n+2)≤1/p-1/q≤α/(n+2-λ), 1/q2=1-μ/q1, where
(6.21)1p1={μq1,ifq1>n+2,μ+1q1-1n+2,ifn+22<q1<n+2.Then, for anyf∈C0∞(ℝn+1)(6.22)∥𝒯2f∥M~q,λ,P0≲∥f∥M~p,λ,P0forp>1,∥𝒯2f∥WM~q,λ,P0≲∥f∥M~1,λ,P0forp=1.



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*Source: 543475-2012-12-24.xml*