Parabolic Fractional Maximal Operator in Modified Parabolic Morrey Spaces

Abstract

We prove that the parabolic fractional maximal operator 𝑀^𝑃_𝛼, 0≤𝛼<𝛾, is bounded from the modified parabolic Morrey space 𝑀_1,𝜆,𝑃(ℝ^𝑛) to the weak modified parabolic Morrey space 𝑊𝑀_{𝑞,𝜆,𝑃}(ℝ^𝑛) if and only if 𝛼/𝛾≤1−1/𝑞≤𝛼/(𝛾−𝜆) and from 𝑀_{𝑝,𝜆,𝑃}(ℝ^𝑛) to 𝑀_{𝑞,𝜆,𝑃}(ℝ^𝑛) if and only if 𝛼/𝛾≤1/𝑝−1/𝑞≤𝛼/(𝛾−𝜆). Here 𝛾=tr𝑃 is the homogeneous dimension on ℝ^𝑛. In the limiting case (𝛾−𝜆)/𝛼≤𝑝≤𝛾/𝛼 we prove that the operator 𝑀^𝑃_𝛼 is bounded from 𝑀_{𝑝,𝜆,𝑃}(ℝ^𝑛) to 𝐿_∞(ℝ^𝑛). As an application, we prove the boundedness of 𝑀^𝑃_𝛼 from the parabolic Besov-modified Morrey spaces 𝐵𝑀^𝑠_{𝑝𝜃,𝜆}(ℝ^𝑛) to 𝐵𝑀^𝑠_{𝑞𝜃,𝜆}(ℝ^𝑛). 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.

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]).

For 𝑥∈ℝ^𝑛 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).

Let 𝑃 be a real 𝑛×𝑛 matrix, all of whose eigenvalues have positive real part. Let 𝐴_𝑡=𝑡^𝑃(𝑡>0) and set 𝛾=tr𝑃. Then, there exists a quasi-distance 𝜌 associated with 𝑃 such that(a)𝜌(𝐴_𝑡𝑥)=𝑡𝜌(𝑥),𝑡>0, for every 𝑥∈ℝ^𝑛;(b)𝜌(0)=0,𝜌(𝑥−𝑦)=𝜌(𝑦−𝑥)≥0, and𝜌(𝑥−𝑦)≤𝑘(𝜌(𝑥−𝑧)+𝜌(𝑦−𝑧));(c)𝑑𝑥=𝜌^𝛾−1𝑑𝜎(𝑤)𝑑𝜌, where 𝜌=𝜌(𝑥), 𝑤=𝐴_𝜌−1𝑥, and 𝑑𝜎(𝑤) is a measure on the ellipsoid {𝑤∶𝜌(𝑤)=1}.

Then, {ℝ^𝑛,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 every 𝑥, 𝑐₁|𝑥|^𝛼1≤𝜌(𝑥)≤𝑐₂|𝑥|^𝛼2(𝑥)𝑐if𝜌≥1,₃|𝑥|^𝛼3≤𝜌(𝑥)≤𝑐₄|𝑥|^𝛼4(𝑥)𝜌()(𝑥)if𝜌≤1,𝜃𝑥≤𝜌for0<𝜃<1. (1.1)

Here 𝛼_𝑖 and 𝑐_𝑖(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∈ℝ^𝑛). In this case, 𝜌(𝑥) is defined by the unique solution of |𝐴_𝑡−1𝑥|=1, and 𝑘=1. This space is just the one studied by Calderón and Torchinsky in [3].(2)Let 𝑃 be a diagonal matrix with positive diagonal entries and let 𝑡=𝜌(𝑥), 𝑥∈ℝ^𝑛 be the unique solution of |𝐴_𝑡−1𝑥|=1.(a) If all diagonal entries are greater than or equal to 1, this space was studied by Fabes and Rivière [4]. More precisely they studied the weak (1,1) and 𝐿^𝑝 estimates of the singular integral operators on this space in 1966.(b)If there are diagonal entries smaller than 1, then 𝜌 satisfies the above (a)–(d) with 𝑘>1.

Thus ℝ^𝑛, 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∈ℝ^𝑛∶𝜌(𝑥−𝑦)<𝑟}, with the Lebesgue measure |ℰ(𝑥,𝑟)|=𝑣_𝑛𝑟^𝛾, where 𝑣_𝑛 is the volume of the unit ellipsoid in ℝ^𝑛. Let also ^∁ℰ(𝑥,𝑟)=ℝ^𝑛⧵ℰ(𝑥,𝑟) be the complement of ℰ(𝑥,𝑟). If 𝑃=𝐼, then clearly 𝜌(𝑥)=|𝑥| and ℰ_𝐼(𝑥,𝑟)=𝐵(𝑥,𝑟). Note that in the standard parabolic case 𝑃₀=diag(1,…,1,2) we have 𝜌(𝑥)=𝑥^′2+𝑥^′4+𝑥²_𝑛2,𝑥=𝑥^′,𝑥_𝑛. (1.2) Let 𝑓∈𝐿₁^loc(ℝ^𝑛). The parabolic fractional maximal function 𝑀^𝑃_𝛼𝑓 and the parabolic fractional integral 𝐼^𝑃_𝛼𝑓 are defined by 𝑀^𝑃_𝛼𝑓(𝑥)=sup_𝑡>0|ℰ(𝑥,𝑡)|^{−1+𝛼/𝛾}_{ℰ(𝑥,𝑡)}𝑓𝑦𝐼𝑑𝑦,0≤𝛼<𝛾._𝑃𝛼𝑓(𝑥)=_ℝ𝑛𝑓𝑦𝜌𝑥−𝑦^𝛾−𝛼𝑑𝑦,0<𝛼<𝛾. (1.3)

If 𝛼=0, then 𝑀^𝑃≡𝑀^𝑃₀ is the parabolic maximal operator. If 𝑃=𝐼, then 𝑀_𝛼≡𝑀^𝐼_𝛼 is the fractional maximal operator, 𝑀≡𝑀^𝐼₀ is the Hardy-Littlewood maximal operator and 𝐼_𝛼≡𝐼^𝐼_𝛼 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. Let 1≤𝑝<∞, 0≤𝜆≤𝛾, [𝑡]₁=min{1,𝑡}. We denote by 𝑀_{𝑝,𝜆,𝑃}(ℝ^𝑛) the parabolic Morrey space, and by 𝑀_{𝑝,𝜆,𝑃}(ℝ^𝑛) the modified parabolic Morrey space the set of locally integrable functions 𝑓(𝑥), 𝑥∈ℝ^𝑛, with the finite norms 𝑓_{𝑀𝑝,𝜆,𝑃}=sup_{𝑥∈ℝ𝑛,𝑡>0}𝑡^−𝜆_{ℰ(𝑥,𝑡)}𝑓𝑦^𝑝𝑑𝑦^1/𝑝,𝑓𝑀_{𝑝,𝜆,𝑃}=sup_{𝑥∈ℝ𝑛,𝑡>0}[𝑡]₁^−𝜆_{ℰ(𝑥,𝑡)}𝑓𝑦^𝑝𝑑𝑦^1/𝑝, (1.4) respectively.
Note that 𝑀_𝑝,0,𝑃ℝ^𝑛=𝑀_𝑝,0,𝑃ℝ^𝑛=𝐿_𝑝ℝ^𝑛,𝑀_{𝑝,𝜆,𝑃}ℝ^𝑛⊂_≻𝑀_{𝑝,𝜆,𝑃}ℝ^𝑛∩𝐿_𝑝ℝ^𝑛,max𝑓_{𝑀𝑝,𝜆,𝑃},𝑓_𝐿𝑝≤𝑓𝑀_{𝑝,𝜆,𝑃}, (1.5) and if 𝜆<0 or 𝜆>𝛾, then 𝑀_{𝑝,𝜆,𝑃}(ℝ^𝑛)=𝑀_{𝑝,𝜆,𝑃}(ℝ^𝑛)=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 ℝ^𝑛.

Definition 1.2 (see [10–14]). Let 1≤𝑝<∞, 0≤𝜆≤𝛾. We denote by 𝑊𝑀_{𝑝,𝜆,𝑃}(ℝ^𝑛) the weak parabolic Morrey space and by 𝑊𝑀_{𝑝,𝜆,𝑃}(ℝ^𝑛) the modified weak parabolic Morrey space the set of locally integrable functions 𝑓(𝑥),𝑥∈ℝ^𝑛 with finite norms: 𝑓_{𝑊𝑀𝑝,𝜆,𝑃}=sup_𝑟>0𝑟sup_{𝑥∈ℝ𝑛,𝑡>0}𝑡^−𝜆𝑦∈ℰ(𝑥,𝑡)∶𝑓𝑦>𝑟^1/𝑝,𝑓_𝑊𝑀_{𝑝,𝜆,𝑃}=sup_𝑟>0𝑟sup_{𝑥∈ℝ𝑛,𝑡>0}[𝑡]₁^−𝜆𝑦∈ℰ(𝑥,𝑡)∶𝑓𝑦>𝑟^1/𝑝, (1.6) respectively.

Note that 𝑊𝐿_𝑝ℝ^𝑛=𝑊𝑀_𝑝,0,𝑃ℝ^𝑛=𝑊𝑀_𝑝,0,𝑃ℝ^𝑛,𝑀_{𝑝,𝜆,𝑃}ℝ^𝑛⊂𝑊𝑀_{𝑝,𝜆,𝑃}ℝ^𝑛,𝑓_{𝑊𝑀𝑝,𝜆,𝑃}≤𝑓_{𝑀𝑝,𝜆,𝑃},𝑀_{𝑝,𝜆,𝑃}ℝ^𝑛⊂𝑊𝑀_{𝑝,𝜆,𝑃}ℝ^𝑛,𝑓_𝑊𝑀_{𝑝,𝜆,𝑃}≤𝑓𝑀_{𝑝,𝜆,𝑃}. (1.7) If 𝑃=𝐼, then 𝑀_𝑝,𝜆(ℝ^𝑛)≡𝑀_{𝑝,𝜆,𝐼}(ℝ^𝑛) is the classical Morrey spaces [15] and 𝑀_𝑝,𝜆(ℝ^𝑛)≡𝑀_{𝑝,𝜆,𝐼}(ℝ^𝑛) 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. Let 𝜑(𝑥,𝑟) be a positive measurable function on ℝ^𝑛×(0,∞) and 1≤𝑝<∞. We denote by 𝑀_{𝑝,𝜑,𝑃}≡𝑀_{𝑝,𝜑,𝑃}(ℝ^𝑛) the parabolic generalized Morrey space, the space of all functions 𝑓∈𝐿_𝑝^loc(ℝ^𝑛) with finite quasinorm: 𝑓_{𝑀𝑝,𝜑,𝑃}=sup_{𝑥∈ℝ𝑛,𝑡>0}𝜑(𝑥,𝑡)⁻¹|ℰ(𝑥,𝑡)|^−1/𝑝𝑓_{𝐿𝑝(ℰ(𝑥,𝑡))}. (1.8) Notice that if we let 𝜑(𝑥,𝑡)=[𝑡]₁^𝜆/𝑝|ℰ(𝑥,𝑡)|^1/𝑝, then we obtain the modified Morrey norm.
The anisotropic result by Hardy-Littlewood-Sobolev states that if 1<𝑝<𝑞<∞, then 𝐼^𝑃_𝛼 is bounded from 𝐿_𝑝(ℝ^𝑛) to 𝐿_𝑞(ℝ^𝑛) if and only if 𝛼=𝛾(1/𝑝−1/𝑞) and for 𝑝=1<𝑞<∞,𝐼^𝑃_𝛼 is bounded from 𝐿₁(ℝ^𝑛) to 𝑊𝐿_𝑞(ℝ^𝑛) if and only if 𝛼=𝛾(1−1/𝑞). Spanne (see [19]) and Adams [1] studied boundedness of the Riesz potential 𝐼_𝛼 for 0<𝛼<𝑛 in Morrey spaces 𝑀_𝑝,𝜆. Later on Chiarenza and Frasca [20] reproved boundedness of the Riesz potential 𝐼_𝛼 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. Let 0<𝛼<𝛾 and 0≤𝜆<𝛾−𝛼, 1≤𝑝<(𝛾−𝜆)/𝛼.(1)If 1<𝑝<(𝛾−𝜆)/𝛼, then the condition 1/𝑝−1/𝑞=𝛼/(𝛾−𝜆) is necessary and sufficient for the boundedness of the operator 𝐼^𝑃_𝛼 from 𝑀_{𝑝,𝜆,𝑃}(ℝ^𝑛) to 𝑀_{𝑞,𝜆,𝑃}(ℝ^𝑛).(2)If 𝑝=1, then the condition 1−1/𝑞=𝛼/(𝛾−𝜆) is necessary and sufficient for the boundedness of the operator 𝐼^𝑃_𝛼 from 𝑀_1,𝜆,𝑃(ℝ^𝑛) to 𝑊𝑀_{𝑞,𝜆,𝑃}(ℝ^𝑛).

If 𝛼=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, for 0<𝛼<𝛾, 𝑀^𝑃_𝛼𝑓(𝑥)≤𝑣_𝑛^{𝛼/𝛾−1}𝐼^𝑃_𝛼𝑓(𝑥), (1.9) hence Theorem A also implies the boundedness of the fractional maximal operator 𝑀^𝑃_𝛼. It is known that the parabolic maximal operator 𝑀^𝑃 is also bounded on 𝑀_{𝑝,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 integral 𝑀^𝑃_𝛼𝑓 in the modified parabolic Morrey space 𝑀_{𝑝,𝜆,𝑃}(ℝ^𝑛). In the case 𝑝=1 we prove that the operator 𝑀^𝑃_𝛼 is bounded from 𝑀_1,𝜆,𝑃(ℝ^𝑛) to 𝑊𝑀_{𝑞,𝜆,𝑃}(ℝ^𝑛) 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 𝑀^𝑃_𝛼 is bounded from 𝑀_{𝑝,𝜆,𝑃}(ℝ^𝑛) to 𝑀_{𝑞,𝜆,𝑃}(ℝ^𝑛) 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 𝑀^𝑃_𝛼 is bounded from 𝑀_{𝑝,𝜆,𝑃}(ℝ^𝑛) to 𝐿_∞(ℝ^𝑛).

The structure of the paper is as follows. In Section 1 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.

By 𝐴≲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. Let 0<𝛼<𝛾, 0≤𝜆<𝛾−𝛼, and 1≤𝑝≤(𝛾−𝜆)/𝛼.(1)If 1<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 𝑀^𝑃_𝛼 from 𝑀_{𝑝,𝜆,𝑃}(ℝ^𝑛) to 𝑀_{𝑞,𝜆,𝑃}(ℝ^𝑛).(2)If 𝑝=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 𝑀^𝑃_𝛼 from 𝑀_1,𝜆,𝑃(ℝ^𝑛) to 𝑊𝑀_{𝑞,𝜆,𝑃}(ℝ^𝑛).(3)If (𝛾−𝜆)/𝛼≤𝑝≤𝛾/𝛼, then the operator 𝑀^𝑃_𝛼 is bounded from 𝑀_{𝑝,𝜆,𝑃}(ℝ^𝑛) to 𝐿_∞(ℝ^𝑛).

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 of 𝑀^𝑃_𝛼 in the parabolic Besov-modified Morrey spaces on ℝ^𝑛 (see [33]), whose norm is given by 𝐵𝑀^𝑠_{𝑝𝜃,𝜆,𝑃}ℝ^𝑛=𝑓∶𝑓𝐵𝑀^𝑠_{𝑝𝜃,𝜆,𝑃}=𝑓𝑀_{𝑝,𝜆,𝑃}+_ℝ𝑛𝑓(𝑥+⋅)−𝑓(⋅)^𝜃𝑀_{𝑝,𝜆,𝑃}𝜌(𝑥)^{𝛾+𝑠𝜃}𝑑𝑥^1/𝜃<∞, (2.1) 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. Let 0≤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 𝑀^𝑃_𝛼 is bounded from the space 𝐵𝑀^𝑠_{𝑝𝜃,𝜆,𝑃}(ℝ^𝑛) to 𝐵𝑀^𝑠_{𝑞𝜃,𝜆,𝑃}(ℝ^𝑛). More precisely, there is a constant 𝐶>0 such that 𝑀^𝑃_𝛼𝑓𝐵𝑀^𝑠_{𝑞𝜃,𝜆,𝑃}≤𝐶𝑓𝐵𝑀^𝑠_{𝑝𝜃,𝜆,𝑃} (2.2) holds for all 𝑓∈𝐵𝑀^𝑠_{𝑝𝜃,𝜆,𝑃}(ℝ^𝑛).

3. Some Several Embeddings

Lemma 3.1. Let 1≤𝑝<∞, 0≤𝜆≤𝛾. Then 𝑀_{𝑝,𝜆,𝑃}ℝ^𝑛=𝑀_{𝑝,𝜆,𝑃}ℝ^𝑛∩𝐿_𝑝ℝ^𝑛,𝑓𝑀_{𝑝,𝜆,𝑃}=max𝑓_{𝑀𝑝,𝜆,𝑃},𝑓_𝐿𝑝. (3.1)

Proof. Let 𝑓∈𝑀_{𝑝,𝜆,𝑃}(ℝ^𝑛). Then from (1.5) we have that 𝑓∈𝑀_{𝑝,𝜆,𝑃}(ℝ^𝑛)∩𝐿_𝑝(ℝ^𝑛), and max{‖𝑓‖_{𝑀𝑝,𝜆,𝑃}, ‖𝑓‖_𝐿𝑝𝑀}≤‖𝑓‖_{𝑝,𝜆,𝑃}.
Let now 𝑓∈𝑀_{𝑝,𝜆,𝑃}(ℝ^𝑛)∩𝐿_𝑝(ℝ^𝑛). Then 𝑓𝑀_{𝑝,𝜆,𝑃}=sup_{𝑥∈ℝ𝑛,𝑡>0}[𝑡]₁^−𝜆_{ℰ(𝑥,𝑡)}𝑓𝑦^𝑝𝑑𝑦^1/𝑝=maxsup_{𝑥∈ℝ𝑛,0<𝑡≤1}𝑡^−𝜆_{ℰ(𝑥,𝑡)}𝑓𝑦^𝑝𝑑𝑦^1/𝑝,sup_{𝑥∈ℝ𝑛,𝑡>1ℰ(𝑥,𝑡)}𝑓𝑦^𝑝𝑑𝑦^1/𝑝≤max𝑓_{𝑀𝑝,𝜆,𝑃},𝑓_𝐿𝑝. (3.2) Therefore, 𝑓∈𝑀_{𝑝,𝜆,𝑃}(ℝ^𝑛) and the embedding 𝑀_{𝑝,𝜆,𝑃}(ℝ^𝑛)∩𝐿_𝑝(ℝ^𝑛)⊂_≻𝑀_{𝑝,𝜆,𝑃}(ℝ^𝑛) is valid.
Thus 𝑀_{𝑝,𝜆,𝑃}(ℝ^𝑛)=𝑀_{𝑝,𝜆,𝑃}(ℝ^𝑛)∩𝐿_𝑝(ℝ^𝑛) and max{‖𝑓‖_{𝑀𝑝,𝜆,𝑃},‖𝑓‖_𝐿𝑝𝑀}=‖𝑓‖_{𝑝,𝜆,𝑃}.

The following statement can be proved analogously.

Lemma 3.2. Let 1≤𝑝<∞, 0≤𝜆≤𝛾, then 𝑊𝑀_{𝑝,𝜆,𝑃}ℝ^𝑛=𝑊𝑀_{𝑝,𝜆,𝑃}ℝ^𝑛∩𝑊𝐿_𝑝ℝ^𝑛,𝑓_𝑊𝑀_{𝑝,𝜆,𝑃}=max𝑓_{𝑊𝑀𝑝,𝜆,𝑃},𝑓_𝑊𝐿𝑝. (3.3)

Lemma 3.3. Let 0<𝛼<𝛾, and 0≤𝜆≤𝛾−𝛼. Then 𝑀_{(𝛾−𝜆)/𝛼,𝜆,𝑃}ℝ^𝑛⊂_≻𝑀_{1,𝛾−𝛼,𝑃}ℝ^𝑛,𝑓_{𝑀1,𝛾−𝛼,𝑃}≤𝑣_𝑛^{1−𝛼/(𝛾−𝜆)}𝑓_{𝑀(𝛾−𝜆)/𝛼,𝜆,𝑃}. (3.4)

Proof. Let 0<𝛼<𝛾, 0≤𝜆≤𝛾−𝛼, 𝑓∈𝑀_{(𝛾−𝜆)/𝛼,𝜆,𝑃}(ℝ^𝑛). By the Hölder's inequality we have _{ℰ(𝑥,𝑡)}𝑓𝑦𝑑𝑦≤_{ℰ(𝑥,𝑡)}𝑓𝑦^{(𝛾−𝜆)/𝛼}𝑑𝑦^{𝛼/(𝛾−𝜆)}_{ℰ(𝑥,𝑡)}𝑑𝑦^{1−𝛼/(𝛾−𝜆)}≤𝑣_𝑛𝑡^{𝛾1−𝛼/(𝛾−𝜆)}_{ℰ()𝑥,𝑡}𝑓𝑦^{(𝛾−𝜆)/𝛼}𝑑𝑦^{𝛼/(𝛾−𝜆)}. (3.5) Moreover, 𝑡^𝛼−𝛾_{ℰ(𝑥,𝑡)}𝑓𝑦𝑑𝑦≤𝑣_𝑛^{1−𝛼/(𝛾−𝜆)}𝑡^{−𝛼𝜆/(𝛾−𝜆)}_{ℰ(𝑥,𝑡)}𝑓𝑦^{(𝛾−𝜆)/𝛼}𝑑𝑦^{𝛼/(𝛾−𝜆)}≤𝑣_𝑛^{1−𝛼/(𝛾−𝜆)}𝑡^−𝜆_{ℰ()𝑥,𝑡}𝑓𝑦^{(𝛾−𝜆)/𝛼}𝑑𝑦^{𝛼/(𝛾−𝜆)}≤𝑣_𝑛^{1−𝛼/(𝛾−𝜆)}𝑓_{𝑀(𝛾−𝜆)/𝛼,𝜆,𝑃}, (3.6) therefore𝑓∈𝑀_{1,𝛾−𝛼,𝑃}(ℝ^𝑛) and 𝑓_{𝑀1,𝛾−𝛼,𝑃}≤𝑣_𝑛^{1−𝛼/(𝛾−𝜆)}𝑓_{𝑀(𝛾−𝜆)/𝛼,𝜆,𝑃}. (3.7)

Lemma 3.4. Let 0<𝛼<𝛾, 0≤𝜆≤𝛾−𝛼. Then for (𝛾−𝜆)/𝛼≤𝑝≤𝛾/𝛼. 𝑀_{𝑝,𝜆,𝑃}ℝ^𝑛⊂_≻𝑀_{1,𝛾−𝛼,𝑃}ℝ^𝑛,𝑓_{𝑀1,𝛾−𝛼,𝑃}≤𝑣^1/𝑝′_𝑛𝑓𝑀_{𝑝,𝜆,𝑃}. (3.8)

Proof. Let 0<𝛼<𝛾, 0≤𝜆≤𝛾−𝛼, 𝑓∈𝑀_{𝑝,𝜆,𝑃}(ℝ^𝑛), and (𝛾−𝜆)/𝛼≤𝑝≤𝛾/𝛼. By the Hölder's inequality we have 𝑓_{𝑀1,𝛾−𝛼,𝑃}=sup_{𝑥∈ℝ𝑛,𝑡>0}𝑡^𝛼−𝛾_{ℰ(𝑥,𝑡)}𝑓𝑦𝑑𝑦≤𝑣^1/𝑝′_𝑛sup_{𝑥∈ℝ𝑛,𝑡>0}𝑡^{𝛼−𝛾/𝑝}[𝑡]₁^𝜆/𝑝[𝑡]₁^−𝜆_{ℰ(𝑥,𝑡)}𝑓𝑦^𝑝𝑑𝑦^1/𝑝≤𝑣^1/𝑝′_𝑛𝑓𝑀_{𝑝,𝜆,𝑃}sup_𝑡>0𝑡^{𝛼−𝛾/𝑝}[𝑡]₁^𝜆/𝑝=𝑣^1/𝑝′_𝑛𝑓𝑀_{𝑝,𝜆,𝑃}maxsup_0<𝑡≤1𝑡^{𝛼−(𝛾−𝜆)/𝑝},sup_𝑡>1𝑡^{𝛼−𝛾/𝑝}=𝑣^1/𝑝′_𝑛𝑓𝑀_{𝑝,𝜆,𝑃}, (3.9) therefore 𝑓∈𝑀_{1,𝛾−𝛼,𝑃}(ℝ^𝑛) and 𝑓_{𝑀1,𝛾−𝛼,𝑃}≤𝑣^1/𝑝′_𝑛𝑓𝑀_{𝑝,𝜆,𝑃}. (3.10)

For the 0≤𝛼<𝛾 we define the following fractional maximal functions: 𝑀^𝑃_𝑝,𝛼𝑓(𝑥)≡𝑀^𝑃_𝛼𝑓^𝑝1/𝑝(𝑥)=sup_𝑡>0|ℰ(𝑥,𝑡)|^{−1+𝛼/𝛾}_{ℰ()𝑥,𝑡}𝑓𝑦^𝑝𝑑𝑦^1/𝑝. (3.11) In the case 𝛼=0 we denote 𝑀^𝑃_𝑝,0𝑓 is simply denoted by 𝑀^𝑃_𝑝𝑓.

Lemma 3.5. Let 1≤𝑝<∞,0≤𝛼<𝛾, and 𝑓∈𝑀_{𝑝,𝛾−𝛼,𝑃}(ℝ^𝑛). Then 𝑀^𝑃_𝑝,𝛼𝑓∈𝐿_∞(ℝ^𝑛) and 𝑀^𝑃_𝑝,𝛼𝑓_𝐿∞=𝑣_𝑛^{(𝛼/𝛾−1)(1/𝑝)}𝑓_{𝑀𝑝,𝛾−𝛼,𝑃}. (3.12)

Proof. We have the following. 𝑀^𝑃_𝑝,𝛼𝑓_𝐿∞=𝑣_𝑛^{(𝛼/𝛾−1)(1/𝑝)}sup_{𝑥∈ℝ𝑛,𝑡>0}𝑡^𝛼−𝛾_{ℰ()𝑥,𝑡}𝑓𝑦^𝑝𝑑𝑦^1/𝑝=𝑣_𝑛^{(𝛼/𝛾−1)(1/𝑝)}𝑓_{𝑀𝑝,𝛾−𝛼,𝑃}. (3.13)

From Lemmas 3.1 and 3.5 we get the following.

Lemma 3.6. Let 1≤𝑝<∞, 0≤𝛼<𝛾 and 𝑓∈𝑀_{𝑝,𝛾−𝛼,𝑃}(ℝ^𝑛). Then 𝑀^𝑃_𝑝,𝛼𝑓∈𝐿_∞(ℝ^𝑛) and 𝑀^𝑃_𝑝,𝛼𝑓_𝐿∞≤𝑣_𝑛^{(𝛼/𝛾−1)(1/𝑝)}𝑓𝑀_{𝑝,𝛾−𝛼,𝑃}. (3.14)

In the case 𝛼=0 from Lemmas 3.5 and 3.6 one gets that for the 𝑀^𝑃_𝑝𝑓 the following property is valid.

Corollary 3.7. Let 1≤𝑝<∞ and 𝑓∈𝐿_∞(ℝ^𝑛). Then 𝑀^𝑃_𝑝𝑓∈𝐿_∞(ℝ^𝑛) and 𝑀^𝑃_𝑝𝑓_𝐿∞=𝑣_𝑛^−1/𝑝𝑓_𝐿∞. (3.15)

In the case 𝑝=1 from Lemmas 3.3 and 3.5 we get for the 𝑀^𝑃_𝛼𝑓 the following property is valid.

Corollary 3.8. Let 0≤𝛼<𝛾, 0≤𝜆≤𝛾−𝛼 and 𝑓∈𝑀_{(𝛾−𝜆)/𝛼,𝜆,𝑃}(ℝ^𝑛). Then 𝑀^𝑃_𝛼𝑓∈𝐿_∞(ℝ^𝑛) and 𝑀^𝑃_𝛼𝑓_𝐿∞=𝑣_𝑛^{𝛼/𝛾−1}𝑓_{𝑀1,𝛾−𝛼,𝑃}≤𝑣_𝑛^{𝛼/𝛾−𝛼/(𝑛−𝜆)}𝑓_{𝑀(𝛾−𝜆)/𝛼,𝜆,𝑃}. (3.16)

From Lemmas 3.4 and 3.6 one gets that for 𝑀^𝑃_𝛼𝑓 the following property is valid.

Corollary 3.9. Let 0≤𝛼<𝛾, 0≤𝜆≤𝛾−𝛼 and 𝑓∈𝑀_{𝑝,𝜆,𝑃}(ℝ^𝑛). Then 𝑀^𝑃_𝛼𝑓∈𝐿_∞(ℝ^𝑛) for (𝛾−𝜆)/𝛼≤𝑝≤𝛾/𝛼 and 𝑀^𝑃_𝛼𝑓_𝐿∞=𝑣_𝑛^{𝛼/𝛾−1}𝑓_{𝑀1,𝛾−𝛼,𝑃}≤𝑣_𝑛^{𝛼/𝛾−1/𝑝}𝑓𝑀_{𝑝,𝜆,𝑃}. (3.17)

4. 𝑀_{𝑝,𝜆,𝑃} Boundedness of the Parabolic Maximal Operator

In this section we study the 𝑀_{𝑝,𝜆,𝑃} boundedness of the maximal operator 𝑀^𝑃.

Theorem 4.1 (see [24]). (1) If 𝑓∈𝑀_1,𝜆,𝑃(ℝ^𝑛), 0≤𝜆<𝛾, then 𝑀^𝑃𝑓∈𝑊𝑀_1,𝜆,𝑃(ℝ^𝑛) and 𝑀^𝑃𝑓_{𝑊𝑀1,𝜆,𝑃}≤𝐶_𝜆𝑓_{𝑀1,𝜆,𝑃}, (4.1) where 𝐶_𝜆 depends only on 𝑛 and 𝜆.
(2) If 𝑓∈𝑀_{𝑝,𝜆,𝑃}(ℝ^𝑛), 1<𝑝<∞, 0≤𝜆<𝛾, then 𝑀^𝑃𝑓∈𝑀_{𝑝,𝜆,𝑃}(ℝ^𝑛) and 𝑀^𝑃𝑓_{𝑀𝑝,𝜆,𝑃}≤𝐶_{𝑝,𝜆,𝑃}𝑓_{𝑀𝑝,𝜆,𝑃}, (4.2) where 𝐶_{𝑝,𝜆,𝑃} depends only on 𝑛, 𝑝, 𝜆, and 𝑃.

Applying Theorem 4.1, one obtains the following result.

Theorem 4.2. (1) If 𝑓∈𝑀_1,𝜆,𝑃(ℝ^𝑛), 0≤𝜆<𝛾, then 𝑀^𝑃𝑓∈𝑊𝑀_1,𝜆,𝑃(ℝ^𝑛) and 𝑀^𝑃𝑓_𝑊𝑀_1,𝜆,𝑃≤𝐶_1,𝜆,𝑃𝑓𝑀_1,𝜆,𝑃, (4.3) where 𝐶_1,𝜆,𝑃 depends only on 𝜆 and 𝑛.
(2) If 𝑓∈𝑀_{𝑝,𝜆,𝑃}(ℝ^𝑛), 1<𝑝<∞, 0≤𝜆<𝛾, then 𝑀^𝑃𝑓∈𝑀_{𝑝,𝜆,𝑃}(ℝ^𝑛) and 𝑀^𝑃𝑓𝑀_{𝑝,𝜆,𝑃}≤𝐶_{𝑝,𝜆,𝑃}𝑓𝑀_{𝑝,𝜆,𝑃}, (4.4) where 𝐶_{𝑝,𝜆,𝑃} depends only on 𝑝, 𝜆, 𝑃, and 𝑛.

Proof. It is obvious that (see Lemmas 3.1 and 3.2) 𝑀^𝑃𝑓𝑀_{𝑝,𝜆,𝑃}=max𝑀^𝑃𝑓_{𝑀𝑝,𝜆,𝑃},𝑀^𝑃𝑓_𝐿𝑝 (4.5) for 1<𝑝<∞ and 𝑀^𝑃𝑓_𝑊𝐿_1,𝜆,𝑃=max𝑀^𝑃𝑓_{𝑊𝑀1,𝜆,𝑃},𝑀^𝑃𝑓_𝑊𝐿1 (4.6) for 𝑝=1.
Let 1<𝑝<∞. By the boundedness of 𝑀^𝑃 on 𝐿_𝑝(ℝ^𝑛) (see, e.g. [3]) and from Theorem 4.1 we get 𝑀^𝑃𝑓𝑀_{𝑝,𝜆,𝑃}≤max𝐶_𝑝,𝐶_{𝑝,𝜆,𝑃}𝑓𝑀_{𝑝,𝜆,𝑃}. (4.7)

Let 𝑝=1. By the boundedness of 𝑀^𝑃 from 𝐿₁(ℝ^𝑛) to 𝑊𝐿₁(ℝ^𝑛) (see, e.g., [3]) and from Theorem 4.1 we have 𝑀^𝑃𝑓_𝑊𝑀_1,𝜆,𝑃≤max𝐶₁,𝐶_1,𝜆,𝑃𝑓𝑀_1,𝜆,𝑃. (4.8)

5. Proof of Main Results

Proof of Theorem 2.1. (1) Sufficiency. Let 𝑟>0, 0<𝛼<𝛾, 0<𝜆<𝛾−𝛼, 𝑓∈𝑀_{𝑝,𝜆,𝑃}(ℝ^𝑛), and 1<𝑝< (𝛾−𝜆)/𝛼. Then 𝑀^𝑃_𝛼𝑓(𝑥)≤sup_𝑡≤𝑟|ℰ(𝑥,𝑡)|^{−1+𝛼/𝛾}_{ℰ(𝑥,𝑡)}𝑓𝑦𝑑𝑦+sup_𝑡>𝑟|ℰ(𝑥,𝑡)|^{−1+𝛼/𝛾}_{ℰ()𝑥,𝑡}𝑓𝑦𝑑𝑦≡𝐴(𝑥,𝑟)+𝐶(𝑥,𝑟). (5.1)

For 𝐴(𝑥,𝑟) we get |𝐴(𝑥,𝑟)|≲sup_𝑡≤𝑟𝑡^𝛼|ℰ(𝑥,𝑡)|⁻¹_{ℰ(𝑥,𝑡)}𝑓𝑦𝑑𝑦≤𝑟^𝛼𝑀^𝑃𝑓(𝑥). (5.2)

By the Hölder inequality 𝐶(𝑥,𝑟)≈sup_𝑡>𝑟𝑡^𝛼−𝛾_{ℰ(𝑥,𝑡)}𝑓𝑦𝑟𝑑𝑦≤min^{𝛼−𝛾/𝑝}𝑓_𝐿𝑝,𝑟^{𝛼−(𝛾−𝜆)/𝑝}𝑓_{𝑀𝑝,𝜆,𝑃}. (5.3)

Thus for all 𝑟>0𝑀^𝑃_𝛼𝑓(𝑥)≲min𝑟^𝛼𝑀^𝑃𝑓(𝑥)+𝑟^{𝛼−𝛾/𝑝}𝑓𝑀_{𝑝,𝜆,𝑃},𝑟^𝛼𝑀^𝑃𝑓(𝑥)+𝑡^{𝛼−(𝛾−𝜆)/𝑝}𝑓𝑀_{𝑝,𝜆,𝑃}. (5.4)

Minimizing with respect to 𝑟, at 𝑟=𝑀^𝑃𝑓(𝑥)⁻¹𝑓𝑀_{𝑝,𝜆,𝑃}^{𝑝/(𝛾−𝜆)},𝑟=𝑀^𝑃𝑓(𝑥)⁻¹𝑓𝑀_{𝑝,𝜆,𝑃}^𝑝/𝑛, (5.5)

we have 𝑀^𝑃_𝛼𝑓(𝑥𝑀)≲min^𝑃𝑓(𝑥)𝑓𝑀_{𝑝,𝜆,𝑃}^{1−𝑝𝛼/(𝛾−𝜆)},𝑀^𝑃𝑓(𝑥)𝑓𝑀_{𝑝,𝜆,𝑃}^{1−𝑝𝛼/𝛾}𝑓𝑀_{𝑝,𝜆,𝑃}. (5.6)

Then 𝑀^𝑃_𝛼𝑓(𝑥)≲𝑀^𝑃𝑓(𝑥)^𝑝/𝑞𝑓^{1−𝑝/𝑞}𝑀_{𝑝,𝜆,𝑃}. (5.7)

Hence, by Theorem 4.2, we have _{ℰ()𝑥,𝑡}𝑀^𝑃_𝛼𝑓𝑦^𝑞𝑑𝑦≲𝑓^𝑞−𝑝𝑀_{𝑝,𝜆,𝑃ℰ(𝑥,𝑡)}𝑀^𝑃𝑓𝑦^𝑝≲[𝑡]𝑑𝑦_𝜆1𝑓^𝑞𝑀_{𝑝,𝜆,𝑃}, (5.8)

which implies that 𝑀^𝑃_𝛼 is bounded from 𝑀_{𝑝,𝜆,𝑃}(ℝ^𝑛) to 𝑀_{𝑞,𝜆,𝑃}(ℝ^𝑛).

Necessity. Let 1<𝑝<𝛾−𝜆/𝛼,𝑓∈𝑀_{𝑝,𝜆,𝑃}(ℝ^𝑛) and assume that 𝑀^𝑃_𝛼 is bounded from 𝑀_{𝑝,𝜆,𝑃}(ℝ^𝑛) to 𝑀_{𝑞,𝜆,𝑃}(ℝ^𝑛).

Define 𝑓_𝑡(𝑥)=∶𝑓(𝑡𝑥), [𝑡]_1,+=max{1,𝑡}. Then𝑓_𝑡𝑀_{𝑝,𝜆,𝑃}=sup_{𝑟>0,𝑥∈ℝ𝑛}[𝑟]₁^−𝜆_{ℰ(𝑥,𝑟)}𝑓_𝑡𝑦^𝑝𝑑𝑦^1/𝑝=𝑡^−𝛾/𝑝sup_{𝑥∈ℝ𝑛,𝑟>0}[𝑟]₁^−𝜆_{ℰ(𝑥,𝑡𝑟)}𝑓𝑦^𝑝𝑑𝑦^1/𝑝=𝑡^−𝛾/𝑝sup_𝑟>0[𝑡𝑟]₁[𝑟]₁^𝜆/𝑝sup_{𝑟>0,𝑥∈ℝ𝑛}[𝑡𝑟]₁^−𝜆_{ℰ(𝑥,𝑡𝑟)}𝑓𝑦^𝑝𝑑𝑦^1/𝑝=𝑡^−𝛾/𝑝[𝑡]^𝜆/𝑝_1,+𝑓𝑀_{𝑝,𝜆,𝑃},𝑀^𝑃_𝛼𝑓_𝑡(𝑥)=𝑡^−𝛼𝑀^𝑃_𝛼𝑓(),𝑀𝑡𝑥_𝑃𝛼𝑓_𝑡𝑀_{𝑞,𝜆,𝑃}=𝑡^−𝛼sup_{𝑥∈ℝ𝑛,𝑟>0}[𝑟]₁^−𝜆_{ℰ(𝑥,𝑟)}𝑀^𝑃_𝛼𝑓𝑡𝑦^𝑞𝑑𝑦^1/𝑞=𝑡^{−𝛼−𝛾/𝑞}sup_𝑟>0[𝑡𝑟]₁[𝑟]₁^𝜆/𝑞sup_{𝑟>0,𝑥∈ℝ𝑛}[𝑡𝑟]₁^−𝜆_{ℰ(𝑡𝑥,𝑡𝑟)}𝑀^𝑃_𝛼𝑓𝑦^𝑞𝑑𝑦^1/𝑞=𝑡^{−𝛼−𝛾/𝑞}[𝑡]^𝜆/𝑞_1,+𝑀^𝑃_𝛼𝑓𝑀_{𝑞,𝜆,𝑃}. (5.9)

By the boundedness of 𝑀^𝑃_𝛼 from 𝑀_{𝑝,𝜆,𝑃}(ℝ^𝑛) to 𝑀_{𝑞,𝜆,𝑃}(ℝ^𝑛)𝑀^𝑃_𝛼𝑓𝐿_{𝑞,𝜆,𝑃}=𝑡^{𝛼+𝛾/𝑞}[𝑡]^−𝜆/𝑞_1,+𝑀^𝑃_𝛼𝑓_𝑡𝐿_{𝑞,𝜆,𝑃}≤𝑡^{𝛼+𝛾/𝑞}[𝑡]^−𝜆/𝑞_1,+𝑓_𝑡𝐿_{𝑝,𝜆,𝑃}≲𝑡^{𝛼+𝛾/𝑞−𝛾/𝑝}[𝑡]^{𝜆/𝑝−𝜆/𝑞}_1,+𝑓𝑀_{𝑝,𝜆,𝑃}. (5.10)

If 1/𝑝<1/𝑞+𝛼/𝛾, then by letting 𝑡→0 we have ‖𝑀^𝑃_𝛼𝑀𝑓‖_{𝑞,𝜆,𝑃}=0 for all 𝑓∈𝑀_{𝑝,𝜆,𝑃}(ℝ^𝑛).

As well as if 1/𝑝>1/𝑞+𝛼/(𝛾−𝜆), then at 𝑡→∞ we obtain ‖𝑀^𝑃_𝛼𝑀𝑓‖_{𝑞,𝜆,𝑃}=0 for all 𝑓∈𝑀_{𝑝,𝜆,𝑃}(ℝ^𝑛).

Therefore 𝛼/𝛾≤1/𝑝−1/𝑞≤𝛼/(𝛾−𝜆).

(2) Sufficiency. Let 𝑓∈𝑀_1,𝜆,𝑃(ℝ^𝑛). From (5.1) we have 𝑦∈ℰ(𝑥,𝑡)∶𝑀^𝑃_𝛼𝑓𝑦>2𝛽≤𝑦∈ℰ(𝑥,𝑡)∶𝐴𝑦,𝑡>𝛽+𝑦∈ℰ(𝑥,𝑡)∶𝐶𝑦,𝑡>𝛽. (5.11)

Then 𝐶𝑦,𝑡≈sup_𝜏>𝑡𝑡^𝛼−𝛾_{ℰ(𝑦,𝜏)}𝑓𝑦𝑡𝑑𝑦≤min^𝛼−𝛾𝑓_𝐿1,𝑡^{𝛼−(𝛾−𝜆)}𝑓_{𝑀1,𝜆,𝑃}.≲[𝑡]^𝜆₁𝑡^𝛼−𝛾𝑓𝑀_1,𝜆,𝑃. (5.12)

Taking into account inequality (5.2) and Theorem 4.2, we have ()∶𝐴≤()𝑦∈ℰ𝑥,𝑡𝑦,𝑡>𝛽𝑦∈ℰ𝑥,𝑡∶𝑀^𝑃𝑓𝑦>𝛽𝐶₁𝑡^𝛼≤𝐶₂𝑡^𝛼𝛽⋅[𝑡]^𝜆₁𝑓𝑀_1,𝜆,𝑃, (5.13) where 𝐶₂=𝐶₁⋅𝐶_1,𝜆,𝑃 and thus if 𝐶₂[2𝑡]^𝜆₁𝑡^𝛼−𝛾𝑀‖𝑓‖_1,𝜆,𝑃=𝛽, then |𝐶(𝑦,𝑡)|≤𝛽 and consequently, |{𝑦∈ℰ(𝑥,𝑡)∶|𝐶(𝑦,𝑡)|>𝛽}|=0.

In the case 2𝑡<1𝐶₂[2𝑡]^𝜆₁𝑡^𝛼−𝛾𝑓𝑀_1,𝜆,𝑃𝑓𝑀=𝛽⟺^1,𝜆,𝑃𝛽≈𝑡^{𝛾−𝛼−𝜆}⟺𝑡^𝛼≈𝑓𝑀_1,𝜆,𝑃𝛽^{𝛼/(𝛾−𝜆−𝛼)}⟺𝑡^𝛼𝑓𝑀_1,𝜆,𝑃𝛽≈𝑓𝑀_1,𝜆,𝑃𝛽^{(𝛾−𝜆)/(𝛾−𝜆−𝛼)}, (5.14)

then 𝑦∈ℰ(𝑥,𝑡)∶𝑀^𝑃_𝛼𝑓𝑦≲1>2𝛽𝛽[𝑡]^𝜆₁𝑡^𝛼𝑓𝑀_1,𝜆,𝑃≈[𝑡]^𝜆₁𝑓𝑀_1,𝜆,𝑃𝛽^{(𝛾−𝜆)/(𝛾−𝜆−𝛼)}. (5.15)

In the case 2𝑡≥1𝐶₂[2𝑡]^𝜆₁𝑡^𝛼−𝛾𝑓𝑀_1,𝜆,𝑃𝑓𝑀=𝛽⟺^1,𝜆,𝑃𝛽≈𝑡^𝛾−𝛼⟺𝑡^𝛼≈𝑓𝑀_1,𝜆,𝑃𝛽^{𝛼/(𝛾−𝛼)}⟺𝑡^𝛼𝑓𝑀_1,𝜆,𝑃𝛽≈𝑓𝑀_1,𝜆,𝑃𝛽^{𝛼/(𝛾−𝛼)}, (5.16)

then 𝑦∈ℰ(𝑥,𝑡)∶𝑀^𝑃_𝛼𝑓𝑦≲1>2𝛽𝛽[𝑡]^𝜆₁𝑡^𝛼𝑓𝑀_1,𝜆,𝑃≈[𝑡]^𝜆₁𝑓𝑀_1,𝜆,𝑃𝛽^{𝛾/(𝛾−𝛼)}. (5.17)

Finally we have 𝑦∈ℰ(𝑥,𝑡)∶𝑀^𝑃_𝛼𝑓𝑦≲[𝑡]>2𝛽_𝜆1𝑓𝑀min^1,𝜆,𝑃𝛽^{(𝛾−𝜆)/(𝛾−𝜆−𝛼)},𝑓𝑀_1,𝜆,𝑃𝛽^{𝛾/(𝛾−𝛼)}≲[𝑡]^𝜆₁1𝛽𝑓𝑀_1,𝜆,𝑃^𝑞. (5.18)

Necessity. Let 𝑀^𝑃_𝛼 be bounded from 𝑀_1,𝜆,𝑃(ℝ^𝑛) to 𝑊𝑀_{𝑞,𝜆,𝑃}(ℝ^𝑛). We have 𝑀^𝑃_𝛼𝑓_𝑡𝑊𝐿_{𝑞,𝜆,𝑃}=sup_𝑟>0𝑟sup_{𝑥∈ℝ𝑛,𝜏>0}[𝜏]₁^−𝜆_{{𝑦∈ℰ(𝑥,𝜏)∶|𝑀𝑃𝛼𝑓𝑡(𝑦)|>𝑟}}𝑑𝑦^1/𝑞=sup_𝑟>0𝑟sup_{𝑥∈ℝ𝑛,𝜏>0}[𝜏]₁^−𝜆_{{𝑦∈ℰ(𝑡𝑥,𝜏)∶|𝑀𝑃𝛼𝑓(𝑡𝑦)|>𝑟𝑡𝛼}}𝑑𝑦^1/𝑞=𝑡^{−𝛼−𝛾/𝑞}sup_𝜏>0[𝑡𝜏]₁[𝜏]₁^𝜆/𝑞sup_𝑟>0𝑟𝑡^𝛼×sup_{𝑥∈ℝ𝑛,𝜏>0}[𝑡𝜏]₁^−𝜆_{{𝑦∈ℰ(𝑥,𝑡𝜏)∶|𝑀𝑃𝛼𝑓(𝑦)|>𝑟𝑡𝛼}}𝑑𝑦^1/𝑞=𝑡^{−𝛼−𝛾/𝑞}[𝑡]^𝜆/𝑞_1,+𝑀^𝑃_𝛼𝑓_𝑊𝑀_{𝑞,𝜆,𝑃}. (5.19)

By the boundedness of 𝑀^𝑃_𝛼 from 𝑀_1,𝜆,𝑃(ℝ^𝑛) to 𝑊𝑀_{𝑞,𝜆,𝑃}(ℝ^𝑛)𝑀^𝑃_𝛼𝑓_𝑊𝐿_{𝑞,𝜆,𝑃}≤𝐶_1,𝑞,𝜆𝑡^{𝛼+𝛾/𝑞−𝑛}[𝑡]^{𝜆−𝜆/𝑞}_1,+𝑓𝑀_1,𝜆,𝑃, (5.20) where 𝐶_1,𝑞,𝜆 depends only on 𝑞, 𝜆, and 𝑛.

If 1<1/𝑞+𝛼/𝛾, then by letting 𝑡→0 we have ‖𝑀^𝑃_𝛼𝑓‖_𝑊𝑀_{𝑞,𝜆,𝑃}=0 for all 𝑓∈𝑀_1,𝜆,𝑃(ℝ^𝑛), which is impossible.

Similarly, if 1>1/𝑞+𝛼/(𝛾−𝜆), then for 𝑡→∞ we obtain ‖𝑀^𝑃_𝛼𝑓‖_𝑊𝑀_{𝑞,𝜆,𝑃}=0 for all 𝑓∈𝑀_1,𝜆,𝑃(ℝ^𝑛), which is impossible.

Therefore 𝛼/𝛾≤1−1/𝑞≤𝛼/(𝛾−𝜆).

Thus Theorem 2.1 is proved.

Proof of Theorem 2.2. By the definition of the parabolic Besov-modified Morrey spaces on ℝ^𝑛 it suffices to show that 𝜏_𝑦𝑀^𝑃_𝛼𝑓−𝑀^𝑃_𝛼𝑓𝑀_{𝑞,𝜆,𝑃}≲𝜏_𝑦𝑀𝑓−𝑓_{𝑝,𝜆,𝑃}, (5.21) where 𝜏_𝑦𝑓(𝑥)=𝑓(𝑥+𝑦).
It is easy to see that 𝜏_𝑦𝑓 commutes with 𝑀^𝑃_𝛼, that is, 𝜏_𝑦𝑀^𝑃_𝛼𝑓=𝑀^𝑃_𝛼(𝜏_𝑦𝑓). Hence we obtain 𝜏_𝑦𝑀^𝑃_𝛼𝑓−𝑀^𝑃_𝛼𝑓=𝑀^𝑃_𝛼𝜏_𝑦𝑓−𝑀^𝑃_𝛼𝑓≤𝑀^𝑃_𝛼𝜏_𝑦𝑓−𝑓. (5.22) Taking 𝑀_{𝑝,𝜆,𝑃}-norm on both sides of the last inequality, we obtain the desired result by using the boundedness of 𝑀^𝑃_𝛼 from 𝑀_{𝑝,𝜆,𝑃}(ℝ^𝑛) to 𝑀_{𝑞,𝜆,𝑃}(ℝ^𝑛).
Thus the proof of the Theorem 2.2 is completed.

6. Parabolic Schrödinger-Type Operators 𝑉^𝜇(𝝏/𝝏𝑡−Δ+𝑉)^−𝛽and 𝑉^𝜇𝛁²(𝝏/𝝏𝑡−Δ+𝑉)^−𝛽

In this section we consider the parabolic Schrödinger operator 𝜕𝜕𝑡−Δ+𝑉onℝ^𝑛+1, (6.1) where 𝑉=𝑉(𝑥,𝑡) is a nonnegative potential which belongs to the parabolic reverse Hölder class 𝐵_𝑞(ℝ^𝑛+1). Examples of such potentials are all positive polynomials but also singular functions like max{|𝑥|,𝑡^1/2}^𝛼 for 𝛼>−(𝑛+2)/𝑞. We prove the modified parabolic Morrey space 𝑀_{𝑝,𝜆,𝑃0}(ℝ^𝑛+1)→ 𝑀_{𝑞,𝜆,𝑃0}(ℝ^𝑛+1) estimates for the operators 𝑉^𝜇(𝜕/𝜕𝑡−Δ+𝑉)^−𝛽 and 𝑉^𝜇∇²(𝜕/𝜕𝑡−Δ+𝑉)^−𝛽, where 𝑃₀=diag(1,…,1,2).

The investigation of Schrödinger operators on the Euclidean space ℝ^𝑛 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 𝐵_𝑞(ℝ^𝑛) for 𝑞≥𝑛/2 and he proved the 𝐿_𝑝 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 𝑉^𝜇(−Δ+𝑉)^−𝛽, 0≤𝜇≤𝛽≤1 and 𝑉^𝜇∇(−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∈𝐵_𝑞(ℝ^𝑛+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 𝐿_𝑝(ℝ^𝑛+1).

The main purpose of this section is to investigate the modified parabolic Morrey space 𝑀_{𝑝,𝜆,𝑃0}(ℝ^𝑛+1)→𝑀_{𝑞,𝜆,𝑃0}(ℝ^𝑛+1) boundedness of the operators 𝒯₁=𝑉^𝜇𝜕𝜕𝑡−Δ+𝑉^−𝛽𝒯,0≤𝜇≤𝛽≤1,₂=𝑉^𝜇∇²𝜕𝜕𝑡−Δ+𝑉^−𝛽1,0≤𝜇≤21≤𝛽≤1,𝛽−𝜇≥2. (6.2) 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 for 𝒯₁, 𝒯₂ and their adjoint operators by using the estimates of fundamental solution for the Schrödinger operator on ℝ^𝑛+1 in [45]. And we prove the modified parabolic Morrey space 𝑀_{𝑝,𝜆,𝑃0}(ℝ^𝑛+1)→𝑀_{𝑞,𝜆,𝑃0}(ℝ^𝑛+1) boundedness of the parabolic fractional maximal operators.

Definition 6.1. (1) A nonnegative locally 𝐿_𝑞 integrable function 𝑉 on ℝ^𝑛+1 is said to belong to the parabolic reverse Hölder class 𝐵_𝑞(ℝ^𝑛+1)(1<𝑞<∞) if there exists 𝐶>0 such that the reverse Hölder inequality 1|𝐾|_𝐾𝑉𝑦,𝜏^𝑞𝑑𝑦𝑑𝜏^1/𝑞≤𝐶|𝐾|_𝐾𝑉𝑦,𝜏𝑑𝑦𝑑𝜏 (6.3) holds for every parabolic cylinder 𝐾=𝐾((𝑥,𝑡),𝑟)=𝑦,𝜏∈ℝ^𝑛+1∶𝑥_𝑖−𝑦_𝑖<𝑟,|𝑡−𝜏|<𝑟²,𝑖=1,…,𝑛 (6.4) of center (𝑥,𝑡) and radius 𝑟 in ℝ^𝑛+1.
(2) Let 𝑉=𝑉(𝑥,𝑡)≥0. We say 𝑉∈𝐵_∞(ℝ^𝑛+1), if there exists a constant 𝐶>0 such that ‖𝑉‖_{𝐿∞(𝐾)}≤𝐶|𝐾|_𝐾𝑉𝑦,𝜏𝑑𝑦𝑑𝜏 (6.5) holds for every parabolic cylinder 𝐾=𝐾((𝑥,𝑡),𝑟) in ℝ^𝑛+1.

Clearly, 𝐵_∞(ℝ^𝑛+1)⊂𝐵_𝑞(ℝ^𝑛+1) for 1<𝑞<∞. But it is important that the 𝐵_𝑞(ℝ^𝑛+1) class has a property of “self-improvement”, that is, if 𝑉∈𝐵_𝑞(ℝ^𝑛+1), then 𝑉∈𝐵_𝑞+𝜀(ℝ^𝑛+1) for some 𝜀>0 (see [46]).

By the functional calculus, we may write, for all 0<𝛽<1, 𝜕𝜕𝑡−Δ+𝑉^−𝛽=1𝜋^∞₀𝜆^−𝛽𝜕𝜕𝑡−Δ+𝑉+𝜆⁻¹𝑑𝜆. (6.6)

Let 𝑓∈𝐶^∞₀(ℝ^𝑛+1). From 𝜕𝜕𝑡−Δ+𝑉+𝜆⁻¹𝑓(𝑥,𝑡)=_ℝ𝑛+1Γ𝑥,𝑡;𝑦,𝜏;𝜆𝑓𝑦,𝜏𝑑𝑦𝑑𝜏, (6.7) it follows that 𝒯₁𝑓(𝑥,𝑡)=_ℝ𝑛+1𝐾₁𝑥,𝑡;𝑦,𝜏𝑉𝑦,𝜏^𝜇𝑓𝑦,𝜏𝑑𝑦𝑑𝜏, (6.8) where 𝐾₁=1𝑥,𝑡;𝑦,𝜏𝜋∫^∞₀𝜆^−𝛽ΓΓ𝑥,𝑡;𝑦,𝜏;𝜆𝑑𝜆for0<𝛽<1𝑥,𝑡;𝑦,𝜏;0for𝛽=1. (6.9)

The following two pointwise estimates for 𝒯₁ and 𝒯₂ which were proved in [42], Lemma 3.2 with the potential 𝑉∈𝐵_∞(ℝ^𝑛+1).

Theorem A. Suppose 𝑉∈𝐵_∞(ℝ^𝑛+1) and 0≤𝜇≤𝛽≤1. Then, for any 𝑓∈𝐶^∞₀(ℝ^𝑛+1)𝒯₁𝑓(𝑥,𝑡)≲𝑀^𝑃0_𝛼𝑓(𝑥,𝑡), (6.10) where 𝛼=2(𝛽−𝜇).

Theorem B. Suppose 𝑉∈𝐵_∞(ℝ^𝑛+1), 0≤𝜇≤1/2≤𝛽≤1 and 𝛽−𝜇≥1/2. Then, for any 𝑓∈𝐶^∞₀(ℝ^𝑛+1)𝒯₂𝑓(𝑥,𝑡)≲𝑀^𝑃0_𝛼𝑓(𝑥,𝑡), (6.11) where 𝛼=2(𝛽−𝜇)−1.

Note that the similar estimates for the adjoint operators 𝑇^∗₁ and 𝑇^∗₂ with the potential 𝑉∈𝐵_𝑞1 for some 𝑞₁>(𝑛+2)/2 are also valid (see [47]).

Theorem C. Suppose 𝑉∈𝐵_𝑞1(ℝ^𝑛+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 𝑇^∗₁𝑓(𝑥,𝑡)≤𝐶𝑀^𝑃0_𝛼𝑞2𝑓^𝑞2(𝑥,𝑡)^1/𝑞2,𝑓∈𝐶^∞₀ℝ^𝑛+1, (6.12) where 𝛼=2(𝛽−𝜇).

Theorem D. Suppose 𝑉∈𝐵_𝑞1(ℝ^𝑛+1) for some 𝑞₁>(𝑛+2)/2, 0≤𝜇≤1/2<𝛽≤1, and 𝛽−𝜇≥1/2. And let 1𝑞₂=𝜇1−𝑞₁,𝑖𝑓𝑞₁>𝑛+2,1−𝜇+1𝑞₁+1𝑛+2,𝑖𝑓𝑛+22<𝑞₁<𝑛+2. (6.13)

Then there exists a constant 𝐶>0 such that 𝑇^∗₂𝑓(𝑥,𝑡)≤𝐶𝑀^𝑃0_𝛼𝑞2𝑓^𝑞2(𝑥,𝑡)^1/𝑞2,𝑓∈𝐶^∞₀ℝ^𝑛+1, (6.14) where 𝛼=2(𝛽−𝜇)−1.

The above theorems will yield the modified parabolic Morrey estimates for 𝒯₁ and 𝒯₂.

Corollary 6.2. Assume that 𝑉∈𝐵_∞(ℝ^𝑛+1), and 0≤𝜇≤𝛽≤1. Let 1≤𝑝≤𝑞<∞, 0≤𝜆<𝑛+2 −2(𝛽−𝜇), and 2𝛽−𝜇≤1𝑛+2𝑝−1𝑞≤2𝛽−𝜇.𝑛+2−𝜆 (6.15)

Then, for any 𝑓∈𝐶^∞₀(ℝ^𝑛+1)𝒯₁𝑓𝑀_{0𝑞,𝜆,𝑃}≲𝑓𝑀_{0𝑝,𝜆,𝑃}𝒯for𝑝>1,₁𝑓_𝑊𝑀_{0𝑞,𝜆,𝑃}≲𝑓𝑀_01,𝜆,𝑃for𝑝=1. (6.16)

Corollary 6.3. Assume that 𝑉∈𝐵_∞(ℝ^𝑛+1), 0≤𝜇≤1/2≤𝛽≤1 and 𝛽−𝜇≥1/2. Let 1≤𝑝≤𝑞<∞, 0≤𝜆<𝑛+1−2(𝛽−𝜇), and 2𝛽−𝜇−1≤1𝑛+2𝑝−1𝑞≤2𝛽−𝜇−1.𝑛+2−𝜆 (6.17)

Then, for any 𝑓∈𝐶^∞₀(ℝ^𝑛+1)𝒯₂𝑓𝑀_{0𝑞,𝜆,𝑃}≲𝑓𝑀_{0𝑝,𝜆,𝑃}𝒯for𝑝>1,₂𝑓_𝑊𝑀_{0𝑞,𝜆,𝑃}≲𝑓𝑀_01,𝜆,𝑃for𝑝=1. (6.18)

Corollary 6.4. Assume that 𝑉∈𝐵_𝑞1(ℝ^𝑛+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 any 𝑓∈𝐶^∞₀(ℝ^𝑛+1)𝒯₁𝑓𝑀_{0𝑞,𝜆,𝑃}≲𝑓𝑀_{0𝑝,𝜆,𝑃}𝒯𝑓𝑜𝑟𝑝>1,₁𝑓_𝑊𝑀_{0𝑞,𝜆,𝑃}≲𝑓𝑀_01,𝜆,𝑃𝑓𝑜𝑟𝑝=1. (6.19)

Corollary 6.5. Assume that 𝑉∈𝐵_𝑞1(ℝ^𝑛+1) for 𝑞₁>(𝑛+2)/2, and 10≤𝜇≤2≤𝛽≤1𝑖𝑓𝑞₁1>𝑛+2,0≤𝜇≤2<𝛽≤1𝑖𝑓𝑛+22<𝑞₁<𝑛+2. (6.20)

Let 𝛼=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 1𝑝₁=𝜇𝑞₁,𝑖𝑓𝑞₁>𝑛+2,𝜇+1𝑞₁−1𝑛+2,𝑖𝑓𝑛+22<𝑞₁<𝑛+2. (6.21)

Then, for any 𝑓∈𝐶^∞₀(ℝ^𝑛+1)𝒯₂𝑓𝑀_{0𝑞,𝜆,𝑃}≲𝑓𝑀_{0𝑝,𝜆,𝑃}𝒯𝑓𝑜𝑟𝑝>1,₂𝑓_𝑊𝑀_{0𝑞,𝜆,𝑃}≲𝑓𝑀_01,𝜆,𝑃𝑓𝑜𝑟𝑝=1. (6.22)

Acknowledgments

The authors would like to express their gratitude to the referees for their very valuable comments and suggestions. The research of V. Guliyev was partially supported by the Grant of Science Development Foundation under the President of the Republic of Azerbaijan Project EIF-2010-1(1)-40/06-1 and by the Grant of 2011-Ahi Evran University Scientific Research Projects (BAP FBA-11-13) and by the Scientific and Technological Research Council of Turkey (TUBITAK Project no. 110T695).