Parabolic Fractional Maximal Operator in Modified Parabolic Morrey Spaces

We prove that the parabolic fractional maximal operator M α , 0 ≤ α < γ , is bounded from the modified parabolic Morrey space ̃ M1,λ,P R to the weak modified parabolic Morrey space W ̃ Mq,λ,P R if and only if α/γ ≤ 1−1/q ≤ α/ γ −λ and from ̃ Mp,λ,P R to ̃ Mq,λ,P R if and only if α/γ ≤ 1/p − 1/q ≤ α/ γ − λ . Here γ trP is the homogeneous dimension on R. In the limiting case γ − λ /α ≤ p ≤ γ/α we prove that the operator M α is bounded from ̃ Mp,λ,P R to L∞ R . As an application, we prove the boundedness of M 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.


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 .

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 −1 x| 1, and k 1.This space is just the one studied by Calder ón 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 −1 x| 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 n r 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 P 0 diag 1, . . ., 1, 2 we have Let f ∈ L loc 1 R n .The parabolic fractional maximal function M P α f and the parabolic fractional integral I P α f are defined by f y dy, 0 ≤ α < γ.

1.3
If α 0, then M P ≡ M P 0 is the parabolic maximal operator.If P I, then M α ≡ M I α is the fractional maximal operator, M ≡ M I 0 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 Calder ón-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 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 all x ∈ 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 W M 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:

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 loc p R n with finite quasinorm: Notice that if we let ϕ x, t t λ/p 1 |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 < ∞, 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, 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 α f 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 W M 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 Schr ödinger type operators on modified Morrey spaces related to certain nonnegative potentials belonging to the reverse H ölder 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.

Main Results
sufficient for the boundedness of the operator M P α from M p,λ,P R n to M q,λ,P R n .

necessary and sufficient for the boundedness of the operator
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 Besovmodified Morrey spaces on R n see 33 , whose norm is given by

3.2
Therefore, f ∈ M p,λ,P R n and the embedding The following statement can be proved analogously.
Journal of Function Spaces and Applications 7 By the Hölder's inequality we have Moreover, By the Hölder's inequality we have

3.10
For the 0 ≤ α < γ we define the following fractional maximal functions:

3.11
In the case α 0 we denote M P p,0 f is simply denoted by M P p f.
Proof.We have the following.

3.13
From Lemmas 3.1 and 3.5 we get the following.

3.14
In the case α 0 from Lemmas 3.5 and 3.6 one gets that for the M P p f the following property is valid.
In the case p 1 from Lemmas 3.3 and 3.5 we get for the M P α f the following property is valid.

3.16
From Lemmas 3.4 and 3.6 one gets that for M P α f the following property is valid. 3.17

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 .
where C λ depends only on n and λ. 4.8

5.1
For A x, r we get

5.3
Thus for all r > 0 M P α f x min r α M P f x r α−γ/p f M p,λ,P , r α M P f x t α− γ−λ /p f M p,λ,P .

5.4
Minimizing with respect to r, at we have Then 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 .

5.9
By the boundedness of f M p,λ,P .

5.10
If 1/p < 1/q α/γ, then by letting t → 0 we have M P α f M 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 α f M q,λ,P 0 for all f ∈ M p,λ,P R n .

5.12
Taking into account inequality 5.2 and Theorem 4.2, we have where In the case 2t < 1

5.15
In the case 2t ≥ 1 5.17 Finally we have 5.18

Journal of Function Spaces and Applications
Necessity.Let M P α be bounded from M 1,λ,P R n to W M q,λ,P R n .We have W M q,λ,P .

5.19
By the boundedness of 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 α f W M 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 α f W M q,λ,P 0 for all f ∈ M 1,λ,P R n , which is impossible.
Proof of Theorem 2.2.By the definition of the parabolic Besov-modified Morrey spaces on R n it suffices to show that where τ y f x f x y .It is easy to see that τ y f commutes with M P α , that is, τ y M P α f M P α τ y f .Hence we obtain 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.

Parabolic Schr ödinger-Type Operators
In this section we consider the parabolic Schr ödinger operator where V V x, t is a nonnegative potential which belongs to the parabolic reverse H ölder 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,P 0 R n 1 → M q,λ,P 0 R n 1 estimates for the operators The investigation of Schr ödinger operators on the Euclidean space R n with nonnegative potentials which belong to the reverse H ölder class has attracted attention of a number of authors cf.40-42 .Shen 41 studied the Schr ödinger operator −Δ V , assuming that the nonnegative potential V belongs to the reverse H ölder class B q R n for q ≥ n/2 and he proved the L p boundedness of the operators Kurata and Sugano generalized Shen's results to uniformly elliptic operators in 43 .Sugano 44 also extended some results of Shen to the operator . 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 bounded- The main purpose of this section is to investigate the modified parabolic Morrey space M p,λ,P 0 R n 1 → M q,λ,P 0 R n 1 boundedness of the operators It is worth pointing out that we need to establish pointwise estimates for T 1 , T 2 and their adjoint operators by using the estimates of fundamental solution for the Schr ödinger operator on R n 1 in 45 .And we prove the modified parabolic Morrey space M p,λ,P 0 R n 1 → M q,λ,P 0 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 H ölder class B q R n 1 1 < q < ∞ if there exists C > 0 such that the reverse H ölder inequality holds for every parabolic cylinder By the functional calculus, we may write, for all 0 < β < 1, Γ x, t; y, τ; λ f y, τ dy dτ, 6.7 it follows that where 6.9 The following two pointwise estimates for T 1 and T 2 which were proved in 42 , Lemma 3.2 with the potential 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 q 1 for some q 1 > n 2 /2 are also valid see 47 .
The above theorems will yield the modified parabolic Morrey estimates for T 1 and T 2 .

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 to32, 34-39 .

Theorem
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.
and 0 < s < 1, then the operator M P α is bounded from the space BM s pθ,λ,P R n .
Let p 1.By the boundedness of M P from L 1 R n to WL 1 R n see, e.g., 3 and from Theorem 4.1 we have λ,P R n and M P f M p,λ,P ≤ C p,λ,P f M p,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 P f ∈ W M 1,λ,P R n andM P f W M 1,λ,P ≤ C 1,λ,P f M 1,λ,P , 4.3where C 1,λ,P depends only on λ and n.2If f ∈ M p,λ,P R n , 1 < p < ∞, 0 ≤ λ < γ, then M P f ∈ M p,λ,P R n and