Anisotropic Fractional Maximal Operator in Anisotropic Generalized Morrey Spaces

In this paper it is proved that anisotropic fractional maximal operator Mα,σ, 0 ≤ α < |σ| is bounded on anisotropic generalized Morrey spaces Mp,φ,σ, where |σ| = ∑n i=1 σi is the homogeneous dimension of R n. We find the conditions on the pair (φ1, φ2) which ensure the Spanne-Guliyev type boundedness of the operator Mα,σ from anisotropic generalized Morrey space Mp,φ1,σ to Mq,φ2,σ, 1 < p ≤ q < ∞, 1/p − 1/q = α/|σ|, and from the space M1,φ1,σ to the weak space WMq,φ2,σ, 1 < q < ∞, 1 − 1/q = α/|σ|. We also find conditions on the φ which ensure the AdamsGuliyev type boundedness of Mα,σ from Mp,φ p ,σ to Mq,φ q ,σ for 1 < p < q < ∞ and from M1,φ,σ to WMq,φ q ,σ for 1 < q < ∞. As applications, we establish the boundedness of some Schödinger type operators on anisotropic generalized Morrey spaces related to certain nonnegative potentials belonging to the reverse Hölder class.


Introduction
In the present paper we will prove the boundedness of the anisotropic fractional maximal operator in the anisotropic generalized Morrey spaces.
By Besov, Il'in and Lizorkin (1966) and Fabes and Rivère (1966), the function F(x, ρ) = n i=1 x 2 i ρ −2σ i , considered for any fixed x ∈ R n , is a decreasing one with respect to ρ > 0 and the equation F(x, ρ) = 1 is uniquely solvable.This unique solution will be denoted by ρ(x).Define ρ(x) = ρ and ρ(0) = 0.It is a simple matter to check that ρ(x − y) defines a distance between any two points x, y ∈ R n .Thus R n , endowed with the metric ρ, defines a homogeneous metric space (Besov, Il'in, & Lizorkin, 1966;Bramanti & Cerutti, 1996;Fabes & Rivère, 1966).Note that ρ(x) is equivalent to One of the most important variants of the anisotropic maximal function is the so-called anisotropic fractional maximal function defined by the formula where |E σ (x, t)| = 2 n t |σ| is the Lebesgue measure of the parallelepiped E σ (x, t).
It coincides with the anisotropic maximal function M σ f ≡ M 0,σ f and is intimately related to the anisotropic Riesz potential operator If σ = 1, then M α ≡ M α,1 and I α ≡ I α,1 is the fractional maximal operator and Riesz potential, respectively.The operators M α , M α,σ , I α and I α,σ play important role in real and harmonic analysis (see, for example Besov, Il'in, & Nikol'skii, 1996;Stein, 1993).
Definition 1.2 (Burenkov, Guliyev, H. V., & Guliyev, V. S., 2007) Let 1 ≤ p < ∞ and 0 ≤ b ≤ 1.We denote by WL p,b,σ ≡ WL p,b,σ (R n ) the weak anisotropic Morrey space as the set of locally integrable functions f The anisotropic result by Hardy-Littlewood-Sobolev states that if 1 < p < q < ∞, then I α,σ is bounded from Spanne, 1966) and Adams (1975) studied boundedness of the Riesz potential I α for 0 < α < n in Morrey spaces L p,λ .Later on Chiarenza and Frasca (1987) was reproved boundedness of the Riesz potential I α in these spaces.By more general results of Guliyev (1994) (see also 1999 & 2009) one can obtain the following generalization of the results in Adams (1975), Chiarenza and Frasca (1987) and Spanne (1966) to the anisotropic case.
|σ| is necessary and sufficient for the boundedness of the operators M α,σ and I |σ| is necessary and sufficient for the boundedness of the operators M α,σ and I It is known that the anisotropic maximal operator M σ is also bounded from L p,b,σ to L p,b,σ for all 1 < p < ∞ and 0 < b < 1 (see, for example Guliyev, 1994Guliyev, & 1999)), which isotropic case proved by Chiarenza and Frasca (1987).
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.

Notations
Everywhere in the sequel the functions ϕ(x, r), ϕ 1 (x, r) and ϕ 2 (x, r) used in the body of the paper, are non-negative measurable function on R n × (0, ∞).
We find it convenient to define the generalized Morrey spaces in the form as follows.
In Nakai (2006) the following statements were proved.
whenever r ≤ τ ≤ 2r, where c does not depend on r, τ and Then for p > 1 the operator M α,σ is bounded from M p,ϕ,σ to M q,ϕ,σ and for p = 1 M α,σ is bounded from M 1,ϕ,σ to W M q,ϕ,σ .
where C does not depend on x and τ.Then the operator M α,σ is bounded from M p,ϕ 1 ,σ to M q,ϕ 2 ,σ for p > 1 and from M 1,ϕ 1 ,σ to W M q,ϕ 2 ,σ for p = 1.
The following lemma is true.
By the continuity of the operator M α,σ : Let y be an arbitrary point from On the other hand, Therefore, for all y ∈ E σ we have By the continuity of the operator M α,σ : Then by (3.3) we get the inequality (3.2).
Applying Hölder's inequality, we get On the other hand, Since by Lemma 3.1 we arrive at (3.4).
Proof.By Lemma we get In the case α = 0 and p = q from Theorem 3.3 we get the following corollary, which proven in Akbulut, Guliyev and Mustafayev (2012) on R n .

Adams-Guliyev Type Result
The following is a result of Adams-Guliyev type for the fractional maximal operator.
p and let ϕ(x, t) satisfy the condition and where C does not depend on x ∈ R n and r > 0. Then the operator M α,σ is bounded from M p,ϕ For M α,σ f 2 (y) for all y ∈ E σ from (3.3) we have Then from conditions (3.9) and (3.10) for all y ∈ E σ we get .
q−p αq for every y ∈ E σ , we have .
Hence the statement of the theorem follows in view of the boundedness of the anisotropic maximal operator provided by Corollary 3.1 in virtue of condition (3.8).
In the case ϕ(x, t) = t (b−1) |σ| p , 0 < b < 1 from Theorem we get the following Adams type result for the fractional maximal operator.
Then for p > 1, the operator M α,σ is bounded from L p,b,σ to L q,b,σ and for p = 1, M α,σ is bounded from L 1,b,σ to WL q,b,σ .

Parabolic Schrödinger Type Operators
In this section we consider the parabolic Schrödinger operator 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 parabolic generalized Morrey M p,ϕ 1 ,σ 0 (R n+1 ) → M p,ϕ 2 ,σ 0 (R n+1 ) estimates for the operators , where σ 0 = (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.Fefferman, 1983;Shen, 1995;Zhong, 1993).Shen (1995) studied the Schrödinger operator −Δ + V, assuming 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 ( 2 and ∇(−Δ + V) −1 .Kurata and Sugano generalized Shens results to uniformly elliptic operators in Kurata and Sugano (2000).Sugano (1998) also extended some results of Shen to the operator V . Following Shen's approach, Gao and Jiang extend the results to the parabolic case.In Gao and Jiang (2005), 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( The main purpose of this section is investigate the parabolic generalized Morrey M p,ϕ 1 ,σ 0 (R n+1 ) → M p,ϕ 2 ,σ 0 (R n+1 ) boundedness of the operators Gao and Jiang (2005) is the special case of T 2 .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 Schrödinger operator on R n+1 in Gao and Jiang (2005).And we prove the parabolic generalized Morrey estimates by using M p,ϕ 1 ,σ 0 (R n+1 ) → M p,ϕ 2 ,σ 0 (R n+1 ) boundedness of the parabolic fractional maximal operators.
The following two pointwise estimates for T 1 and T 2 which proven in Zhong (1993), Lemma 3.2 with the potential V ∈ B ∞ (R n+1 ).