BOUNDEDNESS OF THE FRACTIONAL MAXIMAL OPERATOR IN GENERALIZED MORREY SPACE ON THE HEISENBERG GROUP

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1 Indian J. Pure Appl. Math., 44(2): , April 203 c Indian National Science Academy BOUNDEDNESS OF THE FRACTIONAL MAXIMAL OPERATOR IN GENERALIZED MORREY SPACE ON THE HEISENBERG GROUP V. S. Guliyev and Yagub Y. Mammadov Ahi Evran University, Department of Mathematics, Kirsehir, Turkey Institute of Mathematics and Mechanics of NAS of Azerbaijan, Baku and Nakhchivan Teacher-Training Institute, Nakhchivan, Azerbaijan s: vagif@guliyev.com, yagubmammadov@yahoo.com (Received 9 October 20; after final revision 9 October 202; accepted 0 October 202) In this paper we study the fractional maximal operator M α, 0 α < Q on the Heisenberg group H n in the generalized Morrey spaces M p,ϕ (H n ), where Q = 2n + 2 is the homogeneous dimension of H n. We find the conditions on the pair (ϕ, ϕ 2 ) which ensures the boundedness of the operator M α from one generalized Morrey space M p,ϕ (H n ) to another M,ϕ2 (H n ), < p < <, /p / = α/q, and from the space M,ϕ (H n ) to the weak space W M,ϕ2 (H n ), < <, / = α/q. We also find conditions on the ϕ which ensure the Adams type boundedness of M α from M p,ϕ p (H n ) to M,ϕ (H n ) for < p < < and from M,ϕ (H n ) to W M,ϕ (H n ) for < <. 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-200-()-40/06-, by the Scientific and Technological Research Council of Turkey (TUBITAK Project No: 0T695) and by the grant of 202-Ahi Evran University Scientific Research Projects (PYO-FEN ).

2 86 V. S. GULIYEV AND YAGUB Y. MAMMADOV As applications we establish the boundedness of some Schrödinger type operators on generalized Morrey spaces related to certain nonnegative potentials V belonging to the reverse Hölder class B (H n ). Key words : Heisenberg group; fractional maximal function; generalized Morrey space; Schrödinger operator.. INTRODUCTION Heisenberg group appear in uantum physics and many parts of mathematics, including Fourier analysis, several complex variables, geometry and topology. Analysis on the groups is also motivated by their role as the simplest and the most important model in the general theory of vector fields satisfying Hormander s condition. In the present paper we will prove the boundedness of the fractional maximal operator on the Heisenberg group in generalized Morrey spaces. For g H n and r > 0, we denote by B(g, r) the open ball centered at g of radius r, and by B(g, r) denote its complement. Let B(g, r) be the Haar measure of the ball B(g, r). Given a function f which is integrable on any ball B(g, r) H n, the fractional maximal function M α f, 0 α < Q of f is defined by M α f(g) = sup B(g, r) + α Q f(h) dh. r>0 B(g,r) The fractional maximal function M α f coincides for α = 0 with the Hardy- Littlewood maximal function Mf M 0 f (see [8, 26]). The operator M α play important role in real and harmonic analysis (see, for example [7, 8, 26]). In this work, we prove the boundedness of the fractional maximal operator M α, 0 α < Q from one generalized Morrey space M p,ϕ (H n ) to M,ϕ2 (H n ), < p < <, /p / = α/q, and from M,ϕ (H n ) to the weak space W M,ϕ2 (H n ), < <, / = α/q. We also prove the Adams type boundedness of the operator M α from M p,ϕ p (H n ) to M,ϕ (H n ) for < p <

3 BOUNDEDNESS OF THE FRACTIONAL MAXIMAL OPERATOR 87 < and from M,ϕ (H n ) to W M,ϕ (H n ) for < <. In all the cases the conditions for the boundedness are given it terms of supremal-type ineualities on (ϕ, ϕ 2 ) and ϕ, which do not assume any assumption on monotonicity of (ϕ, ϕ 2 ) and ϕ in r. Let L = Hn +V be a Schrödinger operator on H n, where Hn is the sub-laplacian. As applications we establish the boundedness of some Schrödinger type operators V γ ( Hn + V ) β and V γ Hn ( Hn + V ) β on generalized Morrey spaces related to certain nonnegative potentials V belonging to the reverse Hölder class B (H n ). By A B we mean that A CB with some positive constant C independent of appropriate uantities. If A B and B A, we write A B and say that A and B are euivalent. 2. NOTATIONS We recall some basic facts on the Heisenberg group, which more detailed information can be found in [8, 9, 26] and the references therein. The (2n+)-dimensional Heisenberg group H n is the Lie group with underlying manifold R 2n R and multiplication (x, t)(y, s) = ( n x + y, t + s + 2 (x n+j y j x j y n+j ) ). The inverse element of g = (x, t) is g = ( x, t) and we write the identity of H n as e = (0, 0). The Heisenberg group is a connected, simply connected nilpotent Lie group. We define one-parameter dilations on H n, for r > 0, by δ r (x, t) = (rx, r 2 t). These dilations are group automorphisms and the Jacobian determinant is r Q, where Q = 2n + 2 is the homogeneous dimension of H n. A homogeneous norm on H n is given by g = (x, t) = ( x 2 + t ) /2. This norm satisfies the triangle ineuality and leads to a left-invariant distant d(g, h) = g h. With this norm, we define the Heisenberg ball centered at g = (x, t) with radius r by B(g, r) = {h H n : g h < r}, and we denote by B(g, r) = H n \ B(g, r) its complement. The volume of the ball B(g, r) is C n r Q, where C n j=

4 88 V. S. GULIYEV AND YAGUB Y. MAMMADOV is the volume of the unit ball B : C n = B(e, ) = 2π n+ 2 Γ ( ) n 2 (n + )Γ(n)Γ ( n+ ). 2 Using coordinates g = (x, t) for points in H n, the left-invariant vector fields X,..., X 2n, X 2n+ on H n eual to x,..., x 2n, t at the origin are given by X j = + 2x n+j x j t, X n+j = 2x j x n+j t, j =,..., n, X 2n+ = t, respectively. These 2n + vector fields form a basis for the Lie algebra of H n with commutation relations [X j, X n+j ] = 4X 2n+ for j =,..., n, and all other commutators eual to 0. The sub-laplacian Hn and the gradient Hn are defined respectively by Hn = 2n j= X 2 j and Hn = (X,..., X 2n ). The sub-laplacian operator (which is hypoelliptic by Hormanderfs theorem [6]) is well known to play the same fundamental role on H n as the ordinary does on R n. In the study of local properties of solutions to of partial differential euations, together with weighted Lebesgue spaces, Morrey spaces L p,λ (H n ) play an important role, see [0, 7]. They were introduced by Morrey in 938 [22]. The Morrey space in a Heisenberg group is defined as follows: for p, 0 λ Q, a function f L p,λ (H n ) if f L loc p (H n ) and f Lp,λ := sup r λ p f Lp(B(g,r)) <, u H n, r>0 (If λ = 0, then L p,0 (H n ) = L p (H n ); if λ = Q, then L p,q (H n ) = L (H n ); if λ < 0 or λ > Q, then L p,λ (H n ) = Θ, where Θ is the set of all functions euivalent to 0 on H n.)

5 BOUNDEDNESS OF THE FRACTIONAL MAXIMAL OPERATOR 89 We also denote by W L p,λ (H n ) the weak Morrey space of all functions f W L loc p (H n ) for which f W Lp,λ = sup r λ p f W Lp(B(g,r)) <, u H n, r>0 where W L p (B(g, r)) denotes the weak L p -space of measurable functions f for which Note that f W Lp (B(g,r)) = sup τ {h B(g, r) : f(h) > τ} /p. τ>0 W L p (H n ) = W L p,0 (H n ), L p,λ (H n ) W L p,λ (H n ) and f W Lp,λ f Lp,λ. Everywhere in the seuel the functions ϕ(g, r), ϕ (g, r) and ϕ 2 (g, r) used in the body of the paper, are non-negative measurable function on H n (0, ). We find it convenient to define the generalized Morrey spaces in the form as follows. Definition 2. Let p <. The generalized Morrey space M p,ϕ (H n ) is defined of all functions f L loc p (H n ) by the finite norm f Mp,ϕ = sup ϕ(g, r) B(g, r) p f Lp(B(g,r)). u H n,r>0 According to this definition, we recover the space L p,λ (H n ) under the choice ϕ(g, r) = r λ Q p : L p,λ (H n ) = M p,ϕ (H n ). λ Q ϕ(g,r)=r p In [, 2, 23, 24] there were obtained sufficient conditions on weights ϕ and ϕ 2 for the boundedness of the maximal operator M and the singular integral operators T from M p,ϕ (H n ) to M p,ϕ2 (H n ). In [23, 24] the following condition was imposed on w(g, r): c ϕ(g, r) ϕ(g, τ) c ϕ(g, r) (2.)

6 90 V. S. GULIYEV AND YAGUB Y. MAMMADOV whenever r τ 2r, where c( ) does not depend on t, r and u H n, jointly with the condition: ϕ(g, τ) p dτ r τ C ϕ(g, r)p. (2.2) for the maximal or singular operators and the condition r τ αp ϕ(g, τ) p dτ τ C rαp ϕ(g, r) p. (2.3) for potential and fractional maximal operators, where C(> 0) does not depend on r and u H n. In [24] the following statements were proved. Theorem 2. Let p <, 0 < α < Q p, = p α Q and ϕ(g, τ) satisfy conditions (2.) and (2.3). Then for p > the operator M α is bounded from M p,ϕ (H n ) to M,ϕ (H n ) and for p = M α is bounded from M,ϕ (H n ) to W M,ϕ (H n ). The following statements, containing results obtained in [24] was proved in [] (see also [2, 3, 4, 5]). Theorem 2.2 Let p <, 0 < α < Q p, = p α Q and (ϕ, ϕ 2 ) satisfy the condition r α ϕ (g, r) dr τ r C ϕ 2(x, τ), (2.4) where C does not depend on g and τ. Then the operator M α is bounded from M p,ϕ (H n ) to M,ϕ2 (H n ) for p > and from M,ϕ (H n ) to W M,ϕ2 (H n ) for p =. 3. BOUNDEDNESS OF THE FRACTIONAL MAXIMAL OPERATOR IN THE 3. Spanne type result SPACES M p,ϕ (H n ) We denote by L,v (0, ) the space of all functions g(t), t > 0 with finite norm g L,v (0, ) = ess sup v(t)g(t) t>0

7 BOUNDEDNESS OF THE FRACTIONAL MAXIMAL OPERATOR 9 and L (0, ) L, (0, ). Let M(0, ) be the set of all Lebesgue-measurable functions on (0, ) and M + (0, ) its subset consisting of all nonnegative functions on (0, ). We denote by M + (0, ; )the cone of all functions in M + (0, ) which are non-decreasing on (0, ) and { } A = ϕ M + (0, ; ) : lim ϕ(t) = 0. t 0+ Let u be a continuous and non-negative function on (0, ). We define the supremal operator S u on g M(0, ) by (S u g)(t) := u g L (t, ), t (0, ). The following theorem was proved in [3]. Theorem 3.3 Let v, v 2 be non-negative measurable functions satisfying 0 < v L (t, ) < for any t > 0 and let u be a continuous non-negative function on (0, ) Then the operator S u is bounded from L,v (0, ) to L,v2 (0, ) on the cone A if and only if v 2 S u ( v L (, )) L (0, ) <. (3.) Sufficient conditions on ϕ for the boundedness of M and M α in generalized Morrey spaces M p,ϕ (H n ) have been obtained in [2, 3, 3, 4, 5, 24]. The following lemma is true. Lemma 3. Let p <, 0 α < Q p, = p α Q. Then for p > and any ball B = B(g, r) in H n the ineuality M α f L (B(g,r)) f Lp (B(g,2r)) + r Q holds for all f L loc p (H n ). Moreover for p = the ineuality M α f W L(B(g,r)) f L (B(g,2r)) + r Q sup τ Q+α f L (B(g,τ)) (3.2) sup τ Q+α f L (B(g,τ)) (3.3)

8 92 V. S. GULIYEV AND YAGUB Y. MAMMADOV holds for all f L loc (H n). PROOF : Let < p < < and p = α Q. For arbitrary ball B = B(g, r) let f = f + f 2, where f = fχ 2B, f 2 = fχ and 2B = B(g, 2r). (2B) M α f L (B) M α f L (B) + M α f 2 L (B). By the continuity of the operator M α : L p (H n ) L (H n ) we have M α f L (B) f Lp (2B). Let h be an arbitrary point from B. If B(h, τ) (2B), then τ > r. Indeed, if w B(h, τ) (2B), then τ > h w g w g h > 2r r = r. On the other hand, B(h, τ) (2B) B(g, 2τ). Indeed, w B(h, τ) (2B), then we get g w h w + g h < τ + r < 2τ. Hence M α f 2 (h) = sup τ>0 B(h, τ) α/q f(w) dw B(h,τ) (2B) 2 Q α sup τ>r B(g, 2τ) α/q f(w) dw B(g,2τ) = 2 Q α sup B(g, τ) α/q f(w) dw. B(g,τ) Therefore, for all h B we have M α f 2 (h) 2 Q α sup B(g, τ) α/q Thus M α f L (B) f Lp (2B) + B ( sup B(g, τ) α/q B(g,τ) B(g,τ) f(w) dw. (3.4) f(w) dw ).

9 BOUNDEDNESS OF THE FRACTIONAL MAXIMAL OPERATOR 93 Let p =. It is obvious that for any ball B = B(g, r) M α f W L(B) M α f W L(B) + M α f 2 W L(B). By the continuity of the operator M α : L (H n ) W L (H n ) we have M α f W L(B) f L (2B). Then by (3.4) we get the ineuality (3.3). Lemma 3.2 Let p <, 0 α < Q p, = p α Q. Then for p > and any ball B = B(g, r) in H n, the ineuality M α f L (B(g,r)) r Q sup τ Q f Lp (B(g,τ)) (3.5) holds for all f L loc p (H n ). Moreover for p = the ineuality M α f W L (B(g,r)) r Q sup τ Q f L (B(g,τ)) (3.6) holds for all f L loc (H n). PROOF : Let < p <, 0 α < Q p, = p α Q. Denote M : = B ( sup M 2 : = f Lp (2B). ) B(g, τ) α/q f(w) dw, B(g,τ) Applying Hölder s ineuality, we get M B sup B(g, τ) ( ) p f(w) p dw. B(g,τ)

10 94 V. S. GULIYEV AND YAGUB Y. MAMMADOV On the other hand, ( ) B p sup f(w) p dw B(g, τ) B(g,τ) ( ) B sup f B(g, τ) Lp (2B) M. Since by Lemma 3. we arrive at (3.5). M α f L(B) M + M 2, Let p =. The ineuality (3.6) directly follows from (3.3). Theorem 3.4 Let p <, 0 α < Q p, = p α Q, and (ϕ, ϕ 2 ) satisfies the condition sup t α Q p r<t< ess inf ϕ (g, s) s Q p C ϕ 2 (g, r), (3.7) t<s< where C does not depend on g and r. Then for p >, M α is bounded from M p,ϕ (H n ) to M,ϕ2 (H n ) and for p =, M α is bounded from M,ϕ (H n ) to W M,ϕ2 (H n ). PROOF : By Theorem 3.3 and Lemma 3.2 we get if p (, ) and M α f M,ϕ2 sup ϕ 2 (g, r) sup τ Q f Lp(B(g,τ)) g H n,r>0 τ>r ϕ (g, r) r Q p f Lp (B(g,r)) = f Mp,ϕ, M α f W M,ϕ2 sup ϕ 2 (g, r) sup τ Q f L (B(g,τ)) g H n,r>0 τ>r sup ϕ (g, r) r Q f L (B(g,r)) = f M,ϕ, g H n,r>0 if p =.

11 BOUNDEDNESS OF THE FRACTIONAL MAXIMAL OPERATOR 95 In the case α = 0 and p = from Theorem 3.4 we get the following corollary, which proven in [2] on R n. Corollary 3. Let p < and (ϕ, ϕ 2 ) satisfies the condition sup t Q p r<t< where C does not depend on g and r. ess inf ϕ (g, s) s Q p C ϕ 2 (g, r), (3.8) t<s< Then for p >, M is bounded from M p,ϕ (H n ) to M p,ϕ2 (H n ) and for p =, M is bounded from M,ϕ (H n ) to W M,ϕ2 (H n ). Corollary 3.2 Let p [, ) and let ϕ : (0, ) (0, ) be an decreasing function. Assume that the mapping r ϕ(r) r Q p is almost increasing (there exists a constant c such that for s < r we have ϕ(s) s Q p cϕ(r) r Q p ). Then there exists a constant C > 0 such that Mf Mp,ϕ C f Mp,ϕ if p >, and Mf W M,ϕ C f M,ϕ. 3.2 Adams type result The following is a result of Adams type for the fractional maximal operator (see []). Theorem 3.5 Let p < <, 0 < α < Q p condition and and let ϕ(g, τ) satisfy the sup t Q ess inf ϕ(g, s) r<t< t<s< sq C ϕ(g, r) (3.9) sup t α ϕ(g, τ) p r<τ< where C does not depend on g H n and r > 0. Cr αp p, (3.0) Then the operator M α is bounded from M p,ϕ p (H n ) to M,ϕ (H n ) for p > and from M,ϕ (H n ) to W M,ϕ (H n ).

12 96 V. S. GULIYEV AND YAGUB Y. MAMMADOV PROOF : Let p < <, 0 < α < Q p and f M p,ϕ p (H n). Write f = f + f 2, where B = B(g, r), f = fχ 2B and f 2 = fχ. (2B) For M α f 2 (h) for all h B from (3.4) we have M α (f 2 )(h) 2 Q α sup B(g, τ) α/q B(g,τ) f(w) dw sup τ Q f Lp (B(g,τ)) (3.) Then from conditions (3.0) and (3.) we get M α f(h) r α Mf(h) + sup t α Q p f Lp (B(g,τ)) r α Mf(h) + f M p,ϕ p r α Mf(h) + r αp p sup τ α ϕ(g, τ) p f M. p,ϕ p Hence choose r = ( f Mp,ϕ /p Mf(h) ) p α for every h B, we have M α f(h) (Mf(h)) p f p M p,ϕ p Hence the statement of the theorem follows in view of the boundedness of the maximal operator M in M p,ϕ p (H n ) provided by Corollary 3. in virtue of condition (3.9). M α f M = sup ϕ(g, τ) τ Q M α f L (B(g,τ)),ϕ g H n, τ>0 f p M = f p M = f p M sup p,ϕ p g H n, τ>0 p,ϕ p p,ϕ p f M, p,ϕ p ( ϕ(g, τ) τ Q p Mf L p(b(g,τ)) sup ϕ(g, τ) p τ Q p Mf Lp(B(g,τ)) g H n, τ>0 p Mf M p,ϕ p. ) p

13 BOUNDEDNESS OF THE FRACTIONAL MAXIMAL OPERATOR 97 if < p < < and M α f W M = sup ϕ(g, τ) t Q M α f W L (B(g,τ)),ϕ g H n, τ>0 f M,ϕ = f M,ϕ sup g H n, τ>0 ( = f M,ϕ Mf f M,ϕ, ϕ(g, τ) τ Q Mf W L (B(g,τ)) sup ϕ(g, τ) τ Q Mf W L (B(g,τ)) g H n, τ>0 W M,ϕ ) if < <. In the case ϕ(g, τ) = τ λ Q, 0 < λ < Q from Theorem 3.5 we get the following Adams type result [] for the fractional maximal operator. Corollary 3.3 Let 0 < α < Q, p < Q α, 0 < λ < Q αp and p = α Q λ. Then for p >, the operator M α is bounded from L p,λ (H n ) to L,λ (H n ) and for p =, M α is bounded from L,λ (H n ) to W L,λ (H n ). 4. THE GENERALIZED MORREY ESTIMATES FOR THE OPERATORS V γ( Hn + V ) β AND V γ Hn ( Hn + V ) β In this section we consider the Schrödinger operator Hn + V on H n, where the nonnegative potential V belongs to the reverse Hölder class B (H n ). The generalized Morrey M p,ϕ (H n ) estimates for the operators V γ ( Hn + V ) β and V γ Hn ( Hn + V ) β are obtained. 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. [5, 25, 29]). Shen [25] studied the Schrödinger operator + V, assuming the nonnegative potential V belongs to the reverse Hölder class B (R n ) for n/2 and he proved the L p boundedness of the operators ( + V ) iγ, 2 ( + V ), ( + V ) 2 and ( + V ). Ku-

14 98 V. S. GULIYEV AND YAGUB Y. MAMMADOV rata and Sugano generalized Shens results to uniformly elliptic operators in [8]. Sugano [27] also extended some results of Shen to the operator V γ ( + V ) β, 0 γ β and V γ Hn ( Hn + V ) β, 0 γ 2 β and β γ 2. Later, Lu [2] and Li [9] investigated the Schrödinger operators in a more general setting. The main purpose of this section is investigate the generalized Morrey M p,ϕ - M,ϕ2 boundedness of the operators T = V γ ( Hn + V ) β, 0 γ β, T 2 = V γ Hn ( Hn + V ) β, 0 γ 2 β, β γ 2. Note that the operators V ( Hn + V ) and V 2 Hn ( Hn + V ) in [9] are the special case of T and T 2, respectively. It is worth pointing out that we need to establish pointwise estimates for T, T 2 and their adjoint operators by using the estimates of fundamental solution for the Schrödinger operator on H n in [9]. And we prove the generalized Morrey estimates by using M p,ϕ M,ϕ2 boundedness of the fractional maximal operators. Let V 0. We say V B (H n ), if there exists a constant C > 0 such that V L (B) C V (g)dg B holds for every ball B in H n (see [9]). By the functional calculus, we may write, for all 0 < β <, ( Hn + V ) β = π 0 B λ β ( Hn + V + λ) dλ. Let f C 0 (H n). From ( Hn + V + λ) f(g) = H n Γ(g, h, λ)f(h)dh, it follows that where K (g, h) = { π T f(g) = K (g, h)v (g) γ f(h)dh, H n 0 λ β Γ(g, h, λ) dλ for 0 < β < Γ(g, h, 0) for β =.

15 BOUNDEDNESS OF THE FRACTIONAL MAXIMAL OPERATOR 99 The following two pointwise estimates for T and T 2 which proven in [29], Lemma 3.2 with the potential V B (H n ). Theorem B Suppose V B (H n ) and 0 γ β. Then there exists a constant C > 0 such that T f(g) CM α f(g), f C0 (H n ), where α = 2(β γ). Theorem C Suppose V B (H n ), 0 γ 2 β and β γ 2. Then there exists a constant C > 0 such that T 2 f(g) CM α f(g), f C0 (H n ), where α = 2(β γ). The above theorems will yield the generalized Morrey estimates for T and T 2. Corollary 4.4 Assume that ( V ) B (H n ), and 0 γ β. Let p <, 2(β γ) = Q p and the condition (3.7) be satisfied for α = 2(β γ). Then for any f C0 (H n) T f M,ϕ2 f Mp,ϕ, for p > and T f W M,ϕ2 f M,ϕ for p = Corollary 4.5 Assume that V B (H n ), 0 ( γ ) 2 β and β γ 2. Let p <, 2(β γ) = Q p and the condition (3.7) be satisfied for α = 2(β γ). Then for any f C0 (H n) T 2 f M,ϕ2 f Mp,ϕ, for p > and T 2 f W M,ϕ2 f M,ϕ for p =

16 200 V. S. GULIYEV AND YAGUB Y. MAMMADOV Corollary 4.6 Assume that V( B )(H n ) and 0 γ β. Let p < <, 2(β γ) = Q p and the conditions (3,9), (3.0) be satisfied for α = 2(β γ). Then for any f C 0 (H n) T f M f M, for p >,ϕ p,ϕ p and T f W M,ϕ f M,ϕ for p =. Corollary 4.7 Assume that V B (H n ), 0 ( γ ) 2 β and β γ 2. Let p < <, 2(β γ) = Q p and the conditions (3.9), (3.0) be satisfied for α = 2(β γ). Then for any f C 0 (H n) T 2 f M f M, for p >,ϕ p,ϕ p and T 2 f W M,ϕ f M,ϕ for p =. ACKNOWLEDGMENT The authors would like to express their gratitude to the referees for his (her) very valuable comments and suggestions. REFERENCES. D. R. Adams, A note on Riesz potentials, Duke Math., 42 (975), Ali Akbulut, V. S. Guliyev and R. Mustafayev, On the Boundedness of the maximal operator and singular integral operators in generalized Morrey spaces, Mathematica Bohemica, 36(4) (20), V. Burenkov, A. Gogatishvili, V. S. Guliyev and R. Mustafayev, Boundedness of the fractional maximal operator in local Morrey-type spaces, Complex Var. Elliptic Eu., 55(8-0) (200),

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