Characterizations for the potential operators on Carleson curves in local generalized Morrey spaces

Abstract In this paper, we give a boundedness criterion for the potential operator ℐ α { {\mathcal I} }^{\alpha } in the local generalized Morrey space L M p , φ { t 0 } ( Γ ) L{M}_{p,\varphi }^{\{{t}_{0}\}}(\text{Γ}) and the generalized Morrey space M p , φ ( Γ ) {M}_{p,\varphi }(\text{Γ}) defined on Carleson curves Γ \text{Γ} , respectively. For the operator ℐ α { {\mathcal I} }^{\alpha } , we establish necessary and sufficient conditions for the strong and weak Spanne-type boundedness on L M p , φ { t 0 } ( Γ ) L{M}_{p,\varphi }^{\{{t}_{0}\}}(\text{Γ}) and the strong and weak Adams-type boundedness on M p , φ ( Γ ) {M}_{p,\varphi }(\text{Γ}) .

The main purpose of this paper is to establish the boundedness of potential operator α , < < α 0 1 in local generalized Morrey spaces ( ) x , 0 defined on Carleson curves Γ. We shall give characterizations for the strong and weak Spanne-type boundedness of the operator α from ( ) . Also, we study Adams-type boundedness of the operator α from generalized Morrey spaces , and from the space . We shall give characterizations for the Adams-

Preliminaries
Morrey spaces were introduced by C. B. Morrey [15] in 1938 in connection with certain problems in elliptic partial differential equations and calculus of variations. Later, Morrey spaces found important applications to Navier-Stokes and Schrödinger equations, elliptic problems with discontinuous coefficients, and potential theory.
the Morrey space, and by ( ) the modified Morrey space, the set of locally integrable functions f on Γ with the finite norms respectively.
Note that (see [16,17]) where Θ is the set of all functions equivalent to 0 on Γ. We denote by ( ) WL Γ  : Kokilashvili and Meskhi [18] studied the boundedness of the operator α defined on quasimetric measure spaces, in particular on Carleson curves in Morrey spaces and proved the following: . Then the operator α is bounded from the spaces The following Adams boundedness (see [19]) of the operator α in Morrey space defined on Carleson curves was proved in [20].
(1) If < < is sufficient and in the case of infinite curve also necessary for the boundedness of the operator α from is sufficient and in the case of infinite curve also necessary for the boundedness of the operator α from The following Adams boundedness of the operator α in modified Morrey space defined on Carleson curves was proved in [16], see also [17].
is sufficient and in the case of infinite curve also necessary for the boundedness of the operator α from is sufficient and in the case of infinite curve also necessary for We use the following statement on the boundedness of the weighted Hardy operator: The following theorem was proved in [21].
be bounded outside a neighborhood of the origin. The inequality holds for some > C 0 for all non-negative and non-decreasing g on ( ∞) 0, if and only if

Local generalized Morrey spaces
We find it convenient to define the local generalized Morrey spaces in the form as follows, see [21,22].
Also, the weak generalized Morrey space It is natural, first the set of all, to find conditions ensuring that the spaces ( ) are nontrivial, that is, consist not only of functions equivalent to 0 on Γ.
Remark 3.1. We denote by Ω p,loc the set of all positive measurable functions φ on × ( ∞) Γ 0, such that for all > r 0, In what follows, keeping in mind Lemma 1, for the non-triviality of the space Remark 3.2. We denote by Ω p the sets of all positive measurable functions φ on × ( ∞) Γ 0, such that for all > r 0, and sup , , respectively. In what follows, keeping in mind Lemma 2, we always assume that ∈ φ Ω p .
Let ≤ < ∞ p 1 . Denote by p the set of all almost decreasing functions is almost increasing. Seemingly, the requirement ∈ φ p is superfluous but it turns out that this condition is natural. Indeed, Nakai established that there exists a function ρ such that ρ itself is decreasing, that ( ) .
By elementary calculations we have the following, which shows particularly that the spaces are not trivial, see, for example, [23][24][25]. Let u be a continuous and non-negative function on ( ∞) 0, . We define the supremal operator S¯u on M ∈ ( ∞) g 0, by The following theorem was proved in [26].
Potential operators on Carleson curves in local generalized Morrey spaces  1321 for any > t 0 and let u be a non-negative continuous function on ( ∞) 0, . Then the operator S¯u is bounded from on the cone if and only if The following Guliyev-type local estimate for the maximal operator is true, see for example, [27,28]. and ∈ t Γ 0 . Then for > p 1 and any > r 0 the inequality Moreover, for = p 1 the inequality By the continuity of the operator Let y be an arbitrary point from On the other hand, ( ) ∩ ( ( )) ⊂ ( ) Thus, from Theorem A we have Then by (4.8) we get inequality (4.7). □ Lemma 4.5. Let Γ be a Carleson curve, ≤ < ∞ p 1 and ∈ t Γ 0 . Then for > p 1 and any > r 0 in Γ, the inequality holds for all ∈ ( ) f L Γ p loc . Moreover, for = p 1 the inequality Applying Hölder's inequality, we get On the other hand, we arrive at (4.9). Let = p 1. The inequality (4.10) directly follows from (4.7). □ The following theorem is valid. where C does not depend on r. Then for > p 1 the operator is bounded from Proof. By Theorem 4.2 and Lemma 4.5, we get

Spanne-type results
The following local estimate is true, see for example, [28].
, by the boundedness of the operator α from ( ) where the constant C is independent of f. Observe that the conditions ∈ ( Then for all ∈ ( ) z t r Γ , f z t y f y ν y 3 2 d . and for all ∈ (