Riesz potential in the local Morrey–Lorentz spaces and some applications (original) (raw)
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Journal of Interdisciplinary Mathematics
The aim of this paper is to obtain boundedness conditions for the maximal function M f and to prove the necessary and sufficient conditions for the fractional maximal oparator M α in the Lorentz-Morrey spaces L p,q;λ (R n) which are a new class of functions. We get our main results by using the obtained sharp rearrangement estimates. The obtained results are applied to the boundedness of particular operators such as the Bochner-Riesz operator B δ r and the Schrödingertype operators V γ (−∆ + V) −β and V γ ∇(−∆ + V) −β in the Lorentz-Morrey spaces L p,q;λ (R n), where the nonnegative potential V belongs to the reverse Hölder class B ∞ (R n).
Journal of Mathematical Inequalities, 2011
We prove that the fractional maximal operator M α and the Riesz potential operator I α , 0 < α < n are bounded from the modified Morrey space L 1,λ (R n) to the weak modified Morrey space W L q,λ (R n) if and only if, α/n 1 − 1/q α/(n − λ) and from L p,λ (R n) to L q,λ (R n) if and only if, α/n 1/p − 1/q α/(n − λ). As applications, we establish the boundedness of some Schödinger type operators on modified Morrey spaces related to certain nonnegative potentials belonging to the reverse Hölder class. As an another application, we prove the boundedness of various operators on modified Morrey spaces which are estimated by Riesz potentials.
Boundedness of the maximal operator in the local Morrey-Lorentz spaces
Journal of Inequalities and Applications, 2013
In this paper we define a new class of functions called local Morrey-Lorentz spaces M loc p,q;λ (R n), 0 < p, q ≤ ∞ and 0 ≤ λ ≤ 1. These spaces generalize Lorentz spaces such that M loc p,q;0 (R n) = L p,q (R n). We show that in the case λ < 0 or λ > 1, the space M loc p,q;λ (R n) is trivial, and in the limiting case λ = 1, the space M loc p,q;1 (R n) is the classical Lorentz space ∞,t 1 p-1 q (R n). We show that for 0 < q ≤ p < ∞ and 0 < λ ≤ q p , the local Morrey-Lorentz spaces M loc p,q;λ (R n) are equal to weak Lebesgue spaces WL 1 p-λ q (R n). We get an embedding between local Morrey-Lorentz spaces and Lorentz-Morrey spaces. Furthermore, we obtain the boundedness of the maximal operator in the local Morrey-Lorentz spaces.
Applicable Analysis, 2021
In this paper, we give the definition of local variable Morrey-Lorentz spaces M loc p(•),q(•),λ (R n) which are a new class of functions. Also, we prove the boundedness of the Hardy-Littlewood maximal operator M and Calderón-Zygmund operators T on these spaces including the class of sublinear operators T 0 generated by Calderón-Zygmund operators. Finally, we apply these results to the Bochner-Riesz operator B δ r , identity approximation A ε and the Marcinkiewicz operator µ Ω on the spaces M loc p(•),q(•),λ (R n).
BOUNDEDNESS OF THE RIESZ POTENTIAL IN LOCAL MORREY-TYPE SPACES
Potential analysis 35 (2011), no. 1, 67-87., 2011
The problem of boundedness of the Riesz potential in local Morrey-type spaces is reduced to the problem of boundedness of the Hardy operator in weighted LpL_pLp-spaces on the cone of non-negative non-increasing functions. This allows obtaining sharp sufficient conditions for boundedness for all admissible values of the parameters, which, for a certain range of the parameters wider than known before, coincide with the necessary ones. Abstract. The problem of boundedness of the Riesz potential in local Morrey-type spaces is reduced to the problem of boundedness of the Hardy operator in weighted-spaces on the cone of non-negative non-increasing functions. This allows obtaining sharp sufficient conditions for boundedness for all admissible values of the parameters, which, for a certain range of the parameters wider than known before, coincide with the necessary ones.
Journal of Computational and Applied Mathematics, 2007
The problem of the boundedness of the fractional maximal operator M , 0 < < n, in local and global Morrey-type spaces is reduced to the problem of the boundedness of the Hardy operator in weighted L p-spaces on the cone of non-negative non-increasing functions. This allows obtaining sharp sufficient conditions for the boundedness for all admissible values of the parameters. Moreover, in case of local Morrey-type spaces, for some values of the parameters, these sufficient conditions coincide with the necessary ones.
Burenkov V. I., Guliyev H. V., Guliyev V. S. Necessary and su- cient conditions for boundedness of the fractional maximal operator in the local Morrey-type spaces. J. Comput. Appl. Math. 208, no. 1 (2007), 280-301., 2007
The problem of boundedness of the fractional maximal operator M α , 0 < α < n in local and global Morrey-type spaces is reduced to the problem of boundedness of the Hardy operator in weighted L p-spaces on the cone of non-negative non-increasing functions. This allows obtaining sharp sufficient conditions for boundedness for all admissible values of the parameters. Moreover, in case of local Morrey-type spaces, for some values of the parameters, these sufficient conditions coincide with the necessary ones. Key Words: maximal operator, fractional maximal operator, local and global Morrey-type spaces, weak Morrey-type spaces, Hardy operator on the cone of monotonic functions.