Riesz potential in generalized Morrey spaces on the Heisenberg group (original) (raw)
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Journal of Mathematical Sciences, 2013
We consider the Riesz potential operator I α , on the Heisenberg group H n in generalized Morrey spaces M p,ϕ (H n) and find conditions for the boundedness of I α as an operator from M p,ϕ 1 (H n) to M p,ϕ 2 (H n), 1 < p < ∞, and from M 1,ϕ 1 (H n) to a weak Morrey space W M 1,ϕ 2 (H n). The boundedness conditions are formulated it terms of Zygmund type integral inequalities. Based on the properties of the fundamental solution of the sub-Laplacian on H n , we prove two Sobolev-Stein embedding theorems for generalized Morrey and Besov-Morrey spaces. Bibliography: 40 titles.
Riesz potential on the Heisenberg group and modified Morrey spaces
Analele Universitatii "Ovidius" Constanta - Seria Matematica, 2012
In this paper we study the fractional maximal operator Mα, 0 ≤ α < Q and the Riesz potential operator Iα, 0 < α < Q on the Heisenberg group in the modified Morrey spaces L p,λ (Hn), where Q = 2n + 2 is the homogeneous dimension on Hn. We prove that the operators Mα and Iα are bounded from the modified Morrey space L 1,λ (Hn) to the weak modified Morrey space W L q,λ (Hn) if and only if, α/Q ≤ 1 − 1/q ≤ α/(Q − λ) and from L p,λ (Hn) to L q,λ (Hn) if and only if, α/Q ≤ 1/p − 1/q ≤ α/(Q − λ). In the limiting case Q−λ α ≤ p ≤ Q α we prove that the operator Mα is bounded from L p,λ (Hn) to L∞(Hn) and the modified fractional integral operator Iα is bounded from L p,λ (Hn) to BM O(Hn). As applications of the properties of the fundamental solution of sub-Laplacian L on Hn, we prove two Sobolev-Stein embedding theorems on modified Morrey and Besov-modified Morrey spaces in the Heisenberg group setting. As an another application, we prove the boundedness of Iα from the Besov-modified Morrey spaces B L s pθ,λ (Hn) to B L s qθ,λ (Hn).
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 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.
On some subspaces of Morrey–Sobolev spaces and boundedness of Riesz integrals
Annales Polonici Mathematici, 2013
For 1 ≤ q ≤ α ≤ p ≤ ∞, (L q , l p) α is a complex Banach space which is continuously included in the Wiener amalgam space (L q , l p) and contains the Lebesgue space L α. We study the closure (L q , l p) α c,0 in (L q , l p) α of the space D of test functions (infinitely differentiable and with compact support in R d) and obtain norm inequalities for Riesz potential operators and Riesz transforms in these spaces. We also introduce the Sobolev type space W 1 ((L q , l p) α) (a subspace of a Morrey-Sobolev space, but a superspace of the classical Sobolev space W 1,α) and obtain in it Sobolev inequalities and a Kondrashov-Rellich compactness theorem.
The boundedness of Bessel-Riesz operators on generalized Morrey spaces
2016
In this paper, we prove the boundedness of Bessel-Riesz operators on generalized Morrey spaces. The proof uses the usual dyadic decomposition, a Hedberg-type inequality for the operators, and the boundedness of Hardy-Littlewood maximal operator. Our results reveal that the norm of the operators is dominated by the norm of the kernels.
The boundedness of generalized Bessel-Riesz operators on generalized Morrey spaces
Journal of Physics: Conference Series, 2017
In this paper, we prove the boundedness of Bessel-Riesz operators on generalized Morrey spaces. The proof uses the usual dyadic decomposition, a Hedberg-type inequality for the operators, and the boundedness of Hardy-Littlewood maximal operator. Our results reveal that the norm of the operators is dominated by the norm of the kernels.
Sufficient Conditions for Boundedness of the Riesz Potential in Local Morrey-Type Spaces
2007
The problem of the boundedness of the Riesz potential I , 0 < < n in local Morrey-type spaces is reduced to the problem of the boundedness of the Hardy operator in weighted Lp-spaces on the cone of non-negative non- increasing functions. This allows obtaining sharp sucient conditions for the boundedness for all admissible values of the parameters.