The boundedness of generalized Bessel-Riesz operators on generalized Morrey spaces

Abstract

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.

Norm estimates for Bessel-Riesz operators on generalized Morrey spaces

“…As a counterpart of the results in [10,11], we have the following theorem on the boundedness of I α,γ on Morrey spaces. Note particularly that the estimate holds for p 1 = 1.…”

“…In [10], it is also shown that I α,γ is bounded on generalized Morrey spaces but without a good estimate for its norm as on Morrey spaces. We shall now refine the results, by estimating the norms of the operators more carefully through the membership of K α in Morrey spaces.…”

“…where K α,γ (x) := |x| α−n (1+|x|) γ , is called Bessel-Riesz operator, and the kernel K α,γ is called Bessel-Riesz kernel. The boundedness of these operators on Morrey spaces and on generalized Morrey spaces was studied in [10,11]. Let 1 ≤ p < ∞ and φ : R + → R + be of class G p , that is φ is almost decreasing [∃ C > 0 such that φ(r) ≥ Cφ(s) for r ≤ s] and φ p (r)r n is almost increasing [∃ C > 0 such that φ p (r)r n ≤ Cφ p (s)s n for r ≤ s].…”

“…In [11], we know that for γ > 0, K α,γ is a member of L t (R n ) spaces for some values of t depending on α and γ. It follows from Young's inequality [4] that [10], it is also shown that I α,γ is bounded on generalized Morrey spaces but without a good estimate for its norm as on Morrey spaces. We shall now refine the results, by estimating the norms of the operators more carefully through the membership of K α in Morrey spaces.…”