Boundedness of rough fractional multilinear integral operators on generalized Morrey spaces (original) (raw)

Generalized Local Morrey Spaces and Fractional Integral Operators with Rough Kernel

Journal of Mathematical Sciences, 2013

Let M Ω,α and I Ω,α be the fractional maximal and integral operators with rough kernels, where 0 < α < n. In this paper, we shall study the continuity properties of M Ω,α and I Ω,α on the generalized local Morrey spaces LM {x 0 } p,ϕ. The boundedness of their commutators with local Campanato functions is also obtained.

Boundedness of fractional integral operator with rough kernel on generalized Morrey space

Let Ω be a homogeneous function with zero degree on R n. Fractional integral operator with rough kernel T Ω,α maps function f to convolution between f and kernel function K Ω,α = Ω(x)|x| α−n where 0 < α < n. In this paper, we show Spanne type result and Adam type result for T Ω,α on generalized Morrey spaces. B(z,r) |Ω(z − y)|| f 2 (y)|dy ≤ c sup r>0 B(z,r)∩B c (x,2t) r −n |Ω(z − y)|| f (y)|dy ≤ c sup r>t B(z,r)∩B c (x,2t) |y − z| −n |Ω(z − y)|| f (y)|dy ≤ c sup r>t B(z,r)∩B c (x,2t) |x − y| −n |Ω(z − y)|| f (y)|dy ≤ c B c (x,2t) |Ω(z − y)||x − y| −n | f (y)|dy. By Minkowski inequality, we obtain M Ω f 2 L p (B(x,t)) ≤ c t n/p Ω L p (S n−1) B c (x,2t) |x − y| −n f (y)dy. Note that, inequation (1) also holds for α = 0. Therefore, we have M Ω f 2 L p (B(x,t)) ≤ c t n/p Ω L p (S n−1) ∞ t s −1−(n/p) f L p (B(x,s)) ds.

Vector-Valued Inequality of Fractional Integral Operator with Rough Kernel on Morrey-Adams Spaces

Journal of the Indonesian Mathematical Society, 2022

In 2019, Salim et al proved the vector-valued inequality for maximal operator with rough kernel on Lebesgue spaces and Morrey spaces. This results extend Fefferman-Stein inequality (1971). In 1970's, Adams introduced another variant of Morrey spaces, which called as Morrey-Adams spaces. In this article, we prove vector-valued inequality for maximal operator and fractional integral operator with rough kernel on Morrey-Adams spaces.

Characterizations for the generalized fractional integral operators on Morrey spaces

Mathematical Inequalities & Applications, 1998

We present some characterizations for the boundedness of the generalized fractional integral operators on Morrey spaces. The characterizations follow from two key estimates, one for the norm of some functions in Morrey spaces, and another for the values of the corresponding fractional integrals. We prove three theorems about necessary and sufficient conditions. We show that these theorems are independent by giving some examples. We also obtain counterparts for the weak generalized Morrey spaces.