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Auxiliary Results
Published in A.G. Ramm, A.I. Katsevich, The RADON TRANSFORM and LOCAL TOMOGRAPHY, 2020
Each locally integrable function f defines a distribution, i.e., a functional on C0∞(ℝn), such that the value of this functional on a test function ϕ∈C0∞(ℝn) is given by () <f,ϕ>=∫ℝnf(x)ϕ(x)¯dx.
Maximal and Calderón–Zygmund operators on the local variable Morrey–Lorentz spaces and some applications
Published in Applicable Analysis, 2023
A. Kucukaslan, V. S. Guliyev, C. Aykol, A. Serbetci
Let f be a locally integrable function on . Hardy–Littlewood maximal function Mf is defined by Maximal operators play an important role in the differentiability properties of functions, singular integrals and partial differential equations. They often provide a deeper and more simplified approach to understanding problems in these areas. For the operator M, the rearrangement inequality holds, where c and C are independent of f. Here denotes the right continuous non-increasing rearrangement of f: and is the distribution function of the function f.
Local well-posedness and global existence for a multi-component Novikov equation
Published in Applicable Analysis, 2021
Zhigang Li, Yuxi Hu, Xinglong Wu
Let ρ be a measurable function from to γ is a locally integrable function from to and μ is a continuous and nondecreasing function from to . Assume for some nonnegative real number c, the function ρ satisfies If c is positive, then we have for a.e. If c = 0 and μ satisfies then a.e.
On the Cauchy problem for a generalized Degasperis-Procesi equation
Published in Applicable Analysis, 2020
Let ρ be a measurable function from to γ a locally integrable function from to and μ a continuous and nondecreasing function from to . Assume that for some the function ρ satisfies If c>0, then for a.e. If and μ satisfies then a.e.