Fractional integrals and hypersingular integrals in variable order Holder spaces on homogeneous spaces

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Fractional integrals and hypersingular integrals in variable order Holder spaces on homogeneous spaces N Samko

We consider non-standard H"older spaces Hlb(⋅)(X)Hlb(⋅)(X) of functions ff on a metric measure space (X,d,μ)(X,d,μ), whose H"older exponent lb(x)lb(x) is variable, depending on x∈Xx∈X. We establish theorems on mapping properties of potential operators of variable order al(x)al(x), from such a variable exponent H"older space with the exponent lb(x)lb(x) to another one with a better'' exponent lb(x)+al(x)lb(x)+al(x), and similar mapping properties of hypersingular integrals of variable order al(x)al(x) from such a space into the space with the worse'' exponent lb(x)−al(x)lb(x)−al(x) in the case al(x)<lb(x)al(x)<lb(x). These theorems are derived from the Zygmund type estimates of the local continuity modulus of potential and hypersingular operators via such modulus of their ensities. These estimates allow us to treat not only the case of the spaces Hlb(⋅)(X)Hlb(⋅)(X), but also the generalized H"older spaces Hw(⋅,⋅)(X)Hw(⋅,⋅)(X) of functions whose continuity modulus is dominated by a given function w(x,h),x∈X,h>0w(x,h),x∈X,h>0. We admit variable complex valued orders al(x)al(x), where Ral(x)ℜal(x) may vanish at a set of measure zero. To cover this case, we consider the action of potential operators to weighted generalized H"older spaces with the weight al(x)al(x).

DOI: https://armjmath.sci.am/index.php/ajm/article/view/52 Armenian Journal of Mathematics, 2(2) 38-64

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Keywords

potentials, hypersingular integrals, Matuszewska–Orlicz indices of almost mono-tonic functions, continuity modulus, H ̈older spaces, metric measure space, homogeneousspace growth condition

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