Fractional and hypersingular operators in variable exponent spaces on metric measure spaces
The authors study fractional integrals of variable order in variable exponent Lebesgue spaces on doubling and non-doubling spaces. Let \((X,d,\mu)\) be a (quasi-) metric measure space and \(\Omega\) be an open set in \(X\). The authors study two variants of fractional integral that are defined by \[ \begin{aligned} I^{\alpha(\cdot)}_m f(x) & = \int_\Omega {f(y)d\mu(y)\over d(x,y)^{m-\alpha(x)}},\quad m> 0,\\ J^{\alpha(\cdot)} f(x) & = \int_\Omega {d(x,y)^{\alpha(x)}\over\mu B(x,d(x, y))} f(y) \,d\mu(y),\end{aligned} \] where \(B(x,r)\) is the ball centered at \(x\) with radius \(r\). For the above defined operators, the authors establish certain Sobolev-type theorems on Lebesgue spaces with variable exponents. More precisely, they obtain the boundedness of operators from \(L^{p(\cdot)}(\Omega)\) to \(L^{q(\cdot)}(\Omega)\) with \(p(\cdot)\) and \(q(\cdot): X\to [0,\infty)\) satisfying a certain relationship.
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