rning the boundedness for fractional maximal and potential operators defined on quasi-metric measure spaces from $L^{p),θ}(X,μ)$ to $L^{q),qθ/p}(X,ν)$ (trace inequality), where 1 < p < q < ∞, θ > 0 and μ satisfies the doubling condition in X. The results are new even for Euclidean spaces. For example, from our general results D. Adams-type necessary and sufficient conditions guaranteeing the trace inequality for fractional maximal functions and potentials defined on so-called s-sets in ℝⁿ follow. Trace inequalities for one-sided potentials, strong fractional maximal functions and potentials with product kernels, fractional maximal functions and potentials defined on the half-space are also proved in terms of Adams-type criteria. Finally, we remark that a Fefferman-Stein-type inequality for Hardy-Littlewood maximal functions and Calderón-Zygmund singular integrals holds in grand Lebesgue spaces.
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We investigate weak type estimates for maximal functions, fractional and singular integrals in grand Lebesgue spaces. In particular, we show that for the one-weight weak type inequality it is necessary and sufficient that a weight function belongs to the appropriate Muckenhoupt class. The same problem is discussed for strong maximal functions, potentials and singular integrals with product kernels.
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Necessary and sufficient conditions governing two-weight $L^p$ norm estimates for multiple Hardy and potential operators are presented. Two-weight inequalities for potentials defined on nonhomogeneous spaces are also discussed. Sketches of the proofs for most of the results are given.
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In the setting of spaces of homogeneous type, it is shown that the commutator of Calderón-Zygmund type operators as well as the commutator of a potential operator with a BMO function are bounded in a generalized grand Morrey space. Interior estimates for solutions of elliptic equations are also given in the framework of generalized grand Morrey spaces.
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