### Weighted estimates of a measure of noncompactness for maximal and potential operators.

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Two-sided estimates of Schatten-von Neumann norms for weighted Volterra integral operators are established. Analogous problems for some potential-type operators defined on R are solved.

We derive lower two-weight estimates for the essential norm (measure of noncompactness) for multilinear Hilbert and Riesz transforms, and Riesz potential operators in Banach function lattices. As a corollary we have appropriate results in weighted Lebesgue spaces. From these statements we conclude that there is no $(m+1)$-tuple of weights $(v,{w}_{1},\cdots ,{w}_{m})$ for which these operators are compact from ${L}_{{w}_{1}}^{{p}_{1}}\times \cdots \times {L}_{{w}_{m}}^{{p}_{m}}$ to ${L}_{v}^{q}$.

rning the boundedness for fractional maximal and potential operators defined on quasi-metric measure spaces from ${L}^{p),\theta}(X,\mu )$ to ${L}^{q),q\theta /p}(X,\nu )$ (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...

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.

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.

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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