We study the boundedness of the Hardy–Littlewood maximal operator in abstract extrapolation Banach function lattices and their Köthe dual spaces. The extrapolation spaces are generated by compatible families of Banach function lattices on quasi-metric measure spaces with doubling measure. These results combined with a variant of the integral Coifman–Fefferman inequality imply that every Calderón–Zygmund singular operator is bounded in considered extrapolation spaces. We apply these results to extrapolation spaces determined by compatible families of Calderón–Lozanovskii spaces, in particular to compatible families of Orlicz spaces that are interpolation of weighted Lp-spaces (1 < p < ∞) with Ap weights defined on spaces of homogeneous type.

The boundedness of the Hardy–Littlewood maximal operator, and the weighted extrapolation in grand variable exponent Lebesgue spaces are established provided that Hardy–Littlewood maximal operator is bounded in appropriate variable exponent Lebesgue space. Moreover, we give some bounds of the norm of the Hardy–Littlewood maximal operator in these spaces. As corollaries, we have appropriate norm inequalities and the boundedness of operators of Harmonic Analysis such as maximal and sharp maximal functions; Calderón–Zygmund singular integrals, commutators of singular integrals in grand variable exponent Lebesgue spaces. Finally, applying the boundedness results of integral operators of Harmonic Analysis, we have the direct and inverse theorems on the approximation of 2π-periodic functions by trigonometric polynomials in the framework of grand variable exponent Lebesgue spaces.

Rubio de Francia's extrapolation theorem for new weighted grand Morrey spaces $\mathcal{M}_{w}^{p),\lambda,\theta}(X)$ with weights $w$ beyond the Muckenhoupt $A_p$ classes is established. This result, in particular, implies the one-weight inequality for operators of Harmonic Analysis in these spaces for appropriate weights. Necessary conditions for the boundedness of the Hardy–Littlewood maximal operator and the Hilbert transform in these spaces are also obtained. Some structural properties of new weighted grand Morrey spaces are also investigated. Problems are studied in the case when operators and spaces are defined on spaces of homogeneous type.

In this paper we show the validity of Stein’s interpolation theorem on variable exponent Morrey spaces.

We investigate the multilinear variants of the quantities which measure the noncompactness of multilinear operators taking values in Banach spaces with the uniform approximation property. We show applications to multilinear variant of the Hilbert and Riesz transform on rearrangement invariant spaces. We derive lower estimates of the essential norm of these transforms. Along the way we obtain also as a by-product the lower estimates of the measure noncompactness of these transforms. As a consequence we conclude that the Hilbert and Riesz transforms are not compact multilinear transforms.

We establish a sharp Olsen type inequality $ \big \| g {\mathcal {I}}_{\alpha }(f_{1}, {\dots } , f_{m}) \big \|_{{L^{q}_{r}} } \leq C \big \| g \big \|_{L^{q}_{\ell } } \prod\limits_{j=1}^{m} \big \| f_{j}\big \|_{L^{p_{j}}_{s_{j}}} $ for multilinear fractional integrals $$\mathcal{I}_{\alpha}(\overrightarrow{f})(x)=\int\limits_{(\mathbb{R}^{n})^{m}}\frac{f_1(y_1)⋯f_m(y_m)}{(|x−y_1|+⋯+|x−y_m|)^{mn-\alpha}}d}(\overrightarrow{y}, x∈R^n,$$ $0 < \alpha < mn$, where $L_r^q, L_l^q, L_{s_j}^{p_j}, j = 1,…,m,$ are Morrey space with indices satisfying certain homogeneity conditions. This inequality is sharp because it gives necessary and sufficient condition on a weight function $V$ for which the inequality $ \big \|{\mathcal {I}}_{\alpha }(f_{1}, {\dots } , f_{m}) \big \|_{{L^{q}_{r}}(V) } \leq C \prod\limits_{j=1}^{m} \big \| f_{j}\big \|_{L^{p_{j}}_{s_{j}}} $ holds. Morrey spaces play an important role in relation to regularity problems of solutions of partial differential equations. They describe the integrability more precisely than Lebesgue spaces. We also derive a characterization of the trace inequality $\big \| B_{\alpha } (f_{1},f_{2})\big \|_{{L^{q}_{r}}(d\mu ) } \leq C \prod\limits_{j=1}^{2} \big \| f_{j}\big \|_{L^{p_{j}}_{s_{j}} ({\Bbb {R}}^{n}) }, $ in terms of a Borel measure μ, where Bα is the bilinear fractional integral operator given by the formula $Bα(f_1,f_2)(x)=\int\limits_{R^n}\frac{f_1(x+t)f_2(x−t)}{|t|^{n−α}}dt,0<α

Rubio de Francia’s extrapolation in generalized grand Morrey spaces is derived. This result is applied to the investigation of the regularity of solutions for the second order partial differential equations with discontinuous coefficients in the framework of generalized grand Morrey spaces under the Muckenhoupt condition on weights. Density properties for these spaces are also investigated.

Necessary and sufficient condition governing the boundedness of the multilinear fractional integral operator $T_γ,μ$ defined with respect to a measure μ on a σ-algebra of Borel sets of quasi-metric space X from the product $L_{p}^{1}(X,μ)×⋯×L_{p}^{m}(X,μ)$ to $Lq(X,μ)$ is established. The related weak type inequality is also obtained. The derived results are used to get appropriate boundedness of $T_γ,μ$ in Morrey spaces defined with respect to a measure $μ$.

Weighted extrapolation for pairs of functions in mixed-norm Banach function spaces defined on the product of quasi-metric measure spaces (X,d,μ) and (Y,ρ,ν) are derived. As special cases, we have appropriate results for mixed-norm Lebesgue, Lorentz, and Orlicz spaces. Some of the derived results are applied to get weighted extrapolation in mixed-norm grand Lebesgue spaces.

D. Adams type trace inequalities for multiple fractional integral operators in grand Lebesgue spaces with mixed norms are established. Operators under consideration contain multiple fractional integrals defined on the product of quasi-metric measure spaces, and one-sided multiple potentials. In the case when we deal with operators defined on bounded sets, the established conditions are simultaneously necessary and sufficient for appropriate trace inequalities. The derived results are new even for multiple Riesz potential operators defined on the product of Euclidean spaces.