The problem of boundedness of the Hardy-Littewood maximal operator in local and global Morrey-type spaces is reduced to the problem of boundedness of the Hardy operator in weighted $L_p$-spaces on the cone of non-negative non-increasing functions. This allows obtaining sufficient conditions for boundedness for all admissible values of the parameters. Moreover, in case of local Morrey-type spaces, for some values of the parameters, these sufficient conditions are also necessary.

We collect known and prove new necessary and sufficient conditions for the weighted weak type maximal inequality of the form $${\Phi }_1\left(\lambda \right)\varrho \left(\{x\in X:M_{\mu }f\left(x\right)>\lambda \}\right)\le c{\int }_X{\Phi }_2\left(c\right|f\left(x\right)\left|\right)\sigma \left(x\right)\mathrm{d}\mu \left(x\right),$$ which extends some known results.

We establish new estimates for the Laplacian, the div-curl system, and more general Hodge systems in arbitrary dimension $n$, with data in $L^1$. We also present related results concerning differential forms with coefficients in the limiting Sobolev space $W^{1,n}$.

We prove weighted Littlewood-Paley inequalities for linear sums of functions satisfying mild decay, smoothness, and cancelation conditions. We prove these for general “regular” measure spaces, in which the underlying measure is not assumed to satisfy any doubling condition. Our result generalizes an earlier result of the author, proved on ${ℝ}^d$ with Lebesgue measure. Our proof makes essential use of the technique of random dyadic grids, due to Nazarov, Treil, and Volberg.

Let X be a closed subspace of $L^p\left(\mu \right)$, where μ is an arbitrary measure and 1 < p < ∞. Let U be an invertible operator on X such that $su{p}_{n\in \mathbb{Z}}\left|\right|Uⁿ\left|\right|<\infty$. Motivated by applications in ergodic theory, we obtain (optimal) conditions for the convergence of series like ${\sum }_{n\ge 1}\left(Uⁿf\right)/n^{1-\alpha }$, 0 ≤ α < 1, in terms of $\left|\right|f+\cdots +U^{n-1}f{\left|\right|}_p$, generalizing results for unitary (or normal) operators in L²(μ). The proofs make use of the spectral integration initiated by Berkson and Gillespie and, more particularly, of results from a paper by Berkson-Bourgain-Gillespie....

We revisit the properties of Bessel-Riesz operators and present a different proof of the boundedness of these operators on generalized Morrey spaces. We also obtain an estimate for the norm of these operators on generalized Morrey spaces in terms of the norm of their kernels on an associated Morrey space. As a consequence of our results, we reprove the boundedness of fractional integral operators on generalized Morrey spaces, especially of exponent $1$, and obtain a new estimate for their norm.

Sufficient conditions are derived in order that there exist strong-type weighted norm inequalities for some off-centered maximal functions. The maximal functions are of Hardy-Littlewood and fractional types taken over starlike sets in Rn. The sufficient conditions are close to necessary and extend some previously known weak-type results.

We study the weighted norm inequalities for the minimal operator, a new operator analogous to the Hardy-Littlewood maximal operator which arose in the study of reverse Hölder inequalities. We characterize the classes of weights which govern the strong and weak-type norm inequalities for the minimal operator in the two weight case, and show that these classes are the same. We also show that a generalization of the minimal operator can be used to obtain information about the differentiability of the...

We give norm inequalities for some classical operators in amalgam spaces and in some subspaces of Morrey space.

We study the duality theory of the weighted multi-parameter Triebel-Lizorkin spaces ${\dot{F}}_p^{\alpha ,q}(\omega ;{ℝ}^{n_1}\times {ℝ}^{n_2})$. This space has been introduced and the result $${\left({\dot{F}}_p^{\alpha ,q}(\omega ;{ℝ}^{n_1}\times {ℝ}^{n_2})\right)}^{*}={\mathrm{CMO}}_p^{-\alpha ,q^{\text{'}}}(\omega ;{ℝ}^{n_1}\times {ℝ}^{n_2})$$ for $0<p\le 1$ has been proved in Ding, Zhu (2017). In this paper, for $1<p<\infty$, $0<q<\infty$ we establish its dual space ${\dot{H}}_p^{\alpha ,q}(\omega ;{ℝ}^{n_1}\times {ℝ}^{n_2})$.