In this paper, the boundedness of a large class of sublinear commutator operators $T_b$ generated by a Calderón-Zygmund type operator on a generalized weighted Morrey spaces $M_{p,\varphi }\left(w\right)$ with the weight function $w$ belonging to Muckenhoupt’s class $A_p$ is studied. When $1<p<\infty$ and $b\in \mathrm{BMO}$, sufficient conditions on the pair $({\varphi }_1,{\varphi }_2)$ which ensure the boundedness of the operator $T_b$ from $M_{p,{\varphi }_1}\left(w\right)$ to $M_{p,{\varphi }_2}\left(w\right)$ are found. In all cases the conditions for the boundedness of $T_b$ are given in terms of Zygmund-type integral inequalities on $({\varphi }_1,{\varphi }_2)$, which do not require...

Let $M_{\beta }$ be the fractional maximal function. The commutator generated by $M_{\beta }$ and a suitable function $b$ is defined by $[M_{\beta },b]f=M_{\beta }\left(bf\right)-b{M}_{\beta }\left(f\right)$. Denote by $𝒫\left({ℝ}^n\right)$ the set of all measurable functions $p(·):{ℝ}^n\to [1,\infty )$ such that $$1<p_{-}:=\underset{x\in {ℝ}^n}{\mathrm{ess}\inf }p\left(x\right)\text{and}{p}_{+}:=\underset{x\in {ℝ}^n}{\mathrm{ess}\sup }p\left(x\right)<\infty ,$$ and by $ℬ\left({ℝ}^n\right)$ the set of all $p(·)\in 𝒫\left({ℝ}^n\right)$ such that the Hardy-Littlewood maximal function $M$ is bounded on $L^{p(·)}\left({ℝ}^n\right)$. In this paper, the authors give some characterizations of $b$ for which $[M_{\beta },b]$ is bounded from $L^{p(·)}\left({ℝ}^n\right)$ into $L^{q(·)}\left({ℝ}^n\right)$, when $p(·)\in 𝒫\left({ℝ}^n\right)$, $0<\beta <n/p_{+}$ and $1/q(·)=1/p(·)-\beta /n$ with $q(·)(n-\beta )/n\in ℬ\left({ℝ}^n\right)$.

A sufficient condition for boundedness on Herz-type spaces of the commutator generated by a Lipschitz function and a weighted Hardy operator is obtained.

We investigate weighted norm inequalities for the commutator of a fractional integral operator and multiplication by a function. In particular, we show that, for $\mu ,\lambda \in A_{p,q}$ and α/n + 1/q = 1/p, the norm $\left|\right|[b,I_{\alpha }]:L^p\left({\mu }^p\right)\to L^q\left({\lambda }^q\right)\left|\right|$ is equivalent to the norm of b in the weighted BMO space BMO(ν), where $\nu =\mu {\lambda }^{-1}$. This work extends some of the results on this topic existing in the literature, and continues a line of investigation which was initiated by Bloom in 1985 and was recently developed further by the first author, Lacey, and Wick.

A small perturbation method is developed and employed to construct frames with compactly supported elements of small shrinking support for Besov and Triebel-Lizorkin spaces in the general setting of a doubling metric measure space in the presence of a nonnegative self-adjoint operator whose heat kernel has Gaussian localization and the Markov property. This allows one, in particular, to construct compactly supported frames for Besov and Triebel-Lizorkin spaces on the sphere, on the interval with...

The classical criterion for compactness in Banach spaces of functions can be reformulated into a simple tightness condition in the time-frequency domain. This description preserves more explicitly the symmetry between time and frequency than the classical conditions. The result is first stated and proved for $L²\left({ℝ}^d\right)$, and then generalized to coorbit spaces. As special cases, we obtain new characterizations of compactness in Besov-Triebel-Lizorkin, modulation and Bargmann-Fock spaces.

We describe the complete interpolating sequences for the Paley-Wiener spaces Lπp (1 &lt; p &lt; ∞) in terms of Muckenhoupt's (Ap) condition. For p = 2, this description coincides with those given by Pavlov [9], Nikol'skii [8] and Minkin [7] of the unconditional bases of complex exponentials in L2(-π,π). While the techniques of these authors are linked to the Hilbert space geometry of Lπ2, our method of proof is based in turning the problem into one about boundedness of the Hilbert transform...

For a locally compact Hausdorff group $G$ we investigate what functions in $L_{\infty }\left(G\right)$ give rise to completely continuous multipliers $T_g$ from $L_1\left(G\right)$ into $L_{\infty }\left(G\right)$. In the case of a metrizable group we obtain a complete description of such functions. In particular, for $G$ compact all $g$ in $L_{\infty }\left(G\right)$ induce completely continuous $T_g$.

We characterize, in terms of the Beurling-Malliavin density, the discrete spectra Λ ⊂ R for which a generator exists, that is a function φ ∈ L1(R) such that its Λ translates φ(x - λ), λ ∈ Λ, span L1(R). It is shown that these spectra coincide with the uniqueness sets for certain analytic clases. We also present examples of discrete spectra Λ ∈ R which do not admit a single generator while they admit a pair of generators.

By analyzing the connection between complex Hadamard matrices and spectral sets, we prove the direction "spectral ⇒ tile" of the Spectral Set Conjecture, for all sets A of size |A| ≤ 5, in any finite Abelian group. This result is then extended to the infinite grid Zd for any dimension d, and finally to Rd.

We present the complex interpolation of Besov and Triebel–Lizorkin spaces with generalized smoothness. In some particular cases these function spaces are just weighted Besov and Triebel–Lizorkin spaces. As a corollary of our results, we obtain the complex interpolation between the weighted Triebel–Lizorkin spaces ${\dot{F}}_{p_0,q_0}^{s_0}\left({\omega }_0\right)$ and ${\dot{F}}_{\infty ,q_1}^{s_1}\left({\omega }_1\right)$ with suitable assumptions on the parameters $s_0,s_1,p_0,q_0$ and $q_1$, and the pair of weights $({\omega }_0,{\omega }_1)$.