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.

In the context of variable exponent Lebesgue spaces equipped with a lower Ahlfors measure we obtain weighted norm inequalities over bounded domains for the centered fractional maximal function and the fractional integral operator.

We describe a class O of nonlinear operators which are bounded on the Lizorkin-Triebel spaces Fsp,q(Rn), for 0 < s < 1 and 1 < p, q < ∞. As a corollary, we prove that the Hardy-Littlewood maximal operator is bounded on Fsp,q(Rn), for 0 < s < 1 and 1 < p, q < ∞ ; this extends the result of Kinnunen (1997), valid for the Sobolev space H1p(Rn).

The family of block spaces with variable exponents is introduced. We obtain some fundamental properties of the family of block spaces with variable exponents. They are Banach lattices and they are generalizations of the Lebesgue spaces with variable exponents. Moreover, the block space with variable exponents is a pre-dual of the corresponding Morrey space with variable exponents. The main result of this paper is on the boundedness of the Hardy-Littlewood maximal operator on the block space with...

The author studies the commutators generated by a suitable function a(x) on ℝⁿ and the oscillatory singular integral with rough kernel Ω(x)|x|ⁿ and polynomial phase, where Ω is homogeneous of degree zero on ℝⁿ, and a(x) is a BMO function or a Lipschitz function. Some mapping properties of these higher order commutators on $L^p\left(ℝⁿ\right)$, which are essential improvements of some well known results, are given.

We consider Littlewood-Paley functions associated with a non-isotropic dilation group on ${ℝ}^n$. We prove that certain Littlewood-Paley functions defined by kernels with no regularity concerning smoothness are bounded on weighted $L^p$ spaces, $1<p<\infty$, with weights of the Muckenhoupt class. This, in particular, generalizes a result of N. Rivière (1971).

The $L^p$ boundedness(1 < p < ∞) of Littlewood-Paley’s g-function, Lusin’s S function, Littlewood-Paley’s $g{*}_{\lambda }$-functions, and the Marcinkiewicz function is well known. In a sense, one can regard the Marcinkiewicz function as a variant of Littlewood-Paley’s g-function. In this note, we treat counterparts ${\mu }_S^{\varrho }$ and ${\mu }_{\lambda }^{*,\varrho }$ to S and $g{*}_{\lambda }$. The definition of ${\mu }_S^{\varrho }\left(f\right)$ is as follows: ${\mu }_S^{\varrho }\left(f\right)\left(x\right)=({ʃ}_{|y-x|<t}|1/t^{\varrho }{ʃ}_{\left|z\right|\le t}\Omega \left(z\right)/{\left(\right|z|}^{n-\varrho }{\left)f(y-z)dz\right|}^2\left(dydt\right)/\left(t^{n+1}\right){)}^{1/2}$, where Ω(x) is a homogeneous function of degree 0 and Lipschitz continuous of order β (0 < β ≤ 1) on the unit sphere $S^{n-1}$, and ${ʃ}_{S^{n-1}}\Omega \left(x^{\text{'}}\right)d\sigma \left(x^{\text{'}}\right)=0$. We show that...

We obtain the boundedness of Calderón-Zygmund singular integral operators $T$ of non-convolution type on Hardy spaces $H^p\left(𝒳\right)$ for $1/(1+ϵ)<p\le 1$, where $𝒳$ is a space of homogeneous type in the sense of Coifman and Weiss (1971), and $ϵ$ is the regularity exponent of the kernel of the singular integral operator $T$. Our approach relies on the discrete Littlewood-Paley-Stein theory and discrete Calderón’s identity. The crucial feature of our proof is to avoid atomic decomposition and molecular theory in contrast to what was...

We introduce a new type of variable exponent function spaces $M{\dot{K}}_{q,p(·)}^{\alpha (·),\lambda }\left({ℝ}^n\right)$ of Morrey-Herz type where the two main indices are variable exponents, and give some propositions of the introduced spaces. Under the assumption that the exponents $\alpha$ and $p$ are subject to the log-decay continuity both at the origin and at infinity, we prove the boundedness of a wide class of sublinear operators satisfying a proper size condition which include maximal, potential and Calderón-Zygmund operators and their commutators of BMO...

Let $(X,d,\mu )$ be a metric measure space endowed with a distance $d$ and a nonnegative Borel doubling measure $\mu$. Let $L$ be a non-negative self-adjoint operator of order $m$ on $L^2\left(X\right)$. Assume that the semigroup ${\mathrm{e}}^{-tL}$ generated by $L$ satisfies the Davies-Gaffney estimate of order $m$ and $L$ satisfies the Plancherel type estimate. Let $H_L^p\left(X\right)$ be the Hardy space associated with $L.$ We show the boundedness of Stein’s square function ${𝒢}_{\delta }\left(L\right)$ arising from Bochner-Riesz means associated to $L$ from Hardy spaces $H_L^p\left(X\right)$ to $L^p\left(X\right)$, and also study the boundedness...

In this paper, the boundedness properties for some Toeplitz type operators associated to the Riesz potential and general integral operators from Lebesgue spaces to Orlicz spaces are proved. The general integral operators include singular integral operator with general kernel, Littlewood-Paley operator, Marcinkiewicz operator and Bochner-Riesz operator.

The aim of the present paper is to obtain an inequality of Brézis-Gallouët-Wainger type for Besov-Morrey spaces. We investigate these spaces in a self-contained manner. Also, we verify that our result is sharp.

We study sufficient conditions on the weight w, in terms of membership in the $A_p$ classes, for the spline wavelet systems to be unconditional bases of the weighted space $H^p\left(w\right)$. The main tool to obtain these results is a very simple theory of regular Calderón-Zygmund operators.