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Let $L$ be a linear operator in $L^2({{\mathbb R}^n})$ and generate an analytic semigroup $\{e^{-tL}\}_{t\ge 0}$ with kernels satisfying an upper bound of Poisson type, whose decay is measured by $\theta(L)\in (0,\infty].$ Let $\omega$ on…

This paper investigates the boundedness of a broad class of operators within the framework of generalized Morrey-Banach function spaces. This class includes multilinear operators such as multilinear $\omega$-Calder\'{o}n-Zygmund operators,…

Suppose that $\Omega$ is the open region in $\mathbb{R}^n$ above a Lipschitz graph and let $d$ denote the exterior derivative on $\mathbb{R}^n$. We construct a convolution operator $T $ which preserves support in $\bar{\Omega$}, is…

Let $X$ be a Borel and Borel-regular metric measure space whose closed balls are of positive and finite measure. In this paper, we shall give equivalent conditions for averaging operators on non-reflexive Lebesgue spaces $L^1(X)$ and…

We prove boundedness results for integral operators of fractional type and their higher order commutators between weighted spaces, including $L^p$-$L^q$, $L^p$-$BMO$ and $L^p$-Lipschitz estimates. The kernels of such operators satisfy…

As a class of compact operators on the $\ell^2-$valued Bergman space $A^2_\alpha (\mathbb B_n, \ell^2)$ on the unit ball $\mathbb B_n,$ we study Toeplitz operators with $BMO^1_\alpha (\mathbb B_n, \mathcal L(\ell^2))$ operator-valued…

An RD-space $\mathcal{X}$ is a space of homogeneous type in the sense of Coifman and Weiss with the additional property that a reverse doubling property holds in $\mathcal{X}$. In this setting, the authors establish the boundedness of…

In the paper compact multiplier operators on Banach spaces of analytic functions on the unit disk with the range in Banach sequence lattices are studied. If the domain space $X$ is such that $H_\infty\hookrightarrow X\hookrightarrow H_1$,…

For $p \in (1,N)$ and $\Omega \subseteq \mathbb{R}^N$ open, the Beppo-Levi space $\mathcal{D}^{1,p}_0(\Omega)$ is the completion of $C_c^{\infty}(\Omega)$ with respect to the norm $\left( \int_{\Omega}|\nabla u|^p \right)^ \frac{1}{p}.$…

For $\alpha\in(0, n)$ and a growth function $\varphi:[0,\infty)\rightarrow [0,\infty)$, it is proved that the commutator $[b,I_\alpha]$ generated by fractional integral operator $I_\alpha$ and Orlicz $\mathrm{BMO}$ function $b$ is bounded…

Let $\mathcal{M}$ be the bilinear Hardy-Littlewood maximal function and $\vec{b}=(b,b)$ be a collection of locally integrable functions. In this paper, the authors establish characterizations of the weighted {\rm BMO} space in terms of…

We study mapping properties of the centered Hardy--Littlewood maximal operator $\mathcal{M}$ acting on Lorentz spaces $L^{p,q}(\mathfrak{X})$ in the context of certain non-doubling metric measure spaces $\mathfrak{X}$. The special class of…

Let $\mu$ be a finite Radon measure in $\mathbb{R}^d$ with polynomial growth of degree $n$, although not necessarily $n$-AD-regular. We prove that under some geometric conditions on $\mu$ that are closely related to rectifiability and…

Let X and Y be complex Banach spaces, B_X be the open unit ball of X and HL(B_X,Y) be the Banach space of all holomorphic Lipschitz maps f:B_X->Y such that f(0)=0, endowed with the Lipschitz norm. Given a Banach operator ideal A, we use the…

The Coifman-Fefferman inequality implies quite easily that a Calderon-Zygmund operator $T$ acts boundedly in a Banach lattice $X$ on $\mathbb R^n$ if the Hardy-Littlewood maximal operator $M$ is bounded in both $X$ and $X'$. We discuss this…

This is a continuation of our previous research about an oscillatory integral operator $T_{\alpha, \beta}$ on compact manifolds $\mathbb{M}$. We prove the sharp $H^{p}$-$L^{p,\infty}$ boundedness on the maximal operator $T^{*}_{\alpha,…

We consider local "complementary" generalized Morrey spaces ${\dual \cal M}_{\{x_0\}}^{p(\cdot),\om}(\Om)$ in which the $p$-means of function are controlled over $\Om\backslash B(x_0,r)$ instead of $B(x_0,r)$, where $\Om \subset \Rn$ is a…

In the setting of homogeneous spaces (X,d,{\mu}), it is shown that the commutator of Calder\'on- Zygmund type operators as well as commutator of potential operator with BMO function are bounded in generalized Grand Morrey space. Interior…

We establish the characterization of compactness for the sparse operator (associated with symbol in weighted VMO space) in the two weight setting on the spaces of homogeneous type in the sense of Coifman and Weiss. As a direct application…

Given a bounded Lipschitz domain $D\subset \mathbb{R}^d$ and a Calder\'on-Zygmund operator $T$, we study the relations between smoothness properties of $\partial D$ and the boundedness of $T$ on the Zydmund space $\mathcal{C}_{\omega}(D)$…