Let m: ℝ → ℝ be a function of bounded variation. We prove the $L^p\left(ℝ\right)$-boundedness, 1 < p < ∞, of the one-dimensional integral operator defined by $T_{m}f\left(x\right)=p.v.\int k(x-y)m(x+y)f\left(y\right)dy$ where $k\left(x\right)={\sum }_{j\in \mathbb{Z}}{2}^j{\phi }_j\left(2^{j}x\right)$ for a family of functions ${{\phi }_j}_{j\in \mathbb{Z}}$ satisfying conditions (1.1)-(1.3) given below.

In this paper the notions of uniformly upper and uniformly lower $\ell$-estimates for Banach function spaces are introduced. Further, the pair $(X,Y)$ of Banach function spaces is characterized, where $X$ and $Y$ satisfy uniformly a lower $\ell$-estimate and uniformly an upper $\ell$-estimate, respectively. The integral operator from $X$ into $Y$ of the form $$Kf\left(x\right)=\varphi \left(x\right){\int }_0^{x}k(x,y)f\left(y\right)\psi \left(y\right)\mathrm{d}y$$ is studied, where $k$, $\varphi$, $\psi$ are prescribed functions under some local integrability conditions, the kernel $k$ is non-negative and is assumed to satisfy certain additional...

For 1 ≤ q ≤ α ≤ p ≤ ∞, ${(L^q,l^p)}^{\alpha }$ is a complex Banach space which is continuously included in the Wiener amalgam space $(L^q,l^p)$ and contains the Lebesgue space $L^{\alpha }$. We study the closure ${(L^q,l^p)}_{c,0}^{\alpha }$ in ${(L^q,l^p)}^{\alpha }$ of the space of test functions (infinitely differentiable and with compact support in ${ℝ}^d$) and obtain norm inequalities for Riesz potential operators and Riesz transforms in these spaces. We also introduce the Sobolev type space $W¹\left({(L^q,l^p)}^{\alpha }\right)$ (a subspace of a Morrey-Sobolev space, but a superspace of the classical Sobolev space $W^{1,\alpha }$) and obtain...

In the paper we find conditions on the pair $({\omega }_1,{\omega }_2)$ which ensure the boundedness of the maximal operator and the Calderón-Zygmund singular integral operators from one generalized Morrey space ${ℳ}_{p,{\omega }_1}$ to another ${ℳ}_{p,{\omega }_2}$, $1<p<\infty$, and from the space ${ℳ}_{1,{\omega }_1}$ to the weak space $W{ℳ}_{1,{\omega }_2}$. As applications, we get some estimates for uniformly elliptic operators on generalized Morrey spaces.

The Integral, $I_{\phi }$, and Derivative, $D_{\phi }$, operators of order $\phi$, with $\phi$ a function of positive lower type and upper type less than $1$, were defined in [HV2] in the setting of spaces of homogeneous-type. These definitions generalize those of the fractional integral and derivative operators of order $\alpha$, where $\phi \left(t\right)=t^{\alpha }$, given in [GSV]. In this work we show that the composition $T_{\phi }=D_{\phi }\circ I_{\phi }$ is a singular integral operator. This result in addition with the results obtained in [HV2] of boundedness of $I_{\phi }$ and $D_{\phi }$ or the $T1$-theorems proved...

In this paper we deal with the stationary Navier-Stokes problem in a domain $\Omega$ with compact Lipschitz boundary $\partial \Omega$ and datum $a$ in Lebesgue spaces. We prove existence of a solution for arbitrary values of the fluxes through the connected components of $\partial \Omega$, with possible countable exceptional set, provided $a$ is the sum of the gradient of a harmonic function and a sufficiently small field, with zero total flux for $\Omega$ bounded.

In this paper we show that the fractional integral of order α on spaces of homogeneous type embeds $L^{1/\alpha }$ into a certain Orlicz space. This extends results of Trudinger [T], Hedberg [H], and Adams-Bagby [AB].

Let $A_1,\cdots ,A_m$ be $n\times n$ real matrices such that for each $1\le i\le m,$$A_i$ is invertible and $A_i-A_j$ is invertible for $i\ne j$. In this paper we study integral operators of the form $$Tf\left(x\right)=\int k_1(x-A_{1}y)k_2(x-A_{2}y)\cdots k_m(x-A_{m}y)f\left(y\right)\mathrm{d}y,$$$...$

We establish Lp-boundedness for a class of product singular integral operators on spaces M = M1 x M2 x . . . x Mn. Each factor space Mi is a smooth manifold on which the basic geometry is given by a control, or Carnot-Carathéodory, metric induced by a collection of vector fields of finite type. The standard singular integrals on Mi are non-isotropic smoothing operators of order zero. The boundedness of the product operators is then a consequence of a natural Littlewood- Paley theory on M. This in...