In this paper, we study the the parabolic Marcinkiewicz integral [...] MΩ,hρ1,ρ2 ${ℳ}_{\Omega ,h}^{{\rho }_{1,}{\rho }_2}$ on product domains Rn × Rm (n, m ≥ 2). Lp estimates of such operators are obtained under weak conditions on the kernels. These estimates allow us to use an extrapolation argument to obtain some new and improved results on parabolic Marcinkiewicz integral operators.

We study the maximal regularity on different function spaces of the second order integro-differential equations with infinite delay $\left(P\right)u^{\text{'}\text{'}}\left(t\right)+\alpha u^{\text{'}}\left(t\right)+d/dt\left({\int }_{-\infty }^{t}b(t-s)u\left(s\right)ds\right)=Au\left(t\right)-{\int }_{-\infty }^{t}a(t-s)Au\left(s\right)ds+f\left(t\right)$ (0 ≤ t ≤ 2π) with periodic boundary conditions u(0) = u(2π), u’(0) = u’(2π), where A is a closed operator in a Banach space X, α ∈ ℂ, and a,b ∈ L¹(ℝ₊). We use Fourier multipliers to characterize maximal regularity for (P). Using known results on Fourier multipliers, we find suitable conditions on the kernels a and b under which necessary and sufficient conditions...

We classify weights which map reverse Hölder weight spaces to other reverse Hölder weight spaces under pointwise multiplication. We also give some fairly general examples of weights satisfying weak reverse Hölder conditions.

Let E and F be spaces of real- or complex-valued functions defined on a set X. A real- or complex-valued function g defined on X is called a pointwise multiplier from E to F if the pointwise product fg belongs to F for each f ∈ E. We denote by PWM(E,F) the set of all pointwise multipliers from E to F. Let X be a space of homogeneous type in the sense of Coifman-Weiss. For 1 ≤ p < ∞ and for $\varphi :X\times {ℝ}_{+}\to {ℝ}_{+}$, we denote by $bm{o}_{\varphi ,p}\left(X\right)$ the set of all functions $f\in L_{loc}^p\left(X\right)$ such that $su{p}_{a\in X,r>0}1/\varphi (a,r)(1/\mu \left(B(a,r)\right){ʃ}_{B(a,r)}|f\left(x\right)-f_{B(a,r)}{{|}^{p}d\mu )}^{1/p}<\infty$, where B(a,r) is the ball centered at a and of...

We prove three results concerning convolution operators and lacunary maximal functions associated to dilates of measures. First we obtain an H¹ to $L^{1,\infty }$ bound for lacunary maximal operators under a dimensional assumption on the underlying measure and an assumption on an $L^p$ regularity bound for some p > 1. Secondly, we obtain a necessary and sufficient condition for L² boundedness of lacunary maximal operator associated to averages over convex curves in the plane. Finally we prove an $L^p$ regularity result...