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.

In this paper, the author introduces parabolic generalized local Morrey spaces and gets the boundedness of a large class of parabolic rough operators on them. The author also establishes the parabolic local Campanato space estimates for their commutators on parabolic generalized local Morrey spaces. As its special cases, the corresponding results of parabolic sublinear operators with rough kernel and their commutators can be deduced, respectively. At last, parabolic Marcinkiewicz operator which...

In this paper, we study the singular vortex patches in the two-dimensional incompressible Navier-Stokes equations. We show, in particular, that if the initial vortex patch is C1+s outside a singular set Σ, so the velocity is, for all time, lipschitzian outside the image of Σ through the viscous flow. In addition, the correponding lipschitzian norm is independent of the viscosity. This allows us to prove some results related to the inviscid limit for the geometric structures of the vortex patch.

Our understanding of the interplay between Poincaré inequalities, Sobolev inequalities and the geometry of the underlying space has changed considerably in recent years. These changes have simultaneously provided new insights into the classical theory and allowed much of that theory to be extended to a wide variety of different settings. This paper reviews some of these new results and techniques and concludes with an example on the preservation of Sobolev spaces by the maximal function.[Proceedings...

We prove that if μₙ are probability measures on ℤ such that μ̂ₙ converges to 0 uniformly on every compact subset of (0,1), then there exists a subsequence $n_k$ such that the weighted ergodic averages corresponding to ${\mu }_{n_k}$ satisfy a pointwise ergodic theorem in L¹. We further discuss the relationship between Fourier decay and pointwise ergodic theorems for subsequences, considering in particular the averages along n² + ⌊ρ(n)⌋ for a slowly growing function ρ. Under some monotonicity assumptions, the rate...

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 w be a non-negative measurable function defined on the positive semi-axis and satisfying the reverse Hölder inequality with exponents 0 < α < β. In the present paper, sharp estimates of the compositions of the power means ${}_{\alpha }w\left(x\right):={\left((1/x){\int }_0^x{w}^{\alpha }\left(t\right)dt\right)}^{1/\alpha }$, x > 0, are obtained for various exponents α. As a result, for the function w a property of self-improvement of summability exponents is established.

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...

We prove a necessary condition that has every extremal sequence for the Bellman function of the dyadic maximal operator. This implies the weak-$L^p$ uniqueness for such a sequence.