Let p ∈ (0, 1]. In this paper, the authors prove that a sublinear operator T (which is originally defined on smooth functions with compact support) can be extended as a bounded sublinear operator from product Hardy spacesH(R × R) to some quasi-Banach space B if and only if T maps all (p, 2, s1, s2)-atoms into uniformly bounded elements of B. Here s1 ≥ ⌊n(1/p− 1)⌋ and s2 ≥ ⌊m(1/p− 1)⌋. As usual,...

Abstract We prove $L^p\to L^q$ estimates for local maximal operators associated with dilates of codimension two spheres in Heisenberg groups; these are sharp up to endpoints. The results can be applied improve currently known bounds on sparse domination global operators. also consider lacunary variants and extensions Métivier groups.

In this paper we present a method of studying convolution operator under the Sonin conditions imposed on kernel. The particular case kernel is fractional integral Riemman–Liouville operator, other various types kernels are Bessel-type function, functions with power-logarithmic singularities at origin e.t.c. We pay special attention to study close power type functions. main our aim Sonin–Abel eq...

We study regularity criteria for the d-dimensional incompressible Navier-Stokes equations. We prove in this paper that if u ∈ L ∞ Lxd((0, T ) × R ) is a Leray-Hopf weak solution, then u is smooth and unique in (0, T )× R. This generalizes a result by Escauriaza, Seregin and Šverák [5]. Additionally, we show that if T = ∞ then u goes to zero as t goes to infinity.