We introduce a new family of functional spaces which incorporate Bochner spaces L(R;E), with E being an appropriate Banach space, and to which we extend the H-distributions. We use the developed theory to prove a general version of the velocity averaging lemma in a heterogeneous L, p ≤ 2 setting.

We prove a weighted norm inequality for the maximal Bochner-Riesz operator and the associated square-function. This yields new L(R) bounds on classes of radial Fourier multipliers for p ≥ 2 + 4/d with d ≥ 2, as well as space-time regularity results for the wave and Schrödinger equations.

We give a proof of L-bounds for the quasi-geostrophic equation and other non-local equations. The proof uses mainly tools from functional analysis, notably the product formulas (also known as “operator splitting methods”) and the Bochner-Pollard subordination identities, hence it could be applicable to other equations. 2010 Mathematics Subject Classification: 47D06, 47F05, 35S10, 35Q99.

We prove that any Kähler manifold admitting a flat complex conformal connection is a Bochner-Kähler manifold with special scalar distribution and zero geometric constants. Applying the local structural theorem for such manifolds we obtain a complete description of the Kähler manifolds under consideration.

In this note we find the upper bound for ρ(u,M) = ∫ T 0 ∫ |u(t)|n 0 p(s)dsdt and show that F (y) = y is mφ -Bochner integrable on C(OM ) for 0 ≤ t ≤ T when ∫ OM ‖u0‖ n Mdφ(u0) is finite.

We consider the initial value problem for Navier–Stokes equations over ${\mathbb R}^3 \times [0,T]$ with time $T>0$ in spatially periodic setting. prove that it induces open injective mappings ${\mathcal A}_s\colon B^{s}_1 \to B^{s-1}_2$ where $B^{s}_1$, $B^{s-1}_2$ are elements from scales of specially constructed function spaces Bochner–Sobolev type parametrized smoothness index $s \in \ma...