### 2-microlocal Besov and Triebel-Lizorkin spaces of variable integrability.

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We prove a Calderón-Zygmund type estimate which can be applied to sharpen known regularity results on spherical means, Fourier integral operators, generalized Radon transforms and singular oscillatory integrals.

Let $\mu $ be a nonnegative Radon measure on ${\mathbb{R}}^{d}$ which only satisfies $\mu \left(B(x,r)\right)\le {C}_{0}{r}^{n}$ for all $x\in {\mathbb{R}}^{d}$, $r>0$, with some fixed constants ${C}_{0}>0$ and $n\in (0,d].$ In this paper, a new characterization for the space $\mathrm{RBMO}\left(\mu \right)$ of Tolsa in terms of the John-Strömberg sharp maximal function is established.

This paper is devoted to investigating the properties of multilinear ${A}_{P\u20d7}$ conditions and ${A}_{(P\u20d7,q)}$ conditions, which are suitable for the study of multilinear operators on Lebesgue spaces. Some monotonicity properties of ${A}_{P\u20d7}$ and ${A}_{(P\u20d7,q)}$ classes with respect to P⃗ and q are given, although these classes are not in general monotone with respect to the natural partial order. Equivalent characterizations of multilinear ${A}_{(P\u20d7,q)}$ classes in terms of the linear ${A}_{p}$ classes are established. These results essentially improve and extend...

We prove that for f ∈ L ln⁺L(ℝⁿ) with compact support, there is a g ∈ L ln⁺L(ℝⁿ) such that (a) g and f are equidistributed, (b) ${M}_{S}\left(g\right)\in L\xb9\left(E\right)$ for any measurable set E of finite measure.

We apply a decomposition lemma of Uchiyama and results of the author to obtain good weighted Littlewood-Paley estimates for linear sums of functions satisfying reasonable decay, smoothness, and cancellation conditions. The heart of our application is a combinatorial trick treating m-fold dilates of dyadic cubes. We use our estimates to obtain new weighted inequalities for Bergman-type spaces defined on upper half-spaces in one and two parameters, extending earlier work of R. L. Wheeden and the author....

We prove an approximation lemma on (stratified) homogeneous groups that allows one to approximate a function in the non-isotropic Sobolev space ${\dot{NL}}^{1,Q}$ by ${L}^{\infty}$ functions, generalizing a result of Bourgain–Brezis. We then use this to obtain a Gagliardo–Nirenberg inequality for $$ on the Heisenberg group ${\mathbb{H}}^{n}$.

In the context of spaces of homogeneous type, we develop a method to deterministically construct dyadic grids, specifically adapted to a given combinatorial situation. This method is used to estimate vector-valued operators rearranging martingale difference sequences such as the Haar system.

We present new formulae providing equivalent quasi-norms on Lorentz-Karamata spaces. Our results are based on properties of certain averaging operators on the cone of non-negative and non-increasing functions in convenient weighted Lebesgue spaces. We also illustrate connections between our results and mapping properties of such classical operators as the fractional maximal operator and the Riesz potential (and their variants) on the Lorentz-Karamata spaces.

In the two-parameter setting, we say a function belongs to the mean little BMO if its mean over any interval and with respect to any of the two variables has uniformly bounded mean oscillation. This space has been recently introduced by S. Pott and the present author in relation to the multiplier algebra of the product BMO of Chang-Fefferman. We prove that the Cotlar-Sadosky space $bmo{(}^{N})$ of functions of bounded mean oscillation is a strict subspace of the mean little BMO.

This paper deals with approximation numbers of the compact trace operator of an anisotropic Besov space into some Lp-space,trΓ: Bpps,a (Rn) → Lp(Γ), s > 0, 1 < p < ∞,where Γ is an anisotropic d-set, 0 < d < n. We also prove homogeneity estimates, a homogeneous equivalent norm and the localization property in Bpps,a.

As a natural extension of ${L}^{p}$ Sobolev spaces, we consider Hardy-Sobolev spaces and establish an atomic decomposition theorem, analogous to the atomic decomposition characterization of Hardy spaces. As an application, we deduce several embedding results for Hardy-Sobolev spaces.

Some basic theorems and formulae (equations and inequalities) of several areas of mathematics that hold in Bernstein spaces ${B}_{\sigma}^{p}$ are no longer valid in larger spaces. However, when a function f is in some sense close to a Bernstein space, then the corresponding relation holds with a remainder or error term. This paper presents a new, unified approach to these errors in terms of the distance of f from ${B}_{\sigma}^{p}$. The difficult situation of derivative-free error estimates is also covered.