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In this paper we consider properties of medians as they pertain to the continuity and vanishing oscillation of a function. Our approach is based on the observation that medians are related to local sharp maximal functions restricted to a cube of $R^n$.
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A classical theorem of Kuratowski says that every Baire one function on a G_delta subspace of a Polish (= separable completely metrizable) space X can be extended to a Baire one function on X. Kechris and Louveau introduced a finer gradation of Baire one functions into small Baire classes. A Baire one function f is assigned into a class in this heirarchy depending on its oscillation index beta(f). We prove a refinement of Kuratowskis theorem: if Y is a subspace of a metric space X and f is a real-valued function on Y such that beta_{Y}(f)<omega^{alpha}, alpha < omega_1, then f has an extension F onto X so that beta_X(F)is not more than omega^{alpha}. We also show that if f is a continuous real valued function on Y, then f has an extension F onto X so that beta_{X}(F)is not more than 3. An example is constructed to show that this result is optimal.
We apply wavelets to identify the Triebel type oscillation spaces with the known Triebel-Lizorkin-Morrey spaces $dot{F}^{gamma_1,gamma_2}_{p,q}(mathbb{R}^{n})$. Then we establish a characterization of $dot{F}^{gamma_1,gamma_2}_{p,q}(mathbb{R}^{n})$ via the fractional heat semigroup. Moreover, we prove the continuity of Calderon-Zygmund operators on these spaces. The results of this paper also provide necessary tools for the study of well-posedness of Navier-Stokes equations.
For $l$-homogeneous linear differential operators $mathcal{A}$ of constant rank, we study the implication $v_jrightharpoonup v$ in $X$ and $mathcal{A} v_jrightarrow mathcal{A} v$ in $W^{-l}Y$ implies $F(v_j)rightsquigarrow F(v)$ in $Z$, where $F$ is an $mathcal{A}$-quasiaffine function and $rightsquigarrow$ denotes an appropriate type of weak convergence. Here $Z$ is a local $L^1$-type space, either the space $mathscr{M}$ of measures, or $L^1$, or the Hardy space $mathscr{H}^1$; $X,, Y$ are $L^p$-type spaces, by which we mean Lebesgue or Zygmund spaces. Our conditions for each choice of $X,,Y,,Z$ are sharp. Analogous statements are also given in the case when $F(v)$ is not a locally integrable function and it is instead defined as a distribution. In this case, we also prove $mathscr{H}^p$-bounds for the sequence $(F(v_j))_j$, for appropriate $p<1$, and new convergence results in the dual of Holder spaces when $(v_j)$ is $mathcal{A}$-free and lies in a suitable negative order Sobolev space $W^{-beta,s}$. The choice of these Holder spaces is sharp, as is shown by the construction of explicit counterexamples. Some of these results are new even for distributional Jacobians.
We show that a translation bounded measure has pure point diffraction if and only if it is mean almost periodic. We then go on and show that a translation bounded measure solves what we call the phase problem if and only if it is Besicovitch almost periodic. Finally, we show that a translation bounded measure solves the phase problem independent of the underlying van Hove sequence if and only if it is Weyl almost periodic. These results solve fundamental issues in the theory of pure point diffraction and answer questions of Lagarias.
This paper concerns the smoothness of Tauberian constants of maximal operators in the discrete and ergodic settings. In particular, we define the discrete strong maximal operator $tilde{M}_S$ on $mathbb{Z}^n$ by [ tilde{M}_S f(m) := sup_{0 in R subset mathbb{R}^n}frac{1}{#(R cap mathbb{Z}^n)}sum_{ jin R cap mathbb{Z}^n} |f(m+j)|,qquad min mathbb{Z}^n, ] where the supremum is taken over all open rectangles in $mathbb{R}^n$ containing the origin whose sides are parallel to the coordinate axes. We show that the associated Tauberian constant $tilde{C}_S(alpha)$, defined by [ tilde{C}_S(alpha) := sup_{substack{E subset mathbb{Z}^n 0 < #E < infty} } frac{1}{#E}#{m in mathbb{Z}^n:, tilde{M}_Schi_E(m) > alpha}, ] is Holder continuous of order $1/n$. Moreover, letting $U_1, ldots, U_n$ denote a non-periodic collection of commuting invertible transformations on the non-atomic probability space $(Omega, Sigma, mu)$ we define the associated maximal operator $M_S^ast$ by [ M^ast_{S}f(omega) := sup_{0 in R subset mathbb{R}^n}frac{1}{#(R cap mathbb{Z}^n)}sum_{(j_1, ldots, j_n)in R}|f(U_1^{j_1}cdots U_n^{j_n}omega)|,qquad omegainOmega. ] Then the corresponding Tauberian constant $C^ast_S(alpha)$, defined by [ C^ast_S(alpha) := sup_{substack{E subset Omega mu(E) > 0}} frac{1}{mu(E)}mu({omega in Omega :, M^ast_Schi_E(omega) > alpha}), ] also satisfies $C^ast_S in C^{1/n}(0,1).$ We will also see that, in the case $n=1$, that is in the case of a single invertible, measure preserving transformation, the smoothness of the corresponding Tauberian constant is characterized by the operator enabling arbitrarily long orbits of sets of positive measure.
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