Existence of parabolic minimizers to the total variation flow on metric measure spaces

We give an existence proof for variational solutions u associated to the total variation flow. Here, the functions being considered are defined on a metric measure space $$({\mathcal {X}}, d, \mu )$$ ( X , d , μ ) satisfying a doubling condition and supporting a Poincaré inequality. For such parabolic minimizers that coincide with a time-independent Cauchy–Dirichlet datum $$u_0$$ u 0 on the parabolic boundary of a space-time-cylinder $$\Omega \times (0, T)$$ Ω × ( 0 , T ) with $$\Omega \subset {\mathcal {X}}$$ Ω ⊂ X an open set and $$T > 0$$ T > 0 , we prove existence in the weak parabolic function space $$L^1_w(0, T; \mathrm {BV}(\Omega ))$$ L w 1 ( 0 , T ; BV ( Ω ) ) . In this paper, we generalize results from a previous work by Bögelein, Duzaar and Marcellini by introducing a more abstract notion for $$\mathrm {BV}$$ BV -valued parabolic function spaces. We argue completely on a variational level.

Boundedness of Some Paraproducts on Spaces of Homogeneous Type

Let (X,d,μ) be a space of homogeneous type in the sense of Coifman and Weiss. In this article, the author develops a partial theory of paraproducts {Πj}j=13 defined via approximations of the identity with exponential decay (and integration 1), which are extensions of paraproducts defined via regular wavelets. Precisely, the author first obtains the boundedness of Π3 on Hardy spaces and then, via the methods of interpolation and the well-known T(1) theorem, establishes the endpoint estimates for {Πj}j=13. The main novelty of this paper is the application of the Abel summation formula to the establishment of some relations among the boundedness of {Πj}j=13, which has independent interests. It is also remarked that, throughout this article, μ is not assumed to satisfy the reverse doubling condition.

Some estimates for commutators of Littlewood-Paley g-functions

The aim of this paper is to establish the boundedness of commutator [ b , g ˙ r ] \left[b,{\dot{g}}_{r}] generated by Littlewood-Paley g g -functions g ˙ r {\dot{g}}_{r} and b ∈ RBMO ( μ ) b\in {\rm{RBMO}}\left(\mu ) on non-homogeneous metric measure space. Under assumption that λ \lambda satisfies ε \varepsilon -weak reverse doubling condition, the author proves that [ b , g ˙ r ] \left[b,{\dot{g}}_{r}] is bounded from Lebesgue spaces L p ( μ ) {L}^{p}\left(\mu ) into Lebesgue spaces L p ( μ ) {L}^{p}\left(\mu ) for p ∈ ( 1 , ∞ ) p\in \left(1,\infty ) and also bounded from spaces L 1 ( μ ) {L}^{1}\left(\mu ) into spaces L 1 , ∞ ( μ ) {L}^{1,\infty }\left(\mu ) . Furthermore, the boundedness of [ b , g ˙ r b,{\dot{g}}_{r} ] on Morrey space M q p ( μ ) {M}_{q}^{p}\left(\mu ) and on generalized Morrey L p , ϕ ( μ ) {L}^{p,\phi }\left(\mu ) is obtained.

Effect of vocal versus piano doubling on children’s singing accuracy

Children’s singing development is an important part of the music classroom, where instruction is often assisted by the teacher’s voice or the piano. However, it is unknown whether children sing more accurately when doubled by another voice or instruments. The purpose of this study was to investigate the effect of doubling timbre on children’s singing accuracy. Third- and fourth-grade children ( N = 61) performed pitch matching and song singing tasks doubled by pre-recorded vocal and piano stimuli, counterbalanced to control for order effects. Performance when doubled by voice and doubled by piano was significantly and strongly correlated, r(59) = .81, p < .001. Children performed more accurately on the vocal doubling condition than the piano ( p = .002) on pitch matching tasks, but there was no significant difference on the song singing task.

Local boundedness for minimizers of convex integral functionals in metric measure spaces

In this paper we consider the convex integral functional $ I := \int _\Omega {\Phi (g_u)\,d\mu } $ in the metric measure space $(X,d,\mu )$, where $X$ is a set, $d$ is a metric, µ is a Borel regular measure satisfying the doubling condition, Ω is a bounded open subset of $X$, $u$ belongs to the Orlicz-Sobolev space $N^{1,\Phi }(\Omega )$, Φ is an N-function satisfying the $\Delta _2$-condition, $g_u$ is the minimal Φ-weak upper gradient of $u$. By improving the corresponding method in the Euclidean space to the metric setting, we establish the local boundedness for minimizers of the convex integral functional under the assumption that $(X,d,\mu )$ satisfies the $(1,1)$-Poincaré inequality. The result of this paper can be applied to the Carnot-Carathéodory space spanned by vector fields satisfying Hörmander's condition.

BOUNDEDNESS OF MAXIMAL FUNCTIONS ON NONDOUBLING PARABOLIC MANIFOLDS WITH ENDS

Let $M$ be a nondoubling parabolic manifold with ends. First, this paper investigates the boundedness of the maximal function associated with the heat semigroup ${\mathcal{M}}_{\unicode[STIX]{x1D6E5}}f(x):=\sup _{t>0}|e^{-t\unicode[STIX]{x1D6E5}}f(x)|$ where $\unicode[STIX]{x1D6E5}$ is the Laplace–Beltrami operator acting on $M$ . Then, by combining the subordination formula with the previous result, we obtain the weak type $(1,1)$ and $L^{p}$ boundedness of the maximal function ${\mathcal{M}}_{\sqrt{L}}^{k}f(x):=\sup _{t>0}|(t\sqrt{L})^{k}e^{-t\sqrt{L}}f(x)|$ on $L^{p}(M)$ for $1<p\leq \infty$ where $k$ is a nonnegative integer and $L$ is a nonnegative self-adjoint operator satisfying a suitable heat kernel upper bound. An interesting thing about the results is the lack of both doubling condition of $M$ and the smoothness of the operators’ kernels.

Generalized Riesz potentials of functions in Morrey spaces L(1,ϕ;κ)(G) over non-doubling measure spaces

This paper proves the boundedness of the generalized Riesz potentials {I_{\rho,\mu,\tau}f} of functions in the Morrey space {L^{(1,\varphi;\kappa)}(G)} over a general measure space X, with G a bounded open set in X (or G is {X)}, as an extension of earlier results. The modification parameter τ is introduced for the purpose of including the case where the underlying measure does not satisfy the doubling condition. What is new in the present paper is that ρ depends on {x\in X}. An example in the end of this article convincingly explains why the modification parameter τ must be introduced.

Self-Contracted Curves in Spaces With Weak Lower Curvature Bound

We show that bounded self-contracted curves are rectifiable in metric spaces with weak lower curvature bound in a sense we introduce in this article. This class of spaces is wide and includes, for example, finite-dimensional Alexandrov spaces of curvature bounded below and Berwald spaces of nonnegative flag curvature. (To be more precise, our condition is regarded as a strengthened doubling condition and holds also for a certain class of metric spaces with upper curvature bound.) We also provide the non-embeddability of large snowflakes into (balls in) metric spaces in the same class. We follow the strategy of the last author’s previous paper based on the small rough angle condition, where spaces with upper curvature bound are considered. Here we apply this strategy to spaces with lower curvature bound.

Harmonic measure and Riesz transform in uniform and general domains

Let {\Omega\subsetneq{\mathbb{R}}^{n+1}} be open and let μ be some measure supported on {\partial\Omega} such that {\mu(B(x,r))\leq Cr^{n}} for all {x\in{\mathbb{R}}^{n+1}}, {r>0}. We show that if the harmonic measure in Ω satisfies some scale invariant {A_{\infty}}-type conditions with respect to μ, then the n-dimensional Riesz transform{\mathcal{R}}_{\mu}f(x)=\int\frac{x-y}{|x-y|^{n+1}}f(y)\,d\mu(y)is bounded in {L^{2}(\mu)}. We do not assume any doubling condition on μ. We also consider the particular case when Ω is a bounded uniform domain. To this end, we need first to obtain sharp estimates that relate the harmonic measure and the Green function in this type of domains, which generalize classical results by Jerison and Kenig for the well-known class of NTA domains.